Développement d`un modèle mécanique pour la prédiction des

Transcription

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Institut Supérieur de l’Aéronautique et de l’Espace
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Floriane SOULAS
le mercredi 2 mars 2016
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Development of a lightning strike mechanical
model for the prediction of
damage of aeronautical composite panels
Développement d'un modèle mécanique pour la prédiction des dommages de
panneaux composites aéronautiques soumis à un choc foudre
²DPMF EPDUPSBMF et discipline ou spécialité ED MEGeP : Génie mécanique, mécanique des matériaux
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Institut Clément Ader
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M. Frédéric LACHAUD (Directeur de thèse)
Mme Christine ESPINOSA (Co-directrice de thèse)
Jury :
M. Daniel COUTELLIER, Professeur des Universités - Président
M. Michel BOUSTIE, Directeur de recherches CNRS - Rapporteur
Mme Nadia BAHLOULI, Professeur des universités - Rapporteur
M. Jean-Marc BAUCHIRE Professeur des universités - Examinateur
M. Frédéric LACHAUD, Professeur ISAE - Directeur de thèse
Mme Christine ESPINOSA, Professeur associé ISAE- Co-directrice de thèse
Acknowledgements
This work would not have been possible without the help and smiles of so many people who
accepted me as a member of their team and gave me the opportunity to accomplish this
work and obtain the PhD degree. Directors, colleagues, friends, family, you supported me for
three years of intense work, doubts and small victories; you deserve all my gratitude and
congratulations.
It is an honour to have Pr. Daniel Coutellier and Pr. Jean-Marc Bauchire for accepting to be
jury members. I owe my deepest gratitude to Pr. Nadia Bahlouli and Pr. Michel Boustie for
honouring me by evaluating this work.
First of all I would like to thank the Airbus Group Innovations teams (TX5 and TX3) and Gilles
Peres who gave me the chance to achieve this work by placing their trust in me after my final
year placement and offered me to pursue my work with this PhD. My special thanks go to
Ivan Revel and Stéphane Guinard who supported me during these four years and were
always present to help me when I was stuck or to cheer me up when I needed it. I would not
have succeeded without your implication. I also thank Bruno Lepetit for his support, rigour
and precious advice. I also thank my colleagues at AGI: Emilie, Soukhaina, Mathilde, Nicolas,
Alexandre A., Alexandre H., Olivier, Bruno, Phillipe, Damien, Andy and all the others.
It was a great pleasure for me to work with my directors Frederic Lachaud and Christine
Espinosa, who provided me with amazing supervision, pertinent advice and attention during
all these years. Frederic, thank you for your kindness and happiness, for shouting my name
in the corridors and responding every time I shouted yours desesperatly. Christine, thank
you for your time and jokes, your dedication to my work, you were a second mother to me
during this adventure. You both made me feel that I could do it, and I did!
I would like to thank all the members of DMSM, MSC and ICA who made me feel at home in
the lab, and who helped me until the end: Olivier Cherrier, Thierry Martin, Veronique
Godivier, Xavier Foulquier and all the others. Thanks to all my colleagues for the numerous
coffee breaks that we shared: Loïc, Samuel, Joël, Joseph, Pilou, and all the others. I would
address extra thanks to the two unfortunate who shared my desk: Floran and Arnaud. Thank
you guys for the endless unexpected discussions and topics, for your amazing patience and
help (especially at the end) and our common love for tea time.
Of course, last but not least, I want to thank my parents who always let me follow my path,
my sister who always knows the right thing to say and my friends: Chloé for the
unconditional love and support and the numerous mails we exchanged, Momo for the
uncountable numbers of sandwiches, Chloé my laotong in Paris who knows we have been a
long way since we begin, Laetitia, Alexandre and Claire for their support and for making
these three years fly so fast. We made it.
i
Contents
1
Chapter 1 ................................................................................................................... 1
2
Chapter 2 ................................................................................................................... 5
Objectives ........................................................................................................................................................... 6
2.1
General points on lightning ................................................................................................................... 7
2.2
Lightning and aircrafts: damage and protections .................................................................................. 8
2.2.1
« Zoning » ........................................................................................................................ 8
2.2.2
Direct effects ................................................................................................................. 10
2.2.3
Indirect effects............................................................................................................... 11
2.2.4
Laboratory tests for certification considerations .......................................................... 12
2.3
2.3.1
Metallic materials .......................................................................................................... 15
2.3.2
Composite materials...................................................................................................... 17
2.3.3
Lightning strike protections........................................................................................... 20
2.4
Lightning modelling ............................................................................................................................. 24
2.4.1
Insights into lightning induced phenomena .................................................................. 24
2.4.2
Electro-thermal model .................................................................................................. 27
2.4.3
Mechanical based models ............................................................................................. 31
2.4.4
Energy based model ...................................................................................................... 33
2.5
3
Direct effect damage induced by lightning.......................................................................................... 14
Conclusions and perspectives .............................................................................................................. 35
Chapter 3 ................................................................................................................. 39
Objectives ......................................................................................................................................................... 40
3.1
Preliminary works ................................................................................................................................ 41
3.1.1
Industrial framework ..................................................................................................... 41
3.1.2
Preliminary attempts for an equivalent pressure ......................................................... 41
3.1.3
Compute a magnetic pressure using experimental arc root ......................................... 44
3.1.4
Analysis of lightning induced damage ........................................................................... 49
3.2
Objectives and methodology ............................................................................................................... 54
3.3
From lightning strike to mechanical impacts ....................................................................................... 57
3.3.1
Equivalent impact to lightning strike............................................................................. 57
3.3.2
Creation of a shell model to design the equivalent impact parameters ....................... 64
ii
3.3.3
Validation of the shell model and impact parameters {m, v} ....................................... 65
3.3.4
Rear face displacement profiles .................................................................................... 69
3.3.5
Analytical determination of the variable arc root ......................................................... 71
3.4
4
Conclusion ........................................................................................................................................... 73
Chapter 4 ................................................................................................................. 77
Objectives ......................................................................................................................................................... 78
4.1
4.1.1
Specimen manufacturing .............................................................................................. 79
4.1.2
Canon test: set up and instrumentation ....................................................................... 80
4.2
Test results .......................................................................................................................................... 82
4.2.1
Macroscopic results....................................................................................................... 82
4.2.2
Rear face displacement results ..................................................................................... 84
4.2.3
Conclusions on the structural behaviour ...................................................................... 87
4.3
Analysis of mechanical impact tests damage ...................................................................................... 88
4.3.1
Review of damage in composite materials ................................................................... 88
4.3.2
Analysis of post-mortem examination of mechanical impact tests .............................. 91
4.4
Comparison of lightning strike and mechanical impact damage ......................................................... 99
4.4.1
Rear face displacement ............................................................................................... 100
4.4.2
Total delaminated area ............................................................................................... 103
4.4.3
Influence of surface state ............................................................................................ 104
4.5
5
Mechanical impact tests ...................................................................................................................... 79
Conclusions ........................................................................................................................................ 112
Chapter 5 ................................................................................................................115
Objectives ....................................................................................................................................................... 116
5.1
Introduction ....................................................................................................................................... 117
5.2
Modelling damage in composite materials ....................................................................................... 117
5.2.1
Literature survey on impact modelling ....................................................................... 117
5.2.2
Inter-laminar damage .................................................................................................. 118
5.3
Intra laminar damage: Continuous Diffuse Damage Model .............................................................. 125
5.3.1
CDM framework .......................................................................................................... 125
5.3.2
Damage Modelling: Ilyas former model ...................................................................... 126
5.3.3
Conclusions.................................................................................................................. 130
5.4
Continuous versus discontinuous damage modelling ....................................................................... 131
5.4.1
Review of the different models used .......................................................................... 131
5.4.2
Effects on the sample behaviour ................................................................................. 132
5.5
Final modelling: results and discussion ............................................................................................. 137
iii
5.5.1
From shell to 3D solid element model ........................................................................ 137
5.5.2
Final modelling: results and discussion ....................................................................... 138
5.6
5.6.1
5.6.2
5.6.3
Damage distribution through thickness ...................................................................... 148
5.6.4
Conclusions on the predictive quantities of the model .............................................. 149
5.7
Comparison of lightning damage with numerical simulations .......................................................... 149
5.7.1
5.7.2
Rear face displacement profiles .................................................................................. 153
5.7.3
5.7.4
Damage features ......................................................................................................... 155
5.7.5
Distribution through thickness .................................................................................... 156
5.8
6
Results and discussion ....................................................................................................................... 142
Conclusions ........................................................................................................................................ 158
Chapter 6 ................................................................................................................161
Objectives ....................................................................................................................................................... 162
6.1
Introduction ....................................................................................................................................... 163
6.2
Prospective studies on the effect of the contact surface and stiffness ............................................. 164
6.2.1
Changing the projectile’s shape .................................................................................. 164
6.2.2
Changing the projectile’s nature ................................................................................. 174
6.3
Prospective study on the effect of evolving pressure loading ........................................................... 177
6.4
Conclusions and perspective ............................................................................................................. 187
7
General conclusion..................................................................................................190
8
________________________________ ....................................................................190
9
Bibliography ...........................................................................................................196
10 Appendices .............................................................................................................209
11 Appendix A .............................................................................................................210
12 Appendix B .............................................................................................................212
Appendix C ....................................................................................................................215
Appendix D ....................................................................................................................218
Appendix E ....................................................................................................................221
13 Appendix F..............................................................................................................242
14 Resumé en français .................................................................................................243
iv
v
vi
1
Chapter 1
General Introduction
Since the 1970s, carbon fibre reinforced composites have been widely introduced into
aircrafts. The Airbus A380 includes around 25% of composite material. These new materials
with high strength and stiffness to mass ratios allowed a reduction of the consumed number
of fuel litres per passenger per 100 km from 3 to 5 [117]. Such improvement have pushed
aircraft manufacturers to introduce more and more composite material in their structures
and finally, models such as the A350XWB or the B787 possess up to 50% of composite
material in various parts from vertical stabilizer to primary structures such as fuselage, wings
and fuel tanks.
Figure 1-1: Airbus latest aircraft A350
With previous metallic aircrafts, the lightning threat was naturally handled by the conducting
capacity of the different alloys used in the structures. But composite materials, and
especially those present in primary parts, put back this threat into the light. Lightning strikes
are a common event as a civil aircraft is struck once to twice a year (1/2900 flying hours).
Composite materials are made of carbon fibres embedded into a non-conductive resin. Thus
when lightning strikes, the current delivered by the arc on the structures is not driven by the
materials, inducing extensive damage and burning. To decrease and avoid such damage,
certification tests are lead in laboratory [1, 2]. However these tests are expensive and must
be repeated for coupon specimens and sub-assemblies as well.
In order to protect the composite parts, lightning strike protections have been implemented
upon in service aircrafts. However, those protections, made of expanded metallic wire over
the composite materials, add weight to the lightened structures and thus belittle the
Chapter 1
1
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advantage of the use of low-weight materials. Today, the main issues are to optimize these
protections in terms of efficacy and weight. This involves a better understanding of the
lightning strike phenomenon, of its interaction with the impacted materials and the damage
it induces.
Figure 1-2: Building-Block Approach in composite certification [118]
Traditionally, lightning strike on composite structures is studied from an electro-thermal
point of view. Here, an alternative approach is followed, focusing on the observed damage
due to lightning strike. An equivalent mechanical impact test is designed using a specific
projectile, associated with numerical models, from data extracted from lightning strike
laboratory tests. The mechanical model, which results are validated thanks to actual
mechanical impact test campaign, will be used to predict damage due to lightning strike.
Thus, a predictive model, with various possible loadings such as projectile or numerical
pressure could be used to predict equivalent damage to lightning strike and provide advice
to early design phase on materials and protections.
Chapter 2 outlines all the previous works that have been lead in the field of lightning impact
on composite structures. Chapter 3 offers preliminary works and methodology for the
current work. The design of the equivalent mechanical impacts to lightning strikes is
presented in chapter 4 along with the test campaign that has been run and the comparison
of the mechanical equivalent results with lightning strike ones. The interface behaviour and
damage law for the numerical model are treated in chapter 5. A cohesive interface law is
implemented and coupled with a continuum damage model for ply damage. The final 3D
volume elements numerical model, using these damaging law and of growing complexity, is
presented in chapter 5 along with results comparisons with mechanical impact tests and
Chapter 1
2
__________________________________________________________________________________
lightning strike ones in order to validate the final numerical model in terms of general
behaviour and damage generation, and conclude on the validity of the hypothesis and
methodology. Finally chapter 6 presents prospects and future work.
Chapter 1
3
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Chapter 1
4
2
Chapter 2
State of the Art
2.1
General points on lightning .................................................................................................................. 7
2.2
Lightning and aircrafts: damage and protections ................................................................................ 8
2.2.1
« Zoning » ..................................................................................................................................... 8
2.2.2
Direct effects ............................................................................................................................... 10
2.2.3
Indirect effects ............................................................................................................................ 11
2.2.4
Laboratory tests for certification considerations........................................................................ 12
2.3
Direct effect damage induced by lightning ........................................................................................ 14
2.3.1
Metallic materials ....................................................................................................................... 15
2.3.2
Composite materials ................................................................................................................... 17
2.3.3
Lightning strike protections ........................................................................................................ 20
2.4
Lightning modelling ............................................................................................................................ 24
2.4.1
Insights into lightning induced phenomena ............................................................................... 24
2.4.2
Electro-thermal model ................................................................................................................ 27
2.4.3
Mechanical based models .......................................................................................................... 31
2.4.4
Energy based model.................................................................................................................... 33
2.5
Conclusions and perspectives ............................................................................................................. 35
Chapter 2
5
__________________________________________________________________________________
Objectives
This chapter describes the lightning strike phenomenon and how it is considered from the
point of view of the aeronautical companies.
It describes the damage suffered by the structures that are due to the so-called direct
effects. It explains the point of view that is adopted in this study and how it is important for
an optimal design of the protective layer, to understand and quantify the effect of each
constitutive parameter and their interaction.
Chapter 2
6
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2.1 General points on lightning
The use of composite materials is a technological and an economical challenge for the
aeronautical industry nowadays. Indeed, the mechanical properties as well as the very low
density of such materials allow an important gain of mass over structures as well as reduced
maintenance costs to manufacturers. Unfortunately, composite materials do not possess as
good electrical properties as previous metallic structures. Most of the composite materials
used are Carbon Fibre Reinforced Polymers (CFRP). If carbon fibres can conduct electricity in
a correct way (conductivity of 104 to 105 Ω/m longitudinally), the resin, on the contrary, is a
very poor conductor. Moreover the electrical contact between plies is also of bad
conductivity.
Any aircraft is stroke once to twice a year (1/2900 flying hours). The impact of lightning on
the composite structure induces direct effects (electrical, thermal and mechanical damages)
as well as indirect effects (induced by the current on the electrical on board systems) which
lead to irreversible damages, for instance to the fuselage’s material. As it is often
complicated, not to say impossible, either to repair or to change the impacted panels, it is of
primary interest to foresee the reaction of the structure to a lightning strike in order, if
needed, to change the involved material. Despite the rising interest to this problem, the
damage mechanisms related to lightning strikes and the structural response of the aircraft
still remain largely unknown.
A lightning strike is due to the polarization of a cloud which unloads its excess of charges.
We can distinguish two types of unloading: intra or inter cloud unloading and cloud-to-earth
unloading. The most common ones are of negative polarity. The polarized cloud initiates a
leader (positive or negative) toward the land or another cloud in its vicinity. Under the effect
of this leader, the second cloud will also form a leader as a response to the first cloud, and of
an opposite charge (figure 2.1). When the two leaders connect to each another, the return
stroke happens, which corresponds to a neutralization wave of the current channel (impulse
or initial stroke). Then, a slow discharge can happen (up to one second), corresponding to
the partial or total neutralization of the cloud’s charge (continuing current).
Figure 2-1: Formation of the leaders during a cloud-to-earth lightning strike
Chapter 2
7
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A lightning strike is thus created by the initiation of a leader and is composed of two main
components: an initial impulse and a continuing current. Most of the time, when an aircraft
is struck, it triggers itself this event. Indeed in 90 to 95% of the cases, the aircraft is the one
which initiates the leader: the lightning strike is caused by the aircraft’s presence in a
charged environment (figure 2.2). Thus contrary to what one might think: a flash of lightning
does not strike aircrafts randomly. The rest of the time (5 to 10%), the aircraft intercepts a
forming discharge. Moreover, most of the lightning strikes happen during ascending and
descending phases of the airplane, which corresponds to an altitude inferior to 5km and to
intra-clouds strikes, the less dangerous.
Figure 2-2: Airplane initiating a lightning strike
2.2 Lightning and aircrafts: damage and protections
As lightning strikes are common event during a plane life service several measures have
been introduced to protect the structures against such events. The study of lightning strike
sheds light on two major effects they caused on composite structures: direct and indirect
effects. Both of them are explained in the following section even though this study will be
mainly focused on the direct effects induced by lightning to composite panels that constitute
the structure of aircrafts such as the new A350. Thanks to the study of the damage induced
by those effects, certification processes against the lightning threat have been settled that
will also be presented in the following sections.
2.2.1 « Zoning »
The various parts of an aircraft are not homogeneously exposed to equivalent risks. In order
to protect planes from lightning strikes, studies have been led to establish a map of the
structure, which sheds light on the zones where lightning is often attached. An aircraft is
thus chopped in zones exposed to part of the threat described by the preceding wave of
current. The definition of these zones also constitute a mandatory certification step of the
Chapter 2
8
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international regulations [1, 2] (EUROCAE and SAE). It allows engineers to predict the
“danger zones” and to protect the structure adequately.
These studies end up to what is called the « zoning » which divides the plane into sections
typically impacted by one or another component of the lightning presented before (figure
2.3).
Figure 2-3: Aircraft zoning
Three zones of interest are exposed:
Zone 1 (A and B) corresponds to the parts of the aircraft for which there is a strong
probability of initial attachment of the arc of lightning (lightning component A with or
without component B/C).
Zone 2 (A and B) corresponds to the parts of the aircraft for which there is a strong
probability of being crossed by the arc root initiating in zone 1 and puffing up from the travel
of the aircraft (sweeping zone) (lightning component D with or without component B/C).
Zone 3 corresponds to the other parts of the aircraft. It thus corresponds to the parts of the
aircraft with a lower probability of attachment of the arc of lightning.
The suffixes A and B for zones 1 and 2 correspond to the probabilities of stagnation of the
arc of lightning at the level of the area considered. For suffix A, the time of stagnation of the
arc root in the zone will be short while for the suffix B, the time of stagnation will be long.
The most exposed parts of the aircraft to the attachment of the arc of lightning are the
radome (blister which is located at the level of the aircraft nose section), the ends of the
wings and the tail planes as well as the jet engines. The main parts where the arc will move
are primarily the fuselage and part of the wings.
Chapter 2
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Once the zoning established, each zone is protected with the appropriate protection
considering its type. For example, zones 1 where components A are likely to happen are
always made of complete metallic structures as recommended by the European technical
standards [1]. However, these mandatory studies are not sufficient enough to completely
protect the aircrafts against the lightning threat and especially against its direct effect. An indepth study of the direct effect damage and of their nature is necessary to design and adapt
adequate protection as presented in the following section.
2.2.2 Direct effects
Direct effects due to the interaction of the electrical arc of the lightning and the structure
induce electrical, thermal and mechanical damage to the composite. Most of these effects
occur at the attachment points of the arc on the structure and at the entry as well as exit
points, where the current is the most concentrated and intense. But lightning is also not a
purely electrical phenomenon, it reveals to be far more complex: lightning is a thermal flow,
a source for magnetic pressure and acoustic waves as well as a current source. It induces
numerous effects on the materials impacted with multiple consequences. When it impacts
metallic materials, lightning generally induces the melting and the cutting of the materials as
well as its deflection. For composite materials (up to 1000 times more resistive), lightning
induces surface burning and leads to explosion of the laminate and the break-up of the
fibres around the impact point as well as delamination of the composite material (figure
2.4). It can also provoke the explosion of the fuel tanks in case of sparks.
Direct effects thus mainly concern the structural damage due to the lightning strike on the
impacted parts or materials, and the risk of explosion at the rear face of the impacted parts.
Burnt central
zone: extent of
the damage?
Dark external
burnt mark: limit
of the plasma
action?
Figure 2-4: Visible damage due to lightning impact on a composite plate
Chapter 2
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2.2.3 Indirect effects
Indirect effects are the short name given to all the electromagnetic (EM) effects arising from
a lightning strike on a structure. A lightning strike is caused by two charged leaders
unloading their currents; it is thus an electrical channel through which current flows. This
large amplitude current enters the structure at the so called “entry point” and flows out at
the “exit point”. Direct effects of lightning mainly take place at the entry point while the
current is flowing inside the structure and all the conducting parts in it (external and internal
ones), which include the electrical system on board, until it finds the exit point. This is called
the induced current.
The introduction of composite materials into primary structures of new aircrafts made this
induced current a major concern. Previous metal structures, thanks to their high electrical
conductivity, insured that the plane acted like a Faraday cage, which is not the case anymore
due to the poor electrical conductivity of CFRP. Apart from the structural damage induced by
lightning at the entry point, transient disturbance can be caused to the electronic on board
systems as well as voltage drops and localized breakdowns [3]:
-
-
-
Thermal effects: the continuous current flowing through the structure increases the
local temperature on its way to the exit point. Thus, mechanical parts can endure
intensive heating during the current flow.
Electric discharges: during the circulation of the current, the electric potential of the
system’s parts can vary on very close path which results in the formation of electric
fields large enough to provoke an electrical breakdown and produce electrical
discharge (see figure 2.5). Such discharge can happen in the vicinity of rivets and
junctions between two material parts and can be extremely dangerous when
occurring in fuel tanks [4-8].
Induced effects on electronic inboard equipment: induced current flowing along
cables can reach equipment connectors. A difference of potential on impedant
systems can make them act like source term generating dysfunction in the electronic
equipment on board. These currents are a central matter for all that concerns
electromagnetic compatibility.
Figure 2-5: Electrical discharge induced by machining
Chapter 2
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2.2.4 Laboratory tests for certification considerations
In order to study and certificate aircraft against the lightning threat, aircraft manufacturers
have to define this threat and define proper laboratory means that would allow them to
reproduce it. Conjoint works by the NASA, FAA and industry has led to a recommended
procedure when it comes to lightning: the SAE ARP 5412. This report is internationally
considered as the stole standard concerning the lightning threat [2]. This threat is defined
using data of lightning to the ground [1]. Indeed, it is only in this configuration that
consequent statistical data has been recorded. In addition, the results obtained during inflight test campaigns suggest that lightning level in-flight is undervaluing compared to those
with the ground.
Lightning is composed of an initial stroke, an impulse and a continuing current. In the actual
normalization (AC 20-53A) [1], the natural environment of lightning is represented by the
components A, B, C and D (Figure 2.6). The waveforms thus defined covers 99% of the blows
to the ground in term of severity. The various parameters allowing defining this wave are:
- The intensity peak of the current.
- The rise and descent times of the wave front.
- The electrical load transferred equals to ∫idt (in C).
- The action integral equals to ∫i²dt (in A².s or J/) that represents the ability of the
current to deposit energy on a resistive object.
This current waveform is used by aircraft manufacturers in order to test and certify the good
behaviour of aircraft parts to lightning direct effects and/or indirect effects. Each component
represents a different phase from the current of blasting:
- Component A: Current of the first return stroke: this component has a peak intensity of
200kA ± 10%, an action integral of 2×106 A2.s ± 20% and a total duration of 500μs at the
maximum. The rise time from 10% to 90% of the peak value must be smaller than 50μs
(compared to the component D). This component can cause damage by direct effects or
indirect effects.
- Component B: intermediate current: this component has an average amplitude of 2kA ±
20% and an electrical load transferred of 10C ± 10% during 5ms ± 10%. This component
corresponds to a transition phase of the discharge.
- Component C: continuous current: This component transfers an electrical load of 200C ±
20% during a time between 0.25 and 1 second, with intensity from 200 to 800 A. This
component of limited amplitude can cause very important damages (in particular thermal
ones) because of large quantity of electrical charges that it deposits.
- Component D: current of the subsequent return stroke: this component has a peak
intensity of 100kA ± 10%, an action integral of 0.25 ×106 A2.s ± 20% and a total duration of
500μs at the maximum. The time from 10% to 90% of the peak value must be smaller than
25μs.
Chapter 2
12
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Figure 2-6: Normalized current for lightning strike tests in laboratory [3]
To observe and document the results of a lightning strike on materials, engineers have to
lead test campaigns where they subject a material to this standardized lightning strike.
Although there is not any particular norm about all the parameters to take into account
during testing material resistance to a lightning test, most of the reported tests follow
guidelines and norms defined by EUROCAE or SAE [1, 2]. Tested materials, metallic or
composite ones, are placed in front of an electrode delivering the current impulse (see figure
2.7(b)). This impulse must be controlled and is often a component D (reaching 100 kA in 20
µs) which is considered the most common and dangerous one for composite structures.
Nevertheless components A, B and C can also be used.
In order to generate such intensities or currents, HVG (High Voltage Generator) or HCG (High
Current Generator) are used. For the current work, all the lightning tests were conducted
using the ElectroMagnetic Means for Aerospace (EMMA) platform at the DGA Techniques
Aéronautiques (TA), in Toulouse, France, presented in figure 2.7 (a).
Figure 2-7: Lightning generator (a) and electrode for current delivery on test samples (b)
During those tests and mainly due to the high current intensity and temperatures reached by
the electric arc, very little instrumentation can be disposed to extract quantitative real-time
data. The only available data come from the rear face of the sample: displacement and
Chapter 2
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velocities at chosen location around the centre of the sample during the electric shock. To
measure real-time deflection during the strike, Velocity Interferometer System for Any
Reflector (VISAR) are used, such a technique allowing covering a velocity range of 10 cm/s to
3000 m/s with a precision better than 1 %. In parallel, on several samples, a stereoscopic
imaging technique is used with two Photron SA5 fast cameras at a rate of 262500 framesper-second (which yields a 128 x 128 pixel resolution) which image an optical field of 80 x 80
mm2 around the central portion of the sample, represented by 16 x 16 facets each 5mm
large.
Those tests were led by AGI and Airbus on protected and painted composite samples in
order to validate certain protections and material sequence as well as to study the influence
of several parameters such as the protection and paint thickness.
The deflection of the impacted sample is related to a sudden energy deposit on the surface,
which results from the Joule effect associated to electrical current circulation in the sample
as well as from heat transfer in the air gap between the generator electrode and the sample.
It induces matter vaporization and ejection at the surface, this surface explosion provides
momentum to the sample and subsequently deflects it. This deflection is the only data
available to work with and constitute the basis material for all the following work presented
in this document. The type of results obtained by using VISAR at the rear face of the
impacted samples is presented in figure 2.8.
95
NoECFNoP
5000
4000
ECF195NoP
3000
2000
ECF195P200
1000
0
0
1000
Time (µs)
2000
ECF73P200
NoECF-NoP
75
Speed (m/s)
deflection (µm)
6000
55
ECF195-NoP
35
ECF195-P200
15
ECF73-P200
-5
0
100
200
300 time (µs)
Figure 2-8:(a) Deflection as a function of time. (b) Speed as a function of time of the central point at the rear side of the
sample subjected to a D waveform lightning strike.
2.3 Direct effect damage induced by lightning
The replacement of metallic materials by composite ones led to new challenges. Indeed, the
behaviour of composite materials to lightning is completely different from the one observed
with metallic one, mainly due to their intern composition. In spite of the fact that the zoning
allows a better location of most probable impacts, this method is not sufficient enough to
insure the safety of the new composite structures. Aircraft manufacturers have thus
developed procedures and means of compliance with airworthiness regulations that allow
designing aircraft against lightning. Protections have been studied and integrated to the
composite structures, however such method tend to add weight on the plane, reducing the
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advantages of using low-weight materials such as composite. An optimization of this
protection is then necessary.
2.3.1 Metallic materials
Metallic materials such as aluminium have been largely used in aircraft structures as outer
skins and internal framework. Their high electrical and thermal conductivities provide an
inherent protection against lightning for an aircraft in flight and avoid, most of the time,
critical damage from lightning strikes. Thus, few protective measures had to be added. The
main effects of lightning strike on metallic skins and substructures are the following [11]:
- Melting at the attachment points
- Resistive temperature rise
- Magnetic force effects
Melting and burn through
The main damage resulting from these effects is the apparition of pinhole in the material
(see figure 2.9), mainly due to the action of the continuous component of the current
flowing through the material. However, melting or burn-through of skins is usually not a
safety-of-flight issue unless it occurs in a fuel tank skin: a relatively large amount of time is
needed for melting to occur, and lightning strikes are extremely rapid events.
Resistive heating
Aside from the burning of a hole at the attachment points, the mechanical properties of
metal structures will not be degraded unless the local temperature reaches the melting
point of the considered material [10, 11]. Because they are good electrical conductors,
metals spread out the current sufficiently fast to avoid such increase of temperature, except
within a few centimetres around the entry point (figure 2.9).
Bending due to magnetic forces
Metal skins can also be deformed due to the intense magnetic fields accompanying the
lightning currents near the attachment points. Pinching and crimping may then occur if the
structure is not sufficiently rigid [9-11]. Return stroke (waveform A and D) are the most
responsible component for magnetic force damage, which induce overstress or severe
bending of metals. Even if unusual, such damage can lead some part to severe failure with
potentially complete replacement when not repairable.
Shock wave and overpressure
The lightning arc attaching at the material surface during a stroke delivers current through
an ionized leader and delivers a very large amount of energy in a few microseconds (5 µs to
10 µs). This huge energy deposit causes the leader channel to expand at supersonic speed,
its temperature increasing up to 30 000K° and the channel pressure to 10 atmospheres [911]. At the end of the channel expansion, the arc reaches a diameter of several centimetres
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and the channel pressure is in equilibrium with the surrounding air. A cylindrical shock wave
expands radially from the centre of the arc and, in the case of a lightning strike on a
fuselage, is intercepted by a hard surface. The kinetic energy in the shock wave is thus
transformed into a pressure rise and in the shock wave itself, striking the material. It results
in an overpressure at the surface of the impacted material. Depending on the distance
between the arc and the surface, overpressures can reach several hundred atmospheres at
the surface, resulting in explosion type damage.
Shockwave propagating at the surface and in the thickness of the
material:
The lightning phenomenon is responsible for multiple effects that cause
mechanical damage to the impacted material. These components are yet
impossible to quantify but some of them have been identified (see chapter 3).
One of them, the acoustic shockwave due to the arc expansion propagates in two
directions: radially and in the z-direction of the material. The electric arc thus
damage the material both on the surface and in its core, which are two different
locations enduring two different loading and damage.
Figure 2-9: Lightning damage on aluminium skins (pinhole and magnetic forces action)
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2.3.2 Composite materials
2.3.2.1 Lightning induced damage
When it comes to lightning direct effects on composite thin panels two kinds of damage are
to be observed: the ones taking place in a thin volume near the exposed surface (designated
hereafter as the “surface damage”) and those occurring in the core of the laminate.
Concerning the surface, most of the damages are due to the very high temperature
accompanying the attachment of the arc on the material and result in burning of the first
ply’s carbon fibre, sublimation of the resin [10, 11].
It is important to note that, even if lightning strike on composite materials leads to extensive
damage at the surface of the laminate (see figure 2.10), the more detrimental damages take
place in the core of the material and cannot be seen without specific inspection of the
structure. Such defects can lead to extensive loss of mechanical properties with compressive
loading in particular, leading to a structural failure of an entire zone of the aircraft.
Figure 2-10: Lightning observable damage on a composite sample
The studies of damage mechanisms in composite materials submitted to a lightning strike
have been numerous since 2007 [ref a completer]. In order to identify these mechanisms
and acquire a better understanding of lightning direct effects on composite, two similar
studies, led in 1997 by Chen and al [12] and in 2008 by Y. Hirano and al [13] investigated
those effects on an epoxy matrix and IM600 graphite fibres composite. Tests were made
with different waveforms (component A) on specimens of 350×350 mm.
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These series of test runs shed light on three major failure modes:
-
Fibre breakage and fibre damage trough thickness due to peak current of the
lightning strike,
Resin deterioration,
Extensive delamination of the laminate.
The first mode concerning the fibre breakage is, according to [13], mainly due to the
shockwave that goes along the lightning strike. This result has been reinforced by the work
of T. Ogasawara and al [8], which held a similar study on carbon fibres. The shockwave
transmits a huge energy to the impacted surface; this energy rises and dissipates as heat by
Joule effect, increasing the pressure inside the laminate. The heating provokes the carbon
fibres’ sublimation, which is not limited to the surface but propagates through the laminate’s
thickness.
The second mode deals with resin deterioration. Studies [13, 14] are once again
corroborating about this damage mode, due to resistive heating from Joule effect on the
laminate’s surface and high atmosphere temperature. This temperature can heat up to
30 000K [13] which is fifty times higher than the pyrolysis temperature for conventional
epoxy resin (600K). Resin’s pyrolysis leads to a release of gas inside the composite and its
evaporation is responsible for the characteristic explosive fracture in the vicinity of the
lightning attachment point. This explosion is due to the failure of the dielectric material, the
resin, at the interface with the fibres.
The last mode deals with the observed delamination that propagates in the fibre
direction in each impacted ply. The resin decomposition and the fibre breakage are due to
the impact and to the increase of temperature. These coupled phenomena lead to a
delamination of the first ply on the surface that goes separated from the ply underneath.
This phenomenon leads to a loss of mechanical properties and to a critical failure of the
material which loses its cohesion.
These three failure modes are thus coupled and interact with one another leading to a final
and critical failure of the material.
Lightning strike damage have also been observed and studied by the mean of nondestructive and destructive testing [14-17, 47]. In the case of quasi-iso layup, the lightning
strike damage area presents a very peculiar shape made of two zones articulated around a
90°oriented delamination between the plies 3 (-45°) and 4 (90°).
In addition of the various damage that composite materials undergo when submitted to a
lightning strike, aircraft never fly with nude composite structures. Manufacturers usually add
several layers of paint above the structures for decorative or advertising purpose. As paint is
a non-conductive material, it tends to affect the overall response of the struck parts.
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Several studies [18-20, 23] have been led on the arc root itself, taking into account its
evolution through time and space as a function of the lightning current. They showed that
the arc root radius was a key parameter in the damage zone size for lightning waveform A
and D. They also highlight the fact that the damage zone shape was dependent of the
presence of paint: in the case of unpainted panels, the burnt marks were circular while they
presented a more irregular shape in the case of painted aluminium or composite samples.
Lago and al [18] worked on the influence of paint thickness on the deflection of aluminium
panels, using stereo correlation method. In this study, the type of protection varies but the
injected current (component D) and paint thickness are the same for two panels. They also
tested several paint thicknesses for the same protection. Their results shed light on the
detrimental effects of paint; indeed, the samples with 300µm of paint presented a larger
deflection than those with only 100 µm (see figure 2.11). They interpreted this increase in
rear face deflection as a constriction of the arc root due to the paint that increase the
magnetic and hydrodynamic pressure created by the arc. They also concluded that for
composite panels, an important thickness of paint induced greater damage and that the
vaporized protections transferred important mechanical stresses (loading, pressure) to the
material.
Figure 2-11: Spatial evolution of the rear face of several aluminium panels with various paint thickness [17].
Some authors also estimated the delamination area in the composite material using
ultrasonic C scan [14] or X ray analysis [19] and they reported a damaged area of thousands
of mm². The same tendency is extracted by [19] which tested several paint thickness and
protection on composite material: the thicker the dielectric layer over the protection foil the
greater the obtained damage and the arc penetration in the protection.
Studies [21, 22] have also shed light on the detrimental effects of the paint on the surface of
the panels as aircraft companies always paint their planes. Such dielectric layer on top of the
composite material tends to confine and increase the average size of the damage.
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Lepetit and al [21] studied the influence of the presence of paint above the composite
material tested for lightning strike tests. During these tests an explosion of this paint layer
was clearly observable. They explained this explosion as the result of an internal pressure
build-up under the paint layer and which contributed to mechanical damage. They added to
their analysis that the plasma induced by Joule heating in the material under the paint tends
to be confined by the presence of this dielectric paint layer, enhancing stress and
subsequent damage in the laminate.
This work has been pursued by AGI [22]. They worked on quantifying the influence of paint
thickness by numerically testing several layer thicknesses (figure 2.12) and comparing the
results with experimental data. The conclusions of this model are:
-
-
Presence of paint do confine explosion and increase significantly applied pressure
This applied max pressure and impulse vary as the square root of the paint surface
mass. This suggests to use light paints (with small thicknesses and densities) to
reduce damages.
The applied pressure also approximately increases as the square root of the input
surface energy. This means that the pressure is proportional to the square root of the
surface impedance, and inversely proportional to the arc root radius. A small arc root
will induce higher pressure. A small arc radius is expected for paints with high
dielectric rigidity.
Figure 2-12: Effect of paint thickness on equivalent pressure to lightning strike
2.3.3 Lightning strike protections
In order to protect composite material, in addition to the zoning procedure, several
protections have been designed to help maintaining the structural integrity of impacted
parts [11]. These protections aim to protect and prevent composite parts from serious
damage.
Ideally, a good lightning metal protection should have:
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- A good conductivity in order to evacuate lightning currents.
- A low temperature of fusion and vaporization.
- Light weight in order to minimize the additional weight due to the lightning protection.
- A good behaviour against corrosion (air, rain…).
- Good mechanical properties.
- Easy to maintain and repair.
- Easy to shape for complex structure elements shape.
- Low cost.
The protection should be external because:
- Easier to repair or maintain.
- Safer: the great part of the lightning current will remain external to the structure
Most of the time, these protections consist in an addition of a conductive material upon
the composite. A variety of protections have been investigated (see figure 2.13) such as:
-
Metallic wire mesh, Sheet or Expanded Copper Foil (SCF and ECF)
Paints suitable for conduction (metallic paint)
Metallic fibres in the first ply
Carbon nanotubes
(a)
(b)
Figure 2-13: Protections against lightning strikes (a) ECF and (b) SCF
The protection has to absorb the impact energy in order to limit the mechanical and
thermal effects on the laminate. It also has to be a good conductor to conduct the received
current and its associated energy out of the structure. Unfortunately, these protections are
not always sufficient and the systematic adding of paint (dielectric material) by aeronautical
companies decreases the effects of the protection. Moreover these protections are to cover
the whole structure and it results in an additional weight that reduces the gain made up by
using composite materials. The most common used protections are the following [15].
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Solid Metal Foils (SMF)
SMF are thin layer of various metal deposited on the top surface of the laminate, most of the
time, embedded in a layer of resin in order to insure their adhesion to the material. They
provide an additional conductive layer to the structure. Metal foils of 0.001in (0.025mm) or
greater provide protection for the composite that is about the same as that provided by wire
meshes. However and in spite of the excellent protection abilities of such foils,
manufacturing concerns have limited their application at large scale in industry. It is actually
extremely complicated to drape these foils over large or curved surfaces and structural
parts. To do so, one is obliged to cut the protection to prevent the formation of wrinkles on
which the arc may attach and provoke delamination and damage due to a failure of the
protection layup. The presence of seams, due to the cutting of the protection can also result
in unbounded areas that can confine moisture and thus deteriorate the foils by corrosion.
Because of these difficulties solid metal foils are less commonly employed than other
protection options.
Expanded metal foils (EMF)
Expanded foils are created from SMF by a milling process that perforates and stretches a
solid metal foil. Such foils resemble woven wire mesh but they are made out of a single piece
of metal, which insures a better general conductivity to the foil because of the lack of
contact between the wires. They are typically 0.05 to 0.1mm thick and are widely used in
aeronautics. The expanded foil can be done with aluminium, bronze or copper, which
provides the most effective lightning protection.
Fewer difficulties are encountered for draping EMF over large or curved surfaces than with
SMF, as they can be stretched somewhat. They efficiently promote arc root dispersion and
reduce thermal and shock wave damage. It should be noted that the expanded metal foils
provide a good electromagnetic shielding due to the good contact afforded with the
mechanical fasteners and hard metal surfaces.
However, after a lightning strike, the mesh is generally vaporized at the arc root area. This is
due to the thermal flux coming from the arc root and to the Joule effect inside the metal
wires. The wires diameter varies between 50μm and 250μm and so the current densities
inside the metal wires are very high. Even if the electrical conductivity is good, the energy
due to Joule effect is not negligible inside the metal wires and can damage the metal
meshes. In order to ensure an efficient bonding between the composite and the metal mesh,
the metal mesh is partly embedded in the first resin thickness of the composite materials.
And so, a part of the lightning current could circulate in the carbon fibre and, might damage
the composite material by Joule effect.
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Conductive paints
Another way to ensure more conductivity to a protected structure can be to use the top
layer applied above the composite materials. Paint usually added for purely decorative or
advertising purpose could be “functionalized”. Adding conductive particles to this paint
could be a way to increase the conductivity of the whole plane. This solution seems
attractive because it is necessary for corrosion and esthetical reasons to paint the aircrafts.
Moreover it has been shown [19, 20, 23] that the presence of paint above the metallic
protection has a detrimental effect of the lightning strike damage and tends to increase it
drastically as the dielectric nature of the paint concentrates the arc root in one location and
prevents it from moving along the structure. Most of the time, carbon, aluminium or copper
particles are added to the paint that covers the structures providing a certain amount of
conductivity to the mechanical parts and thus lightning protection. This protection is
marginal, however, since the conductive particles make only random contact with each
other.
Metalized carbon fibres
The point of this internal protection is to increase the conductivity of the carbon fibres
already present in the core of the composite material. To do so, the carbon fibre, prior to
their impregnation with resin are electroplated with metals such as copper or nickel [16].
After this operation the coated fibres can be integrated into epoxy resin in order to create
structural composite materials, as explained previously (section 2.3.2). Nickel is the most
used electroplated metal due to the fact it is relatively cheap, corrosion resistant and a good
electrical conductor. Such protections were primarily used for electromagnetic shielding
rather than lightning protection. The point of the process is thus to increase the conductivity
of the first composite ply, enough to withstand a lightning strike. Unfortunately the lightning
protection level of these coated fibres is lower than the one of Expanded Copper Foil (ECF),
which remains the most used lightning protection.
The presence of protection such as ECF allows reducing significantly the damage due to
lightning strike on composite structures. The extent of damage between a protected and an
unprotected sample is different both at the surface (near the impact point) and in the depth
of the material. Figure 2.14 compares two composite samples of same dimensions and
material. Sample 1 is unprotected while sample two is protected with ECF195 g/m². They
provide, for the same lightning strike, two different delaminated areas, respectively of 1845
and 935 mm², highlighting the beneficial influence of the protection to reduce damage.
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(a)
(b)
Figure 2-14: Visible damage after lightning strike on (a)unprotected sample and (b) protected sample (ECF195)
However these protections are not optimized and represent an additive weight on the
composite structures which were preferred to metal because of their light weight.
Moreover, lightning tests in laboratory are expensive and do not allow to extract a lot of
information. In order to size these protections and to improve the common knowledge of
lightning interaction, other means must be used. One of these means is the numerical
simulation which authorizes to test various configurations in a short time range.
2.4 Lightning modelling
For several decades, the study of direct effects of lightning strike on composite materials has
been led. Historically, lightning direct effects are modelled through coupled electrical and
thermal formalisms. Such models mainly take into account the arc attachment, the
distribution of current and temperature through the composite laminate as a basis to
explain the damage induced by lightning on such structures. However, for some years now,
other studies arise, working on the mechanical effects of the lightning strikes. In the
following section a separation will be made between studies working on the “surface
damage” (which is a thin volume near the exposed surface including the first ply and
potentially the paint and protection of taking into account/or present), and the studies
focused on the damage in the depth of the laminate [13-14, 41-47]. The first part,
concerning the electro-thermal modelling of lightning, offers a classification with growing
complexity of the proposed modelling methodologies.
2.4.1 Insights into lightning induced phenomena
Lightning is known to be a very complex phenomenon, involving multiple physics. As for
now, there are no definitions of the physics involved by a lightning arc or an accepted
chronology of their appearance over time and potential coupling, admitted by all the
scientific community. The following paragraph is a proposition of the lightning phenomenon
that gathers all the potential physics at stake during a strike and their possible influence and
appearance chronology during this strike.
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The presence of metallic protection, in which the delivered current can circulate
perpendicularly to the arc column, can also be responsible for the creation of a magnetic
field. It is admitted that lightning strike involves magnetic forces that can deflect the
impacted sample. Finally, an acoustic shockwave due to the overpressure created by the arc
column associated with internal unknown phenomena are suspected to be responsible for
internal damage such as delamination.
It is difficult to quantify the importance of each of the different physics that are part of the
lightning phenomenon, least of all to quantify their interaction and coupling with one
another. A representation of the possible various physics involved (thermal, electro,
magnetic and mechanical forces) is presented on Figure 2.15 with a first link to the observed
damage on composite material impacted by lightning.
Figure 2-15: Phenomena contributing to lightning direct effect on a fuselage panel [21]
Several phenomena are involved: multiple physics at different time scales, inducing various
phenomena that damage the impacted material.
The first phenomenon is the overpressure created by the arc column that induces a
macroscopic force F1 which can be partly responsible of paint tearing and generation of
delamination in the CFRP due to mechanical crushing of the fibres and the plies and sound
pressure propagating in the material.
Secondly the current I induced by the electric arc in the metallic protection as well as along
and across the carbon fibres and the plies induces itself both a Laplace force F2 and the
heating of the material (heat flux Q).
The third contribution occurs at the surface level, the energy deposited by the electric arc is
responsible of the sublimation of the metallic protection, which induces explosion (inducing
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force F3) and paint detachment. Finally the temperature increasing in the CFRP leads to
resin burning and gas cells explosion which induce delamination as well as overpressure
inside the material that generates the force F4.
The previous figure is a global figure of the lightning interaction with a composite material,
resulting of observations made during lightning test campaign, as it is understood today by
AGI. Such scenario is similar to those proposed by other works [14, 39], as shown on Figures
2.16 and 2.17. This illustration is used as a work basis for the future hypothesis and studies
presented in this section. However, it must be noted that, even though this is a common
figure of the phenomenon, other scenarios highlight the importance of the Joule effects, on
internal damage for example, and do not grant the same importance to mechanical forces
[48, 49].
Figure 2-16:Multi-physical actions of a lightning strike on composite material: thermal, electromagnetic and other
components [39]
Figure 2-17: Possible phenomenon occurring with through-thickness electrical conduction. [14]
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2.4.2 Electro-thermal model
The study of lightning direct effects has mainly been studied by the use of electro-thermal
models focusing on the Joule effects. These models [10-16, 24] thus mainly focus on the
surface damage observed on CFRP panels after laboratory lightning tests. Most of these
studies involve nude panels and it is rare to find in the open literature studies treating of
lightning impacts on protected and painted composite samples [24, 25].
Several studies [23-28] focused on the observed damage following a lightning strike. These
visible damages mainly occur at the “surface” and correspond mainly to removal of paint
and metal protection but may also extend to the first CFRP plies. The damaged surface is the
one which experiences large energy transfer from the arc root (figure 2.18). Surface damage
typically results from thermal effects induced by metal protection layer melting or
vaporization and possibly first carbon ply degradation (tufting, burning) [23, 24].
Figure 2-18: Surface damage after a lightning strike [23]
In such models, energy is injected in the material by the Joule effect induced by the electrical
current flowing through the material and protection, as well as by heat transfer from the arc
root. This energy input results in heating of the protection and material, possibly followed by
melting and vaporization. The extension of the damage is assessed on the basis of the
amount of these liquid or vapour phases. Such an approach is well adapted to describe
damages on metals and has been extended to carbon fibre composite materials for the
continuous C component [26-28].
In his study Lago and al [27] created a numerical model describing the arc root and its
interaction with the composite material, designed as an anode. The point was to quantify
the degradation level of the material as a function of the current intensity and duration,
mainly taking into account the Joule effects in the composite and its degrading effects on the
material. The model simulates the behaviour of a plasma column, representing the lightning
arc, and how it transfers current toward the anode, see figure 2.19.
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Figure 2-19: Temperature field in the plasma column [27]
This model, focused on Joule heating, shows how these effects are responsible for material
degradation. To do so, they compare the area affected by heat generation to the
delamination area obtained during real lightning strike tests. Finally, they investigate the
effects of Laplace forces by applying a convective external force to their plasma column in
order to represent the temporal evolution of the temperature through the material as well
as the deflection of the panels observed during lightning strike tests (figure 2.20). However
the deflections obtained were very low. Studies like this one are very interesting in order to
understand the arc attachment phenomenon and interaction with the impacted material;
however they do not take into account the internal and not visible damage resulting from a
lightning strike and only focus on the attachment problematic.
Figure 2-20: Temporal evolution of the temperature field in the composite material (a) convective force, (b) without
convective force [27]
Ogasawara et al. [14] also presented a coupled electromagnetic/thermal Finite Element (FE)
model to simulate lightning strikes numerically. Their model coupled electromagnetic and
thermal behaviour for specimens without lightning strike protections to compute the
temperature flux in the material. They assumed that the electrical conductivity evolved
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linearly with the temperature varying with matrix decomposition to carbon sublimation.
They showed that the electrical conductivity was effectively dependent of the temperature
growing in the laminate but did not take into account the material’s properties thermal
dependence or the link between electrical properties and thermal decomposition.
Following this first introduction of co-dependence between electrical and thermal
properties, Dong et al also [17] focused on the dependency of the electrical properties along
with temperature. In their study, a chemical reaction of the resin is added to explain the
thermal damage behaviour as a function of time, temperature and other space factors. They
worked on a coupled electrical-thermal-pyrolytic analysis to represent the lightning strike
actions on CFRP panels. They observed that the contours of the temperature field in each
interface drove the pyrolysis degree of the resin, and a strong material properties
dependence on the pyrolysis degree, which they assumed to be a key point in the lightning
strike simulation.
Chemartin and al [24] led a survey of the thermal and mechanical effects of lightning strike
on composite thin panels. Their work was again focused on the simulation, both numerical
and experimental, of the plasma channel, in order to represent the temperature and
conduction profile of this cathode. Their work presents the modelling of the electric arc
physics but without any modelling for the sample to be hit by their arc, neither the
interaction between the arc and its applied force on the resulting panel damage.
Several studies focused on the numerical simulation of the surface current density [29, 30].
Huchette and al [29] proposed an electro-thermal model focused on the pyrolysis
phenomenon induced by the lightning channel and damaging the impacted samples. They
worked on a coupled algorithm linking electrostatic analysis to a transient thermal analysis,
using FE method to estimate the thermal behaviour of the impacted structure. They firstly
compute the current density in the material, taking thus into account the pyrolysis of the
matrix by increasing the electrical and thermal conductivities, which are co-dependent [17].
Then the transient thermal model is used, taking into account the Joule effects applied on
the new material state evaluated after the first electrostatic calculations. The damages due
to thermal action are thus computed and compared to experimental matrix damage and
provide concordant contour areas.
Tholin and al [31], studied the interaction between the lightning arc and various aeronautic
skins, including composite and aluminium ones. They worked on various skins’ conductivity
due to Joule heating, Laplace forces and resistive heating, to study the influence of
material’s skin conductivity on the observed damage mechanisms and arc root spreading at
the surface. Some other works [27, 28] have been led on the arc root itself, taking into
account its evolution through time and space as a function of the lightning current (figure
2.21). They showed that the arc root radius was a key parameter in the damage zone size for
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lightning waveform A and D. They also highlight the fact that the damage zone shape was
dependent of the presence of paint: in the case of unpainted panels, the burnt marks were
circular while they presented a more irregular shape in the case of painted aluminium or
composite samples. Such results have also been reported by Lepetit and al [46], who worked
on an evolving equivalent pressure dependent of the arc radius over time, extracted from
lightning tests observations.
Figure 2-21: Evolution of the arc root radius over time [30]
Finally, in their study Abdelal and Murphy [32] also worked on a coupled electro-thermal
model but this time taking into account the formulation of temperature dependent material
properties, using FE method to model the composite material and the embedded ECF
protection. Contrary to previous study [27, 29-31, 33], which ignored the temperature
dependency of the electrical/thermal material properties, their simulation included the
modelling of various material states such as melting, evaporating or sublimation while
interacting with the composite panel through temperature dependent thermal conductance
properties, in order to predict the thermal damage in the laminate due to a lightning strike.
As for the others, they equate the decomposition area due to thermal effects to the damage
size obtained during effective lightning strikes and numerically computed the evolution of
the temperature profile through the material thickness (figure 2.22), as it is generally
assessed that temperature distribution through laminate material is affected by the
electrical anisotropy due to Joule effects [34].
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Figure 2-22: Decomposed laminate layout [32]
All the presented models are mainly focused on methodologies to represent the lightning
arc root. By doing so they mainly focus on extensive and visible damage occurring on the
first ply and the protection and paint layers when they are considered. They focus on the
attachment of the arc and the surface phenomena happening there, and tried with complex
model taking into account all the physics at stake during a lightning strike to model the arc
and to predict the visible damage, while core damage is not investigated. However, the most
detrimental damages for the composite structures are the ones occurring in the bulk of the
laminate and, most of the time, are not visible after a lightning strike. Such damage as
delaminations and fibre failure belittle the material mechanical properties sometimes up to
complete failure and weaken the structural integrity of the material. In order to understand
the related damage mechanisms, the previous methodology which consist in trying to
reproduce exactly the lightning phenomenon by coupling all the physics at stake, seems to
be too complicated, as it involves complex equations and physics.
In the following sections, several studies are presented that tried to separate the various
parts of the lightning phenomena (thermal, electric, magnetic and mechanical) to focus only
on one of them: the mechanical part of the loading and the associated damage. By doing so,
these studies mainly focused on the mechanical damage occurring inside the laminate and
which are the most detrimental.
2.4.3 Mechanical based models
In spite of the fact that most of the studies on lightning direct effects focus on thermal
damage and arc root interaction with the sample’s surface, there is growing evidence that,
at least for the impulse A or D components, mechanical phenomena can induce damages.
The following methodologies choose to ignore momentarily the electric properties of the
material and the coupling between the electromagnetic and the thermal behaviours. They
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entirely focus on mechanical damage and their potential origin and sources. Thus, they
ignore the decomposition of the material first plies (mainly the epoxy matrix due to high
thermal loads) and its effects on orthotropic electric/thermal conductivity of the material.
Although an extensive literature already exists regarding the impact damage on composite
materials due to foreign objects [35-37, 42, 50], only a few studies have been published on
the mechanical damage due to lightning strike. The first ones focused on residual strength
and structural performance of lightning strike CFRP [23, 25, 38-40].
Featherston and al developed a method to assess the shock effect due to lightning strike
[28]. They worked on the quantification of the different forces applied on CFRP structures by
the lightning arc, discriminating three major ones: acoustic shockwave, magnetic forces and
internal mechanical force due to the vaporization of the metallic protection creating gas
trapped between the material and the layer of paint. They tried to predict the peak
overpressure forces produced by the shockwave by comparing numerical displacement and
velocity of the impacted material with the experimental results. However their work only
focused on aluminium panels and did not take into account the damage resulting from the
lightning strike, but provide interesting methodology.
Hirano and al [13] investigated the mechanical damage observed after a lightning strike in
order to categorize and understand them. They worked on carbon fibre/epoxy composites
using an impulse current test facility, testing several kinds of electrical waveforms (figure
2.23). Their study revealed the different types of damage commonly observed in composite
laminate after a lightning strike, which are presented in section 2.3.2. These damages, of
mechanical origin, are mainly cracks in the matrix resin, fibre breakage, and delamination
and seem to be dependent on the electrical and thermal properties of the laminate.
Figure 2-23: Overview of lightning damage for various lightning strike impacts [13]
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Some other works focus on the impulse waveform of lightning strikes and their effects.
Mechanical momentum induced on samples by lightning strikes has been measured [41-43],
and thermo-mechanical models have been proposed to account for the observed damages
[42, 43]. It has been shown [42] that the transferred momentum can result from magnetic
forces induced by the circulation of current in the sample as well as from air and surface
explosion shock waves, possibly enhanced by the confining effect of paint on the surface.
The thermal model therefore needs to be supplemented by mechanical concepts to provide
a valuable understanding of damage processes. Haigh [41] made a first step in this direction
by focusing on mechanical effects within the material and more particularly on the
mechanical impulse (i.e the force transferred by the electrical arc and integrated over time).
To do so, they mechanically and optically instrumented lightning strike tests in laboratory.
The mechanical impulse is then extracted from deflection measurements and compared
with traditional mechanical impacts. By this mean, they could quantify the mechanical
impulses of order of magnitude around 1 N.s. However they could still make no clear
correlation between mechanical impulse measurements and observed mechanical damage.
Gineste and al [44], on the other hand, pursued the work of Haigh by working on deflection
measurements during lightning strike test at different location on the impacted material,
using high resolution visars. From these measurements, a model was built to compute the
mechanical impulse. Moreover, a thermo-mechanical model was developed to describe
damage in the material.
Karch and al [45] also worked on lightning current pulse responsible for non-thermal damage
on protected CFRP structures. They focused on magnetic forces, the shock waves due to
supersonic channel expansion, and the shock waves due to near surface-explosions related
to the plasma channel expansion over time and the size of the arc root radius as it was
initiated in [42]. They computed the pressure associated to the magnetic forces and
shockwave applied on the sample’s surface in order to quantify the importance of the
separate contribution of electrical, thermal and mechanical loads and extracted several
tendencies such as the small actual contribution of the magnetic forces.
All the above models were focused on the mechanical impact observed after a lightning
strike, which consist in a very different approach compared to what is generally found in the
lightning literature. They tried to find potential explanation for these damages and chose to
separate all the physics present during a lightning strike in order to quantify the importance
of the mechanical part. This is a first step toward a simpler representation of lightning and
another lead for the comprehension of the phenomenon.
2.4.4 Energy based model
In spite of the fact that several studies address the mechanical damage found in CFRP
laminate struck by lightning, only a few of them consider them from a mechanical point of
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view. Feraboli and Kawakami [46] are the first to compare damage from lightning strikes
with traditional mechanical impact. Besides, the design of purely mechanical drop weight
tests which could be considered as equivalent to lightning tests has been investigated in this
study. They focus on low velocity impact with drop tower tests and they investigate a mean
to link the energy deposited by a lightning strike into a composite structure with the one
transmitted via a mechanical impact. They inflicted damage on CFRP plates (unpainted and
unprotected) with both lightning strike and mechanical impacts prior to testing the damage
resistance and tolerance after impact of the samples. Their equivalence criterion is the
transferred energy in the material, comparing the kA of the electric arc with the Joules
transmitted with mechanical impacts, as presented in figure 2.24.
This method allows them equating lightning strike delivered intensity with several “threat
level” corresponding to mechanical impact damage. They use this equivalence to conduct
“equivalent level mechanical impact to lightning strikes”.
Lightning test from
10kA to 200kA
Equivalence between Lightning strike (kA)
and impact damage level (J)
Impacted CFRP panels are tested in tension
& compression for residual strength
Equivalence between lightning threat and mechanical requirements
Figure 2-24: Methodology to couple lightning tests to mechanical impacts [48]
They showed by using non-destructive ultrasonic testing that the mechanical impacts
provided greater damage than equivalent lightning strike but that the damages obtained
were of the same nature: a mix of fibre failure, matrix cracks and extensive delamination, i.e
no burning and no damage that could be attributed to EM effects (see Figures 2.25 and
2.26).
Figure 2-25:Ultrasonic C-scan of post lightning strike specimens at 30, 50 and 70kA [45]
Chapter 2
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Figure 2-26: Ultrasonic C-scan of post impact specimens at several energy (6.78J, 20.34J, 33.89J) [44]
They concluded that the strain energy transmitted by the mechanical impact was much
lower than the energy dissipated during a lightning strike and that such an energy based
comparison was not the good approach. However, the similarities of the damage obtained
by both methods and the residual performance of the two kinds of impact samples were
encouraging enough to them to pursue their research effort.
2.5 Conclusions and perspectives
Lightning is a multi-physical phenomenon involving multiple physical components such as
electromagnetic, thermal and mechanical ones. The focus has been made here on direct
effects that are responsible for structural damage of the impacted panels in order to
optimize the design of adequate lightning strike protections. It is thus necessary to
understand the influence of each parameters (paint thickness, type of paint, type of
protection, mass of the protection and so on) in order to reproduce as faithfully as possible
the event in laboratory and be able to measure, in time and space, these direct effects.
These measurements are not yet possible due to the large temperature and intensity of the
various loading involved in a lightning strike: electric (100 kA), thermal (10 000 to 30 000 °C)
and mechanics (damage generated in 5 to 20 µs). The review of the lightning related
literature, regarding the related damage, showed that historically, lightning strike damage
and direct effects on thin composite panels have been considered through an electrothermal vision. Studies on the subject mainly focused on the representation of the electric
arc and plasma channel and its attachment on the composite surface, modelling the
electrical current as well as the thermal flux transmitted from it. These studies favoured an
ascending methodology by modelling the arc channel in its whole complexity. They thus
focused on the surface damage and interactions, and don’t address core damage.
Nevertheless, some studies tried another approach by working on the damage observed, not
only at the surface and top layers, but also in the depth of the material. They concluded that
these damages were very similar to those observed during common mechanical impact tests
such as drop-tower tests. They then worked on the phenomena responsible for such peculiar
damage. To do so, and contrary to the previous electro-thermal oriented studies, they
decided to investigate the lightning strike by separating the various components of the
loading. They focused on the phenomena that could induce mechanical damage such as the
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Laplace forces or the shockwave resulting from the attachment of the arc. Finally, Feraboli
even compared pure mechanical impacts damage with lightning strikes ones with few
success.
The study put aside the question of the appearance of each loading, their relative
importance and their coupling, which remains unknown. A first step, led by the latest studies
on mechanical damage, was made on trying to discriminate the importance of one of the
physics among the others, through a step by step approach that consisted in considering and
studying only one instead of them all in order to quantify it.
The author intends to give some more insights into this last trend observed in the literature:
it is assumed that the phenomenon remains largely unknown today. In fact, the lightning
strike is today impossible to model in its full complexity as it is yet. The phenomenology, the
quantification of each physics (thermal, electro-magnetic and mechanical), their coupling is
yet unknown, which makes extremely complex and difficult to model the arc channel.
However, the damages are observable and quantifiable. A reverse method is thus being
favoured and the following work will consider the phenomena responsible of these damages
as a black box in order to investigate and quantify the mechanical part of the lightning strike
in the induced damage. This work will help to qualify and give insight of the coupling needed
to fully represent the lightning strike effects on CFRP thin structures.
Chapter 2
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Chapter 2
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Chapter 2
38
3
Chapter 3
Preliminary works and methodology
definition
3.1 Preliminary works .................................................................................................................................. 401
3.1.1 Industrial framework ............................................................................................................................... 41
3.1.2 Preliminary attempts for an equivalent pressure .................................................................................... 41
3.1.3 Compute a magnetic pressure using experimental arc root ................................................................... 44
3.1.4 Analysis of lightning induced damage ..................................................................................................... 49
3.2 Objectives and methodology .................................................................................................................... 54
3.3 From lightning strike to mechanical impacts ............................................................................................ 57
3.4.1 Equivalent impact to lightning strike ....................................................................................................... 57
3.4.2 Creation of a shell model to design the equivalent impact parameters ................................................. 64
3.4.3 Validation of the shell model and impact parameters {m, v} .................................................................. 65
3.4.4 Rear face displacement profiles .............................................................................................................. 69
3.4 Analytical determination of the variable arc root ..................................................................................... 71
3.5 Conclusion ................................................................................................................................................ 73
Chapter 3
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Objectives
This chapter presents an analysis of existing works and results that we have intended to use
as a basis of our work. The objectives of the presented work and the prospects of the
methodology of work are presented as a consequence of the analysis. The analytical
computations performed in our work and the proposed equivalence are presented as well as
analytical computations of the arc root versus time at the end of this chapter.
These theoretical and analytical developments are the basis of the experimental and
numerical modelling presented in chapters 4 and 5.
Chapter 3
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3.1 Preliminary works
The present work on lightning strike damage analysis is the continuation of an activity led by
Airbus Group Innovations (AGI) and Airbus over the last few decades. This field of research
became critical when the use of composite material extended to primary structures. Indeed,
those lighter materials are poor electrical conductors when compared to the previous
metallic ones. New challenges arise, especially in the study of the lightning interaction with
the composite structures and the definition of adequate protections to be used. This chapter
is focused on the researches led by AGI on this particular subject which serve as a basis for
this PhD work.
3.1.1 Industrial framework
AGI has worked on this particular topic for several years. Multiple lightning test campaigns
have been run in order to improve the understanding of the consequences of lightning on
thin composite panels and improve the metallic protections that are already present on
aircrafts and which could be optimized. All the lightning tests performed during the work
presented were part of the lightning tests campaigns led by AGI over the past three years to
improve CFRP protections.
Thanks to the experimental data, the multiple physics involved in the lightning phenomenon
have been investigated and identified and a preliminary chronology of the effects of a strike
has been established. It must be noted though that experimental data is rather complex to
obtain in the case of lightning tests. Indeed, lightning is a very fast event (pulse component
of the current is less than 100 µs) which involves very high temperatures, up to 30 000 K and
very high current intensity up to 200 kA. Lightning strike tests also generate very bright arcs
that tend to saturate traditional cameras disposed for event recording. The rapidity
(lightning current discharged in less than 100µs) and extreme conditions of the tests
(luminosity of the arc and extreme temperature of the plasma) make the implementation of
contact instrumentations on the sample rather tricky. Consequently, during these tests, two
kinds of results only were obtained. Firstly, with the help of an interferometric apparatus
called VISARs, the rear face deflection and velocity of the samples during the impact have
been measured versus time. Secondly, post-mortem analyses were conducted to measure
total delamination area and distribution of these delamination through thickness. With all
these data, several studies are led in order to qualify and quantify the physics at stake during
a lightning strike and thus improve the understanding of the event in order to optimize the
conceived protections.
3.1.2 Preliminary attempts for an equivalent pressure
A first study was led on 2011 by AGI focusing on the rear face displacement and velocity
obtained during test campaign and on how to reproduce it numerically. To reproduce the
global mechanical stress applied to the panel during the current tests, as a combination of
the different forces described before, a constant pressure limited in space and time has
been first considered. The hypothesis to divide the phenomena occurring at the top of the
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sample and their consequence in term of damage is made. Thus, the modelling of the
different potential contributions to this pressure is not engaged, but only the representation
and modelling of what their results may look like. As a first guess, an equivalent pressure
[43] applied at the centre of a numerical model of the sample is sought in order to
reproduce the external measures of rear face displacement and velocity obtained during the
lightning tests. A numerical model using shell elements has been created using the finite
element software ABAQUS 6.11: samples were reproduced as faithful to the test samples as
possible (size, shape, clamping and boundary conditions), and the pressure has been applied
at the centre of the plate.
There were several possible contributors to this pressure with limited clue on their spatial or
temporal distribution. The time dependence of this form of pressure was a step and its
spatial distribution of pressure a Gaussian distributed on a radius r, centred around the
central attachment point of the arc with pressure amplitude A, as shown on Figure 3.1 [50].
Figure 3-1: Spatial and temporal distribution of the equivalent pressure for rear face displacement comparison [50]
The experimental test set up was equipped with five VISARs (IDF1 to IDF5 according to the
figure 3.2) to measure the samples’ displacements versus time at different locations.
Figure 3-2: Scheme of the position of the five VISARs for displacement measurement during lightning tests
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The rear face displacement at the corresponding locations were then computed with the
numerical model and then compared with the experimental results. Iterative computations
were run, calibrating T, r and A until good agreement was obtained when comparing rear
face deflection between experimental and numerical results.
An iterative computation of following parameters is conducted until a good agreement with
experimental displacement is obtained:
T : duration of the pressure application, deduced from the time at maximum speed
(figure 3.3 (b))
35
Vmax
30
Lightning sample 1
Speed (m/s)
25
20
15
10
T
5
0
0
50
100
Time (µs)
150
200
Figure 3-3: Determination of T for the equivalent pressure
Once the pressure applied at the centre of the sample, the sample moved backward, its
speed increased. As the pressure was constant during the application time, the sample
acceleration was a constant too (F=ma), so the speed was linear. When pressure was
released, the sample came back to its original position (and starts oscillating because of
boundaries conditions) and its speed decreased.
r: "radius" of the pressure, half maximum width of the spatial Gaussian distribution, is
approximated according to the delay observed between VISARs signals (large radius 
small delay)
then the amplitude A of the pressure is numerically adjusted.
This first approximation was tested on quasi-isotropic material made of T700/M21 used by
AGI for certification and research campaign. The numerical model used a homogenised
simplified material model to reproduce the lightning sample [50]. Figure 3.4 presents the
comparison between experimental results and numerical computations. Good agreement
have been obtained with the numerical model, mainly for the central VISAR (IDF5 in black)
for which it can be seen that the numerical pressure succeeded in representing the rear face
displacement of the experimental sample. For the other measure points, the model provided
a good approximation but with less accuracy than for the central VISAR. These first results
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evidenced the fact that it was possible to reproduce the mechanical behaviour of lightning
strike plates, without considering the possible physical contributions. The proposed reverse
method took into account the results of a lightning strike on a composite structure in order
to reproduce it in a mechanical way.
Figure 3-4: Example of rear face deflection for both lightning test and numerical pressure comparison (k: 0.25 N.s, T: 40
µs, r: 2.24 cm), Continuous line: experimental result. Dashed line: numerical model [50]
3.1.3 Compute a magnetic pressure using experimental arc root
3.1.3.1 Experimental arc root
By 2012, preliminary results gained from rough simulations with the first guessed equivalent
pressure encouraged to try improving the pressure formulation. This was the main
motivation for a Master Thesis during period 2011-2012 as preparatory work for the PhD.
The study of lightning campaigns highlighted several facts. The samples were placed on a
vertical support perpendicular to the electrode delivering the current (Figure 3.5). This
injected current was delivered by a high current generator visible on Figure 3.6 (a) and
recorded during the lightning strike (Figure 3.6 (b)). The use of real time high-speed camera
(see Figure 3.6 (a)) observations provided insight in the phenomena occurring at the surface
of the impacted samples [47].
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High
current
generator
Figure 3-5: View of the lightning set up and of the electrode delivering the current to the composite samples
Injected current
Intensity (kA)
100
Sample
support
80
60
40
20
0
0
50
Time (µs)
100
(a)
(b)
Figure 3-6: (a) View of the lightning set up and (b) injected current for waveform D lightning tests
Rings of high luminosity appeared with radius increasing over time, as seen on Figure 3.8.
The four images correspond to several instants of the arc attachment as explained on Figure
3.7. These rings were interpreted as the time dependent rim of the heated surface or air.
Burnt
zone
Plasma
Electrode
(a) t=2µs
(b) t=4µs
(c) t=6µs
Figure 3-7:Scheme of the growing lightning arc attachment root
Chapter 3
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(a)t=2µs
(b) t=4µs
Reflection of
the arc in the
paint above
the sample
Electrode
Rim of the
sample
(c) t=6µs
(d) t=8µs
Figure 3-8: Pictures of the strike in its initiation and delivering phases. Ordered from top left to bottom right, the pictures
with a time step of 2 µs. The arc was initiated less than 2 µs before the first picture [43]. Right part of the picture
corresponds to t
Low resistance areas are the preferred path for the injected current, thus it flows mainly
through its rim, which is a better conductor than the composite plate and which is heated by
Joule effect and energy transfer from the arc plasma. This annular portion of the protection
is subsequently consumed, i.e. vaporized or ejected from the sample surface. This process
results in a hollow arc root with an annular shape increasing with time and sublimation of
the metallic protection by the high temperatures generated by the arc. While expanding, the
plasma emits light that provides information on the size and shape of the arc. The size of the
arc root can thus be evaluated directly from the experimental pictures and provide data on
its expansion over time. The final damage area has been taken equal to the burnt area
measured directly on impacted samples.
3.1.3.2 Magnetic forces modelling
Obviously lightning is a complex phenomenon, involving multiple physics. As a consequence,
the preferred path for modelling is a sequential approach: consider only one of the physics
and quantify its influence and action on several results obtained during lightning tests, such
as rear face displacement.
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By separating the complex event, a better understanding and control of the damage
mechanisms related to the event is hoped. A correct model to represent the core
mechanical damages requires the knowledge of the strain and stress state of the material as
a function of time, including failure criteria. A necessary input to the model is some time
dependent externally applied loading which represents a mechanical impact equivalent to
the lightning strike. Different possible contributions to this applied stress have been
considered in previous publications [41-43, 51]. One is the planar shock wave at the surface
of the material, induced by the Joule effect in the metallic protection, enhanced by the
confinement due to the presence of the paint above the protection [43]. A second possible
contribution is the pressure induced by a shock in air resulting from sudden energy deposit
in the lightning arc channel [51]. A third contribution is the Laplace forces resulting from the
interaction between the current flowing through the material and the magnetic field
induced by the arc [41].
Tests and modelling were performed during the thesis work to evaluate the relative weights
of these three contributions. First, deflection measurements were performed for two kinds
of samples. The first one was the standard sample type: quasi-iso (layup: [45°/0°/135°/90°]s)
CFRP (Carbon Fibre Reinforced Plastic) sample made of 8 plies and having a thickness about
1.5 mm. The samples were 450x450 mm² squares with 12 fixations disposed on a circle of
diameter Φ370 mm. The second one was obtained from the previous one by drilling a
central hole (Φ 10mm) in the sample (see Figure 3.9). This allowed insertion of the electrode
in the hole and suppressed as much as possible the arc between electrode and sample.
Differences on the deflection between standard and drilled samples were therefore
expected to result from the air shock contribution induced by the presence of the arc.
However, no significant difference on the defection was observed between drilled and undrilled cases. This suggested that the first contribution to the external pressure, the shock in
the air, is not significant.
Composite
sample with a
central hole
Electrode
placed
inside the
sample
(a)
(b)
Figure 3-9: Composite sample with a central hole (b) before lightning impact, (c) after lightning impact
Chapter 3
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High speed cameras evidenced that a significant surface explosion still occurs, in those
particular drilled cases, and contribute to deflection. Thus, two major contributions may be
responsible for the deflection: Laplace forces and Joule effect induced surface explosion.
The point here was to determine the influence of these contributions. It is possible to
analytically calculate the contribution of the Laplace forces alone, and to implement them in
a numerical model to compute associated rear face displacement due to these forces and
compare them to experimental results.
The formulation of the distribution of magnetic forces using the appropriate laws was
proposed. Magnetic pressure is due to Laplace Forces, resulting from a coupling between a
current into a magnetic field:
𝐹 = 𝑗˄𝐵
(Equation 3.6)
In order to study the impact of the magnetic pressure, a pressure was built resulting from
Laplace forces only.
An order of magnitude of the applied magnetic pressure P(r,t) at distance r from the drilled
centre is given by the product of current density and magnetic field ( Ampere’s law) :
𝑖(𝑡)
𝑗(𝑟, 𝑡) = 2𝜋𝑟𝑒
(Equation 3.7)
With e the thickness of the protection. And magnetic field:
𝐵(𝑟, 𝑡) =
𝜇0 𝑖(𝑡)
2𝜋𝑟
(Equation 3.8)
So the magnetic pressure is expressed as:
𝜇
𝑖(𝑡) 2
)
𝑟
0
𝑃(𝑟) = 4𝜋²
×(
(Equation 3.9)
With u0 the magnetic constant (u0=10-7×4π). A simplifying hypothesis was made to consider
that all the metallic mesh ideally conducts the current and that the magnetic field generating
the Laplace force was present on the entire sample surface. Figure 3.10 shows the computed
deflections resulting from this magnetic contribution at two points of the sample (labeled
visar 1: at the rim of the central hole, and visar 2: 25 mm sideward) and compares them with
the measured one, which includes surface explosion also [41]. There is some delay in the
onset of the deflection at visar # 2 location, as compared to visar # 1. This is due to the fact
that the applied pressure was larger near visar # 1 than near visar # 2 as the arc attached
itself nearer to visar #1. Deflection reaches a stable value near 200 μs. The measured
deflection was typically twice larger than the computed one. This indicates that the magnetic
contribution to the deflection was significant and had a similar order of magnitude to the
contribution than the surface explosion.
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Model visar 1
Model visar 2
Test Visar 1
Test Visar 2
3,5
Deflection (mm)
3
2,5
2
1,5
1
0,5
0
0
100
200
300
400
500
Time (µs)
Figure 3-10: Measured and computed deflections as a function of time.
3.1.4 Analysis of lightning induced damage
All the previous works and observations were based on the external measurements
performed during lightning strike tests. In addition to these tests data some non-destructive
and destructive post-analyses were done in order to assess the damage. Such tests involved
ultrasonic C-scans and micro-cuts of the samples. By using these methods, two damage
types have been identified. The first ones are surface damage mainly in the metallic
protection, the layer of paint and potentially the first ply of CFRP and are essentially due to
thermal effects as shown on Figure 3.11. Surface damages involve sublimation of the
metallic mesh and removal of the paint above the protection as well as burning of the first
ply’s carbon fibres and epoxy resin.
Figure 3-11: Observable damage after lightning strike test
Chapter 3
49
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The second types of damages, called core damage are located in the core of the composite
panel itself. They are suspected to be induced by mechanical stresses propagating in the
depth of the material. Ultrasonic scanning provides information on the size and shape of the
damage in the thickness of the material thanks to a 2D projected view of the damage, as
shown on Figure 3.12.
This figure presents three different lightning impacts and illustrates the variety of damage
profile obtained during experimental tests. Samples (a), (b) and (c) are all protected with
ECF195 and respectively paint with 200, 700 and 160µm of paint. All the samples are made
of composite material (total thickness 1.4 mm) and are constituted of 8 plies with the
following sequence: [45/0/-45/90]s. The ultrasonic analysis is performed from the rear face
of the samples, which is the face opposite to the lightning strike attachment. For more
information on ultrasonic analysis, see Appendix B. The colour scale represents the thickness
of the material, with 1.4 corresponding to the surface of the impacted sample. Thanks to this
scale, it is possible to extract the position of each delamination. For example in sample (b),
the large delamination circled by a black ellipse corresponds to a delamination at the
thickness 0.5 or at the interface between plies 2 and 3. The comparison of these different
lightened samples highlights the variety of damage and the influence of the surface state
(and more particularly of the paint thickness) on the amount of damage generated by
lightning. The observed damages through C-scans greatly remind of the damage features
obtained during classical mechanical impacts which are completely different from those
observed at the “surface” of the impacted samples. These observations led to a separation
of the two different kind of damage found in lightened samples.
(a)
(b)
(c)
Figure 3-12: C-scans for 3 different lightning strikes
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Another argument for substantiating such differentiation between surface and core damage
is the nature of the observed patterns, as presented on Figure 3.14. Indeed, surface
damages seem to be entirely due to thermal and electric effects of the arc root and the
current flowing on the top layers of the samples. They mainly concerns the sublimation of
the copper protection due to extremely high and localized temperatures, burning and
sublimation of the paint and protection layers due to high pressure under the coating and
sometimes of the carbon fibres of the first CFRP ply. In the core of the material there is no
evidence of thermal damage. The observed damages are fibre/resin debonding, transverse
crack, fibre rupture and ply delamination. The study of cross sections seems to confirm that
no thermal damage has affected the core of the material, indeed, the difference between
upper burnt plies and lower ones is quite visible in Figure 3.13.
Figure 3-13: Zoom on micro cuts: difference between thermally damage plies (on the top) and thickness damage.
Microscopic observations of the micro cuts have been made but no further study was led on
the study of thermal damage inside the ply, although none of them was found. Figure 3.14
shows the burnt resin and copper mesh at the top of two samples, both protected with ECF
195 and painted with respectively 200 and 400 µm of paint. Figure 3.15 also presents details
on the surface of the impacted sample. The figure shows microscopic details of the first ply
of the same sample that the one presented in figure 3.13, in the impacted area. The damage
visible at the surface of the impacted samples clearly shows the thermal damage. This kind
of damage is not visible in the micro cuts made in the thickness of the samples but no
microscopic observation of the lower plies was conducted.
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Melted
copper
mesh
Heated
resin
(a)
(b)
Figure 3-14: Microscopic view of (a) heating resin and (b) melted copper mesh
Figure 3-15: External surface of lightning strike samples (microscopic observations)
The presence of large delamination, matrix crack and fibre rupture, is similar to damage
observed in low velocity impacts [52], thus core damages seem to have a mechanical origin
only. Moreover, a comparison has been made between lightning core damage and
traditional damage observed with mechanical impacts such as drop tower or canon gas tests,
as shown in figure 3.16. The two kinds of impacts generate damage of the same kind which
provides another argument in favour of a separation between surface and core damage and
to investigate a mechanical origin for the second ones.
Chapter 3
52
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Figure 3-16: Comparison of core damage induced by both lightning strike and mechanical tests [48]
Ultrasonic scanning also allows extracting information on the damages size and repartition
generated by lightning strike on composite materials. Figure 3.17 presents the damage
repartition through the material’s thickness for two lightning impacts, both protected with
ECF195 and painted with 160 µm of paint. A φ6mm area deprived of paint was placed at the
centre of the sample 107 plate in order to study the influence of the arc root expansion
while consuming the protection. The diagrams present the delamination area versus
material depth. The position 0 stands for the impacted side of the sample. Figure 3.17 shows
that for lightning strike, the major part of the damages is located in the first half of the
material’s thickness. However, it is possible to find damage going through the entire
laminate up to the rear face, as shown of Figure 3.18 which presents a microscopic
examination of another lightning strike test, protected with ECF195 and with a 200 µm paint
layer. On this figure, back face peeling and matrix crack extended to the last ply of the lay-up
are visible after a lightning strike.
delaminated area (mm²)
800
700
Lightning test 107
600
Lightning test 101
500
400
300
200
100
0
Thickness (mm)
Figure 3-17: Damage distribution through thickness for two lightning strikes
Chapter 3
53
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Matrix
crack
Rear
face
peeling
Figure 3-18: Rear face peeling observed on a lightning strike sample, microscopic observations
3.2 Objectives and methodology
Lightning-material interaction involves several physics such as mechanic, electromagnetic
and thermal components as well as probable coupling between them that are not easily
measureable and quantifiable. From post-tests observations, evidence has been made that
the surface explosion due to the attachment of the electric arc on the material’s surface and
the core damage which are a result of the events occurring at the surface are of different
nature and origin (electro-thermal for the surface versus mechanical for the core), see Figure
3.19. It has been shown that those core damages were very similar to those obtained with a
pure mechanical impact such as drop tower or gas gun tests. Their origin seems thus to be of
a mechanical nature as well. Moreover, such damage are the more detrimental one for the
composite structures as they greatly belittle the structural integrity of the materials, the
accent is thus put on these particular damages.
Figure 3-19: Surface and core damages
Chapter 3
54
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The point of the PhD work is the comprehension of these phenomena and more particularly
the damage mechanisms related to the impact of lightning on thin composite structures.
Such comprehension is mandatory for the optimization of the lightning protections that are
already implemented on aircrafts. Lightning strike tests are expensive, complex and have
provided limited information to help the understanding of the associated damage
mechanisms. To overcome these issues, a numerical predictive model is considered to
improve the knowledge of these damage mechanisms.
The assessment of these differences between surface and core damage, by their location,
type and origin, was the starting point of the work hypothesis, as well as the strong similarity
which existed between damage resulting from lightning strikes and traditional mechanical
impacts. Then if lightning strikes result in mechanical damage in CFRP material, it would be
possible to reproduce these damages by using pure mechanical means.
Hypothesis 1: it is possible to separate the damage at the surface and in the core
of the impacted samples and thus focus on the core induced damage and their
mechanical origin
Hypothesis 2: It is possible to mechanically reproduce the rear face deflection of a
sample struck by lightning and the damage caused by it: can a mechanical test
able to do so as well be designed?
The lightning issue is thus divided into two parts: the electro-thermal study that induce a
mechanical pressure on the surface of the sample and the mechanical study that links the
surface pressure to the mechanical damage observed in the core of the sample.
The work presented in this report focuses on the second study, as shown in figure 3.20. The
surface pressure is then considered as an entry data which remains unknown. A mechanical
model has been created and validated through mechanical equivalent impacts to lightning.
These tests are especially justified due to the strong resemblance between lightning damage
with traditional impact damage. Moreover, they are simpler, more reproducible and
mastered than the lightning ones.
Chapter 3
55
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Figure 3-20: Scope of the work flow
The following work will aim to find an equivalence allowing going from the lightning strike
physics to mechanics and then to design and optimize an equivalent mechanical impact to
lightning strike tests and verify the validity of the work hypothesis of an equivalent impact.
The mechanical impact tests are not intended to replicate the surface damage which do not
result from a purely mechanical cause, but presumably the depth damage (delamination)
occurring in the laminate assuming that similarity of deflections implies similarity of
delamination. There are at least two reasons for attempting to use the deflection criterion to
define the equivalence between lightning and impacts. One is that deflection and velocity
are the only quantitative real-time data which is actually available from lightning tests.
Another criterion which could have been used is the amount of energy available in a
mechanical test and in lightning. This has been investigated in [47] and was not successful,
the main reason being that only a part of the electrical energy available in the capacitors
goes into mechanical damages.
Based on the hypothesis that the deflection criterion can lead to equivalence between
lightning and impacts, the process developed in the frame of the thesis is the following one:
• Perform lightning tests where real-time deflections and post-strike damages are
measured,
• Design impact conditions (nature of the impactor mass, size and speed) by both
analytical and simple numerical modelling to reproduce deflections and speeds
similar to those observed in a lightning impact,
• Design and perform mechanical impact tests and numerical model to obtain
mechanical deflections and damages,
• Compare impact deflections and damages to lightning ones,
• Compare core numerical impact induced damage with damage induced by other
equivalent mechanical loadings.
Chapter 3
56
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Mechanical equivalence and thus impact tests will be checked with lightning strike results by
several equivalence parameters that are data extracted from lightning strike tests such as:
rear face displacement and speed of displacement. The validity of the mechanical
equivalence will also be checked by comparison of the damage in the core of the material. At
long term, the following work that consists in a first study of a mechanical representation of
lightning strike coupled with predictive numerical models would be used to also simulate the
surface events
In the future, the mechanical tests and models designed during this work could be used to
reduce the number of lightning strike tests and provide insights on the influence of selected
parameters (such as material, lay-up) and recommendations for early phases of protections
and structures designs.
3.3 From lightning strike to mechanical impacts
It is then proposed to proceed as following:




Propose an experiment of mechanical impact using a projectile that is equivalent to a
lightning strike in terms of the global kinematics of the target, and compute the
projectile characteristics
Propose a corresponding numerical model that reproduces with confidence the
experimental tests
Propose an equivalent surface pressure that do not use any experimental output as input
to determine the loading in time and space
Use the numerical model derived in 2) to simulate an equivalent lighting using only a
mechanical description of the sample behaviour and the applied loading.
3.3.1 Equivalent impact to lightning strike
In order to design an equivalent impact, an equivalent parameter is to be found that allows
to link lightning strike to mechanics. To do so, the information extracted during lightning
strike tests is used. As presented in chapter 2, the only available data from a lightning strike
in laboratory are the rear face displacement and velocity of the sample during the shock,
measured by VISARs. All the lightning samples used for this study are extracted from the
programmed lightning test campaigns led by Airbus Group Innovations with the lightning
apparatus 3.22) disposed in the DGA TA, in Toulouse, France. These campaigns aimed at
testing the performance of several lightning protections and investigating the effects of
several parameters such as paint thickness.
Chapter 3
57
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Results of the VISAR measurements are shown on figure 3.21 presenting deflections and
velocities at the centre of the plate’s rear face for the different cases. For this primary study
several surface states were studied. All the samples were made of the same material and
presented the same sequence [45/0/-45/90]s, they only differ by the type of metallic
protection (ECF73g/m² or ECF195g/m²) and the paint thickness above the protection
(unpainted samples or 200µm paint, see Table 3.1). Two types of material were used,
respectively called CFRP1 and CFRP2.
Material
CFRP1
CFRP1
CFRP1
CFRP1
CFRP2
Samples #
1
2
3
4
5
Surface state
NoECF-NoP
ECF195-NoP
ECF195-P200
ECF73-P200
ECF195-P200
Table 3.1: Presentation of the lightning campaign cases
Analysis of lightning results
The samples 1 and 2, without paint on the surface, exhibit both lower deflections and
velocities than those with a layer of paint, shedding light on the detrimental effect of paint
during the lightning strikes on both rear face displacements and velocity as previously
mentioned in section 2.1.1.2. These tests data also show the influence of the protection
layer used. In fact for the same amount of paint, sample 4 (ECF73) reaches higher deflections
than sample 3 (ECF195). This indicates that the metallic covering plays a beneficial
protection role by decreasing the amount of energy dissipated at the surface and thus
lowering the remaining energy that is transmitted down to the rear face. It is concluded
from figure 3.21 that rear face deflection and speed strongly depend on the surface state
and can be used as representative parameters to characterize the sudden energy deposit on
the surface. This energy results from the Joule effect associated to electrical current
circulation in the sample as well as from heat transfer with the electrical arc located in the
air gap between the generator electrode and sample (see section 2.4.1).
For a lightning strike, the energy delivered corresponds to the sum of the energy actually
dissipated in the material, the one dissipated in the arc formation and the one returned to
the ground. As the quantity to be determined is the energy dissipated in the material, two
configurations arise from the previous analysis. In the case of samples with no surface paint
the damaging energy is the one actually dissipated in the material while for protected
and/or painted samples, a certain amount of energy is dissipated in the metallic mesh
sublimation before entering the composite material. For the study, the mechanical
equivalent test is thus expected to give results similar to strike for the non-painted samples
Chapter 3
58
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at short and large times. However in the case of painted samples, the surface state clearly
influences the structural response of the sample. Regarding painted samples, similarity is
thus expected only during the phase of acceleration of the plate due to the shock as the
surface state has been proved to be a contributor to the deflection of the plate.
(a)
(b)
Figure 3-21:(a) Deflection as a function of time of the central point at the rear side of the sample subjected to a D
waveform lightning strike. (b) Velocities as a function of time for the first 4 samples
Figure 3-22:Lightning strike lab apparatus, DGA TA.
Preliminary approach
The aim of the present analysis is to define impact test conditions, which provide speed and
deflection at the centre of the rear face as close as possible to the ones measured in the
lightning tests with no ECF and no paint. A mechanical impact is defined by a projectile
hitting a sample at a given speed, such a projectile is thus to be defined. As a preliminary
approach, it is assumed that projectiles are spherical, made out of steel, do not deform, and
hit the target normally. Thus, mechanical impacts are fully determined by radius r (thus
mass) and velocity v of the projectile. The strategy is to find r and v by solving an inverse
Chapter 3
59
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problem setting the equivalence of the energy dissipated between the energy deposit and
the kinetic energy recorded at the bottom face. To solve this inverse problem, sudden
character of the surface explosions is used to estimate the delivered strike impulse. As
shown on figure 4.1(b), the sample speed reaches its maximum, in the range 10-80 m/s, with
a rise time ranging from a few μs to a few tens of μs (called “short times”). This short time
represents the time during which the equivalence is valid. The speed decreases to small
values within 100 to 150μs, which corresponds to the duration of a lightning strike (called
“strike time”), and is stabilized near zero at about 300μs (called “large times”).
Lightning Strike characteristics time intervals:
‘Short times’: 10-50 or 10-80 µs, velocity at the rear face centre reaches a
maximum value
‘Strike times’: 50-100 µs or 100-150, velocity drops suddenly
‘Stabilization times’: 100-300µs or 150-500µs
‘Large times’: over 300 or 500µs, peak shock is passed and other phenomena
arise.
On another hand, it is shown on figure 4.1(a) that the deflections of the samples are
dropping down after 1.4ms and tend to reach zero at about 3ms which is consistent with the
natural vibration period T of these samples T=6ms obtained from analytical formulations. In
the time range before T/2, the boundary conditions do not affect the samples’ behaviour
which can be considered as infinite. For large times, the analytical formulation of the
deflection of an infinite plate subjected to a mechanical pressure is therefore used.
Impulse calculation from measured deflections at large times
Calculations are made at large times (about 300μs) which are long enough to authorize
considering that the pressure has been fully delivered, and short enough to emphasize that
the plate is free of boundary. The Green’s function gives the deflection at time t and radius
𝑟 = (𝑥 2 + 𝑦 2 )1/2 for a Dirac delta-function impulse applied at time t=0 at the centre r=0 of
an infinite plate ([53], p. 124):
1
𝜋
𝐺(𝑟, 𝑡) = 4𝜋√𝜇𝐷 ( 2 − 𝑌(𝑡)𝑆𝑖 (
𝑟2
𝐷
𝑡
𝜌ℎ
4√
))
Equation 3.1
Chapter 3
60
__________________________________________________________________________________
Where  is the surfacic mass (kg/m²) of the sample, D (N.m) the bending stiffness. Y(t) is the
𝑧 sin(𝑡)
step function at t=0, Si the sinus integral function: 𝑆𝑖(𝑧) = ∫0
𝑡
𝑑𝑡.
The Green’s function is useful to obtain the deflection d(x,y,t,) at the point (x,y) and time t
for any external applied pressure field P(,,t) with the convolution product:
𝑑(𝑥, 𝑦, 𝑡) = ∫ 𝑑𝜏 ∬ 𝑑𝜉𝑑𝜂𝐺(√(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2 , 𝑡 − 𝜏)𝑃(𝜉, 𝜂, 𝜏)
Equation 3.2
At large times, with respect to the impact duration, the deflection reaches a constant value
called d, which can be obtained from insertion of an asymptotic form of Equation 3.1 into
Equation 3.2.𝐺(𝑟, 𝑡) =
1
4𝜋 √𝜇𝐷
𝜋
(2 − 𝑌(𝑡)𝑆𝑖 (
𝑟2
𝐷
4 √ 𝑡
𝜌ℎ
𝑘
𝑑∞ = 8√𝜇𝐷
))
Equation 3.1
Equation 3.3
Where k is the impulse resulting from the integration of the pressure field applied on the
sample. K is a parameter initially calculated in the Greszczuk theory [54], which allows
determining analytically the mass and velocity of an impactor based on Hertz contact theory,
see Appendix C for details.
𝑘 = ∫ 𝑑𝜏 ∬ 𝑑𝜉𝑑𝜂 𝑃(𝜉, 𝜂, 𝜏)
Equation 3.4
The deflection at large times is extracted from figure 3.24 and the corresponding impulse
from Equation 3.31. With this procedure, the applied pressure P(,,t) and integrated
impulse k are surface state dependent, although no explicit model of the metallic mesh and
paint layer is used. The impulse is thus associated to the momentum mv of a projectile of
mass m, reaching the sample at the speed v, as a consequence of equation.3.
A method to link lightning strike physics to pure mechanical impacts has been set by using
the transferred impulse k. It has been shown that it is possible to equate this impulse k to
1
In practice the following procedure is used: the samples are quasi-isotropic, the equivalent
in-plane Young modulus and Poisson ration are obtained from standard laminate theory
(Exx=46,2 GPa and xy=0,29 for CFRP1 and Exx=57,6 GPa and xy=0,32 for CFRP2). The
𝐸𝑥𝑥ℎ3
corresponding bending stiffness 𝐷𝑥𝑥 = 12(1−𝜈𝑥𝑦 2 )are 33.6 N.m (e=3mm) and 18.0 N.m
(e=1.5mm), surfacic mass are μ=3.1kg/m² and 2.3 kg/m², for CFRP1 and CFRP2 respectively.
Chapter 3
61
__________________________________________________________________________________
the product of the mass by the speed of a projectile (as it was primarily done with the
Greszczuk formulation in Appendix C):
k=m.v
Equation 3.5
As a first approximation the maximum speed of deflection measured during lightning strike
tests is used as the speed of the projectile. Then, the mass is measured as following:
m=k/v*max. The couples (m, v) associated to each lightning strike are thus obtained.2
Presentation of the method
An example of the method is made by using sample 1 from table 3.1. Firstly the maximum
velocity value is extracted from the experimental data obtained during lightning strike test,
as shown on figure 3.23. The maximum value is extracted from velocity vs time curves and
the speed associated with this lightning test #1 is vmax=29m/s. The time duration to this
peak value is also extracted as the “short time” which defines the equivalence time for
further equivalent mechanical impacts. This duration T is of 47µs.
35
Vmax
Lightning sample 1
30
Speed (m/s)
25
20
15
10
5
Short times
0
0
50
time (µs)
100
150
200
Figure 3-23: Velocity vs time curve for lightning sample #1
The study of the displacement curve allows extracting the value of d∞ at large time. This
value is obtained by calculating an average value of the rear face displacement curve up to
almost 2ms, see figure 3.24. For sample #1, d∞ is approximatively 2800µm. the value of the
impulse k is then calculated by using Equation 3. For this sample #1, k=0.228 N.s. Finally, by
2
In practice here, the radius is obtained from its mass using the steel density 7927 kg/m3.
Chapter 3
62
__________________________________________________________________________________
using the maximum velocity of the sample as a fixed value for the projectile speed, the mass
of the associated projectile can be calculated using Equation 3.5. The obtained mass m is of
7.8g. Table 3.2 gathered the values of maximum velocity, short time for equivalence and d∞
for the samples presented in table 3.1. It is seen that the greater the rear face velocity of the
plate, the shorter the equivalence time is and the greater the value of d∞. This method is
applicable as it is but in another fashion as well as presented in the following insert.
3500
d∞
Deflection (µm)
3000
2500
2000
1500
1000
500
Lightning sample 1
0
0
500
Time (µs)
1000
1500
Figure 3-24: Displacement vs time curve for lightning sample #1
Sample #
1
2
3
4
Vmax (m/s)
29
10.6
57
83.5
Short times (µs)
47
34
38
11.5
d∞ (µm)
2800
2000
3500
4400
K (N.s)
0.228
0.16
0.285
0.359
m (g)
7.8
150
5
4.3
Table 3.2: equivalence values for the lightning samples 1 to 4
Using the mechanical equivalence method:
There are two ways to work with the established formula k=m.v:
1) when k is fixed  use Vmax from lightning test  calculate the adequate
mass m
2) when k is fixed  fix a mass m for the projectile  adjust velocity V
The two methods were tested but for the following work presented here, the
second method will be used to adapt to experimental set up and existing set of
projectiles.
Chapter 3
63
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3.3.2 Creation of a shell model to design the equivalent impact parameters
Preliminary estimates of mass and impact speeds for the equivalent projectile have been
defined. In order to refine the method and the equivalence, a simple numerical model is
built to test the projectile hypothesis and compare rear face displacement results with
lightning strikes ones.
The ABAQUS-explicit Finite Element software is used to determine the behaviour of nude
composite plates subjected to mechanical impacts of projectiles. The model represents the
samples, the clamped boundary conditions of the lightning strike tests, and the projectile.
The samples are square plates of dimensions 400x400mm2 clamped by 12 fasteners
disposed along a circle of diameter 370mm. Material input data are summarized in table
3.3.
Exx
(GPa)
154
E22
(GPa)
8.5
E33
(GPa)
8.5
ν12
0.35
ν13
0.35
ν23
0.3
G12
(GPa)
4.2
G13
(GPa)
4.2
G23
(GPa)
4.2
Table 3.3: Material properties used for numerical shell simulations of T800S/M21
The impactor is modelled as an analytical rigid body (analytical sphere). It is assigned an
initial mass and a perpendicular velocity and is positioned above the centre of the plate. A
penalty contact without friction and damping is defined between the projectile and the
target. For this preliminary stage of the study, where impact configurations equivalent to
strikes are designed, composite samples are modelled using shell elements (S4) with four
integration points and a reduced integration (S4R) under the hypothesis of small
perturbations and are set as homogenized elastic orthotropic material following the classical
lamination theory. The mesh (see Figure 3.25) is composed of rectangular elements in
majority. In order to reduce the computation time, a progressive refinement mesh is
prescribed, with density increasing from the boundaries toward the centre of the plate.
Figure 3-25:1⁄4 of the numerical mesh.
Chapter 3
64
__________________________________________________________________________________
3.3.3 Validation of the shell model and impact parameters {m, v}
Several lightning samples are selected in the attempt to define an equivalent impact to
lightning strike. The samples are carbon-epoxy square panels of 400x400 mm2 made of 8
CFRP plies with the following lay-up: [45/0/- 45/90]s. They are all protected by expanded
copper foils, of mass 195 g/m² (ECF195) in lightning strike tests. They differ by their surface
states only: they are covered with expanded copper foils of mass 195 g/m2 (ECF195) and are
painted with different thickness (see section 2.1.2 for protection and paint effects). A small
hole of paint is artificially created on sample 107.
These samples have been lightened on the EMMA platform at DGA-TA Lightning Lab in
Toulouse. The electrical waveform D (peak 100 kA) with rise-times of the order of 20μs and
total duration 100μs was used. Deflections are measured during the test with an
interferometry technique (VISAR).
In parallel to the lightning tests, and according to the range of maximum speeds reached by
samples struck by lightning, a research was led to determine the mechanical apparatus that
would best suit the available experimental data and selected samples. The mechanical
equivalent impact to lightning is defined by several mechanical parameters which are:
-
The impulse k (N.s),
The impact duration Δt,
The electric arc radius.
The point is to find an experimental set up that can satisfy those equivalence parameters.
Several apparatus have been investigated such as:
-
Hyper velocity impact [60, 62, 63, 69, 70]
LASER [58, 60],
SHPB (Split Hopkinson Pressure Bar),
Drop tower/low velocity impact [56, 57, 59, 64-68],
Gas gun [55, 59, 61, 62, 71].
A review has been done on a large basis of test apparatus found in the literature [55-71] to
compare the preceding parameters and calculate the obtained impulse before comparing
the average value of each experimental test to the lightning value to be approached. For
each apparatus, the mass and diameter of standard projectile, velocity range for impact,
material for the projectile, the type of damage generated as well as the impact duration
have been investigated. With the available data on these set up, impulse, energy and
equivalent force was calculated and compared to lightning results.
Chapter 3
65
__________________________________________________________________________________
This preliminary study allows moving aside several technologies. LASER and low velocity
impact set up provide respectively too high and too low impulse value compared to the ones
obtained with lightning. In the case of the LASER, the extremely high temperature and
concentrated force in a small location generated by the impact make this device improper
for the study. Low velocity/high mass impacts provide too high values of impulse while the
opposite configuration, high velocity/small mass such as HSPB, provides too low value of k.
SHPB also works only with very small samples of dimension 1x1cm² while the lightning
samples are very large plates.
It appears that canon gas guns are the most suitable apparatus as they can eject a projectile
of several grams to several hundred in wide range of velocity from 50 up to more than
250m/s. The Institut Clément Ader (ICA) in Toulouse already possessed such a canon gun
with a velocity range of 50-150m/s and a set of projectiles. By comparing the lightning tests
measurements with the available experimental means, a projectile that was already
available at the lab was chosen: a 4g steel ball of diameter 9,8mm. This mass will be used to
compute the associated speed for the selected lightning strikes.
Taking those measurements as inputs for the numerical model, impact conditions leading to
deflections quite similar to the measured strike ones were computed. Table 3.4 provides the
lightning samples used for this study and their original surface state. These samples were
part of an AGI lightning test campaign that aimed to validate several lightning strike
protection and study the influence of paint on the obtained damage.
Lightning
Sample
#
101
102
103
107
110
Surface state (paint +
protection)
ECF195-P160
ECF195-P200
ECFP195-P50
ECF195-P160+
unpainted on a disk ф 6
mm
ECF195-P200 +
unpainted on a disk ф
12 mm
Table 3.4: Lightning samples’ surface states
Figures 3.26 and 3.27 present the lightning data extracted from the experimental tests.
Thanks to this curves, the maximum velocities, short time and value of d∞ are obtained for
sample #103, which is presented here. The maximum velocity is of 42m/s at 18µs. d∞ is
averaged at 3200µm on the rear face displacement curve. Thus by using Equation 3, the
value of the impulse k is calculated such as k=0.164 N.s. As the mass of the projectile is fixed
Chapter 3
66
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at 4g, a first value of velocity is calculated, v=41.2 m/s, which is very similar to the value
obtained when extracting the max velocity peak.
50
45
Lightning sample 103
Velocity (m/s)
40
35
30
25
20
15
10
5
0
0
100
200
Time (µs)
300
400
Figure 3-26: Velocity vs time curve for lightning sample#103
The obtained values of mass and velocity are included in the numerical model in order to
compute the rear face displacement of the plate and compare it with the lightning strike
curves. Several numerical refinement on the projectile velocity are necessary until the
matching couple (m,v) which reproduce at best the lightning results is found.
4000
Displacement (µm)
3500
3000
2500
2000
1500
1000
Lightning sample 103
500
0
0
50
100
150
200
Time (µs)
250
300
350
Figure 3-27: Rear face displacement vs time curve for lightning sample #103
Figure 3.28 presents the final curve of rear face displacement obtained with the numerical
model and compared with lightning strike results. The projectile velocity is 72 m/s. The
computed impact deflection and velocity (slope of the curve) are in good agreement with
measured lightning strike ones at short times (<80μs). At larger times however, the lightning
deflection continues to increase whereas the mechanical impact one reaches the expected
plateau and then decreases. This indicates that some external force continues to apply on
Chapter 3
67
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the material at late time of the strike, which does not occur in the mechanical impact
configuration. Notice that lightning strike ends at 100μs from an electrical point of view. It is
speculated that this extra force must be related to possible thermo-mechanical phenomena
occurring in the metal protection and paint layers at the surface and which are delayed with
respect to the lightning strike. Table 3.5 gathered the final couples (m, v) for several
lightning samples.
4
3,5
Deflection (mm)
3
2,5
2
1,5
Lightning test 103
1
Impact test 65 m/s
0,5
0
0
50
100
150
200
250
300
Time (µs)
Figure 3-28: Comparison of rear face deflections between lightning strike (plain line) and numerical simulation with a
steel projectile (dotted line).
Lightning
Sample
#
101
102
103
107
110
Surface state
(paint + protection)
Calculated
speed (m/s)
Projectile
mass (g)
ECF195-P160
ECFP195-P200
ECFP195-P50
ECF195-P160 +
72
130
65
81
4
4
4
4
unpainted on a disk ф 6
mm
ECF195-P200 +
unpainted on a disk ф
12 mm
87.5
4
Table 3.5: Characteristics of the equivalent mechanical projectile to the associated lightning strikes.
The numerical simulations, by providing satisfying results when compared to lightning strike
ones, show that the equivalence method is valid and can be applied further. It also confirms
the validity of the work hypothesis 2: it is possible to mechanically reproduce the rear face
displacement of lightning strike tests.
Chapter 3
68
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In order to fully validate this hypothesis and the chosen equivalence parameters, an
experimental campaign is led on the basis of the lightning sample presented in table 3.5. The
associated mechanical impact to these lightning samples are tested and presented in the
following sections.
3.3.4 Rear face displacement profiles
In order to better understand the difference between the equivalent impact and the
associated lightning strikes, a study is lead on the surface loading due to both impact. To do
so, several lightning samples are studied and instrumented with stereoscopic technique
using two Photron SA5 fast cameras (figure 3.29(a)) at a rate of 262500 frames-per-second
(which yields a 128*128 pixel resolution) to measure deflection and speed of the 80*80 mm²
central portion of the sample’s rear face, represented by 16*16 facets each 5mm large,
allowing to capture the sample displacement over a profile section, as shown on figure
3.30(b).
Fast cameras
for
displacement
recording
Sample
5
6
7
8
9
10
11
12
13
14
15
16
(a)
17
18
19
20
21
22
23
24
25
26
Stereo-correlation
window for
displacement
profile
measurements
(b)
Figure 3-29:: (a) Stereo-correlation device at the rear face of the impacted sample and (b) speckle marks for stereocorrelation measure of displacement at the rear face of the sample
Chapter 3
69
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Figure 3.31 shows profiles of the impacted sample along segments centred on the impact
point for two samples with different surfaces states called CFRP1 (ECF195-no paint) and 2
(ECF195-200µm paint). Computed profiles were obtained for the corresponding impacts
using the shell model. The surface state appears to be a major parameter controlling the
profile shape. In the absence of paint (Figure 3.30 (a)), the central portion of the sample is
weakly distorted, the material recoils to the shock as a whole, which is an indication that the
surface stress induced by the strike on the surface is rather diffused and has a large base.
(a)
(b)
Figure 3-30: Deflection of the central portion of the samples vs time. Full lines: impact simulations; dotted lines: lightning
strike tests. (a) CFRP1; (b) CFRP2.
By contrast, in presence of paint (figure 3.31 (b)), there is some distortion of the sample, and
the displacement is about twice the value of the unpainted sample, indicating a more
concentrated surface stress. This should be related to a more confined surface explosion due
to the strike in the experiment.
The corresponding impact profiles also induce significant distortion of the sample. This
indicates that the impact is expected to be more representative of the strike in absence of
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70
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paint. Moreover the equivalent mechanical impacts generate a different surface pressure,
sharper than the one observed during lightning strikes, even in absence of paint. Such a
difference is important because it impacts the way the energy is delivered through the
material. The sharp shape of the mechanical pressure, compared to the lightning one, is due
to the shell model which tends to enhance the puncture and surface crushing phenomenon.
However, a steel ball of diameter Φ9.8mm and mass 4 g, is too small and is not the
appropriate choice to represent the surface pressure exerted by the lightning arc. This
difference on the loading applied on the surface of the samples is responsible for the
difference in damage and more particularly on the damage process and distribution
observed between lightning and mechanical impacts.
3.3.5 Analytical determination of the variable arc root
The qualitative arc root growing scenario identified from previous works is implemented into
a quantitative arc root model providing time dependent damaged surface area and possibly
a post-strike damage area prediction.
For the sake of exploratory modelling, a first set of hypothesis was assumed: Joule effect was
the only source of energy deposited into the material and heat transfer from the arc was
neglected, which is typically one order of magnitude smaller is neglected. A metal
protection, with thickness e, surface density δ, volume density ρ, and electrical conductivity
σ of the metallic mesh, was considered. The objective was to reproduce the expansion of the
arc root which express as a growing damage area of the metallic mesh.
The following notations are used: ra(t) the time t dependent arc root radius and ΔH the
vaporization enthalpy. The annular elementary volume is given by:
dV=2πerdr
(Equation 3.1)
The electrical resistance of this annular elementary volume in the protection with radius r is
given by:
1 𝑑𝑟
𝑑𝑅 = 𝜎 2𝜋𝑒𝑟
(Equation 3.2)
This volume disappears at time t(r) when sufficient energy has been dissipated by Joule
effect in the protection:
𝑡(𝑟)
𝜌 ∆𝐻 𝑑𝑉 = 𝑧 ∫0
𝑑𝑅 𝐼(𝜏)2 𝑑𝜏
(Equation 3.3)
z being an empirical correction factor taking into account the shape of the metallic
protection and I(t) the time dependent current waveform. This provides a time dependent
arc root radius:
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1
1
𝑟𝑎(𝑡) = 2𝜋𝛿 (
𝑡1
𝑘𝜌 ∫0 𝐼(𝜏)²𝑑𝜏 2
𝜎
∆𝐻
)
(Equation 3.4)
And time dependent damaged area:
2
1
𝑆(𝑡) = 𝜋𝑟𝑎(𝑡) = 4𝜋𝛿2
𝑡1
𝜎
𝑘𝜌 ∫0 𝐼(𝜏)2 𝑑𝜏
∆𝐻
(Equation 3.5)
This model has been implemented for a solid copper foil (see figure 2.14) with k=1, δ=88
g/m2, ρ=8940 kg/m3, ΔH=620 kJ/kg (fusion enthalpy, heating and phase transition included).
As the resistivity of copper increased linearly with temperature, the time dependent
damaged area was evaluated for 2 temperatures: 20 °C and 1000 °C (close to melting
1
temperature), using the resistivity values 1.68 10-8 Ωm and 13.1 10-8 Ωm such as 𝜌 = 𝜎 with
𝜌 the electrical resistivity and σ the electrical conductivity used in equations 3.4 and 3.5.
Figure 3.31 compares the two computed damaged areas with experimental results, which
were estimated on pictures provided by the high speed cameras, and from the final damage
extension measured on the sample after strike. The model provided a satisfactory bracketing
of the experimental results.
Figure 3-31: Damaged area as a function of time. The experimental points have been evaluated from images acquired
with high speed camera, except for the last point which corresponds to the final damaged area measured on the sample
after strike.
On Figure 3.19, experimental points were similar to the results obtained for the calculation
with 1000°C temperature at short times (<20µs). This was thought as the consequence of a
sudden rise of temperatures during the first 10/15 µs. The start of the “damage plateau”
seems to begin at 20µs as well. It was concluded from these observations that all the
damaging loading was transferred in [0, 20µs] or [0, 50µs] maximum. As only two extreme
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72
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temperatures were tested with the numerical model and as it seemed that the experimental
data fit between them, it would be interesting to test intermediate values of temperature to
obtain a better correlation and information on the approximate temperature to reproduce
the observed surface damage.
It is concluded from this figure that the model provides a satisfactory bracketing of the
experimental results, thus validating the proposed modelling strategy to compute the
evolution of the loading application zone resulting from the complex physics phenomena in
the above parts of the composite. But still remains an issue on the loading amplitude
evaluation.
3.4 Conclusion
The chapter 2 presented the lightning complex electro-thermo-mechanical phenomenon and
its direct effects on aircrafts. In this chapter, we have focused our attention on a much more
simple way of considering the lightning strike event, by considering the aeronautical
structure as the arrangement of a surface and a core parts. It is proposed to look at the
damage in the core part only, considering the complex phenomena arising at the surface as
resulting in a pure mechanical loading applied on the composite core.
Indeed, the analysis of the core damage of impacted composite samples provided a strong
resemblance of the observed damage with the ones obtained with classical mechanical
impacts (drop tower, canon gas gun), such as matrix and fibre cracks and delamination. By
analysing lightning strike damaged samples, it has been stated that the electro-thermal
damage occurring at the surface and the ones obtained in the core were of different types
and origin and can be decorrelated from one another. The focus has been made on the
second ones, excluding the surface events that are at the origin of the mechanical loading
responsible for the mechanical damage observed in the thickness of impacted laminates.
Then, if lightning generates mechanical damage, it can be possible to reproduce them by
using pure mechanical means. From this hypothesis, equivalence has been set, through
analytical computations, between lightning strikes on a complete aeronautical panel
(painted and protected) and mechanical impacts on the composite part (bare material
without paint or protection), based on the transferred impulse and using external
parameters such as rear face displacement and velocity of the impacted samples.
The principle of the equivalence has been proved efficient between lightning strike on a
complete plate and mechanical impacts on bare composite parts through the similarity of
the structural behaviour between aeronautical panels subjected to lightning strike and a
simple numerical shell model of the composite laminate. This simple model also allowed
determining the velocity and mass of a projectile when the impulse is that of a mechanical
impact. With this model preliminary comparative and predictive results are obtained and
compared to lightning and allows designing a mechanical impact test campaign.
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73
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From the analytical and numerical computations, several results have been highlighted. A set
of characteristic times has been evidenced. These characteristic times are important to
understand the equivalence phases and important instants of the chronology of lightning
such as the apparition of the damage. The equivalence method has been validated over the
so-called short times and sometimes up to the stabilization times for particular samples.
Finally, a separation in space between the surface and the core has been established and
first insights on the evolution of the loading surface have been made.
The following chapter focuses on the design of the mechanical campaign and compared the
obtained results with the lightning strike ones.
Chapter 3
74
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Chapter 3
75
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Chapter 3
76
4
Chapter 4
From lightning strikes to mechanical
impacts
4.1
Mechanical impact tests ................................................................................................................... 789
4.1.1
Specimen manufacturing ............................................................................................................ 79
4.1.2
Canon test: set up and instrumentation ..................................................................................... 80
4.2
Test results .......................................................................................................................................... 82
4.2.1
Macroscopic results .................................................................................................................... 82
4.2.2
Rear face displacement results ................................................................................................... 84
4.2.3
Conclusions on the structural behavior ...................................................................................... 87
4.3
Analysis of mechanical impact tests damage .................................................................................... 88
4.3.1
Review of damage in composite materials ................................................................................. 88
4.3.2
Analysis of post-mortem examination of mechanical impact tests ............................................ 91
4.4
Comparison of lightning strike and mechanical impact damage ...................................................... 99
4.4.1
Rear face displacement............................................................................................................. 100
4.4.2
Total delaminated area ............................................................................................................. 103
4.4.3
Influence of surface state ......................................................................................................... 104
4.5
Conclusions ....................................................................................................................................... 112
Chapter 4
77
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Objectives
This chapter presents the results of experimental mechanical impact tests on composite thin
plates. The aim of this work is to validate the equivalence of the designed mechanical impact
with a lightning strike.
The developed experimental setup is described, as well as the experimental plan. Results
and comparison under consideration are structural and material related behaviours. For
structural results, comparisons are made between rear face displacement and velocity
versus time. These results are used to further evaluate the validity of the equivalence
strategy that was first estimated using a numerical shell model. For material behaviour,
comparisons are made between the total delaminated area and the spatial distribution of
the damage in the core of the composite plates.
Chapter 4
78
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4.1 Mechanical impact tests
There are very few studies of such equivalence in open literature, the main one using an
energetic equivalence being the one of Feraboli [39] but with few successes, as presented in
section 2.4.3.
When seeking for mechanical impacts equivalent to lightning strike, we have proposed as
the equivalence parameter to use the transferred impulse k (N.s) which can be equalized to
the product of the mass of a projectile by its velocity when the impulse is delivered through
an impact. In this study the focus is made on small mass and medium to high velocity using a
gas gun (canon) apparatus. The velocity range is v ϵ {50,150} m/s for reproducibility of each
shooting and the projectile mass is 4g.
4.1.1 Specimen manufacturing
The specimens are prepared by hand lay-up of unidirectional material as explained in
Appendix A. The sequence is a quasi-isotropic lay-up of eight plies as following: [45/0/45/90]s. The samples are square plates of dimensions 400×400mm² with a thickness of 2mm
made of T800/M21. Average thickness of T800/M21 plies is 0.25mm. The samples’ curing
cycle and polymerization, with respect to standard procedures, is illustrated and explained in
details in Appendix A. A total of eight samples have been manufactured for the test
campaign. All the plates are then drilled in order to reproduce the lightning samples
clamping system: 12 fasteners disposed in a circle of diameter Φ370mm as shown of figure
4.1 thanks to a special metallic ring fabricated for the campaign.
Figure 4-1: Clamping system for mechanical impacts
Chapter 4
79
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Figure 4-2: Specimen preparation for curing
4.1.2 Canon test: set up and instrumentation
From the literature study [72-76], multiple studies reported the damage on composite
material submitted to impact loading. The amount of damage generated by the launch of a
projectile varies for different impact energy as well as for different couple (mass, velocity)
associated to the projectile.
Following the previous study that associated to a selected bunch of lightning strike tests
equivalent mechanical impacts, actual canon gas tests were run to validate the results of
these impact predicted by the numerical simulations. Mechanical impact tests were
conducted using a stainless steel ball of diameter Φ9.9mm and mass 4g as projectile,
launched by a gas gun apparatus (canon) in a range of velocities from 50 to 150m/s. The
canon is located at the Institut Clément Ader Laboratory (ICA) in Toulouse, France.
Chapter 4
80
__________________________________________________________________________________
(a)
(b)
(c)
Figure 4-3:mechanical impact set up (a) metallic assembly, (b) canon gas gun apparatus, (c) circular aluminium clamping
ring with sample fastened
The mechanical set up was designed to represent the experimental conditions of the
lightning strike tests. A metallic ring has been machined in order to reproduce the peculiar
clocklike clamping system of the plate, with twelve bolts disposed in a circle of diameter
Φ370mm, see figure 4.3. The set-up is equipped with a high speed camera which allows
measuring the projectile velocity at the exit of the gas gun. The rear face displacements are
measured using two displacement sensors (Keyence 20 kHz without contact – figure 4.4(a))
whose position is shown on figure 4.4 (b). Finally three force sensors are placed between the
metallic ring and the assembly (see figure 4.4 (c)) to compute the total impact force at the
centre of the plate.
Chapter 4
81
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(b)
(a)
Figure 4-4: Instrumentation (a) laser displacement sensor, (b) position of the displacement sensors
4.2 Test results
4.2.1 Macroscopic results
(c)
4.2.1.1 Presentation of the lightning/mechanical impacts
The canon apparatus allows ejecting the projectile at several velocities depending on the
pressure applied in the tube. At first, the desired impact velocities are numerically computed
using the simple shell model. Table 4.1 provides the different speeds chosen for the test
campaign and the actual velocities of the projectiles, measured by the high speed camera for
each sample. A discrepancy is to be noted in the obtained velocities. For the test expected at
72 m/s, three launches were emitted, respectively at 75, 75 and 70 m/s. Similarly, for the
expected impact at 60 m/s, the canon gas exited two projectiles at 65 and 50 m/s. These
differences are inherent to the experimental apparatus and concordant with the
repeatability and the dispersion of the test set up, as observed by the lab on similar
campaigns. This dispersion is due to the foam support in which the projectile is placed prior
to its introduction in the canon tube. This expanded foam has not a precise diameter
corresponding to the canon one and thus, the air may not be pushing linearly against it,
adding friction during the ejection and thus slowing down the projectile (see Appendix D).
Lightning strike cases
Associated
mechanical
impacts
Projectile’s speed (m/s)
Chapter 4
82
__________________________________________________________________________________
101 (ECF195-P160)
102 (ECF195-P200)
103 (ECF195-P50)
107 (ECF195-P160,
Φ12mm hole)
110(ECF195-P160,
24mm hole)
4
2
7
1
6
3
75
75
70
124
50
65
8
81
5
87.5
Table 4.1: Lightning strike cases and their equivalent mechanical impacts
The lightning samples presented in table 4.1, which constitute the experimental plan for this
study, have been chosen among several lightning campaigns led by AGI during the past years
and selected among several tens of lightning samples. In order to evaluate the influence of
several parameters such as the presence of paint for the equivalence method, the accent
has been put on choosing samples made of the same material. The same type of protection
has also been selected for the entire samples: ECF195. Finally, two samples with similar
thickness of paint layer have been selected: samples 101 and 102. In order to investigate
further the influence of the paint parameter, one sample with a lower paint thickness has
been added to the plan, sample 103, along with peculiar samples 107 and 110 which
possessed the same amount of paint than sample 101 but with spare zone at the centre.
4.2.1.2 Observable damage
After the mechanical impacts, a visual inspection is led to assess the visible damage on the
impacted samples. Two main features can be extracted from this analysis. For impact at
projectile velocity greater than 75 m/s rear face splinters can be observed whom size
increases with the velocity. These splinters are pretty large for impacts at 85 m/s and 124
m/s as shown on figure 4.5.
Indentation is visible for impacts superior to 65 m/s. Penetration is also to be noted for the
impact case at 124m/s at the impact point on the sample. The presence of such damage
identifies the limits of the equivalence and puts limitation on the speed that can be used for
the mechanical impacts. Indeed, if it is possible in several lightning strike cases to obtain rear
face splinter, marked indentation are never observed during lightning strike. Thus, the
velocity range of the equivalence must be controlled.
Chapter 4
83
__________________________________________________________________________________
Rear face
splinters
(a)
Front face
perforation
initiation
(b)
(c)
Figure 4-5:Observable damage on projectile impacted samples (a) rear face impact at 80m/s, (b) rear face impact at
124m/s, (c) front face impact at 124m/s
4.2.2 Rear face displacement results
Two displacement sensors are positioned at the rear face of the assembly to compute global
displacement of the centre of the impacted panel and a few centimetres below, at 10cm
(see figure 4.13(b)). The displacement sensor measures the deflection of the plate in the zdirection, normal to the impact. Experimental measurements are compared with numerical
ones obtained with the shell finite elements model previously used for the comparison to
lightning strike cases. Velocities are the same for the numerical models than for the impact
conditions in this case.
Experimental results
Figure 4.6(a) presents the raw displacement results for each impact velocity test except the
higher one 125m/s. Indeed, for the test case at 124m/s the large splinter generated by the
impact interfered with the displacement sensor which cannot follow the punctual
displacement, thus, the first peak of deflection is not available for this case. As expected, it
can be seen that the higher the projectile velocities, the higher the first deflection peaks. The
natural vibration of the plate over time is clearly visible at long duration with the particular
shape of the first peak due to the impact of the projectile. It has been presumed that all the
damages due to the impact appear during this first deflection peak and that the natural
vibration of the plate over 100µs does not damage the sample anymore.
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84
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3500
displacement (µm)
2500
1500
500
-500 0
2000
4000
6000
8000
10000
mechanical impact 70m/s
mechanical impact 75 m/s
mechanical impact 75m/s -2
Mechanical impact 81m/s
-1500
-2500
-3500
-4500
time (µs)
(a)
4500
4000
déplacement (µm)
3500
3000
2500
2000
mechanical impact 75 m/s -2
1500
1000
500
0
0
100
200
temps µs
300
400
500
(b)
Figure 4-6:Rear face displacement results (a) long time results for all tests (b) short time results.
From figure 4.6(b), the different characteristic times are derived. 100 µs is obtained as the
mean value for the short times of all experiments. 500µs is obtained to be the large time,
and d∞ can be obtained from the plateau values on each experimental curve.
As the equivalence is made on a short time range (<100µs) a focus on this period and on the
first peak deflection followed by the displacement plateau is made in figure 4.6 (b). The
same observation as before can be made. As the impact velocity increases, a corresponding
increase in the maximum deflection value is observed. For all the cases, this maximum value
is included in the range {2mm, 4mm}.
Numerical predictions
Chapter 4
85
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Experimental results are compared with numerical predictions on figure 4.7. The mechanical
impacts provide, in each case, a slightly higher rear face deflection compared to the
numerical predictions (about 4% at 72 m/s and 8% at 65 m/s). This result is coherent with
the use of a homogenized elastic shell model which does not take into account damage
induced by the impact. It is noticed that impacts at 65 m/s and 72 m/s give approximately
the same maximum value of deflection at short times, and that the deflection is maintained
during a longer period at 65m/s (50μs) than at 72m/s (20μs). While the numerical model is
too simplistic to represent these details, at these two impact velocities, the max peak of
deflection is always approached by the model with a precision that is acceptable and the
general behaviour of the plate also up to large times. This satisfying correlation proved that
the modelling strategy of the mechanical impact is relevant.
3500
3000
Displacement (µm)
2500
2000
1500
Shell model 72m/s
1000
Mechanical test 75m/s
500
0
0
100
200Time (µs)300
400
500
(a)
Displacement (µm)
3000
2500
2000
1500
Shell model 65m/s
1000
500
0
0
100
200
300
Time (µs)
400
500
(b)
Figure 4-7:Comparison of rear face displacements between numerical shell predictions and mechanical impact tests (a)
impact at 75m/s, (b) impact at 65m/s
Post-mortem examinations
In addition to external measurements, non-destructive and destructive post-mortem
examinations have been led in order to extract information related to the damage in the
core of the material. The main non-destructive examinations are X-Ray and C-Scan
observations. Destructive testing mainly involves the cutting of the samples along various
Chapter 4
86
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orientations in order to observe by microscopic examination the damage induced in the
depth of the material.
In chapter 3, the equivalent mechanical impact tests have been presented: test set up,
instrumentation along with first results such as rear face displacements, force sensors results
and first post-mortem examination. Non-destructive testing, mainly ultrasonic analysis (Cscans), and destructive micro-cutting testing for several samples have been presented.
Ultrasonic scanning allows the study of delamination position, size and orientation in the
thickness of the material. The method describing US scans is presented in Appendix B.
Microscopic examinations are led in addition to US scans to complete the study on internal
damage by allowing the study of matrix cracks and delamination. Though not all the C-scans
could be exposed here, the following evidences the main characteristic features observed. A
compilation of the ultrasonic results is detailed in Appendix F.
Ultrasonic examination of impacted samples allows looking at the inside of the material. The
ultrasonic inspection system available in ICA is shown in Figure 4.8. For more details on the
procedure see Appendix B. Ultrasonic testing is a widely used method to detect impact
induced delamination. Specimens are placed in a water cistern and an ultrasonic wave is
sent perpendicular to the sample. The reflection of the wave on internal defect allows
generating a colour coded image of the detected damage in the depth. Delamination can
thus be detected as well as their location, size and shape in the thickness of the material,
from the examined face.
Figure 4-8: Ultrasonic testing system (ICA)
4.2.3 Conclusions on the structural behaviour
The objective of this study was to design an equivalent mechanical impact to lightning strike
damage. Deep insights into lightning experimental results have resulted in identifying a good
candidate for such equivalence criteria: the transferred impulse k was found to be a valuable
equivalence parameter, between lightning strikes and designed mechanical impacts.
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87
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Moreover, characteristic lightning time intervals have been identified. The equivalence
method is derived for short times up to strike times.
In order to check the validity of such equivalence, a simplified shell finite element model was
created to reproduce the behaviour of a homogenized laminate composite submitted to a
projectile impact. The mass and velocity of the projectile are determined for each chosen
lightning strike case using the proposed equivalence method. The simplified shell model
shows good agreement with lightning strike test results for both maximum displacement d∞
and velocity vmax during the equivalence short times (<80µs). The numerical model however
is not able to reproduce the long term displacements (large times). This difference is
attributed to the effect of the surface layers (metallic protection and paint) present on
lightning samples but not modelled in the numerical model, and to the limitation of the
approximation, especially the existence of a stable plateau d∞. Assuming that phenomena
of interest occur at short times, poor correlation at long time is not a critical concern. In
spite of this difference, the equivalence is considered valid on short times range, which are
the ones considered here.
An experimental campaign of impact tests has been led to check the numerical predictions
and compare impact induced displacement results with lightning strike tests.
The first comparison with the equivalent mechanical impacts shows that numerical
simulations adequately reproduce the structural behaviour of impacted samples for both
short and long time as well as displacement and speed of displacement, thus validating the
equivalent mechanical impact model. Additional examinations are led on the mechanical
impacts such as non-destructive testing (ultrasonic scans and X-rays analysis) and destructive
microscopic observations. Comparison of damage through the thickness is not possible with
the simplified shell model, which does not take damage into account. The next chapter is
therefore dedicated to modelling (by different numerical methods) of the post impact
damage and comparison of the predictive damage model with experimental results. Chapter
6 is focused on lightning strike comparisons and discussions with both experimental and
numerical equivalent mechanical impacts.
4.3 Analysis of mechanical impact tests damage
4.3.1 Review of damage in composite materials
Composite materials are made of several components with different physical and/or
chemical properties. By adding various constituents it is aimed to build a new material that
benefit from the best properties of each of its constituents. Most of the time, these new
materials possess properties that suit a particular use. In aeronautics, composite materials
Chapter 4
88
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are preferred to metallic ones because they have higher specific mechanical properties
(modulus, strength).
In aeronautics industry, the most common composite materials used for aircraft structures
and substructures are called CFRP: Carbon Fibre Reinforced Plastics. They are made of two
main components: carbon fibres that act as a reinforcement; and the matrix also called a
resin that acts as a binding agent. For CFRP, epoxy resin is the most used as they possess
good thermomechanical properties and fibre bonding. Epoxy resins are thermosets
polymers with improved mechanical properties, a good temperature related stability… the
more common are the resins named PEEK (Polyether-ether-ketone), PPS (Polysulfide
phenyl), PEI (Polyether imide) and PA (Polyamide).
Figure 4-9: Laminate composite material
The inhomogeneous nature of these materials is responsible for specific damage mechanisms,
completely different from those observed with metallic materials, arising under a variety of different
loadings (impact, electrical and thermal loading) almost all non-quasi-static, non-isotropic and
heterogeneous in their distribution between constituents. These specific damage mechanisms are
due to the complex structural and manufacturing processes that engender new failure mechanisms.
Composite materials are complex structures and stem from a complex manufacturing
process. Extended literature exists on the failure mechanisms related to composite materials
[72-73, 78]. These damages, encountered in composite structures, can be divided in five
major types. Their location and size strongly depend on loading and/or solicitation rate. The
characterization of the loading is thus of primary importance to account for their proportion,
location in the lay-up and evolution.
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Figure 4-10: Typical internal damage under impact (a) [79](b) [80]
The main damage mechanisms involve damage inside a ply, due to failure of the fibres, the
fibre-resin interface debonding or the matrix cracking, and decohesion between plies, figure
4.10.
Fibre failure is due to high tension in the fibre direction and is quantified by standard
uniaxial tension tests [48, 80, 91-96]. When the fibres are solicited in compression in the
fibre direction, fibre buckling takes place. Such structural phenomenon may not induce
failure every time it happens but can trigger other types of damage such as fibre-matrix
debonding and hinges or kink-band creation. Fibre matrix debonding corresponds to a
separation of the fibre and the matrix. It is a complex failure mode that mainly depends on
the fabrication phases [48, 80, 72, 104] of the unidirectional ply used to create the final layup, see figure 4.11.
Figure 4-11:Fractographic features showing (a) longitudinal splitting and fibre-matrix decohesion of a group of fibres, and
(b) fibre radials (GFRP specimen) [81]
Matrix damage can also appear without involving fibre ones. Matrix cracks, another intralaminar damage, takes birth from cracks or voids between fibres within a single composite
ply [72, 80, 81]. Such damages are due to overloads of shear compression or tension in the
direction perpendicular to the fibres direction. The severity of the damage strongly depends
of the loading gradients applied on the structure. Finally, delamination, also called interlaminar damage, consist in the separation of two adjacent plies, more often, of different
Chapter 4
90
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orientations, see figure 4.12. This separation is mainly due to high level of discontinuous
through-thickness stresses [48, 80, 93]. Delamination can be observed through various nondestructive testing [14, 46] such as ultrasonic scanning of the composite structure or X-ray
scanning [19, 48, 72, 77, 104]. Delaminations are a critical damage that can lead to the
complete ruin of the structure [48, 73, 80, 104].
Figure 4-12:Delamination between two plies of different orientation [45/0/-45/90]s (indicated by the black arrow)
4.3.2 Analysis of post-mortem examination of mechanical impact tests
4.3.2.1 C–Scan Examination
The total damaged area due to delamination in the material is obtained thanks to the global
projected view of the damage offered via this technique. This damaged area is obtained
thanks to a rectangular or ellipsoidal box bounding the defects as shown on figure 4.13 for
an impact at 87.5 m/s, or using a crop technique. An ellipsoidal box is used in this study to
measure the total damaged area. Figure 4.14 presents the delaminated area measured for
each mechanical impact as a function of the projectile’s velocity. As expected, the higher the
projectile’s velocity, the higher the damage area as the total observed damage area
increases along with the impact velocity. On this figure, a change of the slope curvature is
observed for velocities greater than 80 m/s which correspond to the beginning of rear face
perforation (observed on samples impacted at 81 and 124 m/s, figure 4.5). However, some
discrepancies are to be observed for samples impacted at 75 m/s as the two impacts
provides quite different delaminated areas. By doubling the scanning, it is possible to
confirm the number, size and position of each delamination but also to reveal smaller ones
that would be hidden by bigger ones. Each delamination is defined by its position in the
laminate, which is identified by the corresponding colour in the colour map associated with
each impact.
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Total delaminated area (mm²)
Figure 4-13: Computation of total delaminated area (mechanical sample 5)
6000
5000
4000
3000
2000
1000
0
50
70
90
110
130
Projectile velocity (m/s)
Figure 4-14: Total delaminated area as a function of the impact velocity
Figure 4.15 and 4.16 present the delamination location and orientation in the core of the
sample impacted at 81 m/s and 65m/s. In figures (a) the scanning is made from the face
opposite to the impact while figures (b) are scanned views from the impact sides of the
plates. The impacts at 65 m/s and 81 m/s generated delamination in each interface, through
all the laminate thickness. The size of the delamination increases as interfaces move away
from the impact side. Delamination orientation is higly dependant on the adjacent plies and
traditionnaly follows the orientation of the lower ply of the interface from the impacted face
[82]. For example, the delamination between plis 3 and 4, respectively oriented at -45 and
90° (interface3), has its long direction oriented at 90°as well.
On both figures 4.15 and 4.16, the largest delamination took place between plies 5 and 6,
respectively at 90° and -45°. A second delamination oriented at -45° is hardly seen between
plies 2 and 3 (interface 2) due to extensive damage in interfaces 1 and 3. For the sample
impacted at 65m/s, Figure 4.27, the sizes of the delaminations are lower than for the impact
at 81 m/s as expected. However, even at a lower impact velocity, the dynamic impact
generated important delamination in all interfaces of the laminate and the larger ones are
the same than for impact at 81m/s. The largest delamination are located at the interfaces 3
(-45/90), 5 (90/-45) and 6 (-45/0).
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(a)
(b)
Figure 4-15:Impact damage measured by ND inspection for QI laminate impacted at 81 m/s : Delamination by ultrasonic
examination & Colour map for delamination cartography (a) recto, (b) verso
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(a)
(b)
Figure 4-16:Impact damage measured by ND inspection for QI laminate impacted at 65 m/s (i) Delamination by ultrasonic
examination (ii) Colour map for delamination cartography
Histograms of delaminated surfaces are extracted from C-scans and reported on figure 4.17
for impacts at 81 and 75 m/s. The biggest delaminations areas are mainly distributed around
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the double central interface 90/90 and tend to be larger on the second half of the laminate
opposite to the impacted side.
Delaminated area (mm²)
600
Impact
side
500
400
300
200
100
0
Thickness (mm)
Figure 4-17:Histogram comparing delamination distribution through thickness for mechanical impact tests at 75 and 81
m/s, layup: [45/0/-45/90]s
4.3.2.2 Destructive inspection
Destructive testing is also led on two impacted samples of T800S/M21, [45, 0, -45, 90]s:
mechanical samples 1 impacted at 124m/s and 2 at 75m/s. The cutting and microscopic
examinations are expected to provide additional information to ultrasonic testing on the
number and position of delamination through the material thickness. The samples are cut
apart through the impact centre point. The cutting planes for the following microscopic
images are presented in Figure 4.18. Two sections are defined: section A–A is along global xdirection and section B–B is along global y-direction of specimen. The cross-sections are
polished with sand paper of several grits from 250 to 600 grits in order to assure a sufficient
quality for microscopic observation.
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95
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Figure 4-18: Cutting plans for microscopic examinations
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96
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97
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98
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Sample 2, impacted at 75 m/s, is cut along the B-B section and sample 1 at 124 m /s along
the A-A section. For sample 2, the microscopic examinations confirm the ultrasonic scanning
results: extensive delamination of interface 3, 4 and 5. The matrix cracks serve as guide to
stop a delamination in the 4th interface (plies oriented at 90/-45) and directed it into the 5th
interface (plies at -45/0). The transition of the delamination from interface 4 to 5 is driven by
a matrix crack in the -45 ply (blue line). The delaminations increase in size when going away
from the impact face. The largest delamination is located in the 4 th interface as predicted by
the ultrasonic scanning.
For sample 1, impacted at 124 m/s, the cut is made along the –x axis, A-A section. Extensive
damages are observed at the impact point. The steel ball initiated a perforation of the first
plies, as shown on figure 4.18, that dragged the first two plies (45 and 0° orientation) inside
the thickness of the material (red arrow). An extensive delamination is visible on the left
side of the impact point, at the interface 5 (-45/0). On the rear face, a large splinter is visible,
due to the violent snatching of the 45° oriented last plies (in blue). Figure 4.19 shows a
specific section of the cut section. The analysis showed that all the interface of the laminate
thickness presented delamination, the larger ones being at the interface 4, 5 and 6, on the
bottom of the sample. All these damages originate from the crushed section induced by the
projectile perforation and the consequent rear face splinters that tend to open the laminate.
4.4 Comparison of lightning strike and mechanical impact damage
In this chapter, the results of the equivalent mechanical impacts, presented in chapter 4, are
firstly compared to the associated lighting strike results. The different lightning strikes cases
selected are remembered in table 4.1 along with the associated mechanical impacts. All
lightning samples were made of T800/M21 with the following staking sequence: [45/0/45/90]s. Samples were protected and painted with various paint thickness. Two samples,
107 and 110 presented spare zones of paint at the centre of the panels, respectively of
6mm and 12mm. The results used for comparison concern global behaviour and local
damage of the samples: time/displacement curves, ultrasonic analysis and microscopic
imaging.
In the third section of this chapter, the comparison is then made between the lightning
strikes and the numerical model presented in chapter 6. The same results are used:
displacement/time curves as well as damage at interfaces compared with C-scan analysis.
These comparisons and analyses will help in the validation process of the mechanical
equivalence of lightning strikes by validating the working hypothesis H0 and H1: “it is
possible to separate surface and damage surface” and “it is possible to mechanically
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reproduce the damage due to a lightning strike”, and in the understanding of the lightning
damage mechanisms.
4.4.1 Rear face displacement
The rear face displacement results for impacts at 65m/s, 75m/s and 81m/s are shown on
figure 4.21, figure 4.22, and figure 4.23 respectively. As a preliminary reminder, two
identical mechanical impact tests were performed for the same configuration 101 on
purpose: they highlight the intrinsically scattered nature of impact induced damage surfaces,
and remind how careful any modelling attempt or quantitative comparison with numerical
results should be interpreted. Indeed, the observed scattering is larger in the investigated
domain of velocity and mass ranges than under low velocity conditions with large mass
projectiles for the same energy level [106, 100].
4000
Displacement (µm)
3500
3000
2500
2000
1500
lightning strike 103
1000
500
0
0
50
100
150
Time (µs)
200
250
Figure 4-21:Displacement versus time for the 65m/s mechanical impact test (full line) and lightning strikes case 103
(dashed line).
For lightning sample 103, which surface is totally covered by paint in the lightning strike, the
slope and values of displacement up to 50µs (‘short times’) are quite close together in the
two cases. Between 50 to 100µs (‘Strike times’), displacements are still of the same order of
magnitude (about 2.5mm), and both slopes change with different amplitude (different
decelerations). After 100µs (‘Stabilization times’), the two curves separate and the
mechanical impact curve tends to decrease while in the lightning strike case the
displacement still rises with a smaller velocity. It must be recalled here that these time scales
are very small compared to the complete displacement signal of the impacted plate so that
in reality, the gaps are small compared to the global plate displacement.
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100
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4000
Displacement (µm)
3500
3000
2500
2000
1500
1000
Lightning test 101
500
0
0
50
100
150
200
250
Time (µs)
Figure 4-22:Displacement versus time for the 70m/s and 75m/s mechanical impact tests and lightning strike cases 101.
For lightning sample 101, the lightning curve is surrounded by the two mechanical tests at 70
m/s and 75 m/s. The impact tests at 75m/s better reproduced the speed of displacement of
lightning; however its maximum displacement seems a bit too high compared to the signal
of the lightning strike (about 3.5mm which has not been obtained properly after what seems
to be the pick). An adequate speed should be found in this range of projectile velocity.
4500
displacement (µm)
4000
3500
3000
2500
2000
Lightning strike test 107
1500
1000
500
0
0
50
100
150
time (µs)
200
250
Figure 4-23:Displacement versus time for the 70m/s and 75m/s mechanical impact tests and lightning strike cases 107.
The abrupt slope of the mechanical impact curve corresponding to the lightning test 107 is
due to acquisition issues during the tests due, because of the high speed of the event. Even
though the values are not really to be used in this case, it is noticeable that the same phases
appear in the lightning strike displacement curve of case 107 than in the 103 case: straight
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101
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slope up to 50µs, then a small stabilization between 50µs and 100-110µs, then a new raise
with a smaller slope. In this case, the stabilization at about 3mm has a bit longer duration,
and the new raise in slope is even smaller than in case 103. These differences are attributed
to the effect of a lack of paint in a central disk of the plate. It is noticeable that the bigger the
central bare disk is, the bigger the rear face displacement is too.
Displacements and velocities at 50μs (mean instant of maximum velocity in lightning strikes)
are compared in table 4.2. These values are obtained directly with the displacement and
velocity curves by extracting the values at 50 µs for the displacement and by measuring the
tangent to calculate the velocity at this same time. Displacement predictions of case 101 are
inferior at 1% for the two cases at 75m/s and get up to 9.4% for the impact at 70m/s. For
cases 103 and 107 errors on displacement predictions are about 30% which is still
acceptable. The mechanical impact fails at reproducing the large time deflections of lightning
and the value of the constant plateau is smaller in the impact cases than in the strike
measurements. This result was expected since the creation of the equivalent mechanical
impact in chapter 4. In this chapter the simple shell model used to design the equivalent
projectile already shew that the mechanical impact was not able to reproduce the long term
behaviour of the lightning strike displacement. This difficulty for the impact equivalent to
represent long time behaviour has been related to thermomechanical surface phenomena
which result from the presence of the metallic protection and dielectric paint layer [43] and
which are delayed with respect to the lightning strike itself. Such delayed phenomena
cannot be represented by the fast mechanical impact processes.
Velocity predictions of case 101 are quite good while for case 103 the error is about 32%.
This again is attributed to the confinement effect of gas generated by the metallic cover
explosion.
Lightning
samples
101
103
107
110
LS
Mech.
displacement
Impact
(µm)
displ. (µm)
(4) 2850
2839
(2) 2843
(7) 2560
(6) 2153
1810
(3) 1863
2626
(8) 3630
2605
(5) 2486
Relative
difference
0.8%
0.5%
-9.4%
+18.9%
+2.9%
+38.2%
-4.6%
LS
velocity
(m/s)
37.4
25.9/30
32.3
37.2
Mech.
Impact vel.
(m/s)
33.6
22.9
35.3
33.8
31.2
27.3
15.2
Relative
difference
-10.1%
-37.8%
-5.8%
+32%
+21.8%
+9.6%
-59.1%
Table 4.2: Displacements and velocities at 50µs
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From the comparisons, it is concluded that the mechanical impacts provide rear face
displacements and velocities in amplitude and evolution quite close to lightning strike ones
from short times to stabilization times. Concerning composite panels covered by layers of
paint that are burnt during lightning tests or small thickness of paint, the mechanical impacts
are considered successful at reproducing the rear face displacement and velocities induced
by lightning strikes at short times. As the representativeness of the equivalent impact tests is
made on the comparison of these two kinematics quantities, and for short times (lightning
equivalence time of 50µs), the method is validated by this experimental campaign.
4.4.2 Total delaminated area
To go further in the equivalence and regarding the limitations of the equivalent mechanical
concept, an analysis of the damage induced by both lightning strike and mechanical impacts
is conducted. To do so, ultrasonic C-scans analyses are conducted on the associated samples.
Table 4.3 presents the total delaminated area measured for each lightning strike and its
associated mechanical impact, and the relative difference. It is reminded here that it is
emphasized that the main damage has been created in the short (50µs) and stabilization
(50µs to 100µs) times since it is the total duration of the lightning strike. As a consequence,
damage induced by the impact is also emphasized to be due to delivery of the equivalent
impulse in this time duration. In both cases damage reported below are post-mortem
measurements.
The lightning sample 102 and 107 and their associated mechanical impacts provide
concordant results for the total delaminated area with a relative difference of 2% in both
cases.
Sample 103 presents zero damage for the lightning C-scans which is relevant with the small
thickness of paint layer on the surface, as it was proven that the bigger the thickness of
paint, the higher the damage due to lightning strike. The corresponding mechanical impact
generates a delamination, as expected, however the damage zone is reduced.
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103
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Lightning
samples
101
102
Associated
mechanical
samples
LS
Mechanical
Projectile’s
delaminated delaminated
speed (m/s)
area (mm²) area (mm²)
4
75
2
75
7
70
1
124
6
50
103
2321
5836
Relative
difference
1626
-29%
1184
-49%
1019
-56%
5732
-2%
223
-
630
-
0
3
65
107
8
81
1711
1746
-2%
110
5
87.5
120
3361
-
Table 4.3: Delaminated area for lightning strikes and associated mechanical impacts
4.4.3 Influence of surface state
Some observations are made on the parameter ‘paint thicknesses. The comparison of
surface damage of samples 101 and 107 shows that in the case of sample 101 the paint was
not removed after the lightning strike while for sample 107 the central zone was unpainted
on a spot zone of a few 6 mm in diameter (see figure 4.24) so that its effect is delayed in
time. The delaminated area is bigger in the case 101 in spite of the same thickness of paint
for the two lightning samples. This difference is attributed to the spare paint area on sample
107 which delays the confining effect of paint and provides only a protected zone for the arc
attachment.
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Zone of
spare paint
Figure 4-24: Spare paint area on lightning sample 107
The absence of paint at the attachment point for sample 107 also changed the dynamic of
the lightning strike as shown on figure 4.25. The spare paint area seems to generate a first
peak of displacement followed by a plateau and a second loading on the sample causing it to
move back again at larger time. The observed difference on displacement results confirms
that in the case of painted samples with a thick layer (samples 101 & 107 with 160 m
paint), the surface states influence the structural response of the impacted panels and that
the equivalent impact are only fully representative in the case of samples with small
thicknesses of paint.
4
Deflection (mm)
3,5
3
2,5
2
1,5
Lightning test 101
Lightning test 107
Lightning test 103
1
0,5
0
0
100
200
300
Time (µs)
400
500
Figure 4-25: Comparison of displacement for lightning tests 101 and 107 (ECF195-160µm paint)
The comparison of samples 101 and 103 (figure 4.25) also shows the influence of the
amount of paint during the impact. Sample 101, with 160µm paint clearly shows a larger
delaminated area as well as a higher displacement peak. The perturbation due to the paint
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also explains this strong difference: in the lightning strike test, the paint is burned on a very
small part at the root attachment and so disturbs very soon the global behaviour of the plate
so that the paint cover is maintained from the first stages of the lightning process, whereas
for sample 103 with lighter paint thickness the surface layer was completely burnt.
Lightning sample 110 presents a very little delaminated area (table 4.10) due to its little
amount of paint and its large central unpainted zone of diameter 12mm which impacted
the overall behaviour of the panel, as shown on figure 4.38. The large unpainted area made
it “seen” by the lightning arc as an unpainted panel during the attachment process, thus the
protection acted very efficiently to protect the structure from lightning damage as it is
designed for and the confinement action of to the paint is delayed. In comparison, the
associated mechanical impact, defined on the basis of the rear face displacement of the
lightning sample, was shot at a high velocity inducing high amount of damage due to large
energy transfer in the material
As a conclusion, it is stated that the equivalent model is able to reproduce lightly and
unpainted panels (see figure 4.26 (a) and (c)) and that the surface state influences the total
surface delamination and the damage mechanisms responsible for it. For unpainted (bare),
slightly painted or painted panels where the paint is removed during the strike, the
equivalent mechanical model is valid for both decoupling and damaging hypothesis H0 & H1,
but it is not yet enough detailed to take into account a quantitative effect
However for heavily painted samples where the paint is not removed during the strike the
hypothesis H0 cannot be totally assumed as it has been evidenced that the surface state
influenced the structural behaviour of the samples. This defines some limits to the
mechanical equivalence: at large time, the effect of paint preservation made the decoupling
between core and surface impossible.
(a)
(b)
(c)
Figure 4-26: Lightning surface damage after impact (a) sample 101, (b) sample 107, (c) sample 103
Chapter 4
106
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4.4.3.1 Damage features
After comparing the delaminated area measured by ultrasonic C-scans, a comparison is
made on the actual C-scans measured from the rear face. For lightning strike samples, the
impacted sides of the plates were too damaged to produce correct ultrasonic scans, and the
protection layer as well as the presence of bare fibres caused perturbations in the data
acquisition. Figure 4.27 thus presents the method used to analyse the scans obtained. The
left scale represents the depth of the interface seen from the analyses side. It starts from 0
on the side opposite to the impact and goes up to 1.44mm which is the total thickness of the
plate. The right scale recalls the numbering of the interfaces in the layup.
Figure 4-27: Two scales for ultrasonic testing method: left the thickness, right the fibre orientation
In traditional impact mechanics, a delamination at the interface between two plies generally
follows the orientation of the ply underneath, as shown on figure 4.28 [72, 73, 104].
Several studies [107, 108] showed that for laminate of unidirectional plies, matrix cracks
appear prior to delamination and tend to direct or close the delamination orientation. These
damages appear parallel to the fibre direction when they reach the interface between two
plies of different orientation they propagate to the lower unidirectional ply plane. This
preferential orientation is due to the fact that the energy delivered by the shock is absorbed
by failure of the weakest link, which is the interface between two plies of different
orientation, leading to the formation, or closure, of delamination at the given interface [48,
109]. It is worth noting that delamination more rarely occurs between plies of same
orientation.
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107
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Figure 4-28: Delamination orientation after mechanical impact
Figure 4.29 presents the results of the mechanical impact case 4 at 75 m/s which
corresponds to the lightning samples 101. The comparison of the two scans shows several
differences. Firstly, the dimensions and shapes of the two damage areas are quite different
but give the same order of total surface, as shown on figure 4.29 and in table 4.10. The
damages obtained with lightning strikes are larger than those obtained with the equivalent
mechanical impacts, except for case 103 and 110 as previously explained.
The lightning strike damage area (figure 4.29 (a)) presents a very peculiar shape made of two
zones articulated around a 90°oriented delamination (circled in red) between the plies 3 and
4 (interface -45/90°). On the contrary, the ultrasonic scan of the mechanical impact (figure
4.29 (b)) presents characteristic features of a mechanical impact (as discussed in the section
5.1):
- A butterfly shape with a central vertical symmetry axis that is the axis of the
projectile displacement, the wings of the butterfly being separated by two distinct
visible cracks;
- The wings are limited by two series of cracks oriented along the fibre’s direction of
the underlying ply (length) and of the upper lying ply (width).
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108
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The large splinter oriented along the last ply orientation is typical of high speed mechanical
impact and is a feature very little experienced in lightning strike C-scans although not rare.
The butterfly shape of the delaminated area with its wings on both sides of a large 90°
oriented delamination between the plies 3 and 4 (-45/90) also is a current feature of this
kind of impact, as shown on figure 4.30.
However the two kinds of impact share two delamination position in common, in interface 2
and 3, respectively oriented at -45° and 90°.
(a)
(b)
(c)
Figure 4-29:Ultrasonic scans for lightning strike (a) and associated mechanical impact (b) at 75m/s for sample 101 at
interface between plies 3 (-45°) and 4 (90°)
Figure 4-30: Ultrasonic testing for mechanical impact at 75m/s
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109
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4.4.3.2 Damage distribution through thickness
The distribution of damage through the thickness is also different between the two impacts.
For both impacts, the position of the latest delamination is highlighted on figure 4.31 under
the respective C-scans.
(a )
(b)
Figure 4-31:Position of the last delamination in Ultrasonic scans for lightning strike (a) and associated mechanical impact
(b) at 75m/s for sample 101 at interface between plies 3 (-45°) and 4 (90°)
For lightning strike, the damage seems always to stop at the double 90°interface and thus
the damage area is confined in the first half of the thickness of the laminate as shown on
figure 4.29 and 4.31 where the dashed lines on the top of the stack correspond to the
metallic lightning protection. The protection has been removed by the strike on the left
upper side, which is the location of the surface damage. The white arrow shows the location
of the main delamination between plies 3 and 4. The delaminated area is the one associated
to the red vertical ellipse on figure 4.29 (a). The mechanical impacts on the other hand
generate damage through all the thickness of the laminate following a classical helicoid
cone, which ends with the large splinter on the rear face of the samples. This difference is
mainly due to the difference in energy deposition through the material. During lightning
strikes, an important part of the energy brought by the arc is used in damaging the metallic
protection and the paint or dispersed into the air while in the mechanical impact, all the
energy of the projectile is transferred directly into the material, generating different damage
mechanisms. Furthermore, the energy deposit is localized in a very small contact zone in the
mechanical impact thus pushing forward the propagation of the compressive dynamic wave
in the depth while it is distributed in time and space during the lightning strike thus
spreading the load on a wider but less deep zone.
Figure 4-32: Micro-cuts of the # 101 lightning sample.
C-scans allow the extraction of delamination per interface. Figure 4.33 presents histograms
showing the relative position of the main delamination through the material thickness and
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110
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their importance for the mechanical impacts corresponding to the lightning strikes 101 and
107. The position zero corresponds to the rear face of the material, opposite to the impacted
side. The histograms clearly show the difference in delamination position through the
thickness for the two kinds of impacts. It confirms that lightning strikes tend to generate
most of the damage up to the half thickness of the material. Mechanical impacts thus create
damage through all the thickness and all interfaces of the material.
700
600
500
Lightning test 101
Mechanical impact
400
300
200
100
Impact
side
0
Thickness (mm)
(a)
800
700
Mechanical impact
600
Lightning test 107
500
400
300
200
100
Impact
side
0
Thickness (mm)
(b)
Figure 4-33:Histograms of the delamination as a function of position in the thickness (as defined in fig. 6) of the material,
for # 101 (a) and # 107 (b) samples.
For the case 101, the study of the general envelope of the delaminated area shows that
lightning and mechanical impacts provided opposed results. The two envelopes present the
same shape but mirror each other, evidencing the difference of damage distribution between
the two impacts. For the lightning case, the first and the last peak are in phase, but the second
notable peak (which stands for the most delaminated area) is not in phase with the others. This
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111
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implies that the pressure responsible for this peak has a larger application time than the first
one. An analytical signal analysis could allow extracting the application time of this lightning
pressure.
For the lightning case 107, there is no phase difference between the first delaminated interface
and the largest one for lightning strike and mechanical impact. It must be noted that the
pressure due to lightning is higher and faster than the mechanical one (higher delaminated
area). The mechanical impulse has the good shape but is not sufficiently fast and intense
enough to match the lightning one.
4.5 Conclusions
The objective of this chapter was to determine the validity of the design of a mechanical
equivalence to a lightning strike in terms of structural and behavioural damage induced to
the material.
From the above analysis, it has been demonstrated that the equivalent mechanical impact
allows the reproduction of lightning strike rear face displacement until the peak of deflection
for nude or slightly painted lightning strike samples, as well as the total damage for slightly
painted samples or painted samples where the paint is removed early during the lightning
strike. These observations are used as a basis to fully validate the proposed equivalence
method. By pushing forward the analysis, the damage distribution of the equivalent
mechanical impact was also investigated. The comparison with lightning strike damage Cscans showed that the mechanical impacts generate damage through the sample thickness
that differ with those obtained after a lightning strike test. Except for interfaces that hide the
next interfaces, C-Scans can be used to quantify properly the difference of damage
distribution for each interface. Direct measures of delamination extent in each interface are
possible using microscopic observations on cut samples. This result has been proved very
useful since it gives qualitative and quantitative information about the mechanical
contribution of the energy in a lightning strike, and also gives an idea of the interaction
between the covering paint versus time. It demonstrates that rear face displacements and
velocities are mainly due to the mechanical part of the loading to the structure, while
damage inside the samples thickness result from both its structural reaction and the
interaction with the paint confinement and the surface behaviour in general.
It is aimed to mainly work on total damaged area for Compression After Impact (CAI) testing
after a lightning strike. The mechanical equivalence being able to reproduce with acceptable
relative difference for both the maximum displacement and the total delaminated area for
protected and painted panels, it is considered validated. Nevertheless, this equivalence is
Chapter 4
112
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not able to reproduce the damage distribution through thickness mainly due to the way the
impulse is delivered.
These results highlight the influence of the paint removal or persistence in the final damage
amount.
“Rear face displacements and velocities, and delamination content,
are mainly due to the mechanical part of the loading that is the
delivered impulse, while damage distribution inside the samples
thickness result from both its structural reaction and the interaction
in particular with the paint confinement at short times and the
surface behaviour in general.”
Differences between mechanical impacts and lightning strikes damage are mainly due to the
type of impulse created by the local contact of a small and hard projectile. Indeed, as a first
attempt of analysis, it has been decided to use a steel ball of 4g mass and Φ9.8mm diameter.
This small projectile launched at relative high speeds (about 70m/s) is known to create
craters and rear face splinters localized around the projectile contact zone: surface craters
appear, then by increasing the impact velocity, penetration and perforation. Even if splinters
also arise during lightning, they are not due to the same load distributions. In fact, the
electrical arc expands radially during the strike and the steel ball is not able to reproduce
such behaviour. At larger times, the deflection plateau observed during lightning strike tests
was not well reproduced by the mechanical impact for the same reasons.
An important result has been highlighted by the experimental plan with various paint
thickness. The major parameter on damage content is not the paint thickness but its
capability to erode or to persist during the lightning strike. In fact sample where paint was
completely consumed by the arc at short times (<20 µs) presented lower damage area than
the other samples with the same amount of paint on the surface.
It is concluded that either the projectile is not adequate if the impulse is that of an impact or
that the impact is not the adequate mean to transfer the impulse and it could be necessary
to consider a smoother contact load in space that could reproduce the peculiar evolution of
the lightning arc root presented in chapter 3. In order to complete the experimental analysis
and to get a predictable and reliable tool, a numerical model is derived in chapter 5. In order
to evaluate better ways to deliver the impulse, numerical tests are conducted in chapter 6.
Chapter 4
113
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Chapter 4
114
5
Chapter 5
Numerical modelling of equivalent
impacts
5.1 Introduction ......................................................................................................................................... 1167
5.2 Modelling damage in composite materials ............................................................................................. 117
5.2.1 Literature survey on impact modelling .................................................................................................. 117
5.2.2 Inter-laminar damage ............................................................................................................................ 118
5.3 Intra laminar damage: Continuous Diffuse Damage Model .................................................................... 125
5.3.1 CDM framework..................................................................................................................................... 125
5.3.2 Damage Modelling: Ilyas former model ................................................................................................ 126
5.3.3 Conclusions ............................................................................................................................................ 130
5.4 Continuous versus discontinuous damage modelling ............................................................................. 131
5.4.1 Review of the different models used ..................................................................................................... 131
5.4.2 Effects on the sample behaviour ........................................................................................................... 132
5.5 Final modelling: results and discussion ................................................................................................... 137
5.5.1 From shell to 3D solid element model ................................................................................................... 137
5.5.2 Results and discussion ........................................................................................................................... 142
5.5.3 Conclusions on the predictive quantities of the model ......................................................................... 149
5.6 Comparison of lightning damage with numerical simulations ................................................................ 149
5.6.1 Rear face displacement ......................................................................................................................... 150
5.6.2 Rear face displacement profiles ............................................................................................................ 153
5.6.3 Total delaminated area .......................................................................................................................... 154
5.6.4 Damage features.................................................................................................................................... 155
5.6.5 Distribution through thickness .............................................................................................................. 156
5.7 Conclusions ............................................................................................................................................ 158
Chapter 5
115
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Objectives
The purpose of this chapter is to propose a numerical model able to replace mechanical
impact tests in order to dispose of a predictive cheap tool that could be used to evaluate the
effect of different parameters of the lightning strikes configuration on composite panels on
the resulting damage.
Comparison of numerical damage with mechanical impact experimental ones and lightning
strike ones are presented in order to analyse the predictability of the model.
Chapter 5
116
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5.1 Introduction
In this chapter, the common damages in composite materials are recalled. Those induced by
mechanical impact tests, presented in chapter 4, are here presented and analysed. Prior to
any numerical modelling, the continuum damage mechanics (CDM) framework will also be
recalled along with the proposed model for damage modelling in composite materials. The
second part of this chapter will present the different laws that are numerically implemented
to reproduce damage in the composite material. These laws stand for two types of damage:
inter and intra-laminar damage. Afterwards, a brief summary of the intermediate models
will be made with an overview of the obtained results, compared with mechanical impact
tests. The final model (M6) will be presented. This model integrates various improvements
of the simplified shell model into a full 3D solid elements damage model via several
enhancement stages, based on the previous work of Ilyas [48] and Cheng [77]. The results
obtained with the final model in terms of rear face displacement and through thickness
damage are then compared to those obtained during the actual mechanical impact tests and
lightning strike results.
5.2 Modelling damage in composite materials
5.2.1 Literature survey on impact modelling
In the literature, many studies can be found on the numerical modelling of impact damage
and their prediction [57, 72-74, 91, 110-113]. Damage under consideration here are
delamination between adjacent plies (even of same orientation), and through thickness
cracks into plies (even not perpendicular to the plies plane). Methods for modelling damage
in composite materials are here categorized into two main families whether it is necessary to
represent delamination and cracks as discontinuities or if it is enough to represent their
effect on the residual strength. If cracks are to be modelled, it is necessary to distinguish
between initiation and propagation to choose the frame of the mechanics which should be
used. If the prediction of the residual strength is the objective, continuous damage
modelling is often used [120]. From these studies two major modelling strategies have
emerged: Continuum Damage Modelling (CDM) and Discrete Modelling (DM).
Discrete models consider the damage as a loss of local continuity and activate either the
initiation or the propagation when a limit load is reached in a given direction (giving the local
normal of the newly created surface). Locations and orientations where the continuity
conditions can be released are chosen a priori and placed into the laminate, like a preexisting crack on which the finite element model would act to represent the opening of a
defect in the material.
Chapter 5
117
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CDM modelling, on the other hands, focuses on damage inside the material by reducing the
material properties such as the material moduli or strength [121-123]. The CDM model for
composite damage behaviour under interest here was initially improved by Matzenmiller
and al [84]. It is based on the hypothesis of a cumulative effect of ruin modes on the load
bearing capability of a Representative Volume Element of matter using a probabilistic
approach. This model was an enhancement of Kachanov model [83] proposed in 1986, who
considered damage as the effect of a distributed population of micro cracks represented and
represent it by an intrinsic variable of state that affects stresses and strains at the
boundaries of a Representative Volume Element of matter. Matzenmiller & al. CDM model
was then widely used to simulate damage in composite material such as fibre failure and
matrix cracks but also for delamination study [85, 86]. However, the use of CDM technique
revealed to be not particularly adapted to model discrete failure such as delamination, which
is a failure mode characterized by highly localized stress areas at material or geometrical
discontinuities. Impact damage results mainly in the interactions of different damage
mechanisms and the modelling of matrix cracks for example and delamination can be
problematic for CDM to achieve, though several studies focused on capturing discrete
aspects in their model [82].
This issue is being resolved by mixing these two modelling techniques. To combine CDM and
discrete method, several studies [82, 88-90] deactivate the delamination criteria in CDM law
and use cohesive elements to model the interface behaviour between two plies while the ply
modelling (matrix and fibre failure) is still insured by the CDM laws. Such a technique
resolved the issue concerning the interaction between matrix crack and delamination
process and the issue concerning delamination initiation, and provides good results in
damage prediction and modelling. Cohesive elements and formulations have then been
widely used in numerical simulations since they are appropriate to model delamination
propagation.
5.2.2 Inter-laminar damage
5.2.2.1 Failure modes
Fracture mechanics is the branch of science that deals with the study of cracks in a structure.
Fracture mechanics is based upon the hypothesis that a structure always contains defects.
These defects may be present in the form of surface cracks or internal cracks. Fracture
mechanics analysis relates parameters coming from loading, geometry and material. Then it
can be inferred under which conditions the cracks may propagate and eventually cause
complete failure of the structure.
Chapter 5
118
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Discrete models of delamination have been developed in the scope of fracture mechanics
dealing with the study of crack initiation and propagation in a structure. Three elementary
modes of failure have been identified as shown on figure 5.1.
Figure 5-1: The three elementary modes of failure.



Mode I (opening mode) corresponds to a tensile stress generated by the
displacement perpendicular to the crack plane.
Mode II (shearing mode) corresponds to a shear stress in the crack plane generated
by the displacement perpendicular to crack front.
Mode III (tearing mode) corresponds to a shear stress generated by a displacement
parallel to the crack front.
In order to model the initiation and propagation of damage throughout structure, specific
elements have been created. Such elements are disposed between two plies and insure the
link between them. They are called cohesive elements.
5.2.2.2 Literature survey on cohesive elements
In their first utilization, cohesive elements were used to simulate single crack propagation in
homogeneous isotropic materials [124, 125]. The growth of this macroscopic defect is
controlled by strain energy release rate [126, 127]. The study of a single macroscopic crack is
analogous to delamination propagation in composites. Today, cohesive elements are widely
used for interface damage modelling in composite materials such as delamination and
debonding. Not only do they take into account both initiation and propagation of damage
but they also can be used without initial damage already implemented in the structure,
which make them an interesting alternative to pure CDM modelling. Adaptive or refined
meshes are necessary to obtain accurate results though.
Cohesive elements have been extensively used to simulate standard characterization tests
such as Double Cantilever Beam (DCB), Mixed-Mode Bending (MMB) and End Notch Flexure
Chapter 5
119
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(ENF) [91-95], respectively presented in figures 5.2, 5.3 and 5.4. But they also have been
used for other applications such as in the work of Johnson et al. [96] who simulated the
penetrating impact of a steel ball in a composite plate.
Figure 5-2: DCB test for material characterization and mode I testing [48]
Figure 5-3: End Notch Flexure (ENF) specimen for Mode II testing [48]
Figure 5-4:Mixed-Mode Bending (MMB) apparatus for Mode I+II testing [48]
Chapter 5
120
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For composite materials, the numerical simulation of delamination is divided into two
phases: (1) initiation and (2) propagation [93]. The initiation phase is generally based on
more or less complex stress criteria and related failure strengths (yields). A quadratic
criterion linking inter-laminar stresses and a characteristic opening displacement (figure 5.5)
is often used as an onset for delamination initiation. The characteristic distance between
two plies is function of several parameters such as material properties and structure
geometry. Delamination initiation can be related to interface stiffness (k) [95].
Figure 5-5: Characteristic distance between two plies [95]
To model delamination, specific elements have been proposed by various authors: (a) zerothickness volumetric elements connecting solid elements [97], (b) finite-thickness volumetric
elements connecting shell elements [88], and (c) line elements [98-100]. Another type of
element is also used, called point decohesion elements, which are identical to non-linear
spring elements connecting nodes [97, 101-102, 114, 115]. A review of the advantages and
disadvantages is made in [48]. It should be noted that for all those elements, once the failure
criterion is met and thus the connection between the plies is broken, the discrete
delamination is initiated and propagates following the delamination law associated to the
elements. Once the connection fully broken, i.e. the distance between two plies being
excessive, the element is deleted from the model.
5.2.2.3 Interface elements
ABAQUS Explicit 6.12 already implements standard cohesive elements with a mixt-bilinear
law (figure 5.6) as presented on eq. 5.1. Decohesion elements are formulated in terms of a
traction vs. relative displacement relationship (figure 5.8) instead of the traditional stress strain relation. This law is based on relative displacement of upper and lower plies. Two
surfaces (top and bottom) are considered, as shown in figure 5.7. Every point in these
surfaces has a corresponding point in the other surface, designated as homologous point.
The opening and closing of delamination are driven by a linear elastic behaviour under
compression, non-linear elastic under traction and shear followed by a linear (in its simplest
form) softening driven by progressive damage development (figure 5.7). In this particular
Chapter 5
121
__________________________________________________________________________________
decohesion element formulation, a sliding mixed mode II is used, which represents both
shear mode II and tear mode III.
Figure 5-6: Constitutive law for cohesive elements
Figure 5-7: Displacement
𝜎 = 𝑘(1 − 𝑑)𝛿
(Equation.5.1)
δI=δ1 represents the normal displacement in mode I and δII = √δ22 + δ23 indicates the inplane shear displacement in mode II. The displacement between two plies is thus given by:
𝛿 = √𝛿𝐼 ² + 𝛿𝐼𝐼2
(Equation. 5.2)
The damage variable d is calculated from the relative displacement of the interface between
two adjacent plies. Damage parameter d is used to scale down the stiffness based on initial
stiffness k of the interface element. The loading and unloading is purely elastic, figure 5.6.
δ (δ−δ0 )
m −δ0 )
m
d = δ(δ
(Equation. 5.3)
Chapter 5
122
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Where δ0 is the initiation critical displacement (eq. 5.4) and δm the propagation critical
displacement (eq. 5.5).
The displacement for damage initiation δ0 under mixed mode is expressed as eq.5.4. When interface
is under compression, its behaviour is only dependent on shear.
1+𝛽2
2
2
)
𝐼𝐼0 +(𝛽𝛿𝐼0 )
𝛿0= 𝛿𝐼0 𝛿𝐼𝐼0 √(𝛿
(Equation. 5.4)
At the inception of delamination, nodes’ displacements are functions of critical
displacements in mode I and mode II (eq. 5.4, 5.6). Failure displacement is a function of
critical displacements and mode I and II critical energy release rates (eq. 5.5). β stands for
coupling between modes I and II, from pure mode I (β=0) to pure mode II (β → ∞, δ0 = δII0,
δ0 = δIIm ). σn and σs are out of plane traction and shear failure stresses; kn and ks are
interfacial stiffness components respectively in the normal and tangential direction of the
surface discontinuity. Once the failure criterion is met, the connection is broken, and a
discrete delamination of the size of the eroded cohesive element is initiated and is ready to
propagate. The displacement corresponding to failure under mixed mode loading is written
as follow:
1+β2
δm = {
δ0
kI
α
kII β2
[(G ) + ( G
Ic
II0
1
α −α
) ]
δIIm
𝛃=
𝛅𝐈𝐈
𝛅𝐈
𝛔
𝛔
, 𝛅𝐈𝟎 = 𝐤𝐧 , 𝛅𝐈𝐈𝟎 = 𝐤𝐬
𝐧
𝐬
<== δI > 0
<== δI ≤ 0
(Equation. 5.5)
(Equation. 5.6)
The propagation of damage is defined by an energy release rate (i.e. the surface energy
available for crack propagation) criterion written as in eq. 5.5. The coupling between mode I
and mode II is controlled by the parameter α. For composite materials, α is commonly taken
as: 1<α<2.
Chapter 5
123
__________________________________________________________________________________
G α
(G I )
Ic
G
α
II
+ (GIIc ) = 1
(Equation. 5.7)
Figure 5-8: traction relative displacement law
Most of the values to be implemented in numerical simulations are obtained through
experimental testing and characterization for each material. In the case of T800/M21, the
material used in the mechanical test campaign, the characterization and determination of
the vital values such as strain energy release rate, out of plane traction and shear failure
stresses, interfacial stiffness components and the parameter α have been determined by
Ilyas [48] during his PhD and his values are used in this work. Strain energy release rate GIc
and GIIc are determined by using Double Cantilever Beam (DCB) and End Notch Flexure tests
(see figures 5.2 and 5.3). Out of plane traction and shear failure stresses are determined by
standard traction tests and the parameter α by using Mixed Mode Bending (MMB) testing
(see figure 5.4). Finally, interfacial stiffness components are numerically determined by
simulating the characterization experiments. The material parameters used in the numerical
simulations are gathered in table 5.1.
KI
(kN/mm3)
KII
(kN/mm3)
100
100
n
(MPa)
60
s
(MPa)
60
GIC
(J/m2)
GIIC
(J/m2)

765
1250
1,0
Table 5.1: material parameters for interface elements
Chapter 5
124
__________________________________________________________________________________
5.3 Intra laminar damage: Continuous Diffuse Damage Model
5.3.1 CDM framework
As a basis for further adaptations and improvements, a pre-existing Diffuse Damage Model
developed formerly in the frame of the continuum damage mechanics by Ilyas [95, 96] is
used. This model is based on the Matzenmiller-Lubliner-Taylor (MLT) formulation [84], which
assumed a preserved anisotropy and elasticity of each ply of the laminate during the damage
evolution by representing each unidirectional ply as a homogenized continuum.
The DDM mainly works on the treatment at a macroscopic level (coupon specimen scale) of
micro crack growth (fibre and matrix level) as a homogenized damage over an elementary
volume. Hashin [103] introduced at the macroscopic level a damage variable d (introduced
by Kachanov [83] in 1958), describing the damaging of the material. The damage variable d
defines the effective stress or the resistance of the material to the damage it undergoes. The
variable d (0<d<1) accounts for the damaged state of the laminate where d=0 represents a
completely sane material and d=1 its complete failure.
The damage variable d is directly linked to the damaged section of the material:
𝑆𝑒𝑓 = 𝑆 − 𝑆𝑑
(Equation.5.8)
Where S is the total section, Sd the damage section and S ef the section of sane material. The
variable d can then be defined as:
𝑑=
𝑆𝑑
𝑆
=
𝑆−𝑆𝑒𝑓
𝑠
(Equation.5.9)
An effective stress 𝜎̂ is defined as the stress transmitted by the intact surface in the crosssection of a representative elementary volume. A relation can be stated between the
nominal and effective stress [82, 87]:
𝜎
𝜎̂ = 1−𝑑
(Equation.5.10)
The deformation of the material being related to the effective stress; the uniaxial elastic law can thus
be written:
𝜖𝑒𝑙𝑎𝑠𝑡𝑖𝑐 =
̂
𝜎
𝐸
(Equation.5.11)
Chapter 5
125
__________________________________________________________________________________
With 𝜖𝑒𝑙𝑎𝑠𝑡𝑖𝑐 the elastic strain. If E0 is the sane material's Young modulus, one can define the modulus
of the damaged material as: 𝐸̂ = 𝐸0 (1 − 𝑑) then the damage variable d can also be written:
𝐸̂
𝑑 =1−𝐸
(Equation.5.12)
𝐸𝑖 = (1 − 𝑑𝑖 )𝐸0
(Equation.5.13)
0
Or in a more general way:
The elastic properties of the damaged material are thus degraded by the damage variable d.
5.3.2 Damage Modelling: Ilyas former model
For the behaviour of T800/M21e laminates under impact conditions, Ilyas [48] derived a
material model inspired from the MLT model. On the basis of elementary testing, Ilyas
identified active ruin modes and selected six damage modes di , i = 1 … 6 which operate on
the six elastic moduli defining the flexibility matrix C−1 (relating strain ε and stress σ by
ε = C−1 σ) as knock-down factors.
1
0
(1−𝑑1 )𝐸11
𝜈12
−𝐸
11
𝜈13
−𝐸
11
𝐶 −1 =
(
𝜈
𝜈
− 𝐸21
− 𝐸31
1
0
(1−𝑑2 )𝐸22
𝜈23
−𝐸
22
𝜈
− 𝐸32
33
1
0
(1−𝑑3 )𝐸33
22
33
1
0
(1−𝑑4 )𝐺12
(Equation.5.14)
1
0
(1−𝑑5 )𝐺23
1
0 )
(1−𝑑6 )𝐺13
Damage Threshold
The six damage variables di are related to five failure modes by threshold functions rj , j =
1 … 5. The region in stress space where damage does not change is bounded by a series of
surfaces fj (σ, dj , rj ) = 0. For a given threshold rj the material does not suffer more damage
if fj (σ, dj , rj ) < 0. When the condition fj (σ, dj , rj ) = 0 is reached, un updated threshold rj
has to be evaluated as a function of σ and dj .
𝑓(𝜎, 𝑑𝑖 , 𝑟𝑖 ) = 𝑓𝑗 (𝜎, 𝑑𝑖 ) − 𝑟𝑗 ² = 𝜎𝑇 ∙ 𝐹𝑗 ∙ 𝜎 − 𝑟𝑗 ² = 0
(Equation.5.15)
Chapter 5
126
__________________________________________________________________________________
For each failure criterion, a specific formulation of f(σ, di , ri ) is obtained from elementary
tests. The five failure criteria used here are those derived by Ilyas (where the < > Macaulay
brackets refer to the positive value):

Fibre failure due to traction along fibre axis:
f1 (σ, d1 , r1 ) = (

〈σ11 〉 2
Xt
σ212 +σ213
) +(
S2fs
) − r12 = 0
(Equation. 5.16)
Fibre failure due to compression along fibre axis:
〈−2σ11 +〈−σ22 −σ33 〉 〉 2
f2 (σ, d2 , r2 ) = (
2Xc
) − r22 = 0
(Equation. 5.17)
Figure 5-9:Tension and compression failure in fibre direction.
Eq. 5.16 presents the tensile failure criterion where XT is the tensile failure stress in fibre
direction. This formulation takes into account the effects of shear loading. Eq. 5.17 describes
the fibre failure due to compression.

Ply failure due to spherical compression
𝒇𝟑 (𝛔, 𝒅𝟑 , 𝐫𝟑 ) = (
〈−𝝈𝟏𝟏 −𝝈𝟐𝟐 −𝝈𝟑𝟑 〉 𝟐
𝟑𝒁𝒄
) − 𝒓𝟐𝟑 = 𝟎
(Equation. 5.18)
Eq. 5.18 presents the failure mode related to spherical compression also called crushing, due
to high stresses in the structure below the projectile during the impact. This criterion is
essentially active in cases of projectile penetration up to perforation.
Chapter 5
127
__________________________________________________________________________________

Ply failure due to shear stresses
𝒇𝟒 (𝛔, 𝒅𝟒 , 𝐫𝟒 ) = (
〈𝝈𝟐𝟐 〉 𝟐
𝒀𝒕
) +(
〈−𝝈𝟐𝟐 〉 𝟐
𝒀𝒄
) + (𝑺
𝝈𝟏𝟐
𝟏𝟐 +〈−𝝈𝟐𝟐
𝟐
) + (𝑺
〉𝐭𝐚𝐧𝛗
𝝈𝟐𝟑
𝟐𝟑 +〈−𝝈𝟐𝟐
𝟐
) − 𝒓𝟐𝟒 = 𝟎
〉𝐭𝐚𝐧𝛗
(Equation. 5.19)
Eq. 5.19 represents the failure due to shear stresses. This equation takes into account the
failure due to tensile and pure compression in transverse directions via the parameters YT
and YC (failure stresses). A coupling between 22 (transverse cracking), 12 (in plane shear)
and 23 (out of plane shear) is made as shown on figure 5.10.
S12 and S23 are the shear failure stresses in 1 − 2 and 2 − 3 planes. The failure criterion takes
into account the difference in mechanical behaviour due to opening or closing of the cracks.
In eq. 5.19 and eq. 5.20, the term tanφ accounts for shear associated frictions once
transverse cracks have been created and are maintained closed in a compressive way.

Interface failure (delamination)
𝒇𝟓 (𝛔, 𝒅𝟓 , 𝐫𝟓 ) = (
〈𝝈𝟑𝟑 〉 𝟐
𝒁𝒕
) + (𝑺
𝝈𝟏𝟑
𝟏𝟑 +〈−𝝈𝟑𝟑
𝟐
) + (𝑺
〉𝐭𝐚𝐧𝛗
𝝈𝟐𝟑
𝟐𝟑 +〈−𝝈𝟐𝟑
𝟐
) − 𝒓𝟐𝟓 = 𝟎
〉𝐭𝐚𝐧𝛗
(Equation.
5.20)
In the criterion responsible for delamination, a coupling is made between out of plane
normal stress 33, out of plane shear stress 13 and 23, respectively parallel and
perpendicular to fibre direction. ZT is the tensile failure stress in out of plane (3-) direction.
Figure 5-10: (a) Compression and (b) tension in transverse direction
Chapter 5
128
__________________________________________________________________________________
Figure 5-11: (a) Compression and (b) tension in out of plane direction.
Damage Evolution Law
In eq. 5.19 and eq. 5.20, the term tanφ accounts for shear associated frictions once
transverse cracks have been opened. The form of the damage variables d has been set by
Matzenmiller and al [84], and can be easily computed from the damage evolution law in
time using some hypothesis for the integration and the initial state:
𝑑 =1− 𝑒
𝑟𝑚
)
𝑚𝑒
(−
with r>0
(Equation. 5.21)
A second expression has been proposed [104,105] which is quite different as depicted by
Ilyas [48]:
1
𝑑 = 1 − exp (𝑚 (1 − 𝑟)𝑚 ) with r >1
(Equation. 5.22)
The parameter m is a stress softening parameter, chosen to model the softening response of
fibre-reinforced composites from the simulation of elementary characterization tests (DCB,
tension …). They can be defined as function of strain rates also, and are responsible of the
brittleness of the rupture, by changing directly the slope of moduli decrease as a function of
the amount of damage. Different values of m can be chosen, one per damage parameter,
but it is then difficult to determine the values through simple tests campaigns. The second
expression of d, given in (eq. 5.22) is the one that will be used in the following work. The
damage temporal changes are described by a coupling matrix [q ij ] and damage growth
functions ϕj (σ, d, ε̇ ) derived from the first principle of thermodynamics and expressed here as
state function (without derivatives):
Chapter 5
129
__________________________________________________________________________________
𝑑̇𝑖 (𝜎, 𝑑, 𝜀̇) = ∑5𝑗=1 𝑞𝑖𝑗 . 𝜙𝑗 (𝜎, 𝑑, 𝜀̇)
1
𝑚
𝜙𝑗 (𝜎, 𝑑, 𝜀̇) = 1 − 𝑒 𝑚(1−𝑟𝑗 ) , 𝑟𝑗 ≥ 1
(Equation. 5.23)
(Equation. 5.24)
The operator [q ij ] contributes to a description of both micro-micro couplings (interaction between
material failure modes) and micro-macro couplings (interactions between material and structural
failure). For example the damage variable d1, expressed in eq. 5.26, is a function of the damage
functions ϕ1 and ϕ2.
1
0
0
[𝑞𝑖𝑗 ] =
1
0
[0
1
0
0
1
0
0
0
1
0
1
0
0
0
1
0
1
0
0
0
0
0
1
0
0]
(Equation. 5.25)
d1 = (1 − 𝜙 1) + (1 − 𝜙 2)
(Equation. 5.26)
5.3.3 Conclusions
This chapter has focused on the numerical tools available in the literature to help reproduce
the damages that are identified as present in the laminates after al lightning strike or after
the impact with a spherical or hemispherical projectile.
The main damage observed on the impacted samples and their relative position were
analysed by using both non-destructive and destructive testing. This analysis highlights the
presence of damage both in the plies and between them under the forms of matrix and fibre
cracks and ply delamination. These two kinds of damage have to be introduced in the
numerical modelling.
To do so, a review of the numerical methods has been presented on the simulation of inter
and intra-laminar damage. Inter-laminar damage involves the discrete modelling of fractures
through the introduction of interface elements, also called cohesive elements, that allows
the separation between two plies of different orientations under a mixt bilinear law and
displacement conditions of the elements of each plies. Intra-laminar damage involves, on
the other hand, the continuum damage mechanics framework and is based on the model of
Matzenmiller-Lubliner-Taylor modified by Ilyas [48]. This model is based on six damage
Chapter 5
130
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variables di are related to five failure modes by threshold functions rj , j = 1 … 5,
representing the various possible damage mechanisms related to composite material.
The following sections will present the design of a full 3D damage model coupling inter and
intra-laminar damage through the improvement of the simplified shell model presented in
chapter 3. This model will be tested for various impact velocities corresponding to the
mechanical tests of table 4.6 that are the equivalent mechanical impacts to the lightning
strikes of table 4. 4.
5.4 Continuous versus discontinuous damage modelling
5.4.1 Review of the different models used
The first developed numerical model was the simplified shell model presented in chapter 3,
called from now on M1. The model was built to represent at best the lightning strike tests
conditions of clamping, size and shape of the sample and boundary conditions.
From this first model, several others are designed that integrate increasing complexity.
These models are presented in table 5.2. The various models presented in table 5.2 have
been compared to the M1 model which serves as a basis for preliminary improvement
comparisons and then to mechanical impact results. A preliminary study is made in order to
check the validity of the numerical model using Abaqus® software. To do so, the exact same
model is built in another Finite Element software, called Samcef®, in order to check the
overall results obtained with the shell model and ensure the choice of numerical integration.
Although the numerical elements and calculation process are not strictly the same in the two
commercial codes, which leads to several differences, the two calculations provide good
agreement in terms of rear face displacements and force versus time. The complete study is
detailed in Appendix E. For each of the created model, the material properties for interface
elements, when implemented, and the plies are presented in table 5.3 and 5.4 respectively.
The final model, called M6, will be presented in the following section 5.5.
Model #
M1
M2
M3
M4
M5
M6
Model
type
shell
volume
volume
volume
volume
volume
Law for the
plies
elastic
elastic
elastic
damageable
damageable
damageable
Presence of
interface definition
none
none
cohesive elements
none
cohesive elements
Cohesive surfaces
Table 5.2: Overview of the designed numerical models
Chapter 5
131
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kn
3
(N/mm )
30
ks
3
(N/mm )
30
σn
(MPa)
65
σs
(MPa)
60
GIC
(J/m²)
700
GIIC
(J/m²)
1300
α
1.0
Table 5.3: Material properties for interface elements
E11
(GPa)
165
XC
(GPa)
1.2
E22
(GPa)
7.64
SFS
(GPa)
1.5
E33
(GPa)
7.64
YT
(GPa)
0.06
ν12
Ν23
ν13
0.35
YC
(GPa)
0.28
0.35
ZT
(GPa)
0.06
0.4
ZC
(GPa)
0.7
G12
(GPa)
5.61
S12R
(GPa)
0.065
G23
(GPa)
2.75
S23R
(GPa)
0.06
G13
(GPa)
5.61
S13R
(GPa)
0.065
XT
(GPa)
2.2
Table 5.4: Material properties for T800/M21
5.4.2 Effects on the sample behaviour
5.4.2.1 Sample kinematics
Figure 5.12 presents the first change brought to the M1 model: the passage to a volume
definition of the elements (model M2) and the addition of damageable interfaces between
the plies (model M3), allowing separation and delamination initiation, by using cohesive
elements.
Figure 5.13 presents the displacement results obtained at the centre of the rear face for two
other models, M4 and M5, with growing complexity and that provided extensive information
on the impact damage generated by the projectile. These models are compared to both M1
model and experimental results for an impact at 65 m/s.
Chapter 5
132
__________________________________________________________________________________
8
model 2D Abaqus
6
Abaqus volumique
Displacement (mm)
4
3D elastic model + interface elements
2
0
-0,1
-2
0,1
0,3
0,5
0,7
0,9
1,1
1,3
1,5
-4
-6
-8
Time (ms)
Figure 5-12: Displacement comparison for several models M1, M2, M3 (spherical projectile Φ16mm, 17g)
3000
Displacement (µm)
2500
2000
1500
Shell model 65m/s
1000
3D model with ply damage 65m/s
3D model with damaging ply and interface elements
65m/s
500
0
0
100
200
Time (µs)
300
400
500
Figure 5-13: Displacement comparison for several models M1, M2, M3 (spherical projectile Φ9.8mm, 4g)
The analysis of the displacement results shows that the numerical models M4 and M5 are
both able to reproduce the displacement curve at both short and large time and provide
according maximum and plateau values of displacement.
5.4.2.2 Sample damage
In order to discriminate the models that provide the best results when compared to
experimental data, the comparison is also made on the damage numerically generated: total
Chapter 5
133
__________________________________________________________________________________
delaminated area and distribution through thickness. The model must be able to reproduce
with accuracy both the size and shape (orientation) of the delaminated interfaces.
Table 5.5 gathers the delaminated area per interface and total results for models M3, M4
and M5, compared with experimental results for mechanical impact at 75 m/s.
interface
Mechanical
test
M3
M4
M5
1
0
69,8
40,
148,96
2
205
109,8
24,
159,4
3
75
725,7
57,
230,6
4
102
2599,6
86,4
237,7
5
701
69,4
161,5
159,4
6
543
48,3
298,3
288,9
Total(mm²)
1626
3622,7
667
1225
Table 5.5: Total delaminated area and distribution through thickness for models M3, M4 and M5 compared with
mechanical impact results
The analysis of the results shows that model M5 is the one that provide the more
concordant results with experimental data. The model M4, even though it provided
concordant displacement results, underestimated the damage in every interface, while
model M3, which did not take into account the damage of the plies, overestimated the
delamination due to the impact.
Finally, the delamination’s’ size, shape and orientation is analysed for each interface for the
three previous models. The cartographies of the delaminated area are presented in figure
5.14.
Chapter 5
134
__________________________________________________________________________________
(a)Interface 1 (45/0) for models M3, M4 and M5
(b)Interface 2 (0/-45) for models M3, M4 and M5
(c)Interface 3 (-45/90) for models M3, M4 and M5
(d)Interface 4 (90/-45) for models M3, M4 and M5
Chapter 5
135
__________________________________________________________________________________
(e)Interface 1 (-45/0) for models M3, M4 and M5
(f)Interface 1 (0-45) for models M3, M4 and M5
Figure 5-14: Cartographies of each damaged interface for the three model: M3, M4 and M5, composite material with the
sequence [45/0/-45/90]s
The study of the numerical delamination shows that except for interfaces 2 and 3 (figure
5.26 (b) and (c)), the cohesive elements do not reproduce the expected orientation, which is
to follow the lower ply direction. Moreover, the large over estimation of interface 3 and 4 is
clearly visible for model M3 and indicates that the use of cohesive elements without damage
in the adjacent plies cannot reproduce accurate delaminated area. Such localization of the
damage is an identified problem of cohesive elements and has been discussed in various
works on the subject [48, 77].
Model M4, without cohesive elements (damage taken into account directly in the plies
damage definition), provide too small delaminated area without any orientation at all. This
model is thus not adapted to the impact velocities and damage that are to be reproduced.
Finally, the study of the delamination by interface for the numerical model M5 confirms that
the cohesive elements are still not able to reproduce the correct orientation following the
lower ply of the interface even when coupled with a damage law for the plies.
From these different models, it is stated that a coupling between damaging plies and
cohesive interface is necessary to represent the damage induced by the expected
mechanical impacts. However, several improvements are to be made in order to solve
Chapter 5
136
__________________________________________________________________________________
several issues related to the numerical simulation. Several solutions will be proposed and
checked with mechanical experimental results in the following section 5.5.
5.5 Final modelling: results and discussion
5.5.1 From shell to 3D solid element model
Several numerical models have been created, based on the initial shell model presented in
chapter 4 and improved into a full 3D damage model. A simple analytical analysis could have
been led in order to check the vibration frequencies of the 3D solid model. However, as the
test samples that are numerically reproduced possessed a peculiar clamping system, such
analysis would have been insufficient. Two different codes were confronted in order to
check these plate frequencies when going from shell to solid elements. The different models
are aimed to reproduce the adequate plate frequencies over a certain time, which is the
chosen equivalence time for lightning, as well as several mechanical quantities: rear face
displacement and velocity which are used for comparison purpose with lightning strike test
results.
The 3D model combining cohesive interface elements and damaging plies provides
satisfactory results concerning rear face displacement and velocity as well as for total
delaminated area. However, several issues arise from the study of the results. Firstly, the
cohesive elements used to model the interface between two plies tend to localize greatly
the damage in the upper part of the thickness avoiding the propagation of delamination
toward lower interface 5, notably. This causes the damage repartition provided by the model
to be different to the one observed during experimental tests.
The cohesive elements tend to localize the damage in the upper part
of the laminate thickness. Moreover, they do not reproduce
accurately the damage orientation.
Moreover, these elements do not reproduce accurately the damage orientation. Typically, in
real tests delamination follows the lower ply direction, guided and confined by matrix cracks
in the laminate [48, 72, 77, 80], but the cohesive elements do not always reproduce this
behaviour. Even though the soft coupling with the damage law for the ply seems to allow a
better representation of the delamination orientation, the effect of localization of the
damage observed previously at the interface 3 and 4 is still present in the new model, but
lowered. This is supposed to be induced by the soft coupling and the absence of interactions
between the two damage models (CDM and DM) that co-exist but do not interact with each
Chapter 5
137
__________________________________________________________________________________
other to generate damage. Finally, improvement can be made on the numerical mesh of the
plate to help the interface elements to better reproduce the damage mechanisms observed
during real tests and especially the direction of the macro cracks for delamination initiation.
Regarding all those observations, several modifications are made on the 3D complete model
that uses simultaneously a damage law for the plies and interface elements for decohesion
between the plies.
The finite element mesh of interface elements must be adapted to
improve the numerical interaction between cracks in the plies’
material and delamination between plies in the laminate structure.
Moreover, the soft coupling between damage evolutions must be
simultaneously adjusted.
5.5.2 Final modelling: results and discussion
Regarding the modelling results obtained in the previous sections, several modifications are
made on the 3D model coupling a damage law for the plies with interface elements
responsible for ply delamination.
A new formulation for the interface elements will be proposed in the following section by
replacing the cohesive elements with cohesive surface in the models. Some improvements
will also be made on the damage law in order to better handle the initiation and propagation
of the damage. Finally, some modifications will be brought to the numerical mesh in the
same purpose of improving the delamination propagation and numerical representation.
The results obtained with this final model are thus extracted and compared directly to the
experimental values.
5.5.2.1 Modification of the interface formulation
Cohesive elements used in our previous models are 3D elements with theoretical non-zero
thickness, disposed between the plies of a laminate and associated with material parameters
such as interfacial stiffness values and failure stresses. Once this failure values are met, the
initiation and propagation of the damage begin until the total failure of the element which is
then deleted from the model.
To solve the localization and disorientation problems, an alternative to these 3D elements is
to replace them by cohesive surfaces. For each ply, top and bottom contact surfaces with
adjacent plies are identified. Each surface is given a contact resistance identical of the
Chapter 5
138
__________________________________________________________________________________
material properties of previously used cohesive elements. Now plies are separate plies,
connected together with links that can break, and the failure is driven by surface stress
criteria on. When the link is broken, the non-penetration contact is automatically activated.
Doing so, all the issues related to the creation of discontinuities and contact between ply
and interface elements are avoided.
In addition to this improvement, the material properties associated to interface surfaces
have also been modified as shown on table 5.10 according to the optimization formulation
presented by Ilyas [48] in his PhD. Table 5.5 presents the material properties used for the
plies and the failure stresses associated to the chosen material which is T800/M21. The layup of the laminate remains unchanged.
The comparison between the two formulations (elements and surfaces) is detailed in
Appendix E in a numerical simulation using the same random generated mesh for the two
simulations.
5.5.2.2 Modification of the coupling matrix q
A major modification is also made in the damage law for the plies. In the numerical law, the
operator [𝒒𝒊𝒋] contributes to a description of both micro-micro couplings (interaction
between material failure modes) and micro-macro couplings (interactions between material
and structural failure). In order to evaluate the respective contributions of different failure
modes and compare with mechanical impact and lightning strike configurations, different
forms of operator [𝒒𝒊𝒋] have been investigated, two main components being explored:


Influence of 𝒇𝟑 for high-speed and localized loadings associated to lightning;
Influence of knock-down factors on coupled flexural shear stiffness terms,
complementary to interfacial damage.
Two variations of [𝒒𝒊𝒋] operator were attempted to adapt mechanical impacts to lightning
strike configurations:
1
0
0
[𝑞𝑖𝑗 ] =
1
0
[0
1
0
0
1
0
0
0
1
0
1
0
0
0
1
0
1
0
0
0
0
0
1
0
0]
1
0
0
[𝑞𝑖𝑗 ] =
0
0
[0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0]
(a)
(b)
Eq. 5.27: Operator[𝐪𝐢𝐣 ], Ilyas [11-12] (q1, (a)) and present (q2, (b))
Chapter 5
139
__________________________________________________________________________________
The definitive form q2 was established by comparing delamination patterns at selected
interfaces with trends emerging from experimental observations of the mechanical impacts.
The case under consideration is the impact at 65m/s. The measured delamination projected
surface is shown on figure 5.15. The red ellipsoid allows us to estimate the delaminated area
at interface 4 (90/-45) which is about 300mm². The red closed line draws the outside limit of
the delamination shape at the same interface. Numerical predictions are shown on figure
5.29
-45/90
0/45
Interface 4:
90/-45
Figure 5-15: Observation of the shape of the delamination at the different interfaces (corresponding to different colours)
for the impact at 65m/s. In particular, the interface 4 (90/-45) delamination is in pale blue and is surrounded by the red
ellipse oriented at
The use of the operator q2 allows a better reproduction of the delamination orientation and
size in accordance with experimental comparisons as shown in figure 5.16. In this example
one can see that the delamination orientation at the interface 4 (90/-45) is well reproduced
when compared to the experimental results presented in figure 5.15. The new formulation
using the operator q2 allows reproducing the shape and orientation of the delaminated
interface in a more precise way than the previous cohesive elements. Sizes of these
delamination areas are identical. For this interface the experimental test (81m/s) provides a
delamination of approximatively 630mm² while the model using q1 and q2 respectively
provides 954mm² and 574mm² of delamination.
Chapter 5
140
__________________________________________________________________________________
(a)
(b)
Figure 5-16: Computed delaminated areas at the interface 4(90/-45) for the impact at 65m/s resulting for q1 (a) and q2
(b) models . The blank surfaces correspond to the deleted interface elements which were totally damaged during the
impact.
5.5.2.3 Modification of the meshing strategy
Meshing is important in numerical simulation. It has been shown that previously used
cohesive interface elements could not reproduce easily the delamination propagation in the
thickness of the laminate and the extent. The numerical mesh used in medium velocity
impact simulation must be able to reproduce correctly the extension of quasi-discrete matrix
crack that lead the opening and closing of delamination [48, 77] and in the same time reduce
the calculation time. In fact, macro cracks have a significant impact in delamination
propagation as they provide guidance for the transition of delamination from one interface
plane to another.
To help macro cracks formation and thus delamination propagation in the appropriate
direction (i.e. lower ply’s fibre direction), a specific mesh is designed for the laminate.
The composite plate consists of 3D volume elements for the plies and cohesive surfaces for
interface between adjacent plies of different orientations, as explain before. Two meshes
are modelled: one for the plies with 0 or 90° orientation and one for the +/- 45° plies. The
elements of each ply are thus configured so that their edges are aligned along the fibre
direction, providing a correct propagation of matrix cracks and delamination whatever the
orientation of the fibres. The first mesh, dedicated to 0° and 90° plies, is composed of a
refined central part of hexahedron elements of dimension 1x1mm² while the second mesh,
for +/-45° plies, is composed of a central area of pentahedron elements of the same
dimensions [77]. The triangles of the second mesh are arranged in the 0° and 90° so that the
plies of different orientations possess the same number of nodes and can be easily bonded,
as shown on figure 5.17, to insure mesh connectivity.
Chapter 5
141
__________________________________________________________________________________
Figure 5-17:Oriented mesh for plies at 45 and -45° in order to model macro cracks and orient delamination propagation
A mesh sensitivity study has been performed and is detailed in Appendix E. As for now and
the rest of the study, the model presented in section 6.1.4 to which the preceding
improvements have been added, is adopted for the numerical simulations. The results
obtained with this model and their comparisons with experimental ones are presented
hereafter.
5.6 Results and discussion
The final numerical model designed for this study includes a specific mesh adapted to the
orientation of each ply in the laminate and predicts damage through a cohesive law for
delamination at plies’ interface and through a damage law for the plies’ damage. The results
of the numerical simulations equivalent to the lightning strike tests of Table 4.6 are
presented in the following sections and compared to mechanical tests in order to validate
the numerical model.
Central rear-face displacements versus time plots are compared on figure 5.18 from
mechanical tests and computational results of i) the shell elastic models used for calibration
of the mechanical impact and ii) the 3D constitutive model allowing estimate of the induced
damage.
All models provide a qualitatively correct description of the displacement, in particular the
long term structural behaviour (>200μs) and post-peak plateau. However, the shell model
systematically underestimates the measurement by 10-30 %. This is partly corrected in the
3D model, which is closer to the measurement than the shell result, especially at short times.
This results from the fact that damage contributes significantly to the rear face
Chapter 5
142
__________________________________________________________________________________
displacement. This suggests that the equivalent mechanical impact definition methodology
should be used in future applications with the 3D method rather than the shell model used
previously.
The 3D model correctly approximates the max peak displacement as well as the long-time
behaviour of the impacted samples. Table 5.6 gathers the maximum displacement results for
both experimental and associated numerical impact. Impacts at 65 m/s and 70 m/s provide
the largest disparity with a relative difference of 25% and 28% respectively, for the other
cases; the numerical model provides satisfactory results.
3500
Displacement (µm)
3000
2500
2000
1500
Shell model 65m/s
1000
500
3D model with ply damage and
cohesive surface 65m/s
0
0
50
100
150
Time( µs)
200
250
300
(a)
4000
Displacement (µm)
3500
3000
2500
2000
1500
1000
Shell model 72m/s
500
3D model ply damage and cohesive surface 75m/s
0
0
100
200
Time (µs)
300
400
500
(b)
Figure 5-18: Comparison of out of plane displacements, for configurations 101-3 (top) and 103-2 (bottom). The 3 curves
correspond to the displacements for: shell model equivalent impact, 3D model equivalent impact, mechanical impact
test (label “mechanical impact”).
Chapter 5
143
__________________________________________________________________________________
Mechanical
samples
Numerical
model max.
displ.
(µm)
3055
Relative
difference
3 (65m/s)
Mech.
impact max.
displ.
(µm)
2656.7
4 (75m/s)
3780
3604
-4.6%
6 (50m/s)
2233.3
2392
7.1%
7 (70m/s)
2740
3235
18%
8 (81m/s)
3866.7
3977
2.8%
15%
Table 5.6: Comparison of the experimental and numerically predicted displacements for several velocity impacts
To go further in the model analysis and to validate the damage obtained by the model, the
total delaminated area is computed and compared to the experimental results. To extract
the total delaminated area from the numerical model, the delamination created in each
interface is measured by the same method used during exploitation of C-scans, through an
ellipsoidal box bounding the delamination. Experimental total delaminated areas are
evaluated by dedicated features within the US analysis software, which approximates the
effective delaminated areas through best-fit ellipses, and sums up throughout the thickness.
Numerical delaminated areas are computed as the summation of elementary surfaces of all
eroded cohesive elements. The results are gathered in table 5.7. The results show
concordant results for simulation with impact velocity superior to 70 m/s, with, for example,
relative difference between simulation and experimental results of about 6% for impacts at
124 and 75 m/s. However, it must be noted that for projectile velocity under this limit, the
numerical model systematically overestimate the total delaminated area. This can be due to
either strain rate effects that are not taken into account in the M6 model or to inaccurate
measurements of the projectile velocity. Indeed, for the canon gas, extensive dispersion is
observed for velocity around 50 m/s, which is the lower limit of the apparatus. At such slow
impact velocity, the contact frictions increase between the tube and the foam in which the
projectile is placed and it is possible that the calculated ejection speed (by mean of high
speed cameras) are overestimated.
Chapter 5
144
__________________________________________________________________________________
Mech.
Impact
delaminated
area
Numerical
model
delaminated
area
(mm²)
(mm²)
1(124 m/s)
3058
3271
6.9%
2 (75m/s)
1184
1729
46%
3 (65m/s)
630
908
44%
4 (75m/s)
1626
1729
6.3%
6 (50m/s)
223
713
200%
7 (70m/s)
1019
1661
63%
8 (81m/s)
1746
2062
18%
Mechanical samples
Relative difference
Table 5.7: Total delaminated areas predicted from the 3D model compared to test results.
As a preliminary reminder, two identical mechanical impact tests were performed for the
same configuration impact at 75m/s (lightning case 101) on purpose: they highlight the
intrinsically scattered nature of impact induced damage surfaces, and remind how careful
any modelling attempt or numerical result should be interpreted; indeed, observed
scattering is larger in the investigated domain of velocity and mass ranges than under low
velocity conditions. However, it is clear that computed delaminated areas tend to overestimate the measurements. This difference may find its origin in sensitivity to high
deformation rates in the fracture process that is still not accounted for in the model (work in
progress of implementation in the model). It could also come from the discrepancies
between a perfect numerical model and an in perfect real material.
In addition to the analysis of total delaminated areas, a key feature when evaluating the
model is its capability to capture the correct orientations of delaminated areas throughout
the thickness: orientations correlate well (see Figures 5.19 and 5.20), which proves that the
model, though still perfectible, captures the essentials of the damage scenario.
Chapter 5
145
__________________________________________________________________________________
Figure 5-19: Comparison of the position and orientation of the delamination for mechanical impact at 75 m/s and the
associated numerical simulation
The same main delamination areas are found in both experimental scans and numerical view
at 90° (interface 3), -45° (interface 4) and 0° (interface 5). Figure 5.20 presents the
delamination per interface in the case of an impact at 65m/s obtained with the numerical
model. The simulations provide the adequate orientation of the damage, with delamination
oriented in the lower ply direction as expected.
Chapter 5
146
__________________________________________________________________________________
(a) (45/0)
(c)(-45/90)
(b)(0/-45)
(d) (90/-45)
(e)(-45/0)
(f) (0/45)
Figure 5-20: Damaged interfaces for the final numerical model, impact at 75 m/s
Chapter 5
147
__________________________________________________________________________________
5.6.3 Damage distribution through thickness
The relative distribution of delamination over the successive interfaces throughout the
thickness is also a significant indicator that the model can reach at least qualitatively
satisfactory agreement. Figure 5.21 shows the experimental measure of delamination
(labelled “mechanical impact”) compared with the 3D constitutive model estimation (see
figure 6.8 for interface numbering, interface 6 is opposite to the impact).
Mechanical impacts and numerical model results as well, provide delamination that goes
through the entire thickness of the laminate following the well-known helicoidally shape.
Mechanical impacts provide a growing delamination size while getting away from the impact
side.
1800
1600
1400
3D numerical model 75 m/s
Mechanical impact test 75m/s
1200
1000
800
600
400
200
0
1
2
3
4
5
Interface #
6
7
(a)
700
600
3D numerical model 81m/s
500
400
300
200
100
0
1
2
3
4
Interface #
5
6
(b)
Figure 5-21:Delamination distribution through thickness for 3D final model for impact at (a) 75m/s and (b) 81m/s
compared to experimental results
The 3D model still provides a slightly different distribution when compared to experimental
results but it provides according tendencies with the mechanical impact as shown in the
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148
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previous sections, in terms of delamination distribution in the sample depth. As previously
mentioned, the analysis of C-scans must be handled carefully, as presented in Appendix B.
The total delaminated area is overestimated by the numerical model for almost all cases.
This is attributed to the unsatisfying prediction of peak values of the rear face vertical
displacement at the centre of the plate; nevertheless, the new numerical model provides
according total delaminated areas as well as delamination orientation an according
distribution through thickness, which is a major improvement when compared to previous
models.
5.6.4 Conclusions on the predictive quantities of the model
This chapter has focused on the design of a 3D numerical model able to reproduce the
mechanical behaviour and damage of a composite plate subjected to medium velocity
impacts. An evolution of a pre-existing Continuum Damage Mechanics models, based on the
work of previous PhD and the available literature, was proposed to the present ranges of
incident masses and velocities.
A new model has been created, taking into account previous modelling [1, 2] and has been
filled with various improvements. The addition of a specific mesh oriented along the ply
direction, coupled with the modification of the coupling matrix q has proved to efficiently
represent the delamination orientation generated by the projectile impact on the composite
samples. The concordant orientation is also supported by the choice to use cohesive surface
instead of cohesive elements which also beneficially reduces the localization of the damage
due to the use of cohesive elements.
The numerical model presents concordant results in term of structural behaviour when
compared to experimental results. Indeed, the slope of the curve, the general displacement
profile and the maximum displacement are well approximated by the numerical simulations.
The analysis of the total delaminated area also provided satisfactory results and the study of
the delaminated interfaces, compared with mechanical impact C-scans, shows that the
model was able to reproduce both the size and orientation of the damage. Those results
validate the numerical simulations and the model results can then be confronted to lightning
strikes data in the following chapter.
5.7 Comparison of lightning damage with numerical simulations
Mechanical impacts have been designed along with a set of numerical models, from simple
dynamic simulations with homogenized shell elements to complete 3D model able to
represent the damage induced in the laminate during and after an impact.
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After the validation of the experimental mechanical tests by comparison with lightning
strikes, a second validation is made between lightning tests results and those obtained with
the numerical 3D model by comparing rear face measurements and interfaces damage as
well. This model provides additional information on the damage mechanisms and their
chronology over time and helps understanding the differences observed in the previous
section between lightning and mechanical damage.
Central rear-face displacements versus time plots are compared for lightning tests,
mechanical tests and computational results of i) the shell elastic models used for calibration
of the mechanical impact and ii) the 3D final constitutive model allowing estimate of the
induced damage. Such comparisons are shown on fig. 5.22 for lightning strikes 103, 101 and
107.
The shell and 3D models are firstly compared to the measurements after mechanical
impacts. All models provide a qualitatively correct description of the displacement, in
particular the long term structural behaviour (> 200μs) and post-peak plateau with
mechanical impact tests. However, the shell model systematically underestimates the
measurement by 10-30 %. This is partly corrected in the 3D model, which is closer to the
measurement than the shell result, especially at short times (≤50µs). This results from the
fact that damage contributes significantly to the rear face displacement, and damage is
taken into account only in the 3D damage model.
3500
Displacement (µm)
3000
2500
2000
1500
Shell model 65m/s
1000
Lightning test 103
500
3D model with ply damage and cohesive surface
65m/s
0
0
50
100
150
Time( µs)
200
250
300
(a)
Chapter 5
150
__________________________________________________________________________________
4000
3500
Displacement (µm)
3000
2500
2000
1500
1000
Lightning test 101
500
Shell model 72m/s
3D model ply damage and cohesive surface 75m/s
0
0
100
200
300
400
500
Time (µs)
(b)
4500
4000
Displacement (µm)
3500
3000
2500
2000
1500
Lightning test 107
1000
Shell model 80m/s
3D model damaging plies&interfaces 81m/s
500
0
0
100
200
Time (µs)
300
400
500
(c)
Figure 5-22:Comparison of out of plane displacements, for configurations 103 (a), 101 (b) and 107 (c). The 4 curves
correspond to the displacements for: lightning test, shell model equivalent impact, 3D model equivalent impact, and
mechanical impact test.
Comparison of the mechanical numerical deflections with the lightning strike measurement
shows that the short time behaviour of the lightning displacement is well described by the
mechanical equivalent model. Both shell and 3D models tend to frame the lightning curves
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151
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for the three different impacts shown on figure 5.35. For lightning sample 103, the 3D model
provides a higher maximal displacement at 65m/s but reproduce adequately the slope of
displacement especially for the simulation at 50m/s even though the maximum
displacement is too low. On the contrary for lightning sample 101, the simulated impact at
75m/s provides excellent results for both speed and displacement results, though still higher
in maximum displacement than lightning strike results. The same observations are made for
lightning sample 107.
As a generality, numerical simulations adequately reproduce the short time displacements
observed during lightning strike but overestimate the maximum deflection values while
underestimating the mechanical impact ones. As expected, the simulations are not able to
reproduce the long-time behaviour of composite thin panels impacted by lightning,
confirming the short time range of the equivalence. There are significant deviations at longer
times (>100μs) which are the result of delayed thermo-mechanical processes and cannot be
accounted for by the present fast hard sphere mechanical impact. These deviations were
expected from the previous shell modelling in chapter 3.
Structural behaviour predicted by the numerical simulations is then compared to the ones
obtained during lightning tests. Table 5.8 gathers the 3D numerical model displacements and
slopes of displacement at 50µs, the lightning equivalence time, compared to the lightning
tests results.
Lightning
samples
LS
displacement
(µm)
3D model
Numerical
displ. (µm)
Relative
difference
LS max
velocity
(m/s)
101
103
107
2861
1810
2626
2724.7
2357
2961.6
-4.7%
+30.2%
+12.7%
37.4
25.9
32.3
3D model
Numerical
max. vel.
(m/s)
32.3
27.2
37.1
Relative
difference
-13.6%
+5%
+14.8%
Table 5.8: Displacements and velocities at 50µs
The displacements and velocities at 50 μs for numerical modelling and lightning strikes are
compared in table 5.8. Displacement predictions of case 101 are 4% less than the numerical
impact at 75 m/s. The comparison for lightning sample 107 also gives satisfactory results
with a relative difference less than 15%. However, in the lightning case 103, the sample with
the least amount of paint at the surface, the numerical model provides a higher overall
displacement for the numerical simulation of about 33% and which is concordant with the
higher displacement curves observed previously in all the cases. This gap is intended to come
from the hole of paint at the plate centre that delays the effect of confinement but does not
invalidate the hypothesis by which core damage comes from a mechanical loading. This case
is then kept for the rest of the analysis with awareness of the disturbed effect of the paint.
Chapter 5
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Velocity predictions of case 101 are quite good (error around 13%) as well for case 103 and
107, which is concordant with the displacement errors observed previously. As for the
previous comparison between lightning tests and mechanical impact ones the surface state
of the lightning strikes is responsible for such difference.
5.7.2 Rear face displacement profiles
In chapter 3, a study of the displacement profile had been led between lightning strike tests
and the shell model in order to study the pressure applied by lightning on the composite
samples and see if the numerical model was able to reproduce this loading. The shell model
provided very sharp profile that punctually matched the maximum displacements but were
very different in shape from the lightning one. The same study is led with the M6 model. The
shell model M1 tends to be more flexible than the 3D model due to its element formulation.
It is also more sensitive to stamping which provided the sharp shape of the displacement
profiles. This default is corrected in the 3D model.
A standard lightning sample made of CFRP2, with the regular quasi-iso lay-up is used. The
sample is covered with ECF195 copper mesh protection and 300 µm of paint. An iterative
simulation with the shell model is conducted in order to obtain the mass and velocity that
suited best the lightning displacement for the projectile, as shown on figure 5.23. The
projectile is a 2g steel ball launched at a velocity of 120 m/s.
Figure 5.23 presents the profiles obtained with the 3D numerical model and compared with
the lightning strike results, centred on the centre of the composite plates. The comparison is
made at several instants. The figure clearly shows that the numerical model is able to
reproduce with accuracy the displacement profile obtained during a lightning strike both in
term of shape and maximum displacement values. Such analysis comforts the numerical
simulation validity to reproduce lightning at short times and even stabilization times
(<100µs) and validate the mechanical equivalence as a projectile is able to reproduce both
the displacement curve’s shape and maximum values for a lightning strike. However, at
larger times, as the numerical simulations is not able to reproduce the displacement plateau
of lightning, the profiles shape and maximum values differ, as expected.
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4
Numerical
simulation
t=19µs
Numerical
simulation
t=34.3µs
Numerical
simulation
t=51.4µs
Numerical
simulation
t=68.5µs
Lightning test
t=22.86µs
3,5
Displacement (mm)
3
2,5
2
1,5
1
Lightning test
t=38.1µs
0,5
Lightning test
t=57.4µs
0
-35
-25
-15
-5
5
15
25
35
45
Lightning test
t=64.76µs
Section (mm)
Figure 5-23: Deflection of the central portion of the samples vs time. Full lines: lightning strike tests; dotted lines: 3D
model impact simulations
After having compared the deflections, the focus is made on the delaminated areas provided
by both lightning tests and 3D diffuse damage material model.
Experimental total delaminated areas are evaluated by dedicated features within the US
analysis software, which approximates the effective delaminated areas through best-fit
ellipses, and sums up throughout the thickness. Numerical delaminated areas are computed
as the summation of elementary surfaces of all eroded cohesive elements. The methods of
measurement can differ from 10% as a consequence. As a result of US and computations
post-processing, table 5.9 synthesizes the obtained surfaces.
The mechanical 3D model provides tendencies in agreement with the mechanical impact as
shown in the previous sections, in terms of delamination distribution in the sample depth.
The total delaminated area is overestimated by the numerical model for all cases. This is
attributed to the unsatisfying prediction of peak values of the rear face vertical displacement
at the centre of the plate for cases 103 and 107. For case 101, the peak value of
displacement predicted by the numerical simulation reaches the peak value of displacement
obtained in the lightning test. It can be seen that for this case, the numerical estimate of the
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total delaminated is a quite accurate prediction of the delaminated area. Indeed for all
cases, the closer the peak displacement in the numerical model predicts the peak
displacement on lightning tests, the more the numerical delamination is an accurate
estimate of lightning tests measurements.
Lightning
samples
Associated
projectile
velocity (m/s)
101-1
101-3
101-2
102
103-1
103-2
107
75
75
70
124
50
65
81
Mechanical
LS delaminated
delaminated
area (mm²)
area (mm²)
2321
4185
0
1711
1085
1626
1019
3058
223
630
1746
Numerical
simulation
delaminated
area (mm²)
Relative
difference
between
simulation and
LS
1729
-25.5%
1661
3271
713
908
2062
-38%
-21.8%
-15%
Relative
difference
between
simulation and
MI
46%
6.4%
63%
6.9%
200%
44%
18%
Table 5.9: delaminated area for lightning strikes, associated mechanical impacts and numerical modelling
However, it is clear that computed delaminated areas systematically over-estimate the
measurements. This difference may find its origin in sensitivity to the difference in the
surface computation or to high deformation rates that is still not accounted for in the model
(work in progress).
5.7.4 Damage features
Total damage area shape and size is compared between the lightning strike and numerical
modelling. Figure 5.24 presents this comparison. The interfaces of same orientation are
circled with the same coloured ellipses. As expected after the comparison between lightning
strike and mechanical impact tests, the observed delamination distributions and shapes are
quite different between lightning strikes and numerical simulations of the equivalent
impacts. As seen before, the same extended damage at interface 3 (between plies at -45 and
90°) is found in both impacts. A common delamination is also visible at interface 2 (0/-45°)
which is not clearly visible for numerical simulation as it is hidden by the larger one at
interface 4 (90/-45°) obtained after the mechanical impacts. However, the extensive rear
face delamination on interface 6 (0/45) is reduced for the lightning strike, that did not
provide as much damage than the mechanical equivalent simulation, even though, for this
lightning case, the delamination went through all the laminate thickness.
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-45/90
90/-45
0/45
(a)
(b)
Figure 5-24: Comparison of the damage features between (a) lightning strike tests and (b) numerical simulation for
composite material [45/0/-45/90]s
5.7.5 Distribution through thickness
As presented in [10], the numerical model tends to reproduce the mechanical impact. The
damage distribution between the numerical model/mechanical tests is not the same as the
lightning strike distribution results. Indeed, lightning strike tends to provide delamination
mainly in the first half of the impacted lay-up, delamination which stops at the double 90/90
interface. On the other hand, mechanical impacts and numerical model results as well,
provide delamination that goes through the entire thickness of the laminate following the
well-known helicoidally shape [82], figure 5.24. As for Figure 5.25, the abscissa corresponds
Chapter 5
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to the depth from the side opposite to the impact, and the dark vertical line marks the
middle depth. Impacts and LS have occurred at the right extremity of the abscissa axis. This
strong difference may be due to the energy delivery during lightning strike and to the
influence of the surface state (presence of protection and paint) on the samples during the
short times period, surface state which is not represented in our mechanical impacts. It has
been shown that the presence of paint above the composite material has a detrimental
effect due to the confining effect of the dielectric layer. This suggests that the surface state
can influence the energy repartition and delivery both in time and space in certain cases.
This had been evidenced in by examining the pressure profile at the rear face of the sample
in section 5.5.2. It was shown that the pressure profile during a lightning strike was
smoother and flatter than during our mechanical impact using a very small steel ball.
700
3D model 75m/s
600
Lightning test 101
500
400
300
200
100
1,52-1,6
1,45-1,52
1,37-1,45
1,29-1,37
1,22-1,29
1,14-1,22
1,07-1,14
Thickness (mm)
0,99-1,07
0,92-0,99
0,84-0,92
0,76-0,84
0,69-0,76
0,61-0,69
0,53-0,61
0,46-0,53
0,39-0,46
0,2-0,3
0,3-0,39
0,1-0,2
0-0,1
0
(a)
800
3D model 81m/s
700
Lightning test 107
600
500
400
300
200
0
0-0,1
0,1-0,2
0,2-0,3
0,3-0,38
0,39-…
0,48-…
0,57-…
0,66-…
0,75-…
0,84-…
0,93-…
1,02-…
1,11-…
1,21-1,3
1,3-1,39
1,39-…
1,48-…
1,57-…
1,66-…
1,75-…
100
Thickness (mm)
(b)
Figure 5-25:Histograms of the delamination as a function of position in the thickness of the material, for # 101 (a) and #
107 (b) samples.
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5.8 Conclusions
Mechanical impacts have been designed as an alternative to lightning tests (chapter 3),
seeking for out of plane displacements and velocities consistent with lighting configurations;
this consistency was established according to i) impulse based equivalence principle and ii) a
first set of dynamics simulations with simple elastic shell models. Then a 3D model was
adapted to present ranges of masses and velocities, in order to compute damage reached
within the laminate. A first step in the validation procedure consisted in checking the validity
of the proposed methodology by comparing rear face displacement measurements between
lightning and mechanical tests, as well as shell and 3D computations. Once established that
mechanical impact characteristics are suitable to the given objective of equivalent out of
plane displacements and velocities, a second step of the validation procedure consists in
comparing delaminated areas reached by i) simulations of mechanical impacts and ii)
mechanical impact tests, in the range of incident masses and velocities.
The present chapter proposed a methodology to approximate a lightning test by a
mechanical impact relying on an equivalence criterion based on rear-face deflection. The
interest is that, if such equivalence is validated, the lightning damage could be simulated by
a mechanical model which would then permit faster lightning protection prototyping and
which would be helpful for both design of improved solutions and understanding of lightning
damage process.
As a conclusion we can say that the numerical model helps understanding that the energy
delivery through a small and rigid impactor is not the best solution to represent lightning as
our model is not able to reproduce large time deflection of the lightning event. Two research
tracks are then possible: in the cases when surface state does not badly influence the
deflection behaviour the study of a more adequate projectile is to be led. On the other hand,
for cases when paint does influence the overall deflection and damaging behaviour of our
lightning samples we must re-think our primary hypothesis of decoupling what happens at
the surface and at the core of the material for both seems to be strongly coupled. In this
perspective a coupled electro-thermal analysis coupled with a mechanical one is to be led (in
progress).
However, the numerical model is able to represent the damage obtained during a
mechanical impact involving a launched projectile as well as the displacement at short times
with certain accuracy (as shown with the displacement profiles in figure 7.15). This model
can now be used to test various configurations and parameters such as the type of material
or the influence of the sequence. Moreover, even if the steel ball is not adapted, other
loading situations have been tested especially during the preliminary works presented in
chapter 3, such as an equivalent pressure. Such loadings are implemented in the numerical
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158
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model and their results in terms of damage and displacement are extracted and compared
with lightning results in preliminary studies, presented in chapter 6.
Though issues related to delamination distribution through thickness remain (chapter 4), the
deflection of the samples and the damage scenario are quite correctly represented so that
the proposed methodology is promising. The steps beyond are i) improvement of the 3D
damage model to better estimate the total delaminated area, and ii) continued calibration of
the equivalent mechanical impact test, based on both displacements of rear-face and
damage observed in specimens subjected to lightning. The method resulting from such
improvements is a useful tool in the design of lightning protected materials, where
modelling would play a significant role in prototyping phases.
As the numerical model has been validated, meaning that it correctly reproduces the
damage due to mechanical impacts, it is possible to modify the projectile or the loading in
the simulation in order to experiment other strategies to reproduce lightning displacement
and damage. Chapter 6 presents several projectiles that have been tested as preliminary
prospects as well as an equivalent pressure to lightning.
Chapter 5
159
Chapter 5
160
6
Chapter 6
Prospective works
6.1 Introduction ......................................................................................................................................... 1623
6.2 Prospective studies on the effect of the contact surface and stiffness .................................................... 164
6.2.1 Changing the projectile’s shape ............................................................................................................. 164
6.2.2 Changing the projectile’s nature............................................................................................................ 174
6.3 Prospective study on the effect of evolving pressure loading ................................................................. 177
6.3 Conclusions and perspective .................................................................................................................. 187
General conclusion ....................................................................................................................................... 190
Chapter 6
161
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Objectives
This chapter presents prospective works using the numerical model validated in chapter 5, in
which the impulse is delivered using other steel projectiles considering an impact delivery of
the impulse or through a pressure applied on the sample’s surface that evolves in time and
space, following the arc root evolution proposed in chapter 3.
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6.1 Introduction
Equivalent mechanical impact tests have been designed and validated by comparison with
lightning strike results, on the lightning equivalence time defined in chapter 3 and chapter 4.
Several results have been presented in chapter 5 on the equivalence of the numerical model
to mechanical real impacts and lighting strikes. The equivalent tests have demonstrated their
ability to reproduce both the kinematic behaviour of the samples (rear face displacement
and velocity) as well as the damage area of lightning for the so called ‘short times’. However,
they cannot reproduce displacement (superior to lightning equivalence time) and fail at
reproducing a concordant damage distribution through laminate thickness in some cases
from the stabilization time, and in all cases at large times. The study of the numerical results
show that even the large time displacement due to lightning could not be obtained with a
small steel ball, and it has been concluded that this way of loading is the clue of the gap
between the current mechanical impact and the lightning strike behaviours. Indeed, the
small size of the projectile led to an energy delivery that differs from the one observed
during lightning strike. Moreover, such an impactor cannot take into account the influence
of the surface state. In chapter 4, the influence of paint over the displacement profile of the
sample has been established, see figure 4.25. Such a temporal evolution indicates that
lightning may proceed in two different steps, depositing energy with a peculiar distribution
over time that explains the observed plate movements. In order to better reproduce this
peculiar evolution over time and the large time behaviour of the plate submitted to
lightning, as a first attempt of improvement the kind of surface loading should be revised.
The following chapter presents several prospective works on this subject which aim to
improve the mechanical equivalence through different shapes and materials for the
projectile in numerical simulations.
A third loading is also tested. As the complete and detailed numerical model has been
validated in chapter 5, and is able to reproduce adequately the damage resulting from a
mechanical impact equivalent to lightning, the damage formulation can be used with
loadings other than a spherical projectile. In this scope, an equivalent pressure has been
created and applied on the numerical model in order to study the damage resulting from a
lightning loading. The amplitude is derived from [43] and the magnetic pressure presented in
section 3.1.3.2. The loading surface evolves in space and in time following the computation
of the arc root proposed in §3.4.
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6.2 Prospective studies on the effect of the contact surface and stiffness
6.2.1 Changing the projectile’s shape
The previous simulation and experimental results involved a spherical projectile. The
impactor was a steel ball of diameter Φ9,8 mm and of mass 4g. With this projectile, the
model provided good displacement and velocity results but the damage distribution through
the thickness was different than the one observed during lightning strike tests. The
difference can be attributed to the surface area of energy deposit. Indeed, in a previous
study on the arc root modelling, presented in section 3.1.3.1, it has been established that
the lightning arc radius grew over the current deposition time, up to several centimetres.
The current is thus deposited over a large portion of the sample’s surface. In order to take
into account this parameter of the surface deposition, another projectile shape is proposed:
a cylinder.
The cylindrical projectile is numerically tested on several lightning strikes that have already
been reproduced with the spherical impactor [128]. Appendix F presents in details the
results comparison between the spherical and the cylindrical projectiles. Several velocities
are tested in this numerical plan. The numerical impacts will be compared to lightning results
for rear face displacements and generated damage.
Definition of the cylindrical projectile
A new projectile is defined which has a flat circular base. By using such a shape, the contact
area between the projectile and the sample is increased. To do so, the masses and radius
previously calculated for the spherical projectile are kept identical. The radius was obtained
thanks to the density of the chosen material (here for steel ρ= 7927 kg/m3). Finally, the
height can easily be calculated from the volume and radius of the cylinder, as shown on
figure 6.1. Values of cylinder’s height are gathered in table 6.3. Firstly, the volume of the
cylinder is obtained:
V = πhr 2
(Equation. 6.1)
The mass is given by:
m = ρV
(Equation. 6.2)
It comes:
V
h = πr2 =
m
ρ
( )
πr²
(Equation. 6.3)
164
__________________________________________________________________________________
Figure 6-1: Cylindrical projectile parameters
Finite element model
The 3D model with elastic properties for the plies and with cohesive elements for the
interfaces, presented in Appendix E, is used for all the following simulations. Cohesive
elements properties are reminded in table 6.1 and composite plies properties in table 6.2.
The numerical sample is a 400*400*2mm3 plate with the circular clamping system of twelve
fasteners presented in section 4.1.2. The usual sequence is used as presented in figure 6.2.
The projectile is positioned at the centre of the plate and an initial velocity is imposed that
corresponds to the maximum velocities obtained during lightning strike tests.
E11
(GPa)
116.5
E22
(GPa)
8.9
E33
(GPa)
8.9
ν12
ν23
ν13
0.3
0.3
0.35
G12
(GPa)
5.7
G23
(GPa)
5.7
G13
(GPa)
3.7
Tableau 6.1: Composite material parameters
KI
(kN/mm3)
100
KII
(kN/mm3)
100
σn
(MPa)
50
σs
(MPa)
50
GIc
(J/m²)
500
GIIc
(J/m²)
1200
α
1.0
Tableau 6.2: Material parameters for the cohesive interfaces
165
__________________________________________________________________________________
v
Figure 6-2: Sequence for the composite laminate
Lightning associated cases
Table 6.3 presents the lightning cases and the associated parameters for the cylinders to be
launched on the numerical samples.
Sample #
1
2
3
4
surface
state
NoECFNoP
ECF195NoP
ECF195P200
ECF73P200
I=mv
(N.s)
vmax (m/s)
m (g)
r (mm)
h (mm)
0.24
30
8
6.2
8.3
0.20
11
18
8.1
11
0.28
58
5
5.3
7.1
0.35
80
4
4.9
6.7
Table 6.3: Lightning cases and equivalent cylinder mechanical impacts parameters
Study of the displacement results
The numerical results are presented in figure 6.3 and compared with lightning strike and
spherical projectile ones. The simulation results provided rear face displacements and
velocities that are concordant for both short and stabilization times for all cases. Different
analyses can be made either on displacements or velocities, for the different time intervals
identified in chapter 3.
Velocities at short times are well predicted by the numerical model, for all cases, and the
application of the loading with the cylinder gives better results than with a ball. This proves
166
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that the methodology of the basis of equivalence using the impulse is valid for short times,
that is for strike times a*up to which damage occur.
In case of sample 1, the displacement is enhanced compared to the ball impact simulation,
and in case of sample 2, the model is even concordant at large times (> 100 µs) with
lightning strike tests. It is noticeable that enhancements are essentially obtained especially
for samples 1 and 2 which are unpainted panels.
The only difference between cases 1 and 2, and cases 3 and 4, is the absence of paint. This
separation between painted and unpainted samples is of the same kind than it was for the
lightning strike loading and the ball impact loading. It is concluded that the absence of paint
is the condition that makes possible the assumption of the decorrelation between the
complex multi-physics in the coating of the laminate and the bulk damage inside. In absence
of paint, the first Newton law can be applied for the laminate which can be isolated from its
environment.
The difference between case 1 and case 2 is the presence of the metallic mesh on sample 2.
In this case, the change in surface between the projectile and the plate which is modelled by
changing the shape of the projectile from a ball to a cylinder gives the right displacement
and velocities up to the stabilization time, and even up to large times. This suggests that the
time delay to apply the loading in the different phases of time (even if the amplitude is
constant over the surface) is better reproduced in case 2 than it is in case 1. This suggests
that in the lightning strike tests the metallic mesh changes the characteristic times of the
loading generated by the multi-physics event acting on the upper surface of the laminate.
Displacements and velocities for the painted samples are similar to those obtained with the
spherical projectile, and the cylindrical projectile cannot reproduce large time displacement
in spite of the bigger contact surface between the projectile and the sample. These
comparisons suggest that the supposed larger contact area’s action may rise later in the
lightning loading on painted samples and has little influence on the short time behaviour of
these samples. The two steps form of the lightning loading may thus involve a sharp first
pressure peak, followed by a second one with a larger contact surface. However, when this
second surface comes to press on the sample, the mechanical projectiles have already
transferred all their energy and are unable to reproduce this second loading.
167
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3000
Deflection (µm)
2500
2000
1500
1000
Lightning test #1
500
Num Simulation steel ball
Num Simulation steel cylinder
0
0
100
200
300
400
500
Time (µs)
2500
Deflection (µm)
2000
1500
1000
Lightning test #2
500
0
0
100
200
300
400
500
Time (µs)
168
__________________________________________________________________________________
4000
3500
Deflection (µm)
3000
2500
2000
1500
Lightning test #3
1000
500
0
0
100
200
300
400
500
Time (µs)
4500
4000
Deflection (µm)
3500
3000
2500
2000
1500
1000
Lightnind test #4
500
0
0
100
200
300
400
500
Time (µs)
Figure 6-3: Comparison of rear face displacement for mechanical impact with cylindrical projectile and lightning results 1
to 4.
Damage distribution
The numerical model allows extracting the damaged areas in the cohesive interfaces and
thus the total delaminated area generated by the impact. The results for the four simulated
cases are presented in table 6.4 and compared with lightning strike results. It is then possible
to draw histograms of the delamination distribution through thickness and compare the
169
__________________________________________________________________________________
results with the lightning strike damage. The histograms for the different cases are
presented in figure 6.4.
Total
2542
90,433
ECF195-NoP
LS test
simulation
182
10,73
25
14,625
0
11,48
0
0
0
0
0
0
207
ECF195-200µ
LS test
simulation
255
204,48
107
261,8
646
1123,08
655
430,92
0
209,92
0
208,5127
36,835
1663
2438,7127
ECF73-200µ
LS test
simulation
306
223,628
1067
252,77
2510
1219,52
722
1159,64
0
604,35
0
205,92
4605
3665,828
Table 6.4: Delaminated interfaces for each couple of lightning/mechanical impacts
The comparison of the damage area presented in the various histograms highlights several
observations. Almost no damage is obtained for the case 2, presumably due to the very low
velocity associated with the impact. For all the other cases, it must be noted that, contrary
to lightning strike tests, and accordingly with the spherical ball previously used, the
cylindrical projectile generates damage in all the thickness of the laminate. Finally, impact
case 3 and 4 provided total delaminated area of the same order of magnitude than the
lightning strikes even though the damage distribution is still different. Lightning case 1
provides larger delaminated area than case 3, even though the associated projectile’s
velocity is smaller, due to the absence of protection on sample #1, inducing more damage
than sample #3 which is protected.
1600
Int1
Int2
Int3
Int4
Int5
Int6
NoECF-NoP
LS test
simulation
228
0
1350
0
964
52,991
0
37,442
0
0
0
0
1400
Lightning test #1
1200
Cylinder projectile
1000
800
600
400
200
0
Interface #
1
2
3
4
5
6
170
__________________________________________________________________________________
200
Lightning test #2
Cylinder projectile
150
100
50
0
Interface # 1
2
3
4
5
6
1200
Lightning test #3
1000
Cylinder projectile
800
600
400
200
0
Interface # 1
2
3
4
5
6
3000
Lightning test #4
2500
Cylinder projectile
2000
1500
1000
500
0
1
2
3
4
Interface #
5
6
Figure 6-4: Histograms for the damage distribution through thickness, comparison between lightning and cylindrical
projectile simulations (sequence [45/0/-45/90]s) for lightning sample 1 to 4.
Damage cartography of each interface is available for the numerical simulation of sample #4
and presented in figure 6.5. The damage areas follow the shape of the cylinder basis, a flat
circular area of delaminated elements. The damages are concentrated on the rim of the
projectile and the centre of the disc does not generate any damage. This damage mechanism
is similar to the one observed during lightning strike tests. Indeed, in chapter 3, the study of
the arc root attachment revealed that the arc root radius expanded over time. It was then
assessed that the magnetic pressure coming from this arc would only press the sample
through the arc root circumference, leaving the central part of the sample free of pressure.
171
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(a)Interface 1
(b) Interface 2
(c)Interface 3
(d) Interface 4
(e)Interface 5
(f) Interface 6
Figure 6-5: Delaminated area for the numerical model for each interface of the laminate of sequence [45/0/-45/90]s
Conclusions
In this section the focus has been put on the variation of the projectile’s shape. Indeed,
previous results obtained with a spherical impactor (see chapter 5) highlighted the act that
172
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this geometry was not the most appropriate to reproduce lightning strike damage and
displacement. The main difference consisted in the energy deposit between the lightning arc
and the spherical steel ball. A prospective study has thus been led to increase the projectile
contact’s surface with the impacted sample, as lightning shows an increasing arc root radius
when delivering its current during a strike. To do so, a cylindrical projectile is proposed as a
first alternative to the steel ball. The cylinder conserves the mass of the previous impactor
and the steel ball radius is applied to the basis of the cylinder. The height is consequently
calculated and several lightning strike cases, that had previously been compared with
spherical impactor simulations, are compared with the new projectile simulation results.
The obtained displacement results for the cylinder simulations were similar to those
obtained with the sphere and provided according results at short times with lightning strike
tests, and even at large time for unpainted sample 1 and 2. However, for painted samples 3
and 4, the cylinder provides similar results than with the previous projectile, and is not able
to reproduce the large time behaviour of the lightning samples. It is concluded from the
analysis of displacements that the presence of the mesh changes the time characteristics of
the multi-physics event in the surface, and that the paint presence changes the interaction
between the laminate and its environment after stabilization times.
It is noticed that velocities are better approximated at short and strike times. This is
noticeable for case 1, and also for cases 3 and 4. The velocities evolutions (accelerations) are
also better approximated. This proves that the equivalence method based on the impulse is
valid.
The cohesive elements used to simulate the interface between the plies allowed extracting
the delaminated area in each interface and compare the total delaminated area with
lightning strike results as well as the damage distribution through thickness. The cylindrical
projectile provides according total delaminated area when compared to lightning strike tests
but, as expected, the distribution through thickness is not well reproduced. As for the
spherical impactor, the cylinder generates delamination in every interface while lightning
tends to generate damage only up to half of the laminate thickness. However, the study of
the cylinder delaminated area for each interface shows that the projectile applied a pressure
on the sample that strongly resemble the hypothesis made on the arc root and magnetic
pressure presented in chapter 3.
Furthermore, it is concluded that using the right loading surface both allowed us to enhance
the rear face velocities and core damage predictions. Thus, the study of the influence of the
loading area should be pursued in the future.
173
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6.2.2 Changing the projectile’s nature
In the previous chapter, it has been stated that the spherical projectile was not the most
suitable geometry to reproduce lightning rear face displacement and damage on CFRP thin
panels. A first preliminary study investigated the influence of the shape of the projectile and
more precisely the influence of the contact area. The study of lightning strike displacement
results also highlighted the fact that the lightning equivalent loading was a two steps
pressure (figure 4.39). In this section, a preliminary study is led to try assessing this loading
and finding another projectile that could reproduce the particular loading of lightning.
New projectile definition
In this scope, a modification is made on the projectile’s material instead of its geometry. The
proposed new impactor is a spherical ball made of rubber, and is thus a deformable
projectile. The point here is to generate a projectile which will deform when hitting the
target sample and thus press the composite plate with both an increasing contact area and
over larger time duration. To do so an Ogden material model [116] (hyper elastic material) is
applied for the rubber projectile. This material law is based on non-linear stress-strain
behaviour and the material is described by means of a strain energy density function, from
which the stress–strain relationships can be derived.
The strain energy density is expressed in terms of principal stretches λj, j=1, 2, 3 such as:
𝜇𝑝
𝛼
𝛼
𝛼
𝑝
𝑝
𝑝
𝑊(𝜆1 , 𝜆2 , 𝜆3 ) = ∑𝑁
𝑝=1 𝛼 (𝜆1 + 𝜆2 + 𝜆3 − 3)
(Eq. 6.4)
𝑝
Where N is the strain energy potential order, μp and αp are material constants. The material
constants for the rubber material are presented in table 6.5 [119]. The projectile is tested on
the configuration of lightning sample #3 (see table 6.3): velocity of 58 m/s and preserved
mass of 5g. The density of the chosen material is ρ=1160 kg/mm3 which provide a radius of
10 mm.
N
3
μ1
-12,09
α1
14,4
μ2
12,11
α2
14,4
μ3
1,75
α3
1,91
D1
0
D2
0
D3
0
Table 6.5: Material constants for the Ogden rubber material law
The composite plate material is the same that the one presented in the previous section
6.2.1. For this preliminary study, the simplified shell model, presented in chapter 3, is used
to reduce the computation time, however, no damage can be extracted from the model and
the comparison is only made on rear face displacement data.
Figure 6.6 presents several views of the projectile impacting the numerical sample at
different times. The several images clearly show the evolution and the crushing of the
rubber projectile on the sample. The figure also shows that after 350 µs (figure 6.7 (d)) the
174
__________________________________________________________________________________
centre of the projectile is not in contact with the composite plate anymore and thus does
not apply any more pressure, which is concentrated on the external surface of the impactor.
(a) t=0 µs
(b) t=150 µs
(c) t=250 µs
(d) t=350 µs
(e) t=400 µs
(f) t=550 µs
Figure 6-6:Numerical impacts of the rubber projectile for several times (shell model)
175
__________________________________________________________________________________
Displacement results
The displacement results are presented in figure 6.7 and compare the results obtained with
both lightning strike test and steel ball projectile simulation. The comparison shows that,
contrary to the non-deformable steel ball projectile, the rubber impactor provides a slower
rear face velocity, the slope of the curve being less abrupt than for lightning strike.
Maximum displacement is of concordant order of magnitude, but this projectile does not
correctly reproduce the short time behaviour of the sample. Yet, it has been established in
chapter 5, that all the damage is generated during the first 50 to 100 µs of the lightning
impact, it can thus be supposed that this configuration would not provide a concordant
damage distribution and total damage area. However, this result must be verified by
conducting the same simulation using the 3D damage model established in chapter 5.
4000
3500
Displacement (µm)
3000
2500
2000
1500
Lightning strike #3
1000
Steel ball projectile simulation
500
Rubber projectile simulation
0
0
200
400
600
800
1000
Time (µs)
Figure 6-7: Comparison between spherical, cylindrical projectiles and lightning
Conclusion
This study was a first tentative to reproduce the peculiar lightning loading with an extensible
load surface by modifying the projectile used as a first approximation. Because of its rigidity,
the contact area was too small, so that the rigid ball seemed to be inappropriate to the
desired modelling. It was demonstrated that using the right value of the contact area with a
cylinder gave the right velocity of the rear face. Since the velocity and the contact surface
seem to be the clue of getting the right delamination distribution in the thickness, as a
second approximation, a rubber material, already used for tyre impacts certification
simulation was used, thanks to its deformable capability, to try another sort of loading on
the usual composite panel. The simulation shows that, for the rear face displacement
176
__________________________________________________________________________________
parameter, this projectile did not provide concordant results. The same impact is to be
simulated with a damage model in order to assess the obtained total damage with such a
projectile, as well as the damage distribution through thickness, in order to obtain more
comparison data on this kind of material. However, this study is just a preliminary work and
other projectiles can be tested such as eroding projectiles. If a deformable impactor is not
sufficient to represent the loading generated by lightning on thin CFRP plate, projectile with
several impedances can be imagined and numerically designed to try reproducing lightning
results. Moreover, other kind of loading can be tested such as variation of the magnetic
pressure presented in section 3.2.2 and applied on the damage numerical model to assess
the obtained damage with such a pressure.
6.3 Prospective study on the effect of evolving pressure loading
In the chapter 3, several studies have been presented. A magnetic pressure has been defined
that is reused here and improved. In order to integrate the various observations made
before, the magnetic pressure is applied on a growing area. In fact, in order to couple both
the observations made on the arc root growing radius and the impact contact area, the
magnetic pressure is applied on a circular cylinder of radius r(t) which is dependent of both
time and current injection.
For this study a lightning sample protected with ECF195 and painted with 170 µm is
considered. The volume of an “eaten” metallic mesh crown by the injected current from the
arc is calculated as follow:
𝑑𝑉 = 2𝜋𝑟𝑑𝑟
(Equation. 6.5)
The metallic resistance is given by:
1 𝑑𝑟
𝑑𝑅 = 𝜎 2𝜋𝑒𝑟
(Equation. 6.6)
With e the thickness of the metallic protection (e=0.179mm). This vanishing crown is
considered completely sublimated at a time t(r) when it would have absorbed a quantity of
energy: ρ ΔH dV.
This energy is induced by Joule effects such as:
𝑡(𝑟)
𝜌 𝛥𝐻 𝑑𝑉 = 𝑘 ∫0
𝑑𝑅 𝐼(𝜏)2 𝑑𝜏
(Equation. 6.7)
177
__________________________________________________________________________________
Then, it is possible to calculate the radius of the crown:
1
𝑟(𝑡) = 2𝜋𝛿 (
𝑡 1
𝐼(𝜏)2 𝑑𝜏
𝜎(𝑇)
𝑘𝜌 ∫0
𝛥𝐻
1
2
)
(Equation. 6.8)
The resistivity σ(T) increases almost linearly with the temperature. For the copper of the
protection, the chosen values are gathered in table 6.6.
1
1
= 𝜎(𝑇 ) (1 + 𝛼(𝑇 − 𝑇0 ))
𝜎(𝑇)
(Equation. 6.9)
0
𝟏
α (K-1)
𝟏
(Ωm)
𝝈(𝑻=𝟐𝟎°𝑪)
-3
6.8 10
1.68 10
𝝈(𝑻=𝟏𝟎𝟎𝟎°𝑪)
-8
13.1 10
(Ωm)
-8
Table 6.6: Resistivity values for the copper mesh
The integral of I(t)² is calculated as follows and corresponds to the Action integral (A².s) of
the injected current corresponding to the component D of lightning (cf. section 2.2.4):
𝑡
∫0 𝐼(𝜏)2 𝑑𝜏 = 𝐼0² ((
1−𝑒 −𝛼𝑡
2𝛼
)+(
1−𝑒 −𝛽𝑡
2𝛽
2(1−𝑒 −(𝛼+𝛽)𝑡 )
)−(
𝛼+𝛽
))
(Equation. 6.10)
The magnetic pressure is then computed for r(t). The position d on the sample is numerically
computed. We can differentiate two regions of application of the magnetic pressure: region
1 and 2 as shown on figure 6.8.
Figure 6-8: Scheme of the sample: region 1 under the arc root and region 2 beyond the arc root [45]
178
__________________________________________________________________________________
 For r > r(t), region 2 the magnetic pressure beyond the arc root radius is obtained as
follow :
The current conservation provides the current density j (A/m2):
I(t)
j(r) = 2πdr
(Equation. 6.11)
In cylindrical coordinate the Ampere’s law ∇ × H = j reads then
1 ∂Hz
r ∂Φ
−
∂HΦ
∂Z
= jr =
I(t)
(Equation. 6.12)
2πrd
For axi-symmetrical applications the magnetic field does not depend on Φ by symmetry,
thus:
∂HΦ
∂Z
=−
I(t)
(Equation. 6.13)
2πrd
Finally integrating the above equation gives:
HΦ (r, z) = −
I(t) z
2πr d
= −jr . z
(Equation. 6.14)
The magnetic volume force fV is defined as:
f v = j × B = j × μ0 H
(Equation. 6.15)
where μ0 is the magnetic permeability. For this axi-symmetrical situation the magnetic
volume force has only one component in z-direction
fzv ⃗⃗⃗
ez = jr e⃗⃗⃗r × BΦ ⃗⃗⃗⃗
eΦ =
μ0 I(t)2 z
e
⃗⃗⃗
4π2 r2 d2 z
= −μ0 j2r . ze⃗⃗⃗z
(Equation. 6.16)
The “effective” magnetic surface pressure acting on the upper side of the sample is given by
integration over the thickness of the sample:
d μ0 I(t)2 z
μ0 I(t)2
4π2 r2 d
8π2 r2
P(r) = ∫0
dz =
2
(Equation. 6.17)
The magnetic pressure will be a maximum at the arc root maximum radius value r(t):
Pmax =
μ0 I(t)2
8π² r(t)²
(Equation. 6.18)
179
__________________________________________________________________________________
With the current given by:
I(t) = A (e−αt − e−βt )
(Equation. 6.19)
with A=6.58.106 A; α=50000 s-1; β=52000 s-1 ;
e= thickness of the metallic protection=δ/ρ [m]
ρ=material density (protection) [kg/m²]
δ=surface mass of the protection [kg/m3]
ΔH=enthalpy of fusion (or vaporization) [J/kg]
k phenomenological coefficient (k=1for SCF, k=1.7 for ECF)
σ electric conductivity, temperature dependent [S.m-1]
 For r < r(t), wich corresponds at the region 1, we calculate the mean surface
magnetic pressure on the arc root area :
In region (1) the magnetic pressure / force depends on the distribution of the impressed
lightning current density, the sample thickness, and the damage (melting and vaporisation)
of the material under and in direct neighbourhood of the arc root.
I(t)
jr (r) = 2πd
r
𝑓𝑜𝑟 𝑟 ≤ 𝑟(𝑡)
r(t)²
(Equation. 6.19)
This radial current density component induces a magnetic field with the component given
by:
HΦ (r, z) = −
I(t)
r
z
𝑓𝑜𝑟 𝑟 ≤ 𝑟(𝑡)
2π r(t)² d
(Equation. 6.20)
Finally the magnetic volume force in z-direction is derived using equations (6.15) and (6.16):
fzv ⃗⃗⃗
ez = −
μ0 I(t)2 r2
4π2
z
e
⃗⃗⃗
𝑟(𝑡)4 d2 z
(Equation. 6.21)
The “effective” magnetic surface pressure under the root radius is given by integration over
the thickness of the sample
d μ0 I(t)2 r2
P(r) = ∫0
4π2
z
𝑟(𝑡)4 d
dz =
2
μ0 I(t)2 r2
8π2
𝑟(𝑡)4
(Equation. 6.22)
180
__________________________________________________________________________________
This magnetic pressure has a maximum at the arc root radius r(t) and approaches to zero as r
goes to zero. To obtain the total force acting under the arc root radius area the magnetic
pressure (equation (6.22)) has to be integrated over this area:
r(t)
F = ∫ P. dA = ∫0
2πr
μ0 I(t)2 r2
8π2
dr =
4
𝑟(𝑡)
μ0 I(t)2
r(t) 3
∫
4π𝑟(𝑡)4 0
r dr =
μ0 I(t)2
16π
(Equation.
6.23)
The total magnetic force under the arc root is independent of value of the arc root radius
r(t). The mean magnetic pressure 𝑃̅ under the arc root area can be defined as ratio of this
2
total force F to the arc root area 𝜋𝑅𝑚𝑎𝑥
:
̅
P=
F
πr(t)²
=
μ0 I(t)2
(Equation. 6.24)
16π2 r(t)²
Final formulation of the magnetic pressure P(r)
For our numerical model we chose a simplified expression of the magnetic pressure which
should provide the same order of magnitude than the complete version of the magnetic
pressure, such as:
 For r < r(t), B=0  P=0
 For r > r(t): 𝐏(𝐫) =
𝛍𝟎 𝐈(𝐭)𝟐
𝟒𝛑𝟐 𝐫 𝟐
(Equation. 6.25)
The numerical intensity reaches a maximum at 19.6µs and a value of 95 kA which is inferior
to the expected value of 100kA but remains acceptable. By taking Imax=95kA and RR=6mm,
we obtain a maximum value of Pmax=40MPa.
Numerical model
The numerical simulation is led on the model presented in chapter 5 that incorporates the
latest modifications: a coupling between a damage law for the plies and the interfaces into a
user defined law called VUMAT in Abaqus. The material properties for the plies and
interfaces are presented in tables 5.10 and 5.11. The usual sequence presented in figure 6.9
is used. The loading is implemented by using a spatial and temporal formulation in a second
user law for the pressure, called VDLOAD.
181
__________________________________________________________________________________
-3
(a)t=1e ms
-3
(b)t=5e ms
(c)t=10e-3 ms
182
__________________________________________________________________________________
(d)t=50e-3 ms
Figure 6-9:Time and spatial evolution of the applied magnetic pressure
Figures 6.10 and 6.11 present the comparison of the rear face displacement and velocity
results for the magnetic pressure and the associated lightning strike test. The numerical
model provides according results for the short time maximum displacement and velocity,
even though the maximum velocity is higher (102 m/s for the magnetic pressure versus 83
m/s for the lightning strike). The displacement follows a plateau and does not decrease as it
does for a rigid ball impact. However, the magnetic pressure does not reproduce the large
time displacement profile and the displacement plateau is lower than the expected one:
2300 mm for the numerical simulation versus 3300 for the lightning strike.
183
__________________________________________________________________________________
4500
4000
Displacement (µm)
3500
3000
2500
2000
1500
Lightning strike test #121
1000
500
Magnetic pressure simulation
Mechanical simulation steel ball projectile impact at 81m/s
0
0
50
100
150
200
250
300
350
Time (µs)
Figure 6-10: Rear face displacement results for magnetic pressure and associated lightning strike
120
Magnetic pressure
simulation
Lightning strike test
Velocity (m/s)
100
80
60
40
20
0
0
50
100
Time (µs)
150
200
Figure 6-11: Rear face velocity results for magnetic pressure and associated lightning strike
Finally, the damage distribution through thickness is numerically computed to be compared
with the lightning strike results. Table 6.7 compares the magnetic pressure total delaminated
area with several lightning test ones. The lightning case used for this study (#121) provided a
total delaminated area of 2112 mm². The magnetic pressure generated 2186.5 mm² of
delaminated area which is concordant with the lightning results.
Figure 6.12 presents the comparison of the damage features obtained with the projectile
simulation at 81 m/s (figure 6.12 (c)), the magnetic pressure (figure 6.12 (b)) and the
lightning strike 107 (figure 6.12 (a)).the two impacts damage and Figure 6.12 the distribution
through thickness comparison
184
__________________________________________________________________________________
Figure 6-2: Comparison of damage distribution between lightning (a), simulated magnetic pressure (b), simulated impact
with equivalent projectile (c)
The comparison shows that numerical magnetic pressure provides extensive damage on
interfaces 3 and 4, concordant with the ones observed on lightning strike scans. More over
the overall shape of the damage generated by the magnetic pressure is more round and
does not present the extensive rear face splinters systematically observed on steel ball
impatcs (figure 6.12 (c)).
Table 6.7 and figure 6.13 present the total delaminated area and damage distribution
through thickness for several lightning strikes and the numerical magnetic pressure. It is
185
__________________________________________________________________________________
shown that the new numerical loading provides concordant total delaminated area with a
relative difference of -5.8%, 25.2%, 3.5% with lightning samples 101, 107, 121 respectively.
Sample
Lightning test 101
Lightning test 107
Lightning test 121
Magnetic pressure
(simulation)
Total
delaminated
area (mm²)
2321
1746
2112
2186
Table 6.7: Comparison of total delaminated areas, lightning samples and magnetic pressure
900
Lightning test 107
800
Magnetic pressure
Lightning test 101
600
Lightning test #121
700
500
400
300
200
100
0
Thickness (mm)
Figure 6-13: Comparison of damage distribution between lightning and simulated magnetic pressure
The numerical magnetic pressure generates large damage in every interfaces of the laminate
and especially in the second half on the sample, contrary to lightning strikes, which, even
though they can induce damage in lower interfaces, provides the main part of them in the
first half of the impacted laminate.
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6.4 Conclusions and perspective
This chapter presented prospective studies that aimed to improve the mechanical
equivalence results presented in chapter 5 and that are based on other modelling choices
also compatible with the results of chapter 3. To do so, several modifications have been
tested on the impulse delivery used to reproduce the lightning strike observed damage.
As a first approximation, the transferred lightning impulse was delivered to the composite
samples via a non-deformable spherical steel ball of diameter Φ9.8mm and mass 4g.
However this projectile shape and the used material did not seem to be the most
appropriate to deliver the energy in the same fashion that lightning. Two studies have thus
been led on the influence of the projectile’s shape and material and presented above. The
first one proposed a cylindrical projectile with a flat circular base of the same diameter than
the previous steel ball. This new impactor has been tested on several lightning cases which
have already been “equivalenced” by the steel ball, in order to investigate the influence of a
larger contact area between the projectile and the target sample. The results showed that
the energy delivery was different and provided according total delaminated area.
Nevertheless, the cylinder impact results, in terms of rear face displacement and damage
distribution, remained close to those obtained with the steel ball and even better at short
times.
The second study proposed to investigate a change in the projectile material. The point was
to try reproducing the peculiar loading of lightning which seems to press on the sample in
two consecutive steps. In order to reproduce this loading, a deformable material is chosen
for the impactor: a tyre material. This new material is implemented in the numerical shell
model presented in chapter 3 and projected on the composite sample. The rear face
displacement results were not concordant with the lightning strike ones. The material being
too light, the rear face velocity of the sample is too slow compared with both lightning and
steel ball simulation results. However, the study of the deformation process of the impactor
suggested that the damage distribution should be investigated and a numerical simulation
using a damage model is to be computed in order to validate or invalidate this kind of
projectile.
Finally, a study has been led, following the preliminary work presented in chapter 3 on a
magnetic pressure obtained from lightning input data. This magnetic pressure has been
improved by the addition of a moving radius for the pressure circumference application and
provided concordant results for both displacements and velocities and total delaminated
area. All this studies are prospective works, led to improve the mechanical equivalence on
the large time reproduction of lightning strike parameters. Several other studies are
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intended, as for example the study of multiple impedance projectiles that could provide the
adequate loading, or at least the closer to the lightning one.
The general conclusion is that the equivalence method proposed in chapter 3 has been
demonstrated as valid in the sense that it is possible to reproduce the structural behaviour
of the core up to short or stabilization times.
Improving the way the impulse is delivered to the core of the plates does not improve in a
significant manner the precision of the structural nor the material behaviour. This is
attributed to the fact that the complete separation of the core damage from the surfaces
behaviour is not always possible. In particular, it has been highlighted in chapter 4 that the
capacity of the paint to resist to the detrimental effects of the surface loading is a crucial
point. It is then emphasized that further studies should be devoted to the role of paint
resistance.
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General conclusion
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7
8
In order to certify aerospace composite structures, manufacturers must be able to
understand and predict the behaviour of these structures to a variety of events. Lightning
strike has become a major concern since composite materials have been integrated in
primary structures as, contrary to their metallic predecessors, they possess lower electrical
conductivity. Consequently, lightning strike on composite structures induces extensive
damage: burning of the composite plies and delamination in the thickness of the laminate.
These damages are also enhanced by the paint coating, thus the necessity to protect these
structures arises and lightning strike protections are already implemented on aircrafts.
Existing protections are currently composed of expanded metallic meshes. But these
protections are not optimal and tend to add weight on the concerned parts where
composite materials were introduced for their lower weight. The need to optimize these
protections has become predominant. To do so, the comprehension of the lightning strike
attachment and direct effect on composite structure is mandatory. The total amount of
inside damage and their detectability from outside are major issues for industries. In fact
these damages, matrix cracks and delamination mostly, are the most detrimental for the
structure’s integrity as they decrease the mechanical properties of the structures and can be
responsible for the uncontrolled complete failure of the mechanical parts. Lightning strike
tests in laboratory are led in order to test protections and increase the knowledge related to
the event. From these tests, norms have been established on the aircrafts’ parts tendency to
be lightened and the level of protection associated.
The scientific community worked on the comprehension of lightning for several decades
now. Lightning is a complex event, involving various physics such as electromagnetic,
thermal and mechanics, which can be coupled with one another. In the literature, studies
mainly focused on the electro-thermal component of lightning to try explaining the
associated damage, and particularly the burning area observed at the top layers of the
stroke laminates. A brief literature overview has been presented on the physics
phenomenon, the laboratory tests, and the different modelling strategies. It is concluded
that, regarding the damage resistance and tolerance, the existing models that intend to
simulate the complex loading and the surface damage are for now too complex. Among
them, however, it has been shown that the damage is not due to electrical or thermal
effects.
General conclusion
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The approach proposed here is then to separate lightning damage into two kinds: the
surface damage (paint and protection layup and first ply of the laminate) and core damage.
These two damages are different both in location and nature. From literature survey and
primary works, it is set that surface damage are mainly due to electro-thermal action of the
lightning arc, while core damage are due to a mechanical loading induced by the event
occurring at the “surface” of the material. Lightning core damage have been analysed and a
strong resemblance with classical impact damage has been established, leading to a
decorrelation hypothesis between the surface and core of the composite material. As
lightning is a complex event whose various physics related components cannot be isolated
nor quantified, it is assumed to separate the arc attachment process on the surface of the
core induced damage and to focus only on the second part. Considering the events of the
surface as a black box, the study is focused on the comprehension of the mechanical core
damage and on how to reproduce them.
From a scientific perspective, the question addressed in this study is: as lightning damage
strongly look alike mechanical ones, is it possible to numerically and mechanically predict
them and is the model able to determine the nature of interactions between the lightning
strike loading and the laminate behaviour?
In this frame, two work hypotheses have been made. The first one is to assume that such
separation between surface and core damage is possible due to their possible origin. The
second one is to assume that it is mechanically possible to reproduce those damage and thus
to define an equivalent loading to lighting strike. The led study aimed to design and validate
a mechanical equivalent load to the lightning phenomenon, as well as a predictive damage
numerical model that is to be validated by comparison with actual equivalent mechanical
tests.
Firstly, an analytical study has been led in order to design equivalence between mechanical
impacts and lightning strikes. This equivalence is based on the transferred impulse k (N.s)
and on experimental data extracted during the laboratory lightning strike tests for several
protected and painted lightning samples. From these tests, several quantities have been
extracted. Firstly a set of characteristic times for the equivalence have been defined. The
“short times” corresponds to the duration during which the equivalence is made, this value
is extracted from time to maximum velocity for lightning tests. “Strike times” refers to the
lightning tests complete duration (time to deliver all the current to the sample).
“Stabilization times” come for the period during which the mechanical sample’s
displacement reaches the displacement plateau. Finally “Large times” refers to moments
where the peak shock is passed and other phenomena arise. Two other quantities have also
been extracted that are used as equivalence parameters: rear face displacement and
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velocity. A simple numerical shell model has been created in order to validate the
equivalence method by comparing predictive values of displacement and velocities with
lightning data. The good correlation allowed developing a mechanical experimental
campaign to assess the validity of the equivalence. A canon gas apparatus has been chosen
that could launch spherical projectiles of several diameters and masses to a range of
velocities from 50 m/s to 150 m/s. For preliminary tests, the chosen projectile was a
spherical steel ball of diameter φ9.8mm and mass 4g. The mechanical campaign has been
led and provided concordant results when compared to the numerical predictive model for
both displacement and velocity comparisons at short time. The same results were obtained
after comparison with lightning data. Nonetheless, the model was not able to reproduce the
large time behaviour of the lightened samples and could not reach the plateau displacement
value. The equivalence is considered validated though, as the mechanical equivalence is
made on short times (in which the displacement results are concordant) and considering
that all the damage are created in the first hundreds of microseconds following the impact.
Damage comparisons have been led between the two kinds of impacts by mean of ultrasonic
scanning. Lightning and equivalent mechanical tests provided concordant values of total
delaminated areas for all the tested cases. However, the study of the damage distribution
through thickness showed that the two impacts provided delamination of different shape
and location. Lightning statistically generates damage only up to the first half of the
impacted laminate while mechanical impact damages extend through all the thickness of the
sample. This difference is mainly due to the energy deposit that differs between lightning
strike, whom energy is also consumed by Joule effects at the surface, and the steel ball
projectile. Moreover, the damage comparison between mechanical impacts and lightning
strikes highlighted the face that the major parameter on damage content is not the paint
thickness but its capability to erode or to persist during the lightning strike. In fact sample
where paint was completely consumed by the arc at short times (<20 µs) presented lower
damage area than the other samples with the same amount of paint on the surface.
In parallel, a 3D volume element model has been created that included inter and intra
laminar damage of the composite material on the basis of previous work for the damage
laws [48, 77]. This model has been confronted to mechanical test results and some
improvements have been made in order to obtain the most accurate predictive damage
model. Then the experimental and numerical mechanical results have been compared to
lightning strike ones. The numerical model provides concordant results with the mechanical
impacts for both short and large time. The comparison has been extended to the damage
distribution through thickness. The numerical simulations provide according values of total
delaminated area for impacts at velocities superior to 75 m/s and tend to overestimate the
damage for lower impact velocities. However, the numerical model is able to reproduce both
the delamination size and orientation for all the damaged interfaces. When compared to
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lightning strike results, the numerical model shows according results at short times, as
expected, but not at larger time and is not able to reproduce the displacement plateau of
lightning. The model provides additional information, as it allows extracting real time events
in the material, and confirms that all the damage created by the projectile, equivalent to
lightning, are generated within the first 50 microseconds following the impact. Moreover,
with the numerical model, it is possible to extract displacement profiles over a section of the
sample, instead of using punctual values. These profiles have been compared to those
measured during lightning strike campaigns by mean of stereo correlation method and
provide according results for short times. Thus, at short time, the steel ball projectile, chosen
as a first approximation, correctly reproduces the sample displacement due to lightning,
both in maximum displacement but also at a larger spatial scale.
From these comparisons, it has been observed that the mechanical equivalent impact was
able to reproduce the lightning displacement and velocity results over the equivalence time
called ‘short times’ (≤50µs) for both painted and unpainted panels. For ‘stabilization times’
(150µs to 500µs), differences appear between painted and unpainted samples. This period
of time is also a stabilization phase in the mechanical impact, and the model fails at
representing the long period for locally unpainted plates. In this stabilization period also, it
was observed that painted sample for which the central painted zone was not burned
suffered higher rear face displacements and velocities, which the model was not able to
reproduce. Indeed the equivalence method does take into account the presence of different
matters and energy dissipation in the coating, but does not for the complex physics induced
by the preservation or the removal of paint. These later are thus considered to be at the
origin of the variability of experimental damage and global behaviours observed for the
composite laminates in a large amount of lightning tests. For large times (>100 µs) the model
is no more able to reproduce the displacement behaviour for any sample. This later
mismatch is due to the difference of energy deposition between lightning and mechanical
impacts with a small projectile. Indeed, the small projectile chosen as a first approximation
does not allow a large contact area and delivers its impact energy on both a small surface
and time range (sharp impulse in space and time). Moreover, the internal damage observed
with the mechanical impacts presented several differences with the lightning one. Even
though the mechanical equivalence allows reproducing an according total delaminated area
when compared to lightning, the distribution of the damage in the thickness is different,
especially for the half part of the laminate. Lightning strike tends to generate damage only in
the first half of the samples, as an unknown part of the current is dissipated in the so-called
surface damage, and not in the whole core of the material; while the mechanical projectile
transfers all its kinematic energy to the entire laminate and thus generates damage through
all the sample’s thickness.
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The equivalence method is thus validated as it satisfied the equivalence criteria established
at the beginning of the study: similar level of rear face displacement and velocity and total
delaminated area. Nevertheless, the differences observed between the two kinds of
impacts, and especially concerning the damage distribution, state that the preliminary
choice of a non-deformable, small diameter spherical projectile was not the best choice to
represent lightning, as expected. Several other possibilities have been considered. Indeed, as
the numerical model’s results have been validated by comparison with experimental data, it
can now be used to predict damage from other kinds of projectiles or loadings. Two
preliminary studies on the projectile have been led. The first one focused on the shape on
the impactor in order to increase the contact area during the impact with a cylindrical
projectile. The second study was related to the material used for the projectile. The study of
the lightning curves suggested that the loading responsible for the samples’ deflection was a
two steps pressure loading. The idea of a deformable projectile, made of rubber, has been
tested on the model and provided interesting first results. Finally a magnetic pressure,
calculated from the lightning current and the magnetic field induced by the arc on the
protected composite sample has been tested. This study was led in order to try quantifying
the importance of one of the physics at stake during a lightning strike by assessing its
contribution to both rear face displacement and internal damage. The magnetic pressure
provides concordant results of rear face displacement at short times and, even though the
overall displacement curve is close to the lightning one, the numerical model does not reach
the lightning displacement plateau. This is due to the fact that the magnetic pressure only
takes into account one of the multiple components of lightning. Numerical damage
distribution has been compared with lightning results and provided according values of total
delaminated area. Moreover, the overall shape of the damage provided by the numerical
pressure is closer to the lightning observations and do not incorporate large back face
splinters, present during projectile simulations. These studies are possible leads and
improvements on certain aspects of the mechanical equivalence. The obtained results also
proved that, in certain cases, the decorrelation between the surface and the core of the
impacted material could not be made and that the event occurring at the surface and
related to the removal or preservation of the paint was strongly linked to the damage
observed in the core. A coupled electro-thermo-mechanical study must be led in this
perspective. The designed predictive damage model can be used to test several surface
loadings and predict their damage.
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________________________________
195
9
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10
Appendices
209
11
Appendix A
Specimens manufacturing process
________________________________________________________
This appendix presents the manufacturing process of the T800/M21 composite specimens
used for both lightning strike and mechanical impact tests.
Pre-impregnated roller carbon/epoxy tape T800S/M21 unidirectional tape (reference
UD/M21/35%/268/T800S/300) is used to made the eight plies specimens of stacking
sequence: [45/0/-45/90]s. The samples are square plates of dimensions 400x400x2mm3
hand made by positioning one unidirectional ply over another along required ply angle as
shown on figure A.1. Once the stacking sequence realized, the laminates are conditioned
under vacuum press SATIM to compact the fabricated lay-up prior to curing (see figure A.2).
Figure A.1: lay-up of the plies to create the laminate
Figure A.2: Compacting lay-ups by vacuum [48]
Appendix A
210
__________________________________________________________________________________
The composite plates are then cured using an auto clave available at the ICA with
programmable curing cycle as shown on figure A.3, respecting the supplier’s
recommendation.
(a)
(b)
Figure A.3: Auto clave for material curing (a) and curing cycle (b)
Appendix A
211
12
Appendix B
Ultrasonic scanning and data extraction
________________________________________________________
Ultrasonic inspection is extensively used to characterize composite damage, especially for
delamination detection, their sizes and positions through the material thickness. An
ultrasonic signal is send to the material and penetrates the sample. The signal crosses the
thickness of the sample and sends a reflection signal when an internal damage is
encountered. The position of the delamination can thus be calculated by comparing the time
between sending and receiving signal with the time of flight of the ultrasonic wave in an
undamaged area. The ultrasonic inspection system available in ICA is shown in Figure B.1.
Figure B.1: ultrasonic apparatus
Appendix B
212
__________________________________________________________________________________
For the samples impact with the projectile, the analysis was made on both sides of the
samples in order to double check the results but for lightning strike impacted ones, the
ultrasonic scanning was only possible on the face opposite to the impact. In fact the bare
fibres due to the lightning arc as well as the presence of metallic protection tend to disrupt
the ultrasonic signal and no image could be retrieve. The ultrasonic analysis of the damage
samples provides images of the damage, projected in 2D. These images, as shown in figure
B.2, are used to extract data such as total delaminated area and damage distribution
through thickness.
Total delaminated area
In order to extract the total delaminated area, the damages are bounded with a rectangle
(yellow) or an ellipse (black). The shape of the bounding box can generate disparity up to
15% (in mm²).
Figure B.2: Different ways of measuring total delaminated area: rectangle or ellipse.
Damage distribution
The results obtained by ultrasonic scanning must not be taken as absolute. AGI designed
special software for data extraction from C-scans, called NDT-kit. With this tool, one can
Appendix B
213
__________________________________________________________________________________
draw histograms of the delaminated area in function of their position in the material
thickness, which allows obtaining a representation of the damage distribution. The
histogram and associated C-scans in NDT-kit interface are shown in figure B.3.
Figure B.3: Ultrasonic scanning results: C-scans and histogram of damage as function of the thickness
1800
1600
Delaminated area
1400
1200
1000
800
600
400
200
0
0 0,77 0,83 0,9 0,96 1,03 1,1 1,16 1,23 1,29 1,36 1,42 1,49 1,56 1,62 1,69 1,75
Interface #
Figure B.4: Histogram of damage, the dotted interface are hidden by the plain one and cannot be taken for valid values of
delamination.
Appendix B
214
Appendix C
Greszczuk theory
________________________________________________________
Greszczuk, [54], by using Hertz contact theory, tries to determine the couples (m, v) relative to a
projectile [9, 41, 129]. The only inputs available with this method are:
-
The impactor : diameter, material (E,,)
The target sample: geometry, dimensions, material (E,,)
Experimental data : impact duration Δt/ lightning impulse
The researched parameters are the mass m and the velocity v of the projectile. The mass is given
by the following equation:
1
1
M=(𝑚𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑙𝑒 + 𝑚𝑡𝑎𝑟𝑔𝑒𝑡)
Two parameters called n’ and 𝛼1 are thus to be calculated, only dependent of material
parameters already known. 𝛼1 corresponds to the maximal deformation of the target samples. It
comes:
1
16
𝐶𝑟 2
𝑛 =(
) ∗ ( 3)
3 ∗ 𝜋 ∗ (𝑘1 + 𝑘2)
𝑠
′
2
5𝑣 2 5
𝛼1 = (
)
4𝑀𝑛′
The lightning impulse, called k, is given by :
𝑡0
′
𝑘 = ∫ 𝑃(𝑡)𝑑𝑡 = 𝑛 ∗
𝑡0
3
3 𝜋𝑡
𝛼12 ∫ sin2 ( ) 𝑑𝑡
0
Variable change
0
𝑥=
𝑡0
𝜋𝑡
𝑡0
𝑘 = 𝑛′ ∗
3
𝛼12
𝜋
3
∫ sin2 𝑥 𝑑𝑥 ∗
0
𝑡0
𝜋
Appendix C
215
__________________________________________________________________________________
𝜋
It comes :
3
∫0 sin2 𝑥 𝑑𝑥 = 1.748
So :
𝑘=(
3
3
1.748
𝛼1
) ∗ 𝑛′ ∗ 𝛼12 ∗ 𝑡0 = 0.556 ∗ 𝑛′ ∗ 𝛼12 ∗ 2.94 ∗ ( )
𝜋
𝑣
A relation between k, m and v can thus be established:
𝑣
𝑀
= 0.367 ∗ 𝑘
Greszczuk theory also allows calculating analytically the contact duration which is taken equals
to the impact time during a lightning strike tests, t0. It comes:
𝛼1
Δt =t0=2.94 ∗ ( 𝑣 )
By replacing the expression of 1:
3
𝑣5
2
2
1
5 5
1
𝑣 5
= ( ) ∗ 2.94 ∗ ( ) ∗ 2 ∗ ( )
𝑡0
3
𝑀
′( )
𝑛 5
Since to this method, one can calculate the velocity v of the projectile, the mass M of the system,
and the value of v/M. The mass of the projectile can thus be easily calculated.
Greszczuk formula also allows determining the maximal deformation 1 as well as the contact
radius between the target and the projectile and the equivalent pressure to the impact, through
the following formula:
Contact radius :
1
4 ∗ 𝐶𝑟
𝜋𝑡 2
∗ 𝛼1 ∗ sin ( ))
𝑎(𝑡) = (
𝑠
𝑡0
Equivalent pressure :
1
3 ∗ 𝑛′ ∗ 𝑠
𝜋𝑡 2
𝑞0(𝑡) =
(𝛼1 ∗ sin ( ))
8𝜋 ∗ 𝐶𝑟 ∗ 𝑚 ∗ 𝑟
𝑡0
Appendix C
216
__________________________________________________________________________________
Appendix C
217
Appendix D
Canon gas apparatus
________________________________________________________
Mechanical impact tests were conducted using a stainless steel ball of diameter Φ9.9mm
and mass 4g as projectile, launched by a gas gun apparatus (canon) in a range of velocities
from 50 to 150m/s. The canon is located at the Institut Clément Ader Laboratory (ICA) in
Toulouse, France and presented in Figure D.1.
(a)
(b)
(c)
Figure D.1: mechanical impact set up (a) metallic assembly, (b) canon gas gun apparatus, (c) circular aluminium clamping
ring with sample fastened
Appendix D
218
__________________________________________________________________________________
The canon apparatus allows ejecting the projectile at several velocities depending on the
pressure applied in the tube. Several velocities (from 60 to 130 m/s) were foreseen for the
mechanical test campaign but a discrepancy was noted in the obtained velocities.
One of the mechanical configurations predicted an impact at 72 m/s. For this case; three
launches were emitted, respectively at 75, 75 and 70 m/s. Similarly, for the expected impact
at 60 m/s, the canon gas exited two projectiles at 65 and 50 m/s. These differences are
inherent to the experimental apparatus and concordant with the repeatability and the
dispersion of the test set up, as observed by the lab on similar campaigns. This dispersion is
due to the foam support in which the projectile is placed prior to its introduction in the
canon tube. This expanded foam has not a precise diameter corresponding to the canon one
and thus, the air may not be pushing linearly against it, adding friction during the ejection
and thus slowing down the projectile (see Appendix D).
A specific campaign was led to determine the dispersion on the ejected velocities due to
friction in the tube. It is possible to calculate analytically, the expected velocity at the exit of
the canon tube as a function of several parameters such as, the projectile mass, reservoir
and canon volume and applied pressure. The theoretical values revealed to overestimate the
measured velocity during the experimental tests, see figure D.2. This means that a friction
coefficient must be taken into account (which also depends on the quality of the foam) and
that the predicted velocities are not concordant with the reality. A friction coefficient of 80%
is applied on the analytically predicted velocity and the new obtained values are closer to
the experimental results, as shown on figure D.2.
600
500
Projectile Velocity (m/s)
400
300
200
Theoretical velocity Ø40
100
Corrected velocity Ø40
Experimental measures Ø40
0
0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930
Pressure in the (bars)
Figure D.2: Theoretical evolution of the projectile velocity as a function of its mass and service pressure:, corrected and
measured values (projectile mass=59g)
Appendix D
219
__________________________________________________________________________________
This study led in the Institut Clément Ader, which hosts the canon gas apparatus, highlights
the importance of friction in the final velocity for the projectile and explains the differences
between desired and obtained values.
Appendix D
220
Appendix E
From shell to 3D solid elements model
________________________________________________________
E.1 Numerical simulation
E.1.1 From shell to 3D model
The first developed numerical model is the simplified shell model presented in chapter 4.
The model is built to represent at best the lightning strike tests conditions of clamping, size
and shape of the sample and boundary conditions. Erreur ! Source du renvoi introuvable.
presents the material properties used for the simulations.
(a)
(b)
Figure E.8 : Numerical mesh of the simplified shell model (a) before and (b) after impact
E11 (GPa)
154
E22 (GPa)
8.5
E33 (GPa)
8.5
ν12
0.35
ν13
0.35
ν23
0.3
G12
(GPa)
4.2
G13
(GPa)
4.2
G23
(GPa)
4.2
Table E.1: material values for T700/M21
Appendix E
221
__________________________________________________________________________________
Comparison between two FE codes
A preliminary study was made in order to check the validity of the numerical model using
Abaqus® software. To do so, the exact same model was built in another Finite Element
software, called Samcef®, in order to check the overall results obtained with the shell model
and ensure the choice of numerical integration. Although the numerical elements and
calculation process are not strictly the same in the two commercial codes, which leads to
several differences, the two calculations provide good agreement in terms of rear face
displacements and force versus time.
Figure E.9 compares the rear face displacement for an impact at 62 m/s in both Abaqus and
Samcef. The obtained results are in very good agreement at both short and large times.
Figure E.10 presents the comparison for the same calculations of the curves force versus
time and shows that the two models provide concordant results as well. The presence of a
peak of force for the ABAQUS formulation (red circle) is due to an overpressure due to the
shell contact formulation. This issue is solved in further improvements of the model.
Such a quick study was a way to insure that the numerical predictions of the shell model
created in Abaqus software were not aberrant as they will be used as a basis for the
upcoming numerical improvements.
8
Shell impact 62 m/s ABAQUS
6
Displacement (mm)
Shell impact 62 m/s SAMCEF
4
2
0
-2
0
0,5
1
1,5
2
-4
-6
-8
Time (ms)
Figure E.9 : Rear face displacement after impact at 62 m/s with two FE codes for shell models: Abaqus and Samcef
Appendix E
222
__________________________________________________________________________________
Force (N)
8000
7000
Shell model ABAQUS
6000
Shell model SAMCEF
5000
4000
3000
2000
1000
0
0
0,1 Time (ms)0,2
0,3
0,4
Figure E.10 : Comparison of force versus time curves for shell models using Abaqus and Samcef softwares
6.1.1.1 3D elastic FE model
A 3D model is then built in which each ply has a thickness and a proper behaviour in the 3D
volume. Plies are continuously linked to the adjacent plies in the lay-up sequence. The
sequence is the same as that of the experimental plates: [45/0/-45/90]s.
The material elastic properties used for each ply are the same than those already presented
in table E.1., except the longitudinal direction which follows the lay-up. The lateral size and
clamping conditions are reproduced from the shell model into the 3D model, as the
projectile and contact conditions. So called C3D8 solid elements are used one element in the
thickness of each ply. They are eight node elements with three degrees of freedom per node
and one integration point in the volume. The mesh is presented in Figure E.4 as well as the
analytical impactor used for the numerical simulations (Figure E.4 (c)). For this study, a
standard impact of 62 m/s is simulated.
Appendix E
223
__________________________________________________________________________________
(a)
(b)
(c)
Figure E.4: 3D model (a) geometry and boundary conditions (b) mesh
It is well know that finite element models of a plate with shell or volumes have different
numerical natural frequencies. This is due to the finite element integration scheme. This
second model was built in order to verify the flexibility of the 3D plate in the time interval of
interest that is for short times duration as regarded by the lightning strike event. Figure E.5
presents the rear face displacement obtained for both 2D and 3D models. At short times
(<0.5ms) comparison provides very good results for the two cases. At longer time (>0.5ms)
the effects of the boundary conditions and the volume behaviour modify the plate vibration
and the difference of formulation between shell and 3D elements can be seen As in the
scope of the lightning study the focus is made on even shorter time (<300µs), the
improvement of the model from shell to 3D elements is considered validated.
Figure E.6 presents the contact force vs time curves for both shell and 3D model. The
maximum force obtained for the two formulations are concordant. The shell model, which is
more flexible than the 3D one tends to provide higher values of force induced on the plate
by the impact (6.4kN for shell model and 5.9kN for the 3D model).
Appendix E
224
__________________________________________________________________________________
10
Shell model impact 62 m/s ABAQUS
8
3D elastic model impact 62 m/s
ABAQUS
Displacement (mm)
6
4
2
0
-2
0
0,5
1
1,5
2
-4
-6
-8
Time (ms)
Figure E.5: Comparison of rear face displacement obtained for shell and 3D elastic models using Abaqus®
9000
8000
3D elastic model ABAQUS
7000
Shell model ABAQUS
Force (N)
6000
5000
4000
3000
2000
1000
0
0
0,2
Time (ms)0,4
0,6
0,8
Figure E.6: Force versus time curves for shell and 3D elastic model using Abaqus® software
Comparison between two FE codes
As for the shell model, a comparison is made between two different finite element software
ABAQUS and SAMCEF in order to validate the obtained results. The same impact at 62 m/s is
reproduced in the two FE codes with the same material definition and boundary conditions.
The purpose of this test is to get an envelope of confidence for the future complex 3D
numerical simulations with Abaqus.
Appendix E
225
__________________________________________________________________________________
Figure E.7 presents the rear face displacements and forces versus time curves obtained in
the two calculations. The same observations as for the comparison with shell model are to
be made. Up until 0.5ms, the two codes provide according results. At larger times, the
difference in the two codes provides a slight disparity in the structural behaviour of the
numerical plates. The SAMCEF model provides about 5% higher values of displacement at
500µs. The SAMCEF model gives a slightly higher value of maximum force of 6.3kN against
6.0 kN for the ABAQUS one. However the overall behaviour tends to be the same, the two
models providing an equal time of vibration for the plate.
8
3D elastic model impact 62 m/s SAMCEF
Displacement (mm)
6
3D elastic model impact 62 m/s ABAQUS
4
2
0
-2
0
0,5
1
1,5
2
-4
-6
-8
Time (ms)
(a)
Force (kN)
7,00
6,00
3D elastic model ABAQUS
5,00
3D elastic model SAMCEF
4,00
3,00
2,00
1,00
0,00
0
0,2 Time (ms)0,4
0,6
0,8
(b)
Figure E.7: (a)Rear face displacement and (b) force versus time curves, after impact at 62 m/s with two FE codes for shell
models: Abaqus and Samcef.
6.1.1.2 3D model: introduction of interface elements
Appendix E
226
__________________________________________________________________________________
After the creation of the complete 3D model, the next step is to introduce damage in the
model. A first improvement consists in keeping elastic properties for the plies but to
introduce cohesive elements between them. Figure 6.8 presents the positioning of each
layer of interface elements and their corresponding orientation. Elastic properties are thus
kept as presented in table E.1. Table E.2 presents the material properties associated to the
interface elements. The sequence of the laminate remains the same: [45/0/-45/90]s.
Interfaces between plies of different orientation are modelled using cohesive elements
which are volume elements with zero-thickness. An identical mesh is reproduced for
cohesive elements and plies so that cohesive elements and plies share nodes. When the
cohesive elements have release all the stored elastic energy, they are eliminated (eroded). A
contact algorithm between the plies and the interface elements manages the nonpenetration of plies that become adjacent and disconnected (penalty based general contact
of Abaqus). Input data used for the cohesive elements and summarized in table E.2 come
from previous works [48].
Figure E.8: Position and orientation of the interface elements between the elastic plies
kn
3
(N/mm )
100
ks
3
(N/mm )
100
σn
(MPa)
60
σs
(MPa)
60
GIC
(J/m²)
500
GIIC
(J/m²)
1200
α
1.0
Table E.2: Interface elements material properties
Figure E.9 presents the rear face displacement and force versus time curves obtained for the
new model integrating interface elements and thus delamination evolution in time,
compared with the pure elastic 3D model. The presence of interface elements, generating
Appendix E
227
__________________________________________________________________________________
delamination and thus opening between the plies is responsible for a higher max peak of
deflection compared to the 3D elastic model without interface elements, see figure E.9(a)
(red circled). The vibration of the plate is also altered by the presence of delamination at
larger times.
0
Displacement (mm)
-1
0
0,2
0,4
0,6
0,8
1
1,2
1,4
-2
-3
-4
-5
-6
3D elastic model
-7
-8
Time (ms)
3D elastic model + interface
elements
(a)
0
-1
0
0,2
0,4
0,6
Force (N)
-2
-3
-4
-5
model_interface_coh_62m/s
-6
model_elastique_62m/s
-7
-8
Time (ms)
(b)
Figure E.9: Rear face displacement (a) and force versus time curves (b) for 3D model with and without interface elements
Figure E.9 (b) shows the comparison of contact force versus time curves for the two 3D
models, with and without interface elements. The presence of interface elements tends to
soften the general behaviour of the model which translated in a slightly higher value of
maximum force of almost 6.7kN against 6.0kN for the model without interface elements.
Appendix E
228
__________________________________________________________________________________
The presence of interface elements allows detecting the position and the number of
damaged interface over time, as presented on figure E.10, showing the 3D model with
cohesive elements during the impact. Such a model provides in situ information on the time
at which delamination appears and their growing rate over time.
Figure E.10: observation of the formation of delamination between ply during the numerical simulation of impact.
Table E.3 presents for three mechanical impacts the total delaminated area and distribution
through thickness obtained with the numerical model and compared with experimental
measures, respectively at 62m/s, 36m/s and 80 m/s, exceptionally for a projectile of mass
16g. The analysis of the results clearly shows that even if the interface elements allow the
opening of the plies they greatly overestimate the size of the delamination and the total
delaminated area. A major over estimation is made at the interface 4 between plies of
respective orientation 90 and -45° as shown on figure E.11.
Exp. Test 1
Delaminated
interface area (mm²)
Simu Test 1
Exp. Test 2
Simu Test 2
Exp. Test 3
Simu Test 3
Delaminated
area (mm²)
Delaminated
area (mm²)
Delaminated
area (mm²)
Delaminated
area (mm²)
Delaminated
area (mm²)
1
0
71,4
0
41,958
0
63,51
2
58
82,8
0
80,84
93
111
3
118
652,7
1
199,68
185
683,4
4
76
3081,5
152
446,213
104
4136,94
5
187
62,4
31
35,504
169
72,738
6
379
83
0
44,714
572
102,51
Total
818
4033,8
184
848,909
1123
5170,098
Table E.3: Comparison of the delaminated area obtained for both mechanical tests and numerical simulations for three
tests respectively at 30, 11 and 80 m/s
Appendix E
229
__________________________________________________________________________________
3500
3000
2500
2000
1500
Exp. test 1
1000
Simu Test 1
500
0
1
2
3
4
5
6
Interface #
Figure E.11: Histogram of the delamination distribution for numerical simulation and experimental results
Figure E.12 presents the delaminated area distribution for each interface after an impact at
75m/s. It has been discussed previously that delamination generally follow the orientation of
the lower ply of an interface. The study of the numerical delamination shows that except for
interfaces 4 and 5 (figure E.12 (d) and (e)), the cohesive elements do not reproduce this
expected behaviour. Moreover, the large over estimation of interface 3 and 4 is clearly
visible on figure E.12 (c) and (d). Such localization of the damage is an identified problem of
cohesive elements and has been discussed in various works on the subject [48, 77].
Even more, the comparison with the analysis of experimental results via C-scans, shows that
the numerical model by localizing damage on the first interfaces does not reproduce
adequately the dynamic response of the plate. C-scans of mechanical impacts shows that the
major delamination were found at interfaces 3, 4 and 5 and the numerical model only
provide delamination up to the 4th interface. The localization of the damage in the interface
3 and 4 induced by the cohesive elements formulation, prevents the damage to progress in
the lower interface and the large damage observed on the interface 5 (-45/0) is not visible in
the numerical simulation. The use of interface elements alone is thus not sufficient enough
to represent the behaviour of the plate submitted to medium velocity impact.
Appendix E
230
__________________________________________________________________________________
(a) Int. 1 (45/0)
(c) Int.3 (-45/90)
(b) Int. 2 (0/-45)
(d) Int.4 (90/-45)
(e) Int.5 (-45/0)
(f) Int.6 (0/45)
Figure E.12: Damaged interfaces for the numerical simulation, impact at 75 m/s.
Appendix E
231
__________________________________________________________________________________
6.1.1.3 3D model: introduction of the damage law
Another alternative to the use of discrete model (introduction of cohesive elements at the
interfaces between plies) is to use solely a CDM model where the plies are damaged
following the five criteria presented in section 5.4.2. Table E.4 presents the material
properties used for the plies and the failure stresses associated to the chosen material which
is T800/M21. The lay-up of the laminate remains unchanged.
E11
(GPa)
165
XC
(GPa)
1.2
E22
(GPa)
7.64
SFS
(GPa)
1.5
E33
(GPa)
7.64
YT
(GPa)
0.06
ν12
Ν23
ν13
0.35
YC
(GPa)
0.28
0.35
ZT
(GPa)
0.06
0.4
ZC
(GPa)
0.7
G12
(GPa)
5.61
S12R
(GPa)
0.065
G23
(GPa)
2.75
S23R
(GPa)
0.06
G13
(GPa)
5.61
S13R
(GPa)
0.065
XT
(GPa)
2.2
Table E.4: Material properties for T800/M21.
Figure E.13 presents the rear face displacement obtained with this CDM formulation,
compared with results from simulation with shell model and mechanical impact results, for
three different impact velocities of 75m/s, 81m/s and 65 m/s respectively, with a projectile
of mass 4g. The initial slope of the curve as well as the speed of deflection are well
represented by this new model compared with the other two, however the plies seem to be
damaged too early which causes the plate to over deflect. The maximum deflection obtained
is accurately reproduced by the numerical model, as presented in Table E.5, compared to
experimental results, with a maximum difference of only 10%.
Mech.
Impact
maximum
deflection
Numerical
model (CDM)
maximum
deflection
(mm)
(mm)
3 (65m/s)
2656.7
2481.3
-6.6%
4 (75m/s)
3780
3701.8
-2.07%
6 (50m/s)
2233.3
2021.2
-9.5%
8 (81m/s)
3866.7
4278.8
10.6%
Mechanical samples
Relative difference
Table E.5: Comparison of the maximum experimental and numerical deflection and relative difference.
Appendix E
232
__________________________________________________________________________________
4000
Displacement (µm)
3500
3000
2500
2000
1500
1000
Mechanical impact test at 75 m/s
Shell model impact at 72m/s
500
0
0
100
200Time (µs)300
400
500
(a)
4500
4000
Displacement (µm)
3500
3000
2500
2000
1500
Shell model 81m/s
1000
Mechanical impact test 81m/s
500
0
0
100
200
300
Time (µs)
400
500
(b)
3000
Displacement (µm)
2500
2000
1500
1000
Shell model 65m/s
500
0
0
100
200Time (µs)300
400
500
(c)
Figure E.13: Rear face displacement for 3D model with and without interface elements and CDM model impact at (a)
75m/s, (b) 81m/s and (c) 65m/s.
Appendix E
233
__________________________________________________________________________________
Table E.6 presents the total delaminated area obtained with the numerical model with the
damaging law for the plies, called CDM, and compared with the results obtained during
experimental tests for several projectile velocities, respectively at 50m/s, 65m/s, 75m/s and
81 m/s. Delamination extent is obtained using the G12 damage distribution. The analysis of
the obtained results clearly shows that the numerical model underestimates the damage
due to the impact. Except for the impact configuration at 50 m/s, all the other tested cases
provide a relative difference greater than 30% when compared with experimental results.
This comes from the fact that the numerical model which implements the delamination
process through the damage law for the material is clearly not able to reproduce the
opening of the plies and also does not take into account a tensile behaviour that is different
than the compressive behaviour in the fibre direction. As a consequence, the volume plate
has a higher stiffness which is the primary cause of both a higher contact force and lower
delamination areas. Moreover, as presented on figure E.14, the damage obtained in every
ply does not follow the adequate orientation.
Mechanical samples
2 (75m/s)
3 (65m/s)
4 (75m/s)
6 (50m/s)
8 (81m/s)
Mech. Impact
total
delaminated
area
(mm²)
1184
630
1626
223
1746
Numerical
model (CDM)
total
delaminated
area
(mm²)
760
416
760
267
956
Relative difference
-35.8%
-33.9%
-53.26%
19.7%
-45.28%
Table E.6: Comparison of the total delaminated area for experimental and numerical results and relative difference
Appendix E
234
__________________________________________________________________________________
(a) Int. 1 (45/0)
(c) Int.3 (-45/90)
(b) Int. 2 (0/-45)
(d) Int.4 (90/-45)
(e) Int.5 (-45/0)
(f) Int.6 (0/45)
Figure E.14: Damaged interfaces for the numerical simulation, CDM model, impact at 65 m/s.
Appendix E
235
__________________________________________________________________________________
6.1.1.4 3D model with damaging plies and interface element
As it has been shown in the previous section that the use of interface elements gives a too
flexible structure whereas the use of the continuous damage law for the plies alone makes
the structure remains too stiff. Alone, any of them does give satisfactory results when
compared to experimental one. A coupling between the two methods to generate damage is
chosen. The coupling between CDM and discrete models should allow a better reproduction
of the damage.
The fourth 3D model uses both the continuous damage law for the plies and interface
elements between them to insure a (soft) coupling between the delamination initiation and
propagation and the in plane damage in the volume of each ply. Among the 5 failure criteria
presented in section 5.4.2, the criterion f5 (Equation. 5.20), corresponding to delamination is
deactivated to avoid competition with the cohesive elements placed at the interfaces
between plies. The criterion f3, corresponding to crushing (compression in z-direction) is also
deactivated in order to limit the effect of punching due to the projectile which interferes
with cohesive elements and increases the delamination.
The cohesive elements properties are kept identical to those presented in Table E.1, as a first
approximation, and material properties for the plies are the ones presented in Table E.4. The
lay-up remains unchanged: [45/0/-45/90]s.
Figures E.15 shows the rear face displacement obtained for shell model, 3D model with ply
damaging law, experimental results and 3D complete model using both ply damage and
interface elements for impact at 65m/s and 75 m/s respectively. The out of plane velocity
(displacement slope) at short times (<50µs) is well reproduced by the complete model. The
rear face displacement at both short and large time is also well approximated, however the
displacement plateau observed from 200µs to 500µs is not well approximated by the
numerical simulation which tends to underestimate it.
Appendix E
236
__________________________________________________________________________________
3000
Displacement (µm)
2500
2000
1500
Shell model 65m/s
1000
500
3D model with damaging ply and interface elements
65m/s
0
0
100
200
time µs
300
400
500
(a)
4000
3500
Displacement (µm)
3000
2500
2000
1500
Mechanical impact test at 75 m/s
1000
Shell model impact at 72m/s
500
3D modle with ply damage and interface elements
75m/s
0
0
100
200
300
Time (µs)
400
500
(b)
Figure E.15: Rear face displacement for 3D model with and without interface elements and CDM model after impact at
(a) 65m/s and (b) 75m/s.
Figures E.16 presents the results of delaminated area per interface and total delaminated
area obtained with the complete model, compared to results obtained with the 3D elastic
model with interface elements only and the associated experimental impacts. The results are
also presented in table E.7 and E.8. Table E.9 gathers the total delaminated area for all the
mechanical samples compared with the results of the two numerical models.
The addition of the damage law for the plies greatly modifies the delamination distribution
through the laminate thickness. The total damage area is greatly reduced and the
localization of the damage at certain interface is slightly reduced as well, however the new
formulation underestimates the delamination size. More importantly, the distribution
through thickness of the delamination proposed by the numerical model still needs
improvements as it is not able to reproduce the experimental observations.
Appendix E
237
__________________________________________________________________________________
Interface
Mechanical test
75m/s
3D CDM elastic model
75m/s
3D complete model
with CDM and
interface elements
75m/s
1
2
3
4
5
6
total
0
205
30
102
701
543
124
24
71
82.5
161,5
298,3
761
48.5
109
804.5
930
12.5
18.7
1923.2
1626
Table E.7: Comparison of the delaminated area in mm² per interface between the mechanical tests and two different
numerical models, impact at 75 m/s
Interface
Mechanical test
65m/s
3D CDM elastic
model 65m/s
3D complete model
with CDM and
interface elements
65m/s
1
2
3
4
5
6
total
0
127
23
131
170.8
174.66
630
70
13,6
9
0
45,8
277,3
415,5
50
96,2
574,6
710,7
13
11
1455,6
Table E.8: Comparison of the delaminated area in mm² per interface between the mechanical tests and two different
numerical models, impact at 65 m/s
1000
900
800
700
600
500
400
300
200
100
0
3D elastic model with
damaging plies 75m/s
3D model with damaging
plies and interface
elements 75m/s
1
2
3
4
Interface #
5
6
Appendix E
238
__________________________________________________________________________________
(a)
800
700
600
3D elastic model with damaging
plies 65m/s
500
3D model with damaging plies
and interface elements 65m/s
400
300
200
100
0
1
2
3
4
Interface #
5
6
(b)
Figure E.16: Delamination distribution through thickness for 3D models with interface elements and with and without
damage law for the plies for impact at (a) 75m/s and (b) 65m/s.
Mech.
Impact total
delaminated
area
Numerical
model (CDM)
total
delaminated
area
(mm²)
(mm²)
Numerical model
with ply damage
and interface
elements
delaminated area
(mm²)
2 (75m/s)
1184
760
1683
3 (65m/s)
630
416
1455.6
4 (75m/s)
1626
760
1683
6 (50m/s)
223
267
395
8 (81m/s)
1746
956
2241
Mechanical samples
Table E.9: Comparison of the total delaminated area for experimental and numerical results: damaging law model and
coupled damage law with interface elements model.
The study of the delamination by interface in the numerical simulations confirms that the
cohesive elements are still not able to reproduce the correct orientation following the lower
ply of the interface, as shown on figure E.17. Even though the soft coupling with the damage
law for the ply seems to allow a better representation of the delamination orientation, the
effect of localization of the damage observed previously at the interface 3 and 4 is still
present in the new model.
Appendix E
239
__________________________________________________________________________________
(a)(45/0)
(c)(-45/90)
(b) (0/-45)
(d) (90/-45)
(e)(-45/0)
(f) (0/45)
Figure E.17: Damaged interfaces for the numerical simulation coupling interface elements and ply damage law, impact at
81 m/s
Appendix E
240
__________________________________________________________________________________
Appendix E
241
13
Appendix F
Cohesive elements and surfaces: a comparison
Lightning
samples
Associated
mechanical
samples
Projectile’s
speed (m/s)
101
4
2
7
1
6
3
8
5
75
75
70
124
50
65
81
87.5
102
103
107
110
LS
delaminated
area (mm²)
2321
5836
0
1711
120
Mechanical
delaminated
area (mm²)
Numerical
delaminated
area (mm²)
1626
1184
1019
5732
223
630
1746
3361
1729
1729
1661
3271
713
908
2062
/
Appendix F
242
LS Max displ.
(mm)
2861
3585
1810
2626
2971
Mechaniacl
impact max
displ.
(mm)
3780
2856
2740
/
2233.3
2656.7
3866.7
/
Shell max
displ.
(mm)
3701.8
3701.8
2609
/
2021.2
2481.3
4278.8
/
14
Resumé en français
________________________________________________________
Développement d'un modèle mécanique pour la prédiction des dommages de
1. Introduction
Depuis les années 70, les matériaux composites ont été introduits de plus en plus massivement dans
les nouvelles générations d’avions. Des aéronefs tels que l’A380 et l’A350 possèdent aujourd’hui
respectivement 25% et 50% de matériaux composites jusque dans des structures dites primaires
(ailes, fuselage, caissons de carburant,…). L’utilisation de ces nouveaux matériaux, aux propriétés
mécaniques supérieures à celles de leurs prédécesseurs métalliques, alliées à leur légèreté, a permis
la réduction de la quantité de carburant par passager de façon notable.
Figure 1.1: Airbus A350
Les avions en service sont soumis régulièrement à la menace d’un impact foudre. En effet, il a été
statistiquement établi qu’un aéronef est foudroyé au moins une fois par an (1/2900 heures de vol).
La menace foudre est donc un évènement courant dans la vie en service d’un avion. Les matériaux
métalliques, précédemment utilisés, sont des conducteurs naturels d’électricité. Cependant, les
matériaux composites, notamment ceux présents dans les structures primaires, ne sont pas d’aussi
bons conducteurs. Les matériaux composites sont constitués de fibres de carbone (conductrices)
noyées dans une résine isolante. Ainsi, lorsque la foudre frappe, les structures composites ne sont
pas à même d’évacuer correctement le courant. Il en résulte des dommages parfois importants tels
que des brûlures et des dommages internes parfois invisibles à l’œil nu. Afin de diminuer voire
d’empêcher l’apparition de ces dommages, des essais de certification ont été instaurés afin de
certifier les matériaux face à cette menace et de mettre au point des protections adaptées [1, 2].
Malheureusement, ces tests sont coûteux et doivent être effectués à la fois sur des coupons et des
sous assemblages, ce qui complexifie et multiplie les étapes de certification.
243
__________________________________________________________________________________
Figure 1.2: Approche building-Block pour la certification des matériaux composites [3]
Afin de protéger les structures en matériaux composites, des protections contre la foudre ont été
mises en place sur les avions en service. Cependant ces protections qui consistent à ajouter une
épaisseur de matériau conducteur (cuivre, argent, bronze) par-dessus le matériau composite,
viennent ajouter du poids sur les structures et diminuent donc l’avantage d’utiliser des matériaux
plus légers que l’acier. Aujourd’hui, le principal objectif réside dans l’optimisation de ces protections,
qui sont largement surévaluées, en termes d’efficacité et de poids. Cette optimisation passe par une
meilleure compréhension du phénomène foudre, de l’interaction de l’arc foudre avec le matériau
impacté et des dommages qui en résultent.
L’étude de l’impact foudre sur matériaux composites est un sujet qui est traditionnellement envisagé
d’un point de vue électrothermique. Dans cette étude, une autre approche est préconisée,
concentrée sur les dommages effectivement observés suite à des chocs foudre. Un impact
mécanique équivalent est proposé, à travers l’utilisation d’un projectile, créé à partir des données
foudre obtenues lors des essais en laboratoire. Des modèles numériques prédictifs sont associés à
une démarche expérimentale. Le modèle numérique, dont les résultats sont validés par les essais
mécaniques menés lors du travail de thèse, sera utilisé pour prédire les dommages résultant d’un
choc foudre. Ainsi, un model prédictif, prenant en compte différents types de sollicitations possibles
telles que des projectiles ou des pressions équivalentes, pourra être utilisé pour prédire
l’endommagement lié à la foudre et fournir des conseils au design des matériaux et des protections.
Le chapitre 2 présente les travaux précédemment menés sur cette problématique de l’impact foudre
sur panneaux composites, dans la littérature. Le chapitre 3 retrace les travaux menés par Airbus
Group Innovations, antérieurs à la thèse et expose les objectifs ainsi que la méthodologie de travail
adoptée pour cette étude. Les mises au point d’un équivalent mécanique à l’impact foudre ainsi que
de la campagne d’essais mécaniques sont relatées au chapitre 4 ainsi que la comparaison entre les
résultats obtenus lors des deux campagnes d’essais foudre et mécanique. La création d’un modèle
244
__________________________________________________________________________________
numérique associé aux essais est présentée au chapitre 5, qui expose les différentes lois de
comportement intégrées au modèle ainsi que les résultats numériques obtenus avec les différents
modèles à la complexité croissante, et compare des résultats numériques et foudres. Finalement, le
chapitre 6 propose des perspectives et des résultats préliminaires sur d’éventuelles pistes de travail à
poursuivre.
2. Etat de l’art
2.1 Généralités sur la foudre
Contrairement aux matériaux métalliques précédemment utilisés, les matériaux composites sont de
mauvais conducteurs électriques. Si les fibres de carbone sont de bons conducteurs (conductivité de
104 à 105 S/m sens fibre), la résine qui leur sert de liant est pour sa part totalement isolante. De plus,
la conductivité entre les différents plis qui constituent les matériaux composites est aussi
relativement faible.
L’impact de la foudre sur les matériaux composites induit deux principaux types d’effets : les effets
indirects (circulation du courant dans les circuits de l’avion et effets électromagnétiques sur le
système électronique de bord) et des effets directs (dommages d’origine électrique, thermique et
mécanique). Les panneaux endommagés sont souvent difficiles à réparer et à remplacer,
principalement à cause de la nature hétérogène des matériaux composites qui consistent en un
empilement de plis d’orientations différentes, les uns sur les autres. Il devient donc capital de
pouvoir prédire et diminuer les dommages liés à la foudre sur ces structures. Cette étude se focalise
principalement sur l’étude des effets directs, effets structuraux qui viennent diminuer les
caractéristiques mécaniques des matériaux endommagés.
Effets Directs
Les effets directs de la foudre résultent de l’attachement de l’arc électrique sur le matériau et
génèrent des dommages électriques, thermiques et mécaniques. La plupart de ces dommages sont
situés aux points d’entrée et de sortie du courant. Dans le cas des matériaux composites ces
dommages peuvent être visibles (brulures, explosion du premier pli, figure 2.1), et sont localisés à la
surface du matériau (fin volume près du point d’impact, comprenant le premier pli), ou invisibles
(délaminages, rupture de fibre et de matrice) dans le cœur du matériau. Ces derniers dommages sont
les plus graves car ils portent atteinte aux propriétés mécaniques du matériau et peuvent mener à
une rupture critique.
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Figure 2.1: Dommage visible suite à un coup de foudre
Plusieurs études portant sur les mécanismes d’endommagement des matériaux composites soumis à
des impacts foudre sont disponibles dans la littérature [6-8]. Ces études se focalisent sur les effets
directs de la foudre. Chen et al [6] et Hirano [7] en 2008 se sont concentrés sur l’étude des
composites à matrice époxy et fibre de carbone. Ils testèrent différents niveau d’impact foudre sur
des spécimens d’assez grande taille 350×350 mm². Les tests menés lors de ces études ont permis de
discriminer trois principaux types de mécanismes d’endommagement: rupture de fibres,
détérioration de la résine et délaminage entre les plis du stratifié. La rupture de fibres est identifiée
[7,8] comme une résultante de l’onde de choc qui se propage depuis l’arc durant le coup de foudre.
Une importante quantité d’énergie est transmise dans le matériau et se dissipe sous forme de
chaleur, due à l’effet Joule. Cette forte augmentation de la température vient causer la sublimation
des fibres de carbone et leur rupture totale ou partielle. Ces dommages se limitent généralement à la
« surface » du matériau (premiers plis du composite) et ne se propagent pas profondément dans le
stratifié. L’effet Joule est également responsable de la pyrolyse de la résine. Cet échauffement libère
des gaz à l’intérieur du composite, qui s’évacuent sous forme d’explosion venant causer d’importants
dégâts dans la zone proche du point d’attachement de l’arc. Le couplage des deux précédents modes
de ruine est responsable de l’apparition de délaminage entre les plis. Les délaminages sont
responsables de l’abattement des propriétés mécaniques des matériaux impactés.
Zoning
Toutes les parties d’un avion ne sont pas exposées aux mêmes risques d’impact foudre. Afin
d’optimiser la protection des aéronefs, une carte statistique des impacts sur la structure a été établie
par les avionneurs et les compagnies. Elle rassemble la tendance de chaque partie de l’avion à être
foudroyée. Cette cartographie, dite « zoning », fait partie de la certification internationale [3, 4]
(EUROCAE and SAE) et permet aux avionneurs de prédire les zones (figure 2.2) les plus affectées et
d’adapter les protections en conséquence.
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Figure 2.2: Zoning d’un avion
Tests de certification en laboratoire
Afin de certifier les aéronefs contre la menace foudre, les constructeurs aéronautiques ont dû mettre
au point des moyens d’essais permettant de reproduire cette menace. Les niveaux de dégâts liés à la
foudre sont édifiés à partir des données recueillies lors des vols [2], qui constituent le seul moyen
d’obtenir des données statistiques réelles.
Une forme normalisée de la foudre (AC 20-53A) [3] a ainsi été établie qui comprend des
composantes impulsionnelles (composantes A et D) et des composantes continues (composante C)
de courant. Cette forme d’onde (figure 2.3) couvre 99% des coups de foudre en terme de sévérité, et
est utilisée par les constructeurs pour tester et certifier le comportement des structures en
matériaux composites et les différentes protections face aux impacts foudre.
Figure 2.3: Courant normalisé pour les essais foudre en laboratoire
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Protections contre l’impact foudre
Suite au zoning des aéronefs, des mesures de protections adaptées sont mises en œuvre. Ces
protections sont créées afin de protéger l’intégrité structurelle des zones impactées et de réduire au
maximum les dommages liés à la foudre. Il existe plusieurs types de protections telles que les
peintures métalliques, l’introduction de fibres métalliques ou de nanotubes de carbone dans le
premier pli de stratifié ou l’ajout d’un grillage métallique (figure 2.4). La dernière solution est la plus
usitée. Elle consiste à ajouter par-dessus le matériau composite, un film ou un grillage de matériau
conducteur tel que le cuivre, le bronze ou l’aluminium, afin de conduire le courant hors de la
structure.
Figure 2.4: Protections contre la foudre. Expanded Copper Foil (à gauche) et Solid Copper Foil (à droite)
2.2 Modélisation de la foudre
2.2.1 Les différentes composantes de la foudre
La foudre est un phénomène complexe qui met en jeu différentes physiques. Aujourd’hui, l’ensemble
de ces composantes et leur couplage restent mal connus et aucun consensus sur leur ordre
d’apparition lors du phénomène n’a été établi par la communauté scientifique. Le paragraphe qui
suit constitue une proposition de chronologie des évènements liés à l’attachement de la foudre sur
panneaux composites, qui rassemble les différentes physiques présentes dans la foudre ainsi que
leurs actions supposées durant un coup de foudre.
Les structures composites sont protégées par des grillages métalliques (ECF) dans lesquels le courant
injecté durant un coup de foudre va circuler, perpendiculairement à la colonne d’arc, et qui peut
ainsi créer un champ magnétique. La circulation du courant dans ce champ est responsable de la
création de forces de Laplace qui peuvent appuyer sur l’échantillon et le faire fléchir
mécaniquement. L’arc foudre est aussi responsable de la création d’une onde de choc acoustique
due à la surpression de la colonne d’arc dans l’air. Cette surpression est suspectée être à l’origine des
dommages internes, tels que le délaminage, observés dans le cœur du matériau impacté.
Il est difficile de quantifier l’importance relative de chacune des composantes de la foudre vis-à-vis
des autres et encore plus de leurs interactions et éventuels couplages. Cependant, la plupart de ces
composantes ont été individuellement identifiées et des hypothèses de chronologie de la foudre ont
été émises. La figure 2.5 est une proposition de représentation de la foudre s’attachant sur une
structure composite, qui prend en compte l’action supposée de chacune des physiques mises en jeu
(thermique, électrique, magnétique, mécanique) sur les dommages observés.
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Figure 2.5: Phénomènes physiques contribuant aux effets directs de la foudre sur des panneaux composites type
fuselage d’avion [21]
Plusieurs phénomènes sont impliqués dans cette représentation : des physiques multiples dont
l’action se déroule sur des échelles de temps spécifiques et différentes et qui induisent des
dommages sur le matériau.
Le premier phénomène supposé agir sur la structure composite est la surpression causée par la
colonne d’arc et qui induit une force macroscopique F1. Cette force est en partie responsable de
l’arrachement de la peinture en surface ainsi que de la création de délaminage dans le CFRP par
écrasement des fibres et des premiers plis. Ensuite, le courant I injecté par l’arc électrique dans la
protection métallique, ainsi que le long des fibres de carbone des premiers plis, engendre des forces
de Laplace F2 et l’échauffement du matériau par effet Joule Q. A la surface du stratifié, l’énergie
déposée par l’arc électrique vient sublimer la protection et éventuellement les fibres du premier pli
et la résine qui les enrobe, créant un relâchement gazeux responsable de l’explosion de la peinture
(force F3) et causant l’arrachement des fibres supérieures (force F4) du à une surpression confinée
entre la peinture et le composite.
Le scénario d’endommagement proposé est le résultat des observations faites lors des diverses
campagnes foudre menées par AGI. Il sert de base de travail à toutes les études qui suivront dans ce
document. Des scénarii similaires sont adoptés dans diverses études [14, 39]. Cependant, il faut
noter que cette interprétation chronologique de la foudre n’est pas admise partout et que d’autres
scénarii sont utilisés, qui mettent notamment en avant les effets thermiques (effets Joule) et
électriques de la foudre et n’accordent pas la même importance aux effets mécaniques [48, 49].
La compréhension du phénomène est primordiale dans l’étude et la modélisation de la foudre car
elle conditionne l’approche numérique et représentative qui en sera faite. Depuis plusieurs
décennies, des études ont visé à reproduire la foudre et à prédire numériquement les
endommagements qu’elle génère. La plupart des modèles que l’on trouve dans la littérature se
concentrent sur les aspects électrothermiques. Ces modèles prennent principalement en compte la
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modélisation de l’attachement de l’arc sur la structure, la distribution du courant et de la
température dans le composite afin d’expliquer les dommages observés suite à l’impact foudre.
Cependant, depuis quelques années, des études focalisées sur l’étude des dommages mécaniques
liés à la foudre commencent à apparaitre. Une classification de la littérature est proposée dans la
section suivante qui fait la différence entre les modélisations électrothermiques et mécaniques.
2.2.2
Modélisation électrothermique
L’étude des effets directs de la foudre est principalement menée d’un point de vue électrothermique
et se concentre sur les effets Joule. Ces modèles [10-16, 24] se concentrent sur les dommages
observés en surface des structures composites impactées par la foudre et utilisent des panneaux nus
pour leurs comparaisons. En effet, peu d’études traitent des dommages induits sur des panneaux
protégés et peints [24, 25].
Plusieurs études [23-28] se centrent sur les dommages qui apparaissent à la « surface ». Ces
dommages correspondent principalement à la sublimation et à l’ablation de la peinture et de la
protection métallique. Ils peuvent éventuellement s’étendre aux premiers plis de CFRP [23, 24]. La
surface endommagée, volumique donc, est une zone qui subit de grands transferts d’énergie depuis
l’arc foudre (figure 2.6).
Figure 2.6: Dommages de surface après un coup de foudre [23]
Dans ces modèles, l’énergie est injectée dans le matériau par effet Joule, induit par la circulation du
courant dans les fibres de carbone et la protection de cuivre. Cette énergie vient chauffer les couches
supérieures et peut entrainer leur vaporisation ou sublimation. L’étendue des dommages est évaluée
à partir de la quantité de phases liquide ou gazeuse générées par effet Joule. Une telle approche est
adaptée à la description des dommages observables sur des matériaux métalliques et a été étendue
à l’analyse des matériaux composites pour des composantes continues de courant, type composante
C [26-28].
Dans cette branche, Lago [27] a créé un modèle numérique qui s’intéresse à la description et à la
reproduction de l’arc foudre (cathode) et à son processus d’attachement et d’interaction avec le
composite (anode). Il a ainsi cherché à quantifier le niveau d’endommagement de l’anode en
fonction de l’intensité et de la durée du courant injecté. Le modèle numérique simule une colonne
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d’arc foudre (plasma) et le transfert de courant vers l’anode, figure 2.7. La zone d’endommagement
numérique, liée aux effets Joule responsables de la dégradation du matériau cible, est comparée à la
zone totale délaminée mesurée après les essais en laboratoire.
Figure 2.7: Champ de température dans la colonne de plasma [27]
De telles études sont intéressantes car elles permettent d’améliorer la compréhension des
phénomènes d’attachement et l’interaction de l’arc avec le matériau. Cependant, ces études ne
prennent pas en compte les dommages internes, invisibles à l’œil nu et se concentre sur la
problématique d’attachement seulement.
D’autres études se sont concentrées sur les propriétés électriques et thermiques et sur la
modélisation des dommages liés principalement aux effets Joule [27-31]. Ogasawara et al. [14] ont
aussi proposé un modèle couplé électromagnétique/thermique pour simuler numériquement l’arc
foudre. Leur modèle se concentre sur des panneaux nus et calcule le flux de température dans le
matériau. Les auteurs font l’hypothèse que la conductivité électrique du composite évolue
linéairement avec la variation de température responsable de la décomposition de la résine et de la
sublimation des fibres de carbone. Ils ont ainsi montré que la conductivité électrique était
effectivement dépendante de l’évolution de la température dans le stratifié sans pour autant
prendre en compte la dépendance des propriétés thermiques du matériau. Dong et al [17] ont
poursuivi cette étude de la dépendance des propriétés électriques à la température. Dans leur
travail, ils ont ajouté un comportement chimique à la résine permettant d’expliquer les dommages
thermiques en fonction du temps, de la température et d’autres facteurs spatiaux. Une analyse
couplée électro-thermo-pyrolytique est ainsi menée pour représenter l’action de la foudre sur les
panneaux de CFRP. Ils ont pu observer que les contours du champ de température dans chaque
interface du stratifié dirigeaient le degré de pyrolyse de la résine, ainsi qu’une forte dépendance du
matériau à l’amplitude de pyrolyse. Ils en ont conclu que ce paramètre était un point clef dans la
modélisation des effets électrothermiques de l’arc foudre. Huchette et al [29] se sont également
intéressés au phénomène de pyrolyse induit par l’arc foudre. Dans leur étude, ils ont couplé des
analyses thermique et électrostatique dans un modèle élément fini (EF) afin d’estimer le
comportement thermique des structures foudroyées. Dans leur modèle, la densité de courant est
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calculée dans le matériau, tout en prenant en compte la pyrolyse de la résine en augmentant les
conductivités électriques et thermiques qui sont co-dépendantes [17]. Les dommages liés à cette
action thermique sont calculés numériquement et comparés à la zone de résine sublimée, mesurée
expérimentalement.
Finalement, Abdelal and Murphy [32], contrairement aux précédentes études [27, 29-31, 33], ont
intégré la dépendance thermique des propriétés matériaux pour le CFRP. Leur simulation intègre
plusieurs états physiques comme la fusion, la vaporisation ou la sublimation pour les matériaux
considérés (CFRP et cuivre). De plus l’arc numérique interagit avec le composite à travers l’utilisation
de propriétés thermiques et conductrices dépendantes donc de la température de façon à prédire
l’endommagement lié à l’augmentation de température dans les premiers plis. Comme les autres
études, ils comparent la zone de décomposition thermique numérique à la zone endommagée
observée sur les éprouvettes foudroyées et calculent ainsi l’évolution du profil de température dans
l’épaisseur du stratifié (figure 2.8). Il est en effet admit que la distribution de température dans
l’épaisseur est dirigée par l’anisotropie électrique due aux effets Joule [34].
Figure 2.8: Composite dégradé (simulation) [32]
Tous les modèles présentés, et la majorité de ceux issus de la littérature, traitent de la foudre d’un
point de vue électrothermique et se concentrent sur la modélisation du pied d’arc foudre. Ce faisant,
ils se concentrent également sur les dommages visibles observables en surface des structures
impactées et qui impactent généralement les couches supérieures (peinture et protection) et les
premiers plis de CFRP. Le phénomène d’attachement et les dégâts qu’il entraine en surface sont ainsi
largement étudiés, au détriment des dommages internes, invisibles à l’œil nu. Pourtant, ces
dommages apparaissant dans l’épaisseur du composite sont aussi les plus graves car ils remettent en
cause l’intégrité structurelle du matériau, en plus de ne pas être détectable sans examen plus
poussé. Afin de comprendre ces mécanismes d’endommagement, la méthodologie employée dans
les précédentes études parait trop complexe. En effet, la foudre est un phénomène encore trop
largement inconnu et la représentation de toutes ses composantes et couplages implique une
connaissance du phénomène qui n’est aujourd’hui pas encore acquise.
Dans le paragraphe suivant, plusieurs études sont présentées qui proposent une autre approche de
la foudre, concentrées sur l’aspect mécanique du chargement et les dommages observés à cœur des
stratifiés.
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2.2.3
Modélisations mécaniques
En dépit du fait que la plupart des études menées sur la foudre adoptent un point de vue thermique
pour expliquer les dommages observés ou se concentrent sur l’attachement du pied d’arc sur la
surface du composite, il semble de plus en plus évident, que pour des composantes de type
impulsionnel A ou D, que les phénomènes mécaniques sont en partie responsables des dommages
observés, notamment dans l’épaisseur des structures foudroyées. Les études qui vont suivre ont
décidé de totalement s’affranchir des propriétés électriques et des couplages électro-thermomagnétiques pour se concentrer exclusivement sur les dommages mécaniques et leur origine. Ainsi,
toute la « surface » du matériau qui subit des dégradations d’ordre thermique est écartée.
Bien qu’il existe une vaste littérature sur les dommages d’impact sur les matériaux composites, dus à
la chute ou la projection d’objets [35-37, 42, 50], peu d’études ont été publiées sur les dommages
mécaniques observés lors d’impacts foudre. Ces études pour une grande partie se sont de plus
uniquement intéressées à la tenue résiduelle des structures impactées [23, 25, 38-40].
Hirano et al [13] se sont penchés sur l’étude des dommages mécaniques résultant d’un impact
foudre dans le but de les catégoriser et de mieux les comprendre. Leur étude, sur panneaux
composites nus soumis à plusieurs types d’impacts (différentes composantes)[41-43] a permis de
montrer que ces dommages à cœur étaient d’origine mécanique : rupture de fibres et de matrice,
délaminage, et qu’ils étaient également dépendants des propriétés thermiques et électriques du
matériau.
Les modélisations thermiques doivent donc être supportées par des concepts mécaniques afin
d’envelopper l’ensemble des dommages et des mécanismes liés à la foudre. Haigh [41] a travaillé
dans ce sens en se concentrant sur l’impulsion mécanique (i.e la force transférée par l’arc électrique
et intégrée sur le temps). Des essais ont été instrumentés, mécaniquement et optiquement, et la
valeur de cette impulsion extraite des données de déplacements des échantillons au cours du temps.
Ces valeurs sont ensuite comparées à des résultats d’impacts traditionnels. De cette façon, il lui a été
possible de quantifier l’impulsion mécanique. L’ordre de grandeur de cette impulsion est d’environ
1N.s. Cependant, le travail des auteurs ne leur a pas permis d’établir une corrélation claire entre
l’impulsion mesurée et les dommages mécaniques observés après foudroiement. De leur côté,
Gineste et al [44], ont poursuivi ce travail et ont travaillé sur les mesures de déplacement obtenues
lors des essais foudre. Avec ces données, obtenues à l’aide de capteurs lasers (VISARs), un modèle a
été mis en place pour calculer l’impulsion mécanique. Cette impulsion a été couplée à un modèle
thermomécanique afin de décrire l’endommagement dans le matériau.
Finalement, Feraboli et Kawakami [46] sont les premiers à comparer les dommages foudre et
mécanique issus d’essais traditionnels. La ressemblance entre les dommages des deux types
d’impacts les poussa à essayer de mettre au point un essai de tour de chute qui reproduirait les
dommages foudre. Ils se sont ainsi focalisés sur des gammes de vitesses assez basses afin
d‘investiguer le lien entre l’énergie déposée par la foudre et celle transmise de façon mécanique à la
structure impactée. Ils ont donc effectué en parallèle des essais foudre et de tour de chute, à des
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niveaux de menace dit équivalents (voir figure 2.9) avant de tester la résistance au dommage et la
tenue après impacts des éprouvettes afin de comparer les résultats obtenus entre essais foudre et
mécanique.
Figure 2.9: Méthode pour coupler essais foudre et impacts mécaniques traditionnels [48]
Des analyses post-mortem sont ensuite menées sur les éprouvettes qui ont montré qu’à menace
“équivalente » les essais mécaniques produisaient des dommages beaucoup plus grands que ceux
obtenus avec les essais foudre mais que les dommages étaient effectivement de même nature. Ils
conclurent de leurs travaux que leur critère d’équivalence, l’énergie transmise dans le matériau,
n’était pas le bon car une importante partie de l’énergie de la foudre est dissipée en effet Joule et en
dommages dits « de surface », de nature électrothermique. Cependant, les similarités entre les faciès
d’endommagement et les performances résiduelles proches des deux types d’impacts suggèrent de
poursuivre des études dans ce genre.
2.3 Conclusions et perspectives
La foudre est un phénomène qui met en jeu de multiples physiques telles que l’électromagnétisme,
la thermique et la mécanique. L’étude de la littérature consacrée à la foudre montre
qu’historiquement le traitement de ce phénomène et de ces dommages induits se fait d’un point de
vue électrothermique. Les études sur le sujet se sont principalement concentrées sur la
représentation de l’arc électrique et du plasma qui vient s’attacher sur le panneau composite. La
modélisation électrique du courant ainsi que du flux thermique qui traverse le composite ont été au
cœur de ces études qui se focalisent sur les dommages observés en surface et l’attachement du pied
d’arc, sans investiguer les dommages à cœur du matériau.
Cependant certaines études se sont penchées sur une autre approche, en travaillant davantage à
partir des dommages observés. Elles en ont conclu que ces dommages étaient très similaires, à cœur
du matériau impacté, à ceux observés lors d’essais d’impacts mécaniques (canon, tour de chute).
Elles ont alors décidé de s‘intéresser aux phénomènes responsables de tels faciès
d’endommagement, en s’écartant des analyses électrothermiques traditionnelles et en
« simplifiant » la foudre en essayant d’isoler l’influence de ses diverses composantes. Les auteurs se
sont intéressés aux phénomènes induits par la foudre qui pourraient être responsables de
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dommages de type mécanique tels que les forces de Laplace ou l’onde de choc liée à l’attachement
du pied d’arc. Enfin, Feraboli pousse même l’étude jusqu’à comparer des dommages purement
mécaniques à ceux issus de la foudre et tente de proposer un système d’équivalence. La méthode
suivie par ces études met complètement de côté la question de la chronologie d’apparition de
chacune des physiques, leur importance relative et leurs possibles couplages. Une première étape
consiste à discriminer l’importance des différentes composantes les unes par rapport aux autres à
travers une méthode simplifiée où chaque physique sera étudiée individuellement, afin de quantifier
son incidence sur les dommages issus de la foudre. Cette méthode est reprise dans ce document afin
d’améliorer la compréhension générale du phénomène et d’optimiser les protections associées.
3. Travaux préliminaires et définition de la méthode de travail
3.1 Travaux préliminaires
Les travaux présentés dans ce chapitre font partis d’un projet d’étude des dommages liés à la foudre
menés par Airbus Group Innovations (AGI) et Airbus depuis plusieurs décennies. L’endommagement
des structures composites dû à la foudre est devenu un sujet de recherche de plus en plus important
depuis l’introduction massive de matériaux composites jusque dans les structures primaires. En effet,
ces nouveaux matériaux allient légèreté et propriétés mécaniques équivalentes à leurs
prédécesseurs métalliques. Cependant, les composites sont de mauvais conducteurs électriques.
Ainsi, l’étude de la foudre et de son interaction avec les composites deviennent des sujets majeurs
pour les constructeurs aéronautiques qui doivent prévoir des protections adaptées à cette menace.
Ce chapitre rassemble une partie des travaux menés par AGI ayant précédé le travail de thèse et lui
ayant servi de support.
3.1.1 Première tentative de chargement équivalent
Une première étude est menée en 2011 par AGI s’appuyant sur les résultats d’essais obtenus lors des
campagnes foudre : les déplacements et vitesses face arrière. Le but de l’étude est de s’affranchir de
toutes les contributions potentiellement responsables du déplacement des échantillons pour essayer
de reproduire de façon simplifiée et mécanique ce comportement du à la foudre. Afin de reproduire
ces résultats, une pression constante limitée en temps et en espace est appliquée dans un modèle
numérique représentant les plaques d’essais. Un modèle numérique élément finis simplifié
(éléments coque et matériau homogénéisé) est créé en utilisant le logiciel ABAQUS 6.11 : les
échantillons sont représentés aussi fidèlement que possible aux éprouvettes utilisées lors des essais
foudre, en respectant les conditions aux limites. La pression est appliquée au centre de la plaque. La
forme temporelle choisie pour cette pression est un Dirac et sa distribution spatiale est représentée
par une Gaussienne, distribuée sur un rayon r, centrée autour du point d’attachement de l’arc
foudre, avec une amplitude A.
Avec les essais foudre, on choisit le temps d’application de la pression équivalente égal au temps
nécessaire pour atteindre la vitesse de déplacement maximum, notée T. Un calcul itératif sur les
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valeurs de A et r est alors mené afin d’obtenir des valeurs de déplacements correspondantes aux
résultats d’essais foudre. Les déplacements face arrière sont mesurés lors des essais à l’aide de 5
VISARs (capteurs lasers). Le modèle numérique permet d’extraire les résultats de déplacement pour
les positions des cinq capteurs et de comparer les résultats numériques aux résultats expérimentaux.
(a)
(b)
Figure 3.1: (a) Comparaison des déplacements face arrière pour les essais foudre et le modèle numérique (k: 0.25 N.s, T:
40 µs, r: 2.24 cm), (b) Vitesse de déplacement au point d’impact comme une fonction du temps et de divers paramètres.
Ligne continue: résultats expérimentaux. Ligne pointillée: modèle [3]
La figure 3.1 présente la comparaison entre les résultats d’essais et ceux obtenus avec le modèle
numérique pour les cinq VISARs. Le modèle permet donc de reproduire correctement les
déplacements observés lors des essais foudre, notamment pour le capteur central (IDF5 en noir). Les
autres capteurs sont également correctement approximés par le modèle.
Cette première étude montre qu’il est possible de reproduire la poussée induite par la foudre sur les
plaques composites en utilisant une formulation purement mécanique. L’étude montre également
qu’il est possible de s’affranchir des événements se déroulant en surface (thermique, électrique,
etc..) et de séparer ce qu’il se passe en surface et dans le reste de la structure.
3.1.2 Modélisation du pied d’arc
Plusieurs campagnes d’essais foudre ont été menées afin d’améliorer la compréhension du
phénomène et l’endommagement qui en résulte. Lors de ces essais, des caméras rapides (1 million
d’images/seconde) ont permis d’enregistrer le phénomène d’attachement et de foudroiement de
l’arc sur les panneaux composites et d’obtenir des informations en temps réel. Ces images ont
notamment permis d’identifier la formation d’anneaux lumineux, donc le rayon semble croitre avec
le temps, au point d’attachement de l’arc. Ces anneaux ont été interprétés comme la signature de
l’évolution temporelle de la surface endommagée par l’arc (protection métallique sublimé et premier
brulé).
Le courant électrique choisit préférentiellement le chemin le plus accessible et le plus simple et a
donc tendance à se diriger vers les bords ou la protection de la plaque impactée qui sont plus
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conducteurs que le composite. La portion circulaire visible sur les images (figure 3.2) correspond
donc à la zone de grillage vaporisé par la circulation du courant. Ce processus se traduit par la
formation d’un arc circulaire creux dont le rayon augmente avec le temps et consomme la protection
métallique due aux hautes températures générées par l’arc. Durant son expansion, l’arc génère une
intense lumière qui renseigne sur sa forme et sa taille, qui peuvent ainsi être évalués directement à
partir des images prises durant l’essai. L’endommagement final, mesurable sur les plaques après
impacts, renseigne également sur le rayon final supposé de l’arc.
Figure 3.2: images de l’initiation de l’arc foudre et de son attachement sur la plaque composite. De gauche à droite,
chaque image est séparée de la suivante de 2 µs. la partie droite correspond au reflet de l’arc électrique dans la peinture
déposée en surface du matériau.
Ce scénario supposé d’augmentation du rayon d’arc est implémenté dans un modèle analytique qui
permet de calculer l’augmentation de la surface endommagée en fonction du temps et de la
température. Une hypothèse exploratoire est faite de ne considérer que les effets Joule comme
source d’énergie déposée sur le matériau impacté. La protection métallique est prise en compte,
puisque sa consommation par l’arc détermine la surface totale endommagée. Cette protection est
définie par son épaisseur e, sa densité de surface δ, sa densité volumique ρ et sa conductivité
électrique σ. L’objectif ici est de voir si il est possible de prédire l’endommagement de la protection
et donc du composite situé en dessous et ainsi de valider l’hypothèse qui est faite sur l’expansion du
rayon d’arc en fonction du temps.
La figure 3.3 compare les zones endommagées obtenues avec le calcul analytique et celles mesurées
lors des essais foudre. Le modèle fournit des résultats satisfaisants et se révèle capable d’encadrer
pour deux extrêmes de températures les résultats expérimentaux. Il est ainsi possible de reproduire
une surface endommagée par l’arc foudre, ce qui valide l’hypothèse d’un rayon d’arc croissant en
fonction du temps et de la surface de protection métallique consommée par effet Joule.
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Figure 3.3: surface endommagée en fonction du temps. Les points expérimentaux sont évalués à partir des images
obtenues avec les caméras rapides.
3.1.3 Modélisation des forces magnétiques
La foudre est un phénomène complexe qui met en jeu de nombreuses composantes physiques. Le
choix qui est fait pour cette étude est de séquencer l’approche de la foudre en ne considérant qu’une
seule des composantes présumées de la foudre pour essayer de quantifier son importance dans la
contribution globale. Cette influence est testée sur différents paramètres tels que, notamment, les
déplacements et vitesses face arrière.
Deux contributions peuvent être à l’origine de la contrainte qui vient fléchir la plaque lors de l’impact
foudre [6-8] : l’onde de choc générée par l’effet Joule dans la protection déposée en surface du
matériau composite et dont l’effet est encore augmenté par la présence de peinture qui tend à
confiner l’explosion générée en surface [8] ; et les forces de Laplace générées par la circulation du
courant foudre dans la protection et le champ magnétique ainsi créé.
Afin d’essayer de quantifier l’importance de ces deux contributions physiques, deux types
d’éprouvettes ont été fabriquées et soumises à des chocs foudres. Les premières sont des
éprouvettes carrées en CFRP (Carbon Fiber Reinforced Plastic) de dimensions 450x450 mm² et avec
un drapage quasi-isotrope [45°/0°/135°/90°]s. Les plaques sont constituées de 8 plis pour une
épaisseur totale de 1,44mm. Les échantillons sont fixés par douze boulons disposés selon un cercle
de diamètre Φ370 mm. Le second type d’éprouvette possède les mêmes caractéristiques que les
précédentes mais sont percées en leur centre d’un trou (Φ 6mm) dans lequel l’électrode délivrant le
courant foudre est placée. Ce dispositif est sensé supprimer autant que possible les interactions
entre l’arc et la surface et ainsi générer la circulation du courant directement dans le matériau,
évitant ainsi l’explosion de surface. Cependant, lors des essais, les déplacements observés entre les
deux types d’éprouvettes restaient très semblables. Les caméras haute définition ont montré que
dans les cas avec les éprouvettes percées, l’explosion de surface avait toujours lieu et contribuait à la
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déflection des échantillons. Ceci suggère donc que la première contribution, le choc du à
l’attachement de l’arc, n’était pas prédominant dans le phénomène d’endommagement lié à la
foudre. Il reste alors deux contributions potentiellement responsables de la déflection des panneaux
foudroyés : les forces de Laplace et l’effet Joule induit par l’explosion de surface.
Il est possible de déterminer analytiquement la contribution magnétique due aux forces de Laplace
et de l’implémenter dans un modèle numérique sous forme de pression externe de surface
appliquée sur une plaque, permettant ainsi de calculer la déflection associée à cette contribution
physique de la foudre et de la comparer aux résultats d’essais.
μ
i(t) 2
)
r
P(r) = 2π0 × (
(Eq. 3.1)
L’hypothèse suivante est faite : l’ensemble du grillage métallique est supposé conducteur et le
champ magnétique responsable des forces de Laplace est présent sur l’ensemble de la surface des
échantillons. La pression issue des forces de Laplace est appliquée sur le même modèle simplifié
évoqué en section 3.1.1.
Figure 3.4: Comparaison des déplacements mesurés et calculés en fonction du temps.
La figure 3.4 présente les résultats de déplacement obtenus par la modélisation numérique et
comparés aux résultats d’essais pour deux VISARs (visar 1 au centre de la plaque et visar 2, placé à 25
mm en dessous du premier). La simulation numérique donne des résultats semblables en termes de
forme et d’amplitude comparés aux essais foudre. L’amplitude totale est cependant environ deux
fois plus petite dans le cas de la simulation (qui ne prend en compte que les forces magnétiques)
comparées aux résultats foudre (qui prennent en compte l’ensemble des contributions physiques de
la foudre). Ce résultat prouve que les forces de Laplace semblent être un contributeur non
négligeable de la foudre et permet une meilleure compréhension de l’influence de chaque
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composante ainsi que de leur importance relative dans le processus d’endommagement lié à la
foudre.
3.1.4 Analyse des dommages foudre
En plus des résultats de déplacement et vitesse face arrière obtenus lors des essais foudre, des
analyses post-mortem sont effectuées sur les éprouvettes impactées : destructives (coupes
microscopiques) et non destructives (analyses ultrason). Ces analyses permettent d’obtenir des
informations sur les dommages générés par la foudre à la surface et à l’intérieur des stratifiés. Les
observations de ces différentes analyses post-mortem permettent d’identifier deux types de
dommages situés dans deux zones distinctes du matériau composite. Premièrement, en surface (ce
qui comprend la peinture, la protection et éventuellement le ou les premiers plis), les dommages
observés sont principalement de nature thermique : sublimation de la peinture, de la protection et
de la résine du premier pli, fibres de carbones mises à nues et brulées (figure 3.5).
Figure 3.5: Dommages dus à la foudre après impact
Le second type de dommage, révélé notamment par l’analyse ultrason et les coupes micro, est
localisé dans l’épaisseur du matériau. Ces dommages comprenant principalement des délaminages
importants, des ruptures de fibres et de matrices, rappellent fortement les faciès d’endommagement
observés lors d’essais d’impacts mécaniques, types petits chocs ou impact basse et moyenne vitesse
(figure 3.6).
Figure 3.6: Comparaison des dommages à cœur pour la foudre et des essais mécaniques [12]
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Des coupes microscopiques effectuées sur des plaques impactées montrent des faciès différents
dans les zones où la peinture et le grillage ont été sublimés et dans l’épaisseur du stratifié, qui
laissent penser que ces derniers ne sont pas d’origine thermique. Cependant une analyse
approfondie reste nécessaire.
La différence observée entre les dommages dits de surface et ceux présents à cœur suggère que ces
dommages sont d’origines différentes et que l’on peut donc dé-corréler leurs origines et les causes
de leur formation.
3.2 Méthode de travail
La foudre est un problème multi-physique qui met en jeu de nombreuses composantes : mécanique,
thermique, électromagnétique. Ces composantes peuvent également être couplées entre elles, et il
est encore impossible de quantifier leur importance relative.
L’étude post-mortem de plaques composites foudroyées en laboratoire a révélé que les dommages
observés à la surface des éprouvettes et ceux obtenus dans l’épaisseur du stratifié sont de natures
différentes et n’engagent pas les mêmes mécanismes d’endommagement. Il a été montré que les
dommages à cœur présentaient des ressemblances évidentes avec ceux obtenus lors d’impacts
mécaniques traditionnels tels que tours de chute ou canon à air comprimé. Ces dommages semblent
donc avoir été générés par une contribution purement mécanique. Ce sont de plus les dommages les
plus graves pour la structure composite car ils diminuent l’intégrité structurelle du stratifié et sont
responsables d’un abattement de ses propriétés mécaniques.
Le but du travail de thèse est la compréhension des phénomènes d’endommagement liés à la foudre.
Cette compréhension est nécessaire afin d’optimiser les protections déjà mises en place sur les
structures primaires aujourd’hui fabriquées à partir de matériaux composites. La séparation des
dommages observés à cœur et en surface, de par leur nature et leur localisation, est à la base du
travail présenté dans ce rapport, de même que le rapprochement qui est fait entre dommages à
cœur après un impact foudre et dommages mécaniques standards.
Ainsi, si les dommages dus à la foudre sont d’origine mécanique, il doit être possible de reproduire
ces dommages en utilisant des moyens purement mécaniques, également. Une hypothèse forte est
faite de dé-corréler les évènements de surface pour se concentrer sur les effets de ces évènements
dans le matériau. Les évènements ayant lieu en surface sont responsables de la poussée mécanique
qui engendre les dommages observés à cœur, cependant la source de cette poussée reste
indéterminée et décision est prise de s’affranchir de son origine pour se concentrer sur les
dommages qui en résultent, leur compréhension et leur reproduction. Deux hypothèses de travail
ont été formulées:

Hypothèse 1 : il est possible de dé-corréler les dommages de surface des dommages à cœur.
L’accent est mis sur les dommages situés dans l’épaisseur du composite et sur leur origine
mécanique.
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
Hypothèse 2 : il est possible de reproduire mécaniquement les déplacements face arrière
générés par un choc foudre et les dommages qui leur correspondent.
Le travail présenté dans la suite de ce rapport s’attache à trouver une équivalence entre la foudre et
une formulation mécanique du phénomène. La mise au point de cette équivalence se traduit par la
création d’un essai mécanique équivalent qui permettra de valider l’hypothèse qui est faite. Ces
essais n’ont pas pour but de reproduire les dommages de surface mais seulement ceux apparaissant
dans le cœur du composite (délaminage).
L’équivalence mécanique et les essais associés seront comparés aux résultats foudre par plusieurs
paramètres : les déplacements et vitesses face arrière, ainsi que les dommages générés par les deux
types d’impact. Des modèles numériques associés aux essais seront également développés et
pourront servir, une fois validés, à la prédiction des dommages, à la réduction du nombre d’essais
foudre et aux conseils sur les paramètres pouvant influer sur les dommages (matériau, empilement,
protections, etc…).
3.3 Équivalence foudre/mécanique
Approche préliminaire
Le but de l’étude présentée ici est de définir un essai d’impact équivalent à la foudre, en ce qu’il
permet de reproduire la vitesse et la déflection observées en face arrière lors des essais foudre. Un
impact mécanique est défini par un projectile, de masse m, lancé contre un échantillon à une vitesse
donnée. En première approximation, le projectile est choisi sphérique, en acier indéformable et
projeté contre l’échantillon à 90° ; ainsi l’impact est défini par le rayon (ou la masse) de l’impacteur
et sa vitesse de projection. Les paramètres r et v sont déterminés par une méthode inverse.
La similitude entre les dommages dus à la foudre et à un impact mécanique ont conduit à proposer
de reproduire les dégâts issus d’essais foudre par des essais virtuels d’impact en laboratoire. Le
travail présenté a pour objectif de mettre en place une démarche de calibrage d’un essai mécanique
considéré équivalent à un impact foudre, et à développer des modèles numériques de substitution.
Notre relation d’équivalence se fait via l’impulsion transférée k (en N.s). Le but de cette équivalence
est de définir des conditions d’impact mécanique reproduisant des vitesse et déflection au centre du
panneau arrière aussi proches que possible des données mesurées lors des impacts foudre. Dans une
étude précédente, un certain nombre d’essais foudre avec des configurations différentes ont été
réalisés en laboratoire. Lors de ces essais le déplacement et la vitesse ont été mesurés au centre des
faces arrière des panneaux impactés. Les mesures ont été effectuées par des VISARS. Ces mesures
seront à la base de l’équivalence que nous voulons établir entre le choc mécanique et le choc foudre,
via l’impulsion transférée. Par définition on peut écrire que l’impulsion transférée à la plaque par un
projectile équivaut à l’intégration d’une pression appliquée sur un temps et une surface. Il vient donc
la relation suivante :
k = ∫ Fdt = ∫
d(mv)
dt
dt
= m×v
(Eq. 3.2)
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En première approximation nous choisissons d’utiliser un projectile sphérique indéformable en acier
venant impacter l’échantillon, ainsi les impacts mécaniques (canon à air comprimé, moyenne vitesse)
sont entièrement déterminés par la masse et la vitesse du projectile. L’étude des déflections en face
arrière des essais foudre montre que ces déplacements atteignent un plateau d∞ que l’on peut relier
à l’impulsion k via la masse surfacique μ et la rigidité de flexion équivalente de la plaque, D.
d∞ =
k
(Eq. 3.3)
8√μD
Avec le calcul de d∞ (Eq.3.2), ces valeurs expérimentales et les mesures de pics de vitesse obtenus
lors des essais foudre, nous sommes capables de calculer l’impulsion k ainsi que la masse associée à
chaque cas foudre sélectionné. Le tableau 3.2 rassemble pour différents cas foudre, présentés
tableau 3.1, les valeurs de déplacements maximum mesurés lors des essais, le temps d’équivalence
appelé « temps court » et la valeur de d∞. On remarquera que plus la vitesse de déplacement face
arrière est élevée, plus la durée d’équivalence mécanique est faible et importante la valeur du
plateau de déplacement. Afin de déterminer les couples {masse, vitesse} équivalents à la foudre,
deux méthodes sont possibles : fixer la vitesse du projectile (égale à celle mesurée pour l’essai
foudre) et déterminer la masse ou à l’inverse, fixer une masse de projectile et ajuster la valeur de
vitesse d’impact. Dans la suite de ce rapport, la seconde méthode sera appliquée.
Essai foudre #
1
2
3
4
Vmax (m/s)
29
10.6
57
83.5
Temps courts (µs)
47
34
38
11.5
d∞ (µm)
2800
2000
3500
4400
K (N.s)
0.228
0.16
0.285
0.359
m (g)
7.8
150
5
4.3
Table 3.1: Valeurs des paramètres d’équivalence pour les essais foudre 1 à 4
Des modèles numériques ont été réalisés avec ABAQUS explicite afin de reproduire au mieux les
conditions d’essais foudre et calculer la déflection pour nos différents cas avant de les comparer aux
valeurs d’essais choisis figure 3.7. Des premiers modèles simples sont réalisés, dans le but de
déterminer les caractéristiques du projectile permettant de reproduire la déflection de la plaque due
à la foudre. Pour le composite, des éléments de type coque (S4R) homogénéisés sont utilisés sous
hypothèse de petites déformations avec utilisation d’un maillage raffiné de l’extérieur vers le centre
de la plaque. Le projectile est modélisé sous la forme d’un solide indéformable (rigid body) non
maillé, disposé au centre de la plaque à impacter, et un déplacement autorisé dans la direction
normale à la plaque. Une masse fixe et une vitesse initiale lui sont assignées.
Les modèles simples montrent une bonne corrélation avec les essais foudre aux temps dits ‘temps
courts’ (50µs) et ‘de stabilisation’ (<100µs) dans tous les cas ainsi que pour des temps allant jusqu’à
300µs pour les cas non peints. Le modèle échoue à reproduire le plateau de déflection pour les cas
peints.
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Fig. 3.7 : Comparaison des déflections en face arrière pour les essais foudre et la simulation numérique
Cela prouve dans un premier temps que le type d’impact choisi permet effectivement de reproduire
de façon satisfaisante la déflection d’une plaque soumise à un impact foudre et donc l’intérêt de la
méthode développée ici. Ceci valide l’hypothèse 2 : il est possible de reproduire mécaniquement les
déplacements face arrière dus la foudre. Cela établit ensuite un cadre de validation de la méthode, et
met en évidence une première forme d’interaction entre les phénomènes de surface et les
dommages internes que nous avons associés à la partie mécanique du chargement. Les simulations
numériques, en fournissant des résultats satisfaisants comparés à la foudre, ont montré que la
méthode d’équivalence était valide et pouvait être appliquée à des essais réels.
Essai foudre
#
101
102
103
107
110
Etat de surface
(peinture + protection)
ECF195-P160
ECFP195-P200
ECFP195-P50
ECF195-P160 + épargne
peinture disque ф de 6 mm
ECF195-P200 + épargne
peinture disque ф de 12 mm
Vitesse d’impact
calculée (m/s)
72
130
65
81
Masse du
projectile (g)
4
4
4
4
87.5
4
Table 3.2: Caractéristiques du projectile équivalent pour les essais foudre choisis.
Afin de valider totalement cette méthodologie d’équivalence et son cadre de validité, des essais
d’impact ont été menés en laboratoire. Ils sont listés au tableau 3.2., est présentés ci-après.
4. Essais mécaniques équivalents
Des essais d’impacts équivalents ont été menés à l’Institut Clément Ader. Pour ces essais, un canon à
air comprimé a été utilisé (figure 4.1). Ce montage et ce canon permettent de projeter des projectiles
dans la gamme de vitesse 50 à 200m/s. L’hypothèse de départ de notre méthode spécifiait
l’utilisation d’un projectile indéformable, aussi des billes en acier de masse 4 grammes, déjà
disponibles avec le canon, ont été utilisées. L’ensemble des cas foudre et de leurs impacts
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mécaniques associés sont présentés dans le tableau 4.1. Les échantillons à impacter ont également
été fabriqués à l’ICA. Il s’agit de plaques minces de plis unidirectionnels CFRP de 400*400 mm².
Figure 4.1: Canon à air comprimé
Essais
mécaniques
associés
Vitesse
projectile
attendue
(m/s)
Vitesse
projectile
réelle (m/s)
2
72
75
4
72
75
7
72
70
1
120
124
3
60
65
6
60
50
107
8
80
81
110
5
85
87.5
Essais foudre
#
101
102
103
Table 4.1: Vitesses d’impact des essais mécaniques équivalents
Le montage a été dimensionné afin d’être le plus représentatif possible des essais foudre menés
précédemment. En particulier, un anneau rigide a été fabriqué pour maintenir les plaques lors des
essais d’impact, en suivant le schéma de fixation par douze boulons disposés en horloge sur un cercle
de diamètre Φ370mm des essais foudre.
Le montage est équipé d’une caméra rapide qui mesure la vitesse du projectile en sortie de canon.
Trois capteurs de force sont disposés entre l’anneau de fixation et le bâti principal afin de mesurer
l’effort induit par le choc sur la plaque à des fins de validation des modèles numériques. Enfin, deux
capteurs de déplacement laser (Keyence 20kHz sans contact) sont disposés à l’arrière de la plaque
afin de mesurer sa déflection au cours du temps.
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4.1 Déplacements face arrière
Les résultats de déplacements face arrière obtenus lors des essais mécaniques sont comparés avec
les résultats foudre. La figure 4.35 présente la comparaison pour le cas 103-3 (65 m/s).
3500
Displacement (µm)
3000
2500
2000
1500
lightning strike 103
1000
500
0
Time (µs)
0
50
100
150
200
250
Figure 4.2: Déplacement temporel pour l’impact mécanique à 65 m/s (ligne pleine) et le choc foudre (ligne pointillée)
L’essai foudre 103-3 était une éprouvette entièrement peinte. La pente de la courbe de déplacement
jusqu’à 50 µs (« temps courts ») présente une bonne concordance avec les essais mécaniques, les
déplacements sont du même ordre de grandeur. Entre 50 µs et 100 µs, les déplacements sont
toujours du même ordre de grandeur pour les deux essais et on observe un même changement de
pente dans les deux cas avec des amplitudes variables (décélérations différentes). Apres 100µs
(« temps de stabilisation »), les deux courbes s’éloignent et l’essai mécanique tend à décroitre alors
que la foudre voit ses déplacements augmenter à nouveau avec une vitesse plus faible. Il faut noter
que l’échelle de temps pour la comparaison foudre/mécanique est très courte comparée au signal
complet obtenu lors des essais.
Pour les différents cas foudre testés, les résultats des impacts équivalents sont rassemblés dans le
tableau 4.2 : déplacements et vitesses à 50µs (instant moyen de vitesse maximum obtenu lors des
essais foudre). Ces valeurs sont obtenues directement à partir des courbes de déplacements et
vitesses face arrière des deux essais en extrayant les résultats à 50µs et en mesurant la tangente à
cet instant pour obtenir la vitesse. Pour le cas 101, les prédictions de déplacement présentent un
écart inférieur à 1% pour les deux cas à 75 m/s et montent jusqu’à un écart de 9.4% pour l’essai à
70m/s. On mesure un écart d’environ 30% pour les cas foudre 103 et 107 ce qui reste acceptable.
L’essai mécanique équivalent échoue à reproduire le plateau de déplacement. Ce résultat était
cependant attendu car prédit par le modèle coque présenté au chapitre 3, déjà incapable de
reproduire le comportement à long terme des essais foudre. Cette difficulté rencontrée par le
modèle équivalent est mise sur le compte des phénomènes électrothermiques de surface, dû à la
présence de peinture et de protection métallique [43], et qui sont retardés par rapport à l’instant de
l’impact. Ce retard ne peut être représenté par les processus mécaniques très rapides mis en jeu ici.
Les prédictions sur les vitesses face arrière sont également satisfaisantes pour le cas 101 bien qu’une
266
__________________________________________________________________________________
erreur de 32% soit obtenue pour l’essai 103. Cette différence est attribuée aux effets de confinement
liés à l’épaisseur de peinture et à l’explosion de la protection.
Essais foudre
Déplacement
foudre max.
(µm)
101
2839
103
1810
107
110
2626
2605
Déplacement
essai
mécanique
(µm)
(4) 2850
(2) 2843
(7) 2560
(6) 2153
(3) 1863
(8) 3630
(5) 2486
Différence
Relative
0.8%
0.5%
-9.4%
+18.9%
+2.9%
+38.2%
-4.6%
Vitesse
foudre
(m/s)
Vitesse essai
mécanique
(m/s)
Différence
Relative
33.6
22.9
35.3
33.8
31.2
27.3
15.2
-10.1%
-37.8%
-5.8%
+32%
+21.8%
+9.6%
-59.1%
37.4
25.9/30
32.3
37.2
Table 4.2: Déplacement et vitesse à 50µs
A partir de ces comparaisons, il est conclu que l’essai mécanique équivalent fournit des valeurs de
déplacements et vitesses face arrière satisfaisants à la fois en amplitude et évolution temporelle,
comparés aux essais foudre pour des temps courts jusqu’aux temps de stabilisation. Dans le cas de
panneaux foudre peints avec de faibles épaisseurs ou dans le cas où la peinture est complètement
brûlée durant l’essai, le modèle mécanique reproduit correctement les courbes de déplacements et
vitesses aux temps courts également. Comme l’équivalence se fait sur la comparaison de ces deux
quantités cinématiques et sur les durées dite « temps courts », la méthode est donc validée par la
campagne expérimentale.
4.2 Surface délaminée totale
Afin d’aller plus loin dans la validation de la méthode d’équivalence et dans l’optique d’identifier les
limites de la méthode, une analyse ultrason des plaques impactées en essais foudre et mécaniques
équivalents est menée. Le tableau 4.3 présente les surfaces totales délaminées pour chaque essai
foudre et les essais mécaniques associés, ainsi que la différence relative entre les deux essais.
Les essais foudre 102 et 107 et les essais mécaniques associés fournissent des valeurs concordantes
de surface totale endommagée, avec un écart relatif de 2%. L’échantillon 103 ne présente aucun
dommage foudre. Ce résultat est concordant avec la très faible épaisseur de peinture déposée sur la
plaque, en effet il a été montré [128] qu’une importante couche de peinture générait des dégâts plus
importants. L’essai mécanique associé fournit quant à lui une zone endommagée réduite.
Essais foudre
Essais Méca.
associés
Vitesse
projectile
(m/s)
4
75
2
75
Surface
Surface
délaminée
délaminée
foudre (mm²) Méca. (mm²)
Différence
Relative
1626
-29%
1184
-49%
101
2321
267
__________________________________________________________________________________
102
7
70
1
124
6
50
103
5836
1019
-56%
5732
-2%
223
-
630
-
0
3
65
107
8
81
1711
1746
-2%
110
5
87.5
120
3361
-
Table 4.3: Surface délaminée pour les essais foudre et leurs essais mécaniques associés
4.3 Faciès de rupture
Après avoir comparé les surfaces totales délaminées, une étude est menée sur les endommagements
observés grâce aux C-scans. Les échantillons ont été analysés des deux côtés pour les essais
mécaniques et seulement côté opposé à l’impact pour les essais foudre. En effet, les fibres brulées et
mises à nues viennent perturber le traitement du signal ultrason et empêchent d’obtenir des
données. La figure 4.3 présente la méthode utilisée pour analyser les images obtenues. L’échelle de
gauche représente la profondeur de l’interface depuis le côté opposé à l’impact, cette échelle
commence à 0 et s’échelonne jusqu’à 1.44mm (épaisseur de la plaque composite). L’échelle de
droite donne les différentes interfaces et les orientations des plis.
8
7
6
5
4
3
2
1
Figure 4.3: Double échelle pour l’analyse ultrason: à gauche l’épaisseur, à droite l’orientation des plis
Traditionnellement, en impact mécanique, un délaminage apparait à l’interface entre deux plis
d’orientations différentes et ce délaminage est plus long dans la direction du pli inférieur [72, 73,
104]. Plusieurs éludes [107, 108] ont montré que pour des stratifiés confectionnés à partir de plis
unidirectionnels, des fissures matricielles apparaissent en amont des délaminages et semblent diriger
l’ouverture longitudinale et la fermeture latérale de ces dommages. Les fissures matricielles
apparaissent parallèlement à la direction des fibres et se propagent jusqu’à atteindre l’interface
entre deux plis d’orientations différentes avant de se propager dans la direction du pli inférieur
268
__________________________________________________________________________________
(figure 4.4). Cette orientation préférentielle est due au fait que l’énergie délivrée par le choc est
absorbée par la rupture du lien le plus faible, ici l’interface entre deux plis d’orientations différentes,
ce qui mène à l’initiation, ou la fermeture, d’un délaminage [48, 109].
Figure 4.4: Orientation préférentielle des délaminages après un choc
La comparaison des scans foudre et mécanique présente des différences entre les deux essais.
Premièrement, les formes générales des dommages sont très différentes entre les deux essais
comme le montrent la figure 4.5 et le tableau 4.3. De manière générale, bien que la surface totale
délaminée soit du même ordre de grandeur, la foudre tend à provoquer des dommages plus
importants que les impacts mécaniques.
Dans les cas foudre, les dommages représentent une forme particulière en deux zones centrées sur
un large délaminage orienté à 90° (cercle rouge) entre les plis 3 et 4 (interface -45/90). Les essais
mécaniques, dans leur cas, présentent des caractéristiques classiques des impacts :
-
Une forme dite « en aile de papillon » avec un axe de symétrie central qui est l’axe de
déplacement du projectile. Les ailes du papillon sont séparées par deux fissures bien visibles
Les ailes sont bornées par deux séries de fissures orientées selon les fibres du pli inférieur (en
longueur) et du pli supérieur (profondeur).
Une large écharde est visible en face arrière, orientée selon la direction du dernier pli (45°). Ce
dommage est typique des impacts haute vitesse et ne se retrouve qu’assez peu dans les scans
d’endommagement foudre. Le large papillon avec ses ailes de part et d’autre d’un large délaminage à
90° est également un faciès couramment observé dans ce genre d’impact. Malgré leurs différences,
les deux essais partagent des interfaces délaminées communes : les interfaces 2 et 3 respectivement
orientées à -45° et 90°.
Figure 4.5: Analyse ultrason pour l’essai foudre 101 (gauche) et l’essai mécanique associé (droite) impact à 75 m/s
269
__________________________________________________________________________________
4.4 Distribution des dommages dans l’épaisseur
La distribution des dommages dans l’épaisseur du stratifié est également différente pour les deux
impacts foudre et mécanique. Pour les deux essais, la figure 4.6 présente la position du délaminage
le plus éloignée de la face impactée, pour un essai à 75 m/s.
Figure 4.6: Position du délaminage le plus éloigné de la surface d’impact, foudre (gauche) et impact mécanique (droite)
Dans les cas foudre, les dommages semblent toujours s’arrêter à la double interface centrale
orientée à 90°. Les dommages sont ainsi confinés à la première moitié de l’épaisseur (figure 4.6). Sur
la figure 4.7, on peut voir que la peinture et la protection métallique (pointillés blancs) ont été
sublimées par l’impact de la foudre. La flèche blanche indique la position du délaminage le plus
important, entre les plis 3 et 4.
Les impacts mécaniques, quant à eux, génèrent des dommages dans toute l’épaisseur du stratifié, en
suivant un cône hélicoïdal classique qui se termine par une large écharde au niveau du dernier pli. La
différence entre les dommages générés par les deux essais est principalement due à la manière dont
l’énergie est déposée dans le matériau. Durant les essais foudre, une importante partie de l’énergie
amenée par l’arc électrique est utilisée pour endommager la protection et la peinture ou dispersée
dans l’air. Lors des impacts mécaniques, toute l’énergie cinétique du projectile est transférée au
matériau, générant différents mécanismes d’endommagement. De plus dans le deuxième cas,
l’énergie est déposée sur une très petite surface, de manière très localisée, ce qui vient augmenter la
propagation d’une onde de choc dynamique en compression dans la profondeur du matériau alors
que dans le cas de la foudre, cette énergie est distribuée aussi bien temporellement que
spatialement à la surface de l’échantillon. Le chargement foudre est donc appliqué sur une surface
radialement plus large et plus petite dans la profondeur.
Figure 4.7: Coupe microscopique de l’essai foudre 101
L’analyse par ultrason permet également d’extraire des données de distribution des dommages dans
l’épaisseur et par interface. La figure 4.8 présente des histogrammes montrant la position relative de
chaque délaminage dans l’épaisseur des stratifiés pour les essais foudre et mécanique 101 et 107.
La position zéro correspond à la face opposée à l’impact de l’échantillon. Les histogrammes mettent
clairement en évidence la différence de distribution des délaminages entre les deux essais et
270
__________________________________________________________________________________
confirment que la foudre tend à générer des dommages jusqu’à la moitié seulement de l’épaisseur
du matériau quand les impacts foudre en génèrent à chaque interface.
700
Lightning test 101
600
Mechanical impact
500
400
300
200
100
0-0,1
0,1-0,2
0,2-0,3
0,3-0,39
0,39-0,46
0,46-0,53
0,53-0,61
0,61-0,69
0,69-0,76
0,76-0,84
0,84-0,92
0,92-0,99
0,99-1,07
1,07-1,14
1,14-1,22
1,22-1,29
1,29-1,37
1,37-1,45
1,45-1,52
1,52-1,6
0
Thickness (mm)
(a)
delaminated area (mm²)
800
Mechanical impact
700
Lightning test 107
600
500
400
300
200
100
0-0,1
0,1-0,2
0,2-0,3
0,3-0,38
0,39-0,48
0,48-0,57
0,57-0,66
0,66-0,75
0,75-0,84
0,84-0,93
0,93-1,02
1,02-1,11
1,11-1,21
1,21-1,3
1,3-1,39
1,39-1,48
1,48-1,57
1,57-1,66
1,66-1,75
1,75-1,84
0
thickness (mm)
(b)
Figure 4.8: Histogrammes des dommages dans l’épaisseur pour les plaques foudre 101 et 107 et leurs essais mécaniques
associés.
5. Modèle numérique équivalent
5.1 Modèles numériques
La stratégie de modélisation numérique comprend une loi d’endommagement des plis composites
volumiques enrichie par l’introduction d’interfaces entre les plis, et le développement de modèles
mécaniques et de programmes utilisateurs numériques utilisant ces lois. Les interfaces,
endommageables, sont modélisées à l’aide d’éléments cohésifs, permettant la décohésion due aux
délaminages des plis par des ouvertures, et gérant les re-fermetures par contact.
271
__________________________________________________________________________________
5.2 Modèle d’endommagement des plis
Le modèle d’endommagement continu utilisé pour les plis composites est initialement développé par
Ilyas [4-8]. Ce modèle prend en compte la dégradation du matériau au travers de 5 critères de ruine à
effet cumulatif, et définis par les seuils {ri} i=1,5 suivants :
Rupture traction sens fibre
〈σ11〉 2
f1(σ, ω, r) = (
Xt
σ122 +σ132
) − r12
Sfs2
) +(
=0
(Eq. 5.1)
) − r22 = 0
(Eq. 5.2)
Compression sens fibre
〈−2σ11+〈−σ22−σ33〉 〉 2
f2(σ, ω, r) = (
2Xc
Compression sphérique
〈−σ11−σ22−σ33〉 2
f3(σ, ω, r) = (
Rupture cisaillement
〈σ22〉 2
f4(σ, ω, r) = (
Délaminage
Xc
Zt
〈−σ22〉 2
) +(
〈σ33〉 2
f5(σ, ω, r) = (
) − r32 = 0
3Zc
Yc
(Eq. 5.3)
σ12
2
σ23
2
) + (S12+〈−σ22〉tanφ) + (
) − r42 = 0
S23+〈−σ22〉tanφ
σ13
2
σ23
2
) + (S13+〈−σ33〉tanφ) + (
) − r52 = 0
S23+〈−σ23〉tanφ
(Eq. 5.4)
(Eq. 5.5)
Le paramètre tan φ prend en compte les effets de friction dus au cisaillement lorsque les plis sont
soumis à une compression sens transverse.
A partir de ces critères de ruine, 6 variables d’endommagement {di} i=1,6 sont calculées grâce à une
matrice de couplage {qij} et viennent abattre progressivement les rigidités élastiques du matériau.
Les variables {rj} j=1,5 telles que rj≥1 représentent la perte de potentiel mécanique associée au mode
de ruine considéré. Les variables d’endommagement {di} i=1,6 telles que 0d1 définissent quant à
elles l’état de dégradation d’un élément du pli consécutif à la perte cumulée des potentiels via la
fonction d’évolution ϕj. Lorsque d=0, l’élément concerné est intact et lorsque d atteint la valeur 1,
l’élément est totalement détruit. On peut ainsi lier les variables d’endommagement d et la matrice
de couplage comme suit:
𝒅̇ = [𝒒]𝝋̇
(Eq. 5.6)
Ou encore :
m
1
1r j  


i ( , ,  )   qij . j ( , ,  )   qij . 1  e m
 , rj  1
j 1
j 1


5
5
(Eq. 5.7)
272
__________________________________________________________________________________
La loi d’endommagement est donc une loi orthotrope à endommagement en (1-d) comme le montre
l’équation 5.8, et donc les dommages est obtenue par une loi dérivée.
1
0
(1−𝑑1 )𝐸11
𝜈
− 𝐸12
11
𝜈13
−𝐸
11
𝐶 −1 =
(
𝜈
𝜈
− 𝐸21
− 𝐸31
1
0
(1−𝑑2 )𝐸22
𝜈23
−𝐸
22
𝜈
− 𝐸32
33
1
0
(1−𝑑3 )𝐸33
22
33
1
0
(1−𝑑4 )𝐺12
(Equation.5.8)
1
0
(1−𝑑5 )𝐺23
1
0 )
(1−𝑑6 )𝐺13
Le processus de dégradation des rigidités structurelles étant à la fois conséquence et cause de la
perte de potentiel au cours du temps, la matrice des contributions {qij} conditionne également les
couplages micro-micro des modes de ruine entre eux, et micro-macro des modes de ruine du
matériau et de la structure. Afin d’évaluer la contribution des différents modes de ruine dans la
distribution des dommages de la structure foudroyée par rapport à une structure impactée,
différentes matrices de couplage {qij} ont été testées autour de deux variations fondamentales : prise
en compte ou non d’une ruine en compression sphérique significative dans les chargements très
rapides et localisés d’impact, prise en compte ou non de la perte de rigidité de la structure en flexioncisaillement en complément des éléments d’interface. Sont présentées ici la matrice complète
d’origine obtenue par Ilyas [48] (Eq 5.9, à gauche) et la matrice contenant les modifications pour une
structure foudroyée (Eq. 5 .10, à droite) [10].
(Eq. 5.9)
et
(Eq. 5.10)
5.3 Modèle de comportement des éléments cohésifs d’interface
Le modèle développé est basé sur un modèle cohésif classique en mode mixte bilinéaire (fig. 5.1). Le
modèle a été implémenté dans Abaqus Explicit 6.12. Le principe repose sur le calcul des
déplacements relatifs entre couches. Le comportement en ouverture est piloté par une loi élastique
linéaire en compression et élastique non linéaire en traction et en cisaillement.
273
__________________________________________________________________________________
Figure 5.1. Loi de comportement des éléments cohésifs.
Le déplacement des nœuds initiant le délaminage est déterminé en fonction des déplacements
critiques en mode I et en mode II (Eq. 5.11 et 5.12). Le déplacement à rupture est donné en fonction
du déplacement critique et des taux de restitution d’énergie en mode I et II (Eq. 5.13).
1+β2
δ0= δI0 δII0 √(δII0 )2 +(βδI0 )2
δr =
β=
2(1+β2 )
δII
δI
δ0
(Eq. 5.11)
1
[(k
n
GIc
σ
α
) +
α −α
k β2
( Gs ) ]
IIc
(Eq. 5.12)
σ
; δI0 = kn ; δII0 = ks
n
(Eq. 5.13)
s
Le terme de couplage mode I/mode II est donné par β, rapport entre l’ouverture en mode II et
l’ouverture en mode I. Quand β tend vers zéro, le modèle décrit le mode I pur. Pour le mode II pur,
une valeur de β très grande est choisie; δ0 et δr sont alors respectivement égaux à δII0 etδIIm . Les
raideurs en traction et en cisaillement des éléments d’interfaces sont respectivement k n et k s . σn
et σs sont les contraintes à rupture de l’interface en traction hors plan et cisaillement.
5.4 Modélisation numérique
Le modèle numérique utilisé sous Abaqus-explicite est un modèle volumique (3ddl par nœud, 1 point
d’intégration par élément) représentant chaque pli par 1 couche d’éléments. Le maillage est raffiné
au centre de la plaque (Fig.5). Chaque interface entre les plis d’orientations différentes est modélisée
par des éléments cohésifs (COH3D8). Ces éléments sont supprimés lorsqu’ils atteignent leur valeur
de déplacement à rupture. La plaque est fixée sur un cercle de diamètre 370mm. Un contact sans
frottement est défini entre le projectile et la plaque. Le projectile est un volume 3D positionné au
centre de la plaque avec une vitesse initiale et les caractéristiques du matériau acier élastique. Les
propriétés matériaux et les paramètres de la loi cohésive sont répertoriés dans le tableau 5.1.
On teste les deux matrices de couplage q1 et q2 (Eq.5.9 et Eq.5.10). La matrice q2 ne prend pas en
compte la variable d’endommagement d3 qui correspond à la compression sens transverse. La
variable d’endommagement d5 qui abat le module de cisaillement transverse G23, n’est pas
désactivé. La conservation de cette variable d5 permet la propagation de microfissures via la matrice
de couplage avec l’évolution des potentiels de ruine, ce qui n’est pas possible si elle était désactivée,
et permet donc une meilleure propagation des délaminages [5]. Ceci est illustré sur la figure 5.4 qui
274
__________________________________________________________________________________
présente le délaminage obtenu pour l’interface 4 (90/-45) pour les deux matrices de couplage.
L’utilisation de la matrice q2 (fig5.4 (b)) donne une orientation du délaminage plus cohérente. La
matrice de couplage q2 est par suite retenue dans notre loi d’endommagement et dans nos futurs
calculs.
E11
E22
E33
ν12
ν13
ν23
G12
G13
G23
165 GPa
7.6 GPa
7.6 GPa
0,35
0,35
0,4
5.6 GPa
5.6 GPa
2.7 GPa
S11RT
S11RC
S22RT
S22RC
S33RT
S33RC
S12
S13
S23
0.065
GPa
0.065
GPa
0.06
GPa
2.2 GPa
1.2 GPa
0.06 GPa 0.28 GPa 0.06 GPa 0.7 GPa
Sfs
Sd
mi
ϕ
dmax
C1
εref
1.5 GPa
1
10
10
0,99
4,7
0,75
kn
ks
σn
σs
Gic
GIIc
α
60MPa
60MPa
500J/m²
1200J/m²
1.0
3
100N/mm
3
100N/mm
Tableau 5.1. Caractéristiques du matériau utilisé et des paramètres de la loi cohésive
(a)
(b)
Figure 5.2 : (a) modèle numérique; (b) localisation des éléments cohésifs
275
__________________________________________________________________________________
3500
deplacement (µm)
3000
2500
2000
1500
modele pli et interface endom loi1
1000
modele plis et interfaces endom loi2
500
0
0
50
100
150
temps (µs)
200
250
Figure 5.3 : Comparaison des deux matrices de couplage q1 et q2
(a)
(b)
Figure 5.4 : Comparaison des délaminages avec les matrices de couplage q1 et q2
5.5 Comparaison essais mécaniques –simulations d’impact
On compare ici l’historique des déplacements face arrière obtenus par essai d’impact, avec les
résultats de calcul d’un modèle avec endommagement volumique des plis seuls et la matrice q2, et
ceux d’un modèle avec endommagement des plis et des interfaces. Les deux modèles utilisent la
matrice de couplage q2. Les déplacements maximaux sont résumés dans le tableau 5.2 pour les trois
essais foudre de référence. La figure 5.5 illustre typiquement les graphes déplacement/temps, pour
un impact à 65 m/s. Dans tous les cas, le modèle sans interface se révèle un peu trop rigide, et sousestime d’environ 4% le déplacement face arrière. Le modèle avec interfaces est un peu plus souple.
L’écart de vitesse maximale est d’environ 10% à basse vitesse, et se réduit lorsque la vitesse d’impact
augmente. D’un point de vue général, les deux modèles approchent de façon correcte le
déplacement maximal face arrière obtenu lors des essais. On note en revanche que le modèle avec
endommagement volumique seul conserve une rigidité élastique résiduelle après impact supérieure
au modèle avec interfaces et à l’essai réel car le déplacement décroit et n’atteint pas la forme
plateau observable sur les autres graphes à 200µs. Dans les trois cas, on retrouve bien un plateau de
276
__________________________________________________________________________________
déflection environ 200µs après le pic de déflection pour les modèles avec interfaces. Les valeurs
plateaux restent un peu éloignées des valeurs réelles avec un écart de 10% à 20% environ (voir
tableau 5.3). Le modèle avec endommagement volumique et interfaces est utilisé pour la suite de
l’étude.
3500
Displacement (µm)
3000
2500
2000
1500
Shell model 65m/s
1000
500
3D model with ply damage and cohesive surface
65m/s
0
0
50
100
150
Time( µs)
200
250
300
(a)
4000
Displacement (µm)
3500
3000
2500
2000
1500
1000
Shell model 72m/s
500
3D model ply damage and cohesive surface
75m/s
0
0
100
200temps µs300
400
500
(b)
Figure 5.5: Comparaison des déplacements entre essais mécaniques et simulation pour les configurations 101-3 (a) et
103-2 (b).
Essais méca
Déplacement
max
essai
impact (µm)
Calcul
modèle
endom. vol. plis
(µm)
Ecart
modèle
vol / essai
Calcul modèle vol.
plis et interfaces
(µm)
1 : 65m/s
2 : 75m/s
3 : 81m/s
2663
3780
3863
2481
3442
-
-6.8%
-8.9%
-
2914
3542
3690
Ecart modèle
vol.
+
interfaces
/
essai
+9.4%
-6.3%
-4.5%
Tableau 5.2 : Ecarts relatifs essais/calculs numériques pour le déplacement maximal (en µm)
277
__________________________________________________________________________________
Essais méca
Déplacement
plateau
essai
impact (µm)
Calcul
modèle
endom. vol. plis
(µm)
Ecart
modèle
vol / essai
Calcul modèle vol.
plis et interfaces
(µm)
1 : 65m/s
2 : 75m/s
3 : 81m/s
2270
3000
3056
inexistant
inexistant
inexistant
-
2038
2368
2580
Ecart modèle
vol.
+
interfaces
/
essai
-10.2%
-21%
-15.6%
Tableau 5.3 : Ecarts relatifs essais/calculs numériques pour le déplacement plateau d∞ (en µm)
La surface délaminée dans le modèle volumique avec interfaces est la surface des éléments cohésifs
supprimés au cours du calcul comme illustré sur la figure 5.4 (érosion à rupture). Dans les essais et
dans les calculs la mesure de la surface totale délaminée est la surface d’une ellipse contenant ‘au
mieux’ la surface délaminée projetée dans le plan de la plaque.
Le tableau 5.4 présente les valeurs de surface totale délaminée pour les essais mécaniques et leurs
simulations numériques associées. Les résultats du modèle numérique présentent un écart maximum
de l’ordre de 20% pour l’impact à 65m/s, et l’écart augmente jusqu’à environ 30% à 81m/s.
Essais mécaniques
Surface délaminée essais
mécanique (mm²)
Surface délaminée
Simulation numérique (mm²)
Différence
Relative
2 (75m/s)
1184
1683
42%
3 (65m/s)
630
765
21,40%
4 (75m/s)
1626
1683
3,50%
6 (50m/s)
223
395
77%
7 (70m/s)
1019
1439
41.2%
8 (81m/s)
1746
1453
-16,80%
Tableau 5.4 : Ecarts relatifs essais/calculs numériques pour la surface délaminée (en mm²)
L’analyse des résultats et des différentes comparaisons montre que le modèle d’endommagement
retenu est à même de reproduire correctement les déplacements face arrière mesurés lors des essais
mécaniques menés en laboratoire, autant en terme de déplacement maximal (écart maximal de 10%)
qu’en terme de comportement général de la plaque et de valeur plateau (écart maximal de 20%). La
loi d’endommagement, couplée à des éléments d’interfaces endommageables, permet également
d’approcher de façon correcte la surface totale délaminée mesurée lors des essais avec un écart
acceptable de l’ordre de 20% pour les vitesses basses. On note que le modèle prédit de manière plus
précise le déplacement maximal pour des vitesses d’impact plus élevées, mais plus précisément le
déplacement plateau d∞ et la surface totale délaminée à des vitesses plus basses.
5.6 Distribution des dommages dans l’épaisseur
La distribution relative des délaminages dans les différentes interfaces est également un indicateur
important afin de vérifier la validité du modèle numérique. L’analyse plus précise des
endommagements numériques interface par interface montre que les éléments cohésifs restituent
278
__________________________________________________________________________________
de manière cohérente l’orientation des délaminages, avec une direction longue orientée selon le pli
inférieur de l’interface endommagée. Par exemple, concernant l’interface 90/-45 (élément cohésif 4),
l’orientation est à -45° (Fig.5.6). La cartographie interface par interface permet de retrouver des
faciès similaires, notamment la présence d’un important délaminage orienté à 90° et qui correspond
à l’interface 3 (-45/90). On retrouve également un dommage important à l’interface 4 (-45/90) ainsi
que l’écharde face arrière orientée à 45° observée dans les essais (Fig.5.6).
(a) Interface 4(90/-45)
(b) Interface 3(-45/90)
Figure 5.6 : (a) Délaminage dans l’interface 4 dans la simulation numérique d’impact à 81m/s, et (b) délaminage dans
l’interface 3 dans la simulation à 75m/s
La figure 5.7 présente les mesures expérimentales de délaminage de l’essai à 81 m/s comparé avec la
simulation numérique associée. Le modèle numérique et les essais présentent des délaminages
semblables tant en taille qu’en disposition. On retrouve pour les deux résultats la forme hélicoïdale
et des dommages qui grossissent à mesure que l’on s’éloigne de la surface d’impact.
700
3D numerical model 81m/s
600
500
400
300
200
100
0
1
2
3
4
Interface #
5
6
Figure 5.7: Distribution des délaminages dans l’épaisseur pour le modèle 3D final et l’essai associé
Le modèle 3D présente une distribution légèrement différente de l’essai mais fournit des tendances
concordantes au niveau de la distribution dans l’épaisseur. Comme mentionné précédemment,
l’analyse des C-scans doit être faite avec des précautions, voir l’Annexe B.
279
__________________________________________________________________________________
Le modèle a tendance à surestimer le délaminage total dans la majorité des cas. Ceci est attribué
principalement à une mauvaise estimation des pics de déplacements en face arrière, cependant le
modèle fournit une surface délaminée totale et une distribution dans l’épaisseur concordantes avec
les essais expérimentaux.
5.6.1
Comparaison des dommages foudre avec les résultats de simulation numérique
Les essais mécaniques ont été mis au point en parallèle d’un jeu de modèle numériques
équivalents, depuis un modèle simplifié coque jusqu’à un modèle 3D complexe capable de
reproduire les dommages créés durant un impact de projectile.
Après la validation des essais expérimentaux par comparaison avec les essais foudre, une seconde
validation est effectuée entre les essais foudre et les résultats numériques, en comparant les
mesures face arrière et les dommages dans les différentes interfaces. Le modèle 3D fournit des
informations supplémentaires sur les mécanismes d’endommagement et leur chronologie
d’apparition et permet ainsi une meilleure compréhension des différences observées précédemment
entre la foudre et les impacts mécaniques.
5.6.2
Déplacements face arrière
On compare les courbes de déplacement face arrière en fonction du temps pour les essais foudre et
mécanique et les simulations numériques (modèles coques et 3D complet). Les résultats sont
présentés figure 5.8 pour les essais foudre 101 et 103, respectivement impacts à 65 et 75 m/s.
La comparaison des résultats numériques avec la foudre montre que le modèle est capable de
reproduire correctement les déplacements foudre aux temps courts, comme prédit par le modèle
coque. Les résultats des modèles coque et 3D encadrent les courbes issues de la foudre, comme
montré sur la figure 5.8. Pour l’essai 103, le modèle numérique fournit un maximum de déplacement
légèrement supérieur à la valeur foudre (impact à 65 m/s) mais reproduit de façon adéquate la pente
de la courbe, notamment pour la simulation à 50 m/s, bien que dans ce cas le déplacement maximal
soit un peu faible.
Comparaison des dommages foudre avec les résultats de simulation numérique
280
__________________________________________________________________________________
Figure 5.8 : Comparaison des déplacements face arrière pour les essais foudre 103 et 101, le modèle numérique et l’essai
mécanique associé.
Pour l’essai 101 au contraire, le modèle numérique à 75 m/s donne d’excellents résultats tant sur les
déplacements que sur la vitesse, bien qu’une fois encore le modèle surestime la valeur de
déplacement maximum.
De manière générale, le modèle numérique reproduit correctement les déplacements aux temps
courts observés lors d’un coup de foudre mais surestime le maximum de déflection, là où il sousestime celui des essais d’impacts mécaniques. Comme attendu, les simulations ne permettent pas de
reproduire le comportement à long terme des panneaux fins impactés par la foudre, confirmant ainsi
la validité de l’équivalence sur des temps relativement restreints. Le tableau 5.5 rassemble les
valeurs de déplacement et vitesse à 50µs pour le modèle numérique et les résultats foudre.
Essais
foudre
Déplacements
foudre (µm)
Déplacements
modèle
numérique 3D
(µm)
101
103
107
2861
1810
2626
2749
2410
3030
Différence
Relative
Vitesse
max.
foudre
(m/s)
-3.17%
+33.1%
+15.4%
37.4
25.9
32.3
Vitesse
max.
modèle
numérique
3D (m/s)
32.5
28.8
36.6
Différence
Relative
-13.1%
+11.2%
+13.3%
Table 5.5 : Déplacements et vitesses à 50 µs
Les prédictions numériques de déplacements du cas 101 sont satisfaisantes, 4% d’erreur comparées
à la foudre pour l’essai à 75 m/s. La simulation équivalente à l’essai 107 fournit également des
résultats concordants, avec une différence relative de 15% seulement par rapport à la foudre.
Cependant, pour le cas 103, l’échantillon avec la plus faible épaisseur de peinture en surface, la
simulation numérique fournit un écart à l’essai d’environ 33%, concordant avec la surestimation des
valeurs de déplacements maximum observée précédemment.
En ce qui concerne les prédictions des vitesses en face arrière, la simulation pour le cas 101 est
satisfaisante, seulement 13% d’erreur, de même que pour les cas 103 et 107. Cette différence est
Déplacements face arrière
281
__________________________________________________________________________________
due à la présence d’une épargne de peinture au centre de la plaque, qui vient retarder les effets de
confinement due à la peinture mais n’invalide pas l’hypothèse que les dommages dans le cœur du
composite sont d’origines mécaniques. Ces résultats sont concordants avec les erreurs de
déplacements observées précédemment, l’effet de l’état de surface étant ainsi jugé responsable des
différences notées.
5.6.3
Etude des profils de déplacement face arrière
Afin de valider le modèle numérique, une nouvelle analyse est menée sur le comportement de recul
des plaques impactées par la foudre. Pour les essais foudre précédemment utilisés dans ce rapport,
les valeurs de déplacement et vitesse ont été mesurés à l’aide de capteurs de déplacements laser, dit
VISARs, qui sont des moyens de mesure ponctuels. Une campagne foudre ultérieure à celle utilisée
dans ce rapport a été menée, en utilisant un dispositif de stéréo-corrélation d‘images afin de
mesurer les déplacements sur une section entière de l’échantillon, couvrant ainsi une plus grande
surface.
Des échantillons standards de CFRP2 sont utilisés avec la même stratification que pour les essais
foudre 101, 103, 107 etc... Les plaques sont protégées à l’aide d’un ECF195 et peinte avec 300µm de
peinture. Un calcul avec le modèle coque permet d’établir rapidement la masse et la vitesse
permettant d’approcher la courbe de déplacement de l’essai foudre choisi. Le projectile de masse 2g
est ensuite intégré au calcul 3D et projeté sur la plaque numérique à une vitesse de 120 m/s.
4
Numerical
simulation
t=19µs
Numerical
simulation
t=34.3µs
Numerical
simulation
t=51.4µs
Numerical
simulation
t=68.5µs
Lightning test
t=22.86µs
3,5
Displacement (mm)
3
2,5
2
1,5
1
Lightning test
t=38.1µs
0,5
Lightning test
t=57.4µs
0
-35
-15
5
Section (mm)
25
45
Lightning test
t=64.76µs
Figure 5.9: Déplacement sur la section centrale de l’éprouvette au cours du temps. Lignes pleines : essai foudre, Lignes
pointillées : simulation d’impact numérique
La figure 5.9 présente les profils obtenus avec le modèle 3D, comparés avec les résultats de l’essai
foudre. La comparaison est effectuée à plusieurs instants de 19µs après l’impact jusqu’à 68µs.
L’étude des courbes montre que le modèle numérique est capable de reproduire le déplacement de
Etude des profils de déplacement face arrière
282
__________________________________________________________________________________
la plaque sur toute une section de l’échantillon et non seulement en un seul point, tant en terme de
forme temporelle que de maximum de déplacement. Cette analyse vient confirmer la validité du
modèle numérique et de la méthode d’équivalence aux temps courts (jusqu’aux temps de
stabilisation) et valide l’équivalence mécanique. Un projectile mécanique peut correctement
reproduire les déplacements face arrière observés lors d’un impact foudre.
5.6.4
Surface totale délaminée
Après avoir comparé les déflections, une étude comparative des surfaces totales délaminées est
menée entre la foudre et le modèle numérique.
Les surfaces délaminées expérimentales sont obtenues par détourage des images obtenues par
l’analyse ultrason, cette méthode permet d’approximer la zone totale endommagée au moyen
d’ellipses. Les surfaces numériques sont obtenues par somme des dommages élémentaires obtenus
dans chaque interface du modèle. Un écart d’environ 10%, suivant la méthode utilisée (carrée,
ellipse, détourage) est courant dans ce genre de mesure post-mortem. Le tableau 5.6 rassemble les
surfaces totales délaminées obtenues pour les essais foudre 101, 103 et 107 et comparées à leurs
impacts numériques équivalents.
Le modèle 3D fournit des tendances concordantes avec les essais mécaniques en termes de
distribution des dommages comme montré précédemment. La surface totale délaminée est
surestimée par le modèle, comparée à la foudre, pour tous les cas présentés. Cette surestimation est
attribuée à une mauvaise prédiction des pics de déplacement au centre des échantillons pour les
panneaux 103 et 107. Dans le cas 101, le modèle numérique fournit une valeur de déplacement très
proche de celle obtenue lors de l’essai foudre et la valeur de surface totale délaminée, dans ce cas,
est celle qui comporte le moins d’erreur (cas 101-3, 3.5%). On constate que plus le pic de
déplacement foudre est correctement approximé par le modèle, plus la surface totale délaminée est
également bien estimée par celui-ci.
Essais
foudre
Vitesse
projectile
associée
(m/s)
101-1
101-3
101-2
103-1
103-2
107
75
75
70
50
65
81
Surface
délaminée
foudre
(mm²)
2321
0
1711
Surface
délaminée
essais
mécanique
(mm²)
1085
1626
1019
223
630
1746
Surface
Différence
Différence
délaminée
relative essai relative essai
simulation
foudre et
Méca. et
numérique
simulation
simulation
(mm²)
55%
1683
-27%
3,50%
1439
-38%
41%
395
77%
765
21,40%
1453
-15%
-17%
Tableau 5.6 : comparaison des surfaces totales délaminées entre essais mécaniques et foudre et le modèle numérique
283
__________________________________________________________________________________
6. Travaux prospectifs
Les essais d’impacts mécaniques équivalents ont été mis au point et validés par comparaison avec les
résultats foudre. Il a été prouvé que l’équivalence était valable sur des temps dits courts et que les
essais ainsi que le modèle numérique associé étaient capables sur ces durées d’équivalence de
reproduire la vitesse et le déplacement face arrière observés pendant les essais foudre. De plus, il a
été montré que la méthode d’équivalence permettait de prédire de façon correcte la surface totale
délaminée obtenue après les impacts foudre, bien que la distribution et l’allure des dommages dans
l’épaisseur des matériaux testés ne soient pas adéquatement reproduits par la mécanique. Il a été
établi également que l’utilisation d‘un impacteur sphérique très petit (bille de diamètre 9.8mm et de
masse 4g) n’était pas le projectile le plus adéquat pour représenter la foudre. Dans cette optique,
plusieurs études préliminaires, portant sur des modifications possibles du dit projectile ont été
menées.
Le chapitre qui suit présente trois études qui se sont intéressées à l’influence de la forme du
projectile (augmentation de la surface de contact) de deux façons différentes ainsi qu’à l’application
d’une autre forme de chargement plus proche de la foudre. Ce chargement est dérivé de la pression
magnétique [43] présentée au chapitre 3 et possède à présent une évolution spatiale et temporelle
représentative du pied d’arc foudre.
6.1 Etudes préliminaires sur l’augmentation de la surface de contact
6.1.1 Modification de la forme du projectile
Les simulations présentées précédemment utilisaient un projectile sphérique: une bille en acier de
diamètre 9.8 mm et de masse 4g. Ce projectile fournissait de bonnes prédictions de déplacement et
vitesse face arrière mais était incapable de reproduire la distribution des dommages dans le stratifié.
Cet écart peut être attribué à la surface de dépôt d’énergie. En effet la bille a tendance à fortement
localier l’énergie délivrée, or il a été montré (chapitre 3) que l’arc électrique de la foudre avait
tendance à s’étendre au cours du temps et ainsi augmenter sa surface de contact avec le matériau.
Afin de palier à ce problème, une nouvelle géométrie de projectile est proposée : le cylindre.
Le cylindre est formulé à partir du diamètre des billes utilisées ultérieurement, son poids est
également conservé et la hauteur du cylindre calculée en conséquence. Plusieurs vitesses d’impacts
sont testées et les résultats du projectile cylindrique sont comparés aux résultats avec la bille d’acier
et de la foudre. Le tableau 6.1 présente les différents cas foudre testés et les paramètres du cylindre
numérique associé (vitesse, masse, hauteur).
Cas
foudre #
1
2
3
4
Etat de surface
I=mv (N.s)
vmax (m/s)
m (g)
r (mm)
h (mm)
NoECF-NoP
ECF195-NoP
ECF195-P200
ECF73-P200
0.24
0.20
0.28
0.35
30
11
58
80
8
18
5
4
6.2
8.1
5.3
4.9
8.3
11
7.1
6.7
TABLE 6.1: PARAMETRES DU PROJECTILE CYLINDRIQUE POUR DIFFERENTS CAS FOUDRE
284
__________________________________________________________________________________
Analyse des résultats de déplacement
Les résultats numériques obtenus pour les échantillons foudre 1 et 2 sont présentés figure 6.1. Les
simulations fournissent des valeurs de déplacement et de vitesses pour le cylindre concordant avec
les résultats foudre aux temps courts et de stabilisation dans tous les cas. Les vitesses aux temps
courts sont bien prédites par le nouveau modèle qui donne de meilleurs résultats que ceux obtenus
avec la bille. Ceci confirme que la méthode d’équivalence, basée sur le transfert d’impulsion, est bien
valide aux temps courts.
3000
Deflection (µm)
2500
2000
1500
Lightning test #1
1000
500
0
0
100
200
300
Time (µs)
400
500
2500
Deflection (µm)
2000
1500
1000
Lightning test #2
500
0
0
100
200
300
Time (µs)
400
500
Figure 6.1 : Comparaison des déplacements face arrière obtenues pour le projectile cylindrique avec la foudre et le
projectile sphérique
Dans le cas de l’échantillon 1, le déplacement obtenu est plus faible que pour la bille mais plus
proche de la foudre aux temps courts. Pour l’échantillon 2, le nouveau projectile fournit de meilleurs
résultats jusqu’aux temps longs (<100µs) comparé à la foudre. Ces améliorations sont visibles sur
toutes les comparaisons mais plus particulièrement pour les cas 1 et 2 qui sont des panneaux foudre
non peints. Les cas 3 et 4 sont eux peints et présentent des résultats un peu différents. Il en est
285
__________________________________________________________________________________
conclu que l’absence de peinture est une condition nécessaire pour la dé-corrélation des événements
à cœur et en surface. En l’absence de peinture, la première loi de Newton peut s’appliquer au
stratifié qui peut alors être isolé de son environnement.
Les déplacements et vitesses obtenus avec le cylindre sont similaires à ceux de la sphère. De la même
manière, le cylindre n’arrive pas à reproduire les déplacements aux temps longs en dépit de sa plus
grande surface de contact. Cette observation suggère que l’influence de cette surface agit plus
tardivement dans le chargement foudre pour les panneaux peints mais n’a que très peu d’influence
aux temps courts concernant le comportement de ces échantillons. Le chargement foudre en deux
temps semble donc comprendre une première phase de pression très rapide, suivie par une seconde
qui s’effectue sur une zone de contact plus large que la première. Néanmoins, lorsque cette seconde
phase de pression s’applique sur l’échantillon, le projectile mécanique a, lui, déjà transféré toute son
énergie et ne peut donc reproduire cette seconde phase.
Distribution des dommages
Le modèle numérique permet d’extraire les dommages générés par l’impact du cylindre dans les
différentes interfaces du stratifié. Les résultats pour les quatre cas sont présentés au tableau 6.2 et
comparés avec les résultats foudre. On peut voir que pour les cas non peints, le cylindre, projeté à
faible vitesse, sous-estime grandement les dommages. Dans les cas peints, au contraire, à grande
vitesse, le modèle surestime l’endommagement total. De plus, comme pour la sphère, la distribution
des dommages dans l’épaisseur ne correspond pas à celle observée en foudre.
Int1
Int2
Int3
Int4
Int5
Int6
Total
NoECF-NoP
Essai
foudre simulation
228
0
1350
0
964
52,991
0
37,442
0
0
0
0
2542
90,433
ECF195-NoP
Essai
foudre simulation
182
10,73
25
14,625
0
11,48
0
0
0
0
0
0
207
ECF195-200µ
ECF73-200µ
Essai
Essai
foudre simulation foudre simulation
255
204,48
306
223,628
107
261,8
1067
252,77
646
1123,08
2510
1219,52
655
430,92
722
1159,64
0
209,92
0
604,35
0
208,5127
0
205,92
36,835
1663
2438,7127
4605
3665,828
Table 6.2: Interface délaminée pour chaque couple foudre/simulation cylindre
6.1.2 Changement de matériau pour le projectile
Afin d’essayer de reproduire au mieux le chargement foudre, une seconde géométrie de projectile
est envisagé. La première étude, qui visait à augmenter la surface de contact entre le projectile et le
286
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matériau, a confirmé que la foudre semblait appuyer en deux temps sur les plaques composites
impactées. Le nouveau projectile choisi va essayer de se rapprocher de ce chargement atypique.
Nouveau projectile
Le nouveau projectile proposé est une sphère de caoutchouc, type pneu, déformable. Le but ici et
d’utiliser un projectile qui se déformera au contact de la cible, produisant ainsi une première phase
de poussée liée à sa projection et une seconde liée à sa déformation contre l’échantillon. Pour
représenter numériquement ce projectile, une loi matériau de type Ogden [116] (matériau hyperélastique) est utilisée. Cette loi se base sur une définition non linéaire de la relation contraintedéformation. Le matériau est défini en fonction d’une fonction en densité d’énergie de
déformation telle que λj, j=1, 2, 3:
𝜇
𝛼
𝛼
𝛼
𝑝
𝑝
𝑝
𝑝
𝑊(𝜆1 , 𝜆2 , 𝜆3 ) = ∑𝑁
𝑝=1 𝛼 (𝜆1 + 𝜆2 + 𝜆3 − 3)
(Eq. 6.1)
𝑝
N est l’ordre du potentiel d’énergie de déformation, μp et αp sont des constantes matériau,
présentées dans le tableau 6.3 [119]. Ce nouveau projectile est testé sur l’échantillon foudre 3
(tableau 6.1) : vitesse d’impact 58 m/s, masse préservée de 5g. La densité choisie est ρ=1160
kg/mm3, ce qui permet de calculer le rayon de la sphère (r=10mm).
N
3
μ1
-12,09
α1
14,4
μ2
12,11
α2
14,4
μ3
1,75
α3
1,91
D1
0
D2
0
D3
0
Table 6.3: Constantes matériaux pour la loi Ogden
Le matériau utilisé est le même que celui présenté au chapitre 6. Pour cette étude préliminaire, le
modèle coque simplifié, présenté au chapitre 3 est utilisé afin de réduire le temps de calcul.
Cependant ce choix ne permet pas d’extraire des données concernant les dommages, la comparaison
se fera donc uniquement sur les résultats de déplacement face arrière.
Résultats de déplacement
Les résultats de déplacement obtenus avec le nouveau projectile sont présentés figure 6.2 avec les
résultats foudre associés. La comparaison montre que, contrairement aux précédents projectiles
rigides, l’impacteur déformable fournit une vitesse de déplacement plus faible que celle de la foudre.
Le déplacement maximum obtenu numériquement est du même ordre de grandeur mais la sphère
de caoutchouc ne permet pas de reproduire aux temps courts le comportement du à la foudre.
Cependant, il a été établi au chapitre 5 que tous les dommages foudre étaient générés durant les
premières 50 à 100 µs suivant l’impact foudre. On peut donc supposer que cette nouvelle
configuration ne permettra pas d’obtenir une distribution des dommages concordante avec la
foudre. Néanmoins ce résultat devra être vérifié en effectuant une simulation semblable avec le
modèle 3D endommageable établi au chapitre 5.
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4000
Displacement (µm)
3500
3000
2500
2000
1500
Lightning strike #3
1000
Steel ball projectile simulation
500
Rubber projectile simulation
0
0
200
400
600
Time (µs)
800
1000
Figure 6.2: Comparison between spherical, cylindrical projectiles and lightning
Conclusion
Cette étude était une première tentative de reproduire le chargement particulier de la foudre en
utilisant un projectile déformable. A cause de leur rigidité et leur forme sphérique, les précédents
projectiles utilisés ne transféraient leur énergie que sur une zone extrêmement réduite et n’étaient
ainsi pas capables de reproduire les dommages de façon adéquate. De plus l’étude avec le cylindre a
montré qu’une surface de contact correcte génèrerait des vitesses plus proches de celle de la foudre.
La vitesse de déplacement et la surface de contact semblant être les paramètres clefs pour obtenir la
bonne distribution des dommages dans l’épaisseur, en seconde approximation un projectile fait de
matériau caoutchouc, typique de ceux utilisés pour la certification de tenue aux impacts pneu, a été
simulé. Sa capacité à se déformer lors de l’impact semble une piste intéressante pour reproduire le
chargement en deux temps de la foudre.
La simulation a montré que ce projectile ne permettait pas de reproduire les déplacements face
arrière de la foudre et que l’impact générait des vitesses trop faibles. La même simulation doit
cependant être effectuée sur le modèle endommageable afin de valider ou d’invalider les dommages
générés par un tel projectile et ainsi d’affiner la comparaison de ce type d’impact avec la foudre.
Cette étude reste préliminaire et pose simplement les bases de la suite des travaux envisagés
concernant l’amélioration du projectile équivalent. D’autres projectiles sont également envisagés,
tels que les projectiles érodables. Dans le cas où un projectile déformable ne serait pas suffisant pour
représenter les dommages attendus, des essais de création numérique de projectiles à plusieurs
impédances sont également envisageables afin de créer un impacteur numérique approprié.
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6.2 Etude préliminaire des effets d’une pression évolutive
Dans le chapitre 3, une pression magnétique a été définie. Cette pression est reprise et améliorée
dans cette étude. Afin d’intégrer les observations précédentes, la pression magnétique initiale est
maintenant appliquée sur une surface évolutive. La pression s’applique sur un rayon qui grandit au
cours du temps. Cet ajout permet de satisfaire une cohérence avec les observations faites sur
l’évolution spatiale et temporelle du rayon d’arc ainsi que la nécessité d’appliquer le chargement sur
une zone de contact plus large. La nouvelle pression magnétique est donc appliquée sur un disque de
rayon r(t) qui dépend à la fois du temps et du courant injecté.
Pour cette étude, une plaque foudre protégée par de l’ECF195 et recouverte d’une couche de
peinture d’épaisseur 170 µm est utilisée afin de d’utiliser les données de courant des essais. La
pression magnétique est calculée pour r(t) et appliquée en fonction de la position d sur l’échantillon.
Pour d < r(t) P=0
Pour d > r(t) on définit :
I(t) 2
P = fe = μ0 (2πd)
(Eq. 6.2)
Avec I(t) le courant tel que :
I(t) = A (e−αt − e−βt )
(Eq. 6.3)
e= épaisseur de la protection métallique =δ/ρ [m]
ρ=densité de la protection) [kg/m²]
δ=masse surfacique de la protection [kg/m3]
ΔH=enthalpie de fusion [J/kg]
k coefficient phénoménologique (k=1for SCF, k=1.7 for ECF)
σ conductivité électrique [S.m-1]
L’intensité numérique du courant atteint son maximum en 19.6µs pour une valeur de 95kA ce qui est
cohérent avec les valeurs expérimentales de 100 kA en 20 µs.
Résultats numériques
La simulation numérique est effectuée avec le modèle 3D endommageable présenté au chapitre 5.
Les propriétés matériaux utilisées pour les plis et les interfaces sont présentées aux tableaux 5.10 et
5.11. La séquence d’empilement reste inchangée : [45/0/-45/90]s. Le chargement équivalent est
implémenté via une formulation temporelle et spatiale intégrée dans une seconde routine utilisateur
nommée VDLOAD.
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La figure 6.3 présente la comparaison des résultats de déplacements face arrière pour le nouveau
chargement de pression équivalente avec plusieurs essais foudre ainsi que les résultats du précédent
projectile sphérique en acier. Le nouveau modèle donne de très bons résultats de déplacement et
vitesse aux temps courts, bien que le maximum de vitesse soit plus élevé que pour l’essai foudre
associé (102 m/s contre 83 m/s pour l’essai foudre #121). La courbe de déplacement atteint un
plateau mais ne diminue pas comme le fait la courbe du projectile sphérique se rapprochant ainsi
encore un peu plus des courbes foudres. Cependant, la pression magnétique ne reproduit pas
correctement les déplacements à long terme et le plateau obtenu est lui aussi plus bas que les
valeurs obtenus lors des essais foudre : 2300 mm pour la simulation contre 3300 pour le cas foudre
#121.
Finalement, la distribution des dommages dans l’épaisseur est comparée entre la foudre et la
simulation de pression magnétique équivalente. Le tableau 6.4 rassemble les surfaces totales
endommagées pour plusieurs cas foudre proches (en termes de protection et épaisseur de peinture)
et la simulation. Le cas foudre de définition #121 donne une surface totale délaminée de 2112 mm².
La pression magnétique donne elle une zone délaminée de 2186.5 mm². De même, les autres essais
foudre donnent des valeurs concordantes avec la simulation numérique. En effet, le nouveau
chargement fournit des résultats très proches de ceux observés en foudre avec un écart relatif de
respectivement 5.8%, 25.2%, 3.5% avec les essais foudre 101, 107 et 121.
4500
4000
Displacement (µm)
3500
3000
2500
2000
1500
Magnetic pressure simulation
Mechanical simulation steel ball projectile impact at 81m/s
1000
500
0
0
50
100
150
200
Time (µs)
250
300
350
Figure 6.3: Déplacement face arrière pour la pression magnétique comparée à divers essais foudre et à la simulation avec
projectile bille
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Echantillon #
Essai foudre 101
Essai foudre 107
Essai foudre 121
Pression magnétique
(simulation)
Surface totale
délaminée (mm²)
2321
1746
2112
2186
Table 6.4: Comparaison des surfaces délaminées totales pour les essais foudre et la pression magnétique
La figure 6.4 présente la comparaison des dommages obtenus pour trois cas d’impact: l’essai foudre
107 (a), la simulation numérique avec projectile équivalent (impact à 81 m/s) (b) et la simulation de
pression magnétique (c). La comparaison montre que la pression magnétique génère des
délaminages importants aux interfaces 3 et 4, concordants avec les observations des C-scans foudre.
De plus, la forme générale de la zone endommagée générée par la pression magnétique est plus
« ronde » et ne présente pas d’écharde en face arrière comme on les retrouve sur les simulations et
essais avec le projectile. Cependant, la pression magnétique continue de générer des dommages
dans toutes les interfaces contrairement à la foudre dont la majorité des délaminages se concentre
dans la première moitié du composite comme le montre la figure 6.5.
(a)
Figure 6.4: Comparaison de la forme des dommages obtenus (a) foudre, (b) pression magnétique, (c) impact bille
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900
Lightning test 107
800
Magnetic pressure
700
Lightning test 101
600
Lightning test #121
500
400
300
200
0
0-0,1
0,1-0,2
0,2-0,3
0,3-0,38
0,39-…
0,48-…
0,57-…
0,66-…
0,75-…
0,84-…
0,93-…
1,02-…
1,11-…
1,21-1,3
1,3-1,39
1,39-…
1,48-…
1,57-…
1,66-…
1,75-…
100
Thickness (mm)
Figure 6.5: Distribution des dommages: comparaison entre la pression magnétique et divers essais foudre
7. Conclusion générale
Afin de certifier les structures composites, les compagnies aéronautiques doivent être capables de
comprendre et prédire le comportement de ces composites face à un panel d’évènements. La
menace foudre est devenue un enjeu majeur depuis que les matériaux composites ont été intégrés
aux structures primaires des aéronefs, notamment à cause du fait que, contrairement à leurs
prédécesseurs métalliques, ces nouveaux matériaux possèdent une mauvaise conductivité
électrique. La foudre génère de nombreux dommages visibles et invisibles : premiers plis brulés et
fibres mises à nues, délaminages dans l’épaisseur. De plus, la taille de ces dommages est encore
augmentée par la présence de peinture ajoutée par les compagnies aériennes. Ainsi, il devient
nécessaire de protéger ces nouvelles structures et des protections spécifiques contre la foudre ont
déjà été mises en place. Ces protections consistent principalement à ajouter au-dessus du matériau
composite, un pli métallique pour mieux évacuer le courant issu de la foudre. Ces protections ne sont
cependant pas optimisées et viennent ajouter un poids non négligeable sur les structures
concernées, où les matériaux composites avaient été choisis pour leur faible masse comparée aux
métaux. Le besoin se fait d’optimiser ces protections. Dans cette optique, la compréhension du
phénomène foudre, de son attachement sur les matériaux composites et des dommages qu’il en
résulte sont nécessaires. Des essais en laboratoires ont donc été menés depuis quelques décennies
afin de tester plusieurs types de protections et améliorer la connaissance du phénomène. A partir de
ces campagnes d’essai, des normes ont été établies afin de protéger les zones les plus à même d’être
foudroyées de manière adéquate.
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La communauté scientifique a travaillé sur la représentation de la foudre pendant plusieurs
décennies. Le foudroiement d’un aéronef est un évènement complexe, qui met en jeu de
nombreuses physiques telles que l’électromagnétisme, la thermique et la mécanique, qui peuvent de
plus être couplés entre elles. Dans la littérature, les études sur le sujet se sont principalement
concentrées sur la composante électrothermique de la foudre afin d’expliquer les dommages visibles
(zone brulée en surface). Une rapide revue de la littérature a été présentée sur la physique du
phénomène, les essais menés en laboratoire et les différentes stratégies de modélisation que l’on
peut trouver. Il a été conclu de cette étude bibliographique que, compte tenu de la résistance au
dommage, les modèles déjà existants et disponibles ne sont pas suffisants pour simuler la complexité
du chargement foudre. La plupart de ces modèles se concentrent uniquement sur les dommages
thermiques observables en surface. Cependant, certaines études se sont focalisées sur les dommages
internes causés par la foudre. Ces dommages, de nature mécanique (délaminage, fissure matricielle,
rupture de fibres) sont les plus dangereux car ils remettent en cause l’intégrité structurelle du
matériau et peuvent être responsables de la rupture totale de la structure composite. De plus, ces
dommages sont invisibles à l’œil nu et peuvent ne pas être détectés lors d’examens visuels simples.
Ces études, ainsi que l’observation approfondie des dommages à cœur des matériaux foudroyés sont
à la base du travail présenté dans ce rapport.
Les dommages dus à la foudre ont été séparés en deux catégories distinctes : les dommages de
surface (peinture et protection sublimées, éventuellement premier pli brulé) et les dommages à
cœur. Ces deux types de dommages différents par leur localisation et leur nature. A partir de la
littérature et des travaux préliminaires effectués en début de thèse, il a été établi que les dommages
de surface sont principalement d’origine électrothermique, dus à l’attachement de l’arc foudre. Les
dommages à cœur sont, pour leur part, principalement dus à un chargement mécanique résultant
des événements de surface. Les dommages foudre ont été analysés et une forte ressemblance avec
des dommages issus d’impacts purement mécaniques a également été établie, menant à une
hypothèse de dé-corrélation entre la surface et le cœur du matériau foudroyé. La foudre étant un
phénomène complexe dont les différentes composantes ne peuvent, aujourd’hui encore, être isolées
ni quantifiées, nous faisons l’hypothèse de travail de séparer le processus d’attachement de l’arc
foudre à la surface des dommages générés en profondeur et de nous concentrer sur ces dommages
internes. Considérant ainsi la surface du matériau comme une boite noire, nous concentrons l’étude
qui fait suite sur la compréhension des dommages mécaniques à cœur et leur reproduction.
D’un point de vue scientifique nous posons la question suivante : les dommages liés à la foudre étant
très proches de ceux observés durant un impact mécanique classique, est-il possible de les
reproduire de façon purement mécanique (numériquement et par essais réels) ? Le modèle
numérique associé peut-il aider à déterminer la nature des interactions entre le chargement foudre
et le comportement du matériau impacté ?
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Dans cette optique, nous nous attachons à proposer puis valider un essai mécanique équivalent à la
foudre, ainsi qu’un modèle numérique prédictif des dommages et du comportement global qui est
lui-même validé par comparaison avec les résultats d’essais mécaniques.
Premièrement, une étude analytique a été menée afin de mettre au point une équivalence entre
impact mécanique et essai foudre. Cette équivalence est basée sur l’impulsion transférée k (N.s) et
sur les données expérimentales extraites durant les essais foudre en laboratoire, menés sur plusieurs
éprouvettes protégées et peintes. De ces essais, plusieurs quantités sont extraites. Tout d’abord nous
proposons d’introduire plusieurs valeurs de temps caractéristiques. Le « temps foudre » fait
référence à la durée totale de délivrance du courant foudre dans le matériau. Le « temps courts »
correspond à la durée sur laquelle l’équivalence est valable et correspond au temps nécessaire pour
atteindre le maximum de vitesse cinématique face arrière lors des essais foudre. Le « temps de
stabilisation » correspond à la durée nécessaire à l’échantillon mécanique pour atteindre un plateau
de déplacement. Finalement, le « temps longs » fait référence à une durée où le pic de déplacement
est dépassé et où d’autres phénomènes se manifestent. Les deux principales quantités extraites des
essais foudre sont les déplacement et vitesse centre face arrière. Un modèle numérique simplifié,
utilisant des éléments coques a été créé afin de valider cette équivalence en comparant les valeurs
prédictives issues du modèle avec les données foudre. La bonne corrélation obtenue a permis de
développer un essai correspondant afin de valider définitivement l’équivalence. Un canon à air
comprimé a été choisi, permettant de projeter des projectiles de différents diamètres à des vitesses
allant de 50 à 150 m/s. Pour les tests préliminaires, le projectile choisi est une bille sphérique de
masse 4g et de diamètre φ 9.8 mm. La campagne d’essai mécanique a fourni des résultats
concordants avec les prédictions numériques en termes de déplacement et vitesse. Les essais
mécaniques équivalents ont également fourni des résultats concordants comparés aux résultats
foudre aux temps courts, cependant le comportement à long terme n’a pas pu être approché.
L’équivalence est cependant considérée valide comme les résultats sur la période dites d’équivalence
sont concordants, et considérant que l’ensemble des dommages foudre sont créés dans les cents
microsecondes qui suivent l’impact. Une comparaison des dommages générés a été effectuée entre
les deux types d’impacts à l’aide d’analyses ultrasons. Les essais foudre et mécaniques ont fournis
des valeurs de surfaces totales délaminées proches pour tous les cas testés avec plus ou moins de
succès. Les écarts les plus importants entre le modèle et les essais sont obtenus pour les cas foudre
où la peinture est épaisse ou n’a pas été brûlée. La répartition des dommages dans l’épaisseur a
montré une différence significative entre les deux essais. La foudre tend à générer des dommages
seulement jusqu’à la première moitié de l’épaisseur du matériau quand les impacts mécaniques
créent des dommages dans toutes les interfaces en suivant une forme conique croissante. Cette
différence est principalement due à la différence dans la méthode de dépôt d’énergie entre les deux
essais et notamment au fait que lors de l’impact foudre une partie de cette énergie est consommée
en effet Joule à la surface tandis que le projectile délivre toute son énergie cinétique.
En parallèle des essais, un modèle numérique 3D a été mis au point qui inclue la formation et la
propagation de dommages inter et intra laminaire, fondé sur des travaux précédents [48, 77]. Les
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résultats de ce modèle sont confrontés aux données expérimentales mécaniques et foudre afin
d’obtenir des prévisions de dommages les plus précises possibles. Le modèle numérique fournit des
résultats concordants avec la mécanique et la foudre tant pour les déplacements et vitesses face
arrière, que pour les surfaces totales délaminées. La comparaison a été étendue à la distribution des
dommages dans l’épaisseur. Le modèle fournit des valeurs concordantes pour les vitesses d’impacts
supérieures à 75 m/s mais tend à surestimer les dommages totaux pour les vitesses inférieures. Le
modèle reste cependant capable de reproduire avec précision la forme, la taille et l’orientation des
délaminages dans les interfaces endommagées. Comparés à la foudre, les résultats de simulation
fournissent les mêmes résultats que les essais mécaniques. Le modèle permet d’obtenir des
informations supplémentaires. Il a permis de confirmer que l’ensemble des dommages étaient bien
créés dans les premières 80 µs suivant l’impact. De plus, il est possible d’extraire du modèle des
profils de déplacements sur une section, ce qui fournit plus d’information que des résultats ponctuels
en un seul point. Ces profils ont été comparés à ceux obtenus lors des essais foudre obtenus grâce à
des caméras de stéréo corrélation d’images. La comparaison a fourni des résultats très proches,
montrant que le modèle était capable de reproduire le comportement global de la plaque et pas
seulement un point isolé et particulier.
A partir de ces comparaisons, il a été observé que l’équivalent mécanique permettait de reproduire
les déplacements et vitesses face arrière sur les temps courts pour les panneaux composites peints et
non peints. Pour des temps supérieurs à 150 µs, des différences sont visibles suivant la présence ou
non de peinture sur les échantillons. Dans les cas peints avec peu d’épaisseur de peinture, ou dans
les cas où la peinture est entièrement brûlée par la foudre, le modèle permet de reproduire le
comportement de plaque observé durant les essais foudre. Dans les cas où l’épaisseur de peinture
est plus importante (> 200 µm) et où cette peinture ne disparait pas au point d’impact, le modèle est
incapable de reproduire correctement le comportement à long terme du à la foudre. La présence de
peinture en surface est à l’origine de la variabilité des surfaces totales endommagées observées en
foudre.
L’équivalence est donc validée puisqu’elle satisfait aux critères de définition établis au début de
l’étude : niveaux de déplacement et vitesse face arrière similaires et surfaces totales délaminées
concordantes. Cependant des différences sont observées entre les deux impacts et notamment en ce
qui concerne la distribution des dommages. Ces différences viennent remettre en cause le choix
initial fait pour le projectile équivalent : une petite sphère en acier. Plusieurs autres projectiles sont
alors envisagés et testés dans des études prospectives préliminaires afin d’améliorer le modèle
équivalent. Un cylindre numérique est créé qui conserve le diamètre et la masse de la bille initiale
afin de jouer sur la surface de contact entre le projectile et la cible. Cette étude a montré qu’une
surface de contact plus grande permettait d’obtenir des résultats de vitesse face arrière meilleurs
que la bille quand on les compare à la foudre. La surface de contact est donc un paramètre à prendre
en compte. Cependant, ce nouvel impacteur ne permet toujours pas de reproduire le déplacement à
long terme de la foudre. Des essais avec un projectile déformable (type pneu) ont fourni de premiers
résultats intéressants. Enfin, une pression numérique, calculée à partir des données de courant des
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essais foudre a été testée. Cette étude a été menée afin de quantifier l’importance d’une des
composantes de la foudre et sa contribution aux déplacements et aux dommages finaux dus à la
foudre. Cette pression magnétique a fourni des résultats concordants comparés à la foudre en terme
de déplacements face arrière aux temps courts et bien que les temps longs ne soient toujours pas
bien représentés, le plateau de déplacement obtenu est plus proche des valeurs foudre que celui
obtenu avec le projectile sphérique. Cependant ce résultat était attendu dans la mesure où les
résultats foudre englobent toutes les composantes là où le modèle n’en intègre qu’une seule, les
forces de Laplace. La surface totale délaminée fournit par le modèle est concordante avec les essais
foudre. De plus, la forme générale des dommages est plus proche de celle observée en foudre et
certains faciès typiques de l’impact (échardes face arrière) ne sont plus présents.
Ces différentes études sont des pistes possibles dans le processus d’optimisation du modèle
d’équivalence. Les résultats obtenus ont également prouvé que l’hypothèse de dé-corrélation entre
les événements à cœur et en surface ne peut pas toujours être faite. Une analyse couplé électrothermo-mécanique devra être menée.
Bibliographie
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300
Thèse de Doctorat
Titre: Développement d'un modèle mécanique pour la prédiction des dommages de
Auteur: Floriane SOULAS
Résumé:
Dans un contexte industriel où l’utilisation de matériaux composites s’est généralisée jusqu’à
atteindre les structures primaires, la menace foudre se révèle être une problématique majeure.
Avec un coup de foudre en moyenne par an et par avion en service, les nouvelles structures
composites, moins bonnes conductrices que leurs prédécesseurs en métal, doivent être
protégées. Les protections mises en œuvre par les fabricants et les équipementiers sont des
couches minces ajoutées à l’empilement composite, initialement choisi pour le compromis
optimal qu’il offre entre résistance et légèreté. L’optimisation et le conseil concernant les
protections foudre deviennent alors un enjeu industriel d’importance. Dans ce cadre, le travail
de thèse a porté sur l’étude et la compréhension des dommages issus des chocs foudre sur des
structures protégées dans le but ultérieur d’optimiser ou de créer des protections adaptées.
Nous proposons une méthode qui permet de déterminer les caractéristiques d’un impact
mécanique sur une plaque composite nue, équivalent à un choc foudre sur une structure
protégée. Une campagne d’essais d’impacts avec un canon du laboratoire couplé à une campagne
numérique ont permis de conclure que la stratégie et la méthode d’équivalence sont fondées, et
permettent de prendre en compte les paramètres constitutifs de la protection de surface. Les
modèles proposés permettront d’aborder les questions de conception des protections.
Mots clés: Foudre, Equivalence mécanique, Matériaux composites, Modèle numérique,
Délaminage, Impact, Endommagement, Dynamique rapide non linéaire
Abstract:
In an industrial context where more and more composite materials are integrated into primary
structures, the lightning threat has become a major issue for aircraft manufacturers. As lightning
strikes in service airplane about once a year, the new composite structures, with a lower
electrical conductivity than their metallic predecessors, must be protected. The protections
already integrated by manufacturers are mainly made of expanded metallic foil layered above
the composite lay-up, thus adding weight on the low density structures and reducing the gain of
weight. The optimization of such structures and counsel concerning lightning protection become
a major industrial stake. In the scope of the PhD work, the proposed work focused on the study
of the damage mechanisms due to lightning strikes on protected composite panels in order to
optimize or offer adequate protections against this threat. A methodology is proposed to
determine a mechanical impact on a bare composite plate equivalent to a protected and even
painted structure submitted to a lightning impact. An experimental campaign of mechanical
impacts using a canon gas gun coupled to a numerical plan is led and allows concluding on the
strategy and its validity by taking into account the state surface of the lightning samples.
Keywords: Lightning, Mechanical equivalent impact, Composite material, Numerical modelling,
Delamination, Impact, Damage, Transient non-linear Dynamics

Développement d`un modèle mécanique pour la prédiction des

Transcription

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