The propagation of fracture in aeolotropic materials | SpringerLink

THE
PROPAGATION
OF FRACTURE
MATERIALS
IN AEOLOTROPIC
C. Atkinson*
ABSTRACT
The theory is developed for the propagation of a semi-infinite plane crack in an aeolotropic
material where the direction of propagation lies in a symmetry plane. The rate of work done
by the applied forces in maintaining a constant velocity is calculated. This work is dissipated in the surface energy of the extending crack. This gives a relation between the applied
force, the surface energy and the velocity of propagation as a function of direction. Such a
relationship could afford insight into preferred directions of cracking in aeolotropic materials.
INTRODUCTION
This is an application
o f t h e m e t h o d s u s e d b y C r a g g s {1960)
on the propagation
of a crack in an isotropie
material
with a
constant
s p e e d V 1. W e w i l l , h o w e v e r ,
consider
more general
materials
than that considered
by Craggs and we can use this
special
case (isotropy,
where
the equations of motion can be
factorised
into two equations in two separate
potentials,
as is
well known) as a useful check on the algebra.
The general method of solution of the equations of motion is
similar to that used by Stroh (1962) and has the advantage that
it applies for all values of V 1 whether subsonic or supersonic.
Although we deal quite generally with an aeolotropie body with
plane strain, the assumption
of plane strain is not always justified
unless
the edge of the crack runs perpendicular
to a plane of
symmetry.
Therefore,
for the more general aeolotropic
body we
would have a displacement
in the z direction;
however we can
modify this by supposing a force in the z direction distributed
along the surface
of the crack and chosen so as to make the
displacement
in the z direction zero. With such a modification
the energy
calculations
as in section
6 would be unaffected,
for in the direction
of this applied force we are allowing no
displacement
and hence no work would be done by it.
The analysis concerning the case of general anisotropy
where
we allow a displacement
in the z direction will be considered
in a subsequent
paper.
From
this rather idealised problem,
that of a semi infinite
crack
propagating
with a constant velocity, we hope, by con:
siderations
of energy
to find an upper
bound
to the velocity
of the crack
and also the possibility of preferred
directions of
cracking.
We give some
indication in the simpler
cases as to why only
the case of elliptic equations
seems
to be significant; furthermore,
it seems
likely that a rigorous proof could be constructed
using the property
deduced
by Saenz (1953), giving upper and
* Department of Mathematics, University of Leeds; at present in the University of Melbourne.
48
c. Atkinson
lower bounds for the velocity of propagation which reduces the
equations to elliptic form, and thence finding the condition for
the crack to close.
Nevertheless, sufficient indication has been given in the past
that the elliptic equations are the suitable ones for these
p h e n o m e n a that w e will content ourselves by illustrating this in
one or two simple eases.
MA THEMA
TICA
L
SPE CIFICA
TION
Consider
an infinite elastic body. Two
systems
of rectangular
cartesian
co-ordinates
are used. The co-ordinates
(x', y', z')
are referred to axes chosen with respect to the symmetry
properties of the body,
but only cracks
moving
in a given plane
are considered.
The edge of the crack is straight and is taken
as the z axis, the crack then lies in the (x, z) plane, and plane
strain in planes (x, y) is assumed.
Hence
we have the following
conditions
a cut over the half
plane y = 0, x < vlt where vj is constant and t is the time. We
also assume
that ~,., ~x. on the two sides of the cut are equal,
and t h a t the c r a c k o p e n s s y m m e t r l c a l l y u n d e r
boundary conditions
of the f o r m
YY
(A)
Y
~yy = - f(x - v l t ) ;
.
.
.
~xy = g(x - v l t ) f o r - ~ < x - v l t < 0,
w h e r e f and g are differentiable any n u m b e r of times except,
perhaps, at a finite n u m b e r of isolated points; and that
(B)
Raxx,
RCrxy, R~yy--* 0
as R = {(x -vlt) 2 + y2}~ _. ** uniformly in the
upper half space y > 0. (That is no load at infinity. )
W e shall obtain the equations of motion in the (x,y, z) coordinate s y s t e m for general values of the elastic coefficients.
ANALYSIS
OF
MOTION
L e t (u, v) be the c a r t e s i a n c o m p o n e n t s of the d i s p l a c e m e n t
an e l e m e n t in the ( x , y , z) a x e s .
T h e e q u a t i o n s of m o t i o n a r e [ M u s g r a v e (1954)]
8t 2 = H n u
pO2u
xx
of
+ 2H16Uxy + H66Uyy + H16Vxx + (H12 + H 66)Vxy + H26Vyy
82v
PDt--~ = H16uxx + (H12 + H66)Uxy + H26Uyy + H66Vxx + 2H26Vxy + H22Vyy.
(1)
w h e r e the elastic coefficients Hij are m e a s u r e d
crack axes (x', y', z').
Now
write
X
= x -Vlt,
{Yij = aij(X,Y) and
relative to the
82
at 2
= Vl 2
82
8x 2
,
C. Atkinson
49
and put
(u,v)
= (A x,Ay)~(x÷Ty)
(2)
in the e q u a t i o n s of m o t i o n a b o v e .
~ " (A x {(H 11 - PV12) + 2H 16 T + H 66T 2} + Ay {H 16 + (H 12 + H66) T + H26T2}) = 0
~' '(AxlH16+ (HI2+H66)T + H26T2}+Ay[(H66-PVI2)+ 2H26T+ H22T2})=0
The use of this substitution is associated with a wave function
and embodies the ideas expressed by Sneddon (1952) which have
since been exploited by Craggs (1960).
Thus, for non zero Ax, Ay and ~" ~ 0
( H l l - P V 1 2 ) + 2H16T + H66T2
HI6+(HI2+H66)T + H26T2
=
H16 + (H12+H66)T + H26 T 2
(H66-PV12)+2H26 T + H22 T2
0
(3)
T h i s i s a q u a r t i c in T , a n d the n a t u r e of t h e r o o t s v a r i e s with
V 1 s o t h a t f o r a c o m p l e t e d i s c u s s i o n we w o u l d n e e d to l o o k a t
a n u m b e r of p o s s i b i l i t i e s . H o w e v e r , t h e r e a r e p h y s i c a l r e a s o n s
f o r e x p e c t i n g the p o s s i b l e r a n g e of c r a c k v e l o c i t i e s to be t h o s e
f o r w h i c h all f o u r r o o t s a r e c o m p l e x , and t h i s c a s e i s the one
treated here (see Appendix).
N e v e r t h e l e s s ; i f we w e r e to f o r m u l a t e a m o r e g e n e r a l p r o b l e m ,
s a y t h a t of a m o v i n g f o r c e of s u p e r s o n i c s p e e d o v e r the s u r f a c e
of a n a e o l o t r o p i c s o l i d , t h e n the m o r e g e n e r a l p r o b l e m of the
n a t u r e of the r o o t s of (3) w o u l d be a p p r o p r i a t e . ~ T h i s will b e
c o n s i d e r e d in a s u b s e q u e n t d i s c u s s i o n . *
T H E CASE O F C O M P L E X
R O O T S O F (3)
L e t T 1 , ~ i , T ~ , ~2 be t h e c o m p l e x r o o t s of e q u a t i o n (3) and
l e t (Ax, A v); (-~x, --y ), (Bx, By), (Bx, By) be the c o r r e s p o n d i n g v a l u e s
of the cor~stants in e q u a t i o n (2). T h e n , the c o m p l e t e f o r m s f o r the
displacements
are
u = A x ¢ ( X + T l Y ) + A x ~ ( X + T l y ) . + BxqJ(x+T2y) + N x ~ ( x + T 2 y ) ]
v
AY ~ ( x + T l y ) + AY ~ ( x + T l y ) + BY ¢j(x+T2y) + B / y ( x + ~ 2 y ) ~
(4)
T h e s e e x p r e s s i o n s s a t i s f y the e q u a t i o n s of m o t i o n and a r e r e a l ,
t h e r e f o r e t h e y a r e s u i t a b l e v a l u e s f o r the d i s p l a c e m e n t s . We n o w
n e e d to c h o o s e t h e m t o s a t i s f y t h e b o u n d a r y c o n d i t i o n s .
Let
z 1 = x + TlY,
z-~ = x + ~llY,
z2
z2
x + T2y,
x+T2y
3
(5)
(6)
~(zl)
-- u 2 + i v 2
~(~'~)
u 2 - iv 2 J
* We do, of course, have certain inequalities between the H' s, such as the condition of a
positive strain energy function and others, but it is the opinion of the auth6r that these a r e
not sufficient to restrict the general nature of the roots.
C. Atkinson
50
zl
= xl +iYl,
TI
= ~i + i~l,}
(7)
whe re
z 2 = x 2 + iy 2,
T 2 = ot2 + i~ 2
X 1 = X+OtlY
Yl
= ~31Y
X 2 = X+ce2y
Y2
= /32Y
Now write,
d#
Then
w1 =
±
i z l 2 = ~1 + i t / 1
= ~1
iz2 2 = ~2+i~2
= ~2
8u 1
8Vl
d¢(z I)
-+i
; w* 8x I
8x I
1
dz I
dz 1
d@
8u 2
8v 2
w2 - -- + i
; w*
dz 2
8x 2
8x 2
2
Similarly
8v 1
8u I
-
i
I
d~(z 2)
-
-
8x I
8x 1
8U 2
8v 2
8x 2
8x 2
(8)
-
d~'~2
I n t h e s e e q u a t i o n s t h e w l ' s e a n be r e g a r d e d a s f u n c t i o n s o f ~1 a n d
t h e w 2 ' s a s f u n c t i o n s o f ~2.
T h e b o u n d a r y c o n d i t i o n s (8) a s x, y - - ** i m p l y w 1 (~1), w2 (~2) -" 0
u n i f o r m l y a s ~1, ~2 -~**. T h e v a l u e o f e a c h f u n c t i o n i n t h e u p p e r
half plane may therefore be given in terms of the values on the
real axis, by Cauchy's theorem
in the form
w(~) = ~ 1
/** w(t)
d_**t
- ~ dt,
0= ~ 1
0 --
f . : t-w(t)
/ ~,:, dt; t h e n
dt
where
in each ease convergence
is assumed
along the real axis, with indentations
wherever
singularities.
(9)
and integration
necessary
due
is
to
CONDITIONS
BOUNDARY
T h e b o u n d a r y c o n d i t i o n s ¢ryy = - f ( x ) , Crxv = g(x) f o r -** < x < 0
on
- 0 n o w correspond to conditions on the real axis of the ~i,
~2planes and m a y be written
Cryy = AW 1 + A W l * + B w 2 + B w 2 * = - f ( t 2)
axy
at
AlWl + ~lWl*
~i = ~2 = t,
A
where
l
+ BlW 2 + ~1w2 * = g(t2)J
t is real,
(io)
and
= H I 2 A x + H 2 2 A y T l + H 2 6 ( A x T 1 + Ay),
A 1
HI6A x +H26AyTI
+H66(AxT l +Ay),
B
H I 2 B x + H 2 2 B y T 2 + H 2 6 ( B x T 2 ~- By),
B1
H I 6 B x + H 2 6 B y T 2 + H 6 6 ( B x T 2 + By).
Applying the C a u c h y formulas (9) in (I0), w e obtain
(Ii)
C.Atkinson
-i/2~ri f :
51
f(t2) dt = A w I + B w 2 ,
I/2ri I: g ( t 2 ) d t = A l W l + B l W 2 ,
---=T
t
which
give
wi(~l)
= 2~ri(AiB
W2(~2)
-i
= 2~i(AIB
" ABi)
Craggs,
AB1)
we
PROBLEM
consider
f(x - v l t ) = 0
a specific
x - vit
problem
<-a
)
0 >x - v i t >-a
= P
g(x - v i t ) = 0
(13)
x - vlt <-a
= S
where a is a positive real
stants.
S u b s t i t u t i n g in e q u a t i o n s
(12)
f ~ Aif(t 2) + Ag(t 2)
~-~-- ~2
dt.
A SPECIFIC
Following
dr,
~ - ~I
0 >x - vlt >-a
constant,
and P and S are
(12) t o c a l c u l a t e
both con-
w I a n d w2,
we find
1
BI P + BS
1
W l ( ~ l ) = 2~ri(Al B - A}~I) ,J-a2
Then,
after
t - ~1
dt.
integrating,
I
{ B i P + BS}
1
"~
a~ - iziT
W 1 (Z 1) =
log
2vi(AIB
- ABI)
a½ + izl½
-{AiP + AS}
w2(z2 )=
a½ - iz2½
log
2~i(AIB - ABI)
1"
i
a2 + iz2T
Using the appropriate singularity to ensure that the applied
forces are resisted bythe material at the end of the crack, w e have
{ B i P + BS}
Wl(Zl) = +
2~i(AiB -ABI)
/.
a~ - izlT
l
i +i~
~1og aZ.+ izlT
i
{ A i P + AS}
w 2 ( z 2) = _
/
2 iiA- :7 i) \log aT + iz~T
THE
FLOW
OF
ZlT /
(14)
i
a T - iz2T
1
2i
2i
I +i7~"
z2 T / j
ENERGY
We will now consider the balance of energy in the problem.
Here the arguments expressed in [I] follow almost word for word,
52
C.Atkinson
y e t f o r c o m p l e t e n e s s we will e x p r e s s t h e m h e r e . We c a n v e r i f y
t h a t the s t r e s s and v e l o c i t y c o m p o n e n t s a r e O(R-3/2) a s R - - - .
T h i s i m p l i e s t h a t t h e e n e r g y d e n s i t y i s O(R -3) a n d t h e r e i s no
t r a n s p o r t of e n e r g y to i n f i n i t y . T h e e n e r g y w h i c h is c r e a t e d in
the m a t e r i a l
b y P a n d S m u s t t h e r e f o r e e s c a p e t h r o u g h the
singularities.
We c a n t h e r e f o r e d e d u c e ( s e e [1]) t h a t the o n l y
l o s s of e n e r g y i s a s s o c i a t e d w i t h the s i n g u l a r i t y at X = y = 0
a n d a l l t h e w o r k d o n e b y P a n d S is l o s t h e r e .
Furthermore,
u s i n g the c r i t e r i o n f o r c r a c k i n g e x p r e s s e d b y
G r i f f i t h (1921), i t i s n a t u r a l t h a t we e q u a t e the r a t e of l o s s of
m e c h a n i c a l e n e r g y a t t h i s s i n g u l a r i t y to the r a t e of i n c r e a s e of
s u r f a c e e n e r g y of the m a t e r i a l w h i c h c a n be w r i t t e n a s 2TV1
w h e r e T is t h e e n e r g y p e r u n i t a r e a of the s u r f a c e .
We c a n n o w w r i t e
~u _
au
)
at
Vz ~ = - V1 f a x w l + AxWl;:" + BxW2 + BxW2*}
8va_t_=
replacing
-
vz axBV-
d~
~
by
V l{Ayw I +Aywl*
W1
+ B y W 2 + Byw2*},
in (8) and s i m i l a r l y
as
T h e r a t e a t w h i c h the e n e r g y f o r c e s
U =
f_o((
a
P
88-~[1y=o - S
Put T = -X, then T is positive, and
= (Cx +~)
(au)
v)
b-~
where
a
+
y=o+
Y
Cx =
(~logFa½+d-
CY)
+
T 2
-
La-~
T½
f o r w 2.
do w o r k i s t h e n
()}
~- y=o
~)u
I}
2a ~
~½
dx;
+ (c~ + ~ )
2a½ }
- .-~-_~ + (c~ + ~ )
(15)
V
[ ( A 1 P + A S ) B x _ ( B 1 p + BS)Ax] ,
2$i(A 1 B - A B 1)
C_y is s i m i l a r t o Cx w i t h s u f f i x x r e p l a c e d b y y,
C x is t h e c o m p l e x c o n j u g a t e of C x a n d s i m i l a r l y f o r the o t h e r s . ,
and
c~ = ~ i c
X.
(16)
Hence
U = - 2 a ( P { C y + Cy} - S{C x + C % ] ) + a ( P { C ~ + Cy} - S{C x + C'x} ).
If n o w we m a k e the a s s u m p t i o n t h a t T i s the s a m e , w h e t h e r t h e
s u r f a c e i s f o r m e d b y d i r e c t o r t a n g e n t i a l s t r e s s , we c a n t h e n
e q u a t e t h e r a t e of w o r k i n g to t h e r a t e of c r e a t i o n of n e w e n e r g y
a n d h e n c e get
U = 2TV 1 .
(17)
C. Atkinson
53
This gives a relation between the applied force, the surface energy
and the velocity of propagation
as a function of direction.
If we
make the above assumption
about the surface
energy,
we could
deduce from
(17) t h e v a r i a t i o n
of P with velocity (that is the
v a r i a t i o n o f P f o r d i f f e r e n t v a l u e s c f Vj. a n d d i f f e r e n t d i r e c t i o n s
o f p r o p a g a t i o n w i t h S = 0). W e W o u l d t h e n f i n d t h a t f o r a g i v e n
V 1 there would be a direction in which the value of P, required
to maintain
the motion
w i t h S = 0, w o u l d b e a m i n i m u m .
We
would then deduce that in such a direetion the crack would more
easily propagate.
Similarly we could, of course,
consider the variation of S with
P = 0.
Acknowledgements.
My grateful
thanks are due to Professor
J.W. Craggs
suggesting
the work and for many helpful criticisms.
Received
June
15,
for
1964.
REFERENCES
1.
2.
3.
4.
5.
6.
CraggsJ. W. LMech. Phys. Solids, 8, 1960, 66-75.
Stroh,A.N. /.Math. and Phys. 61, 2, 1962, 1.
Saenz, A.N.J.Rational Mech. and Anal. 2, 1953, 83.
Sneddon,I.N. Rendiconti Cir. Math. di Palermo Series II, 1952. 1.
Musgrave,M°J.P. Proc.Roy.Soc. A 22_.~6, 1954, 339-355.
Griffith,A.A. Phil.Trans.Roy.Soc. A 221, 1921, 163.
A PPENDIX
In this appendix we intend to give s o m e indication as to w h y
propagation with a speed greater than the speed of shear waves
is unlikely. W e proceed to consider the d o s i n g of the cracks.
Media
of Hexagonal Symmetry
dimensions).
From
Musgrave
(5),
used here we find that
and
(equivalent
changing
OV2 - C l l
T l
and
T2
are
both
complex
ell
After
a little
algebra
to that
(°11 - %2)
0v 2
is
the nomenclature
in two
20V 2 - ( e l z - cZ2 )
eu
Thus
to isotropy
<
,
provided
- c
12
2
we have
< ell
the
same
"
result
as
[1].
which
C.Atkinson
54
2
2½
V}C~2(I - V}C~2)~P l o g
ag +
1
aT -
/ ~V~
D 7t, ~,~-~-/y=o+ = 7rSV1 { 2 - V 2 / C : - 2(1 - VI / C 1) (1 - V } / C : ) ½ }
t
-
[
Tg
2ag
1 --1_---
T~
(18)
T2
w h e r e ~- = - X .
T o g i v e s o m e i d e a of t h e p r a c t i c a b i l i t y o f a c r a c k t o p r o p a g a t e
w i t h a v e l o c i t y > C 2 ( < C 1 ) , w e p u t V 1 = C 2 i n (18) a n d c o n s i d e r .
DV
t h e s i g n o f (8-t-)y=o a t t h e e n d o f t h e c r a c k X = 0.
We find
DTr~
(~V)
~ - y=o =~VI
1
where
D
:
4(1
-
2
V 1/C
22 ) Y ( I
-
V2/C})½
(2 - V 2 / C 2 ) 2
w h i c h g i v e s D = - 1 f o r V 1 = C 2 H e n c e ~_yV = -_SS w h i c h m e a n s
the crack is closing.
"
8t
pC 2
This gives us the necessary
indication, at least for materials
with these symmetries,
t h a t t h e e l l i p t i c e q u a t i o n s a~re s u f f i c i e n t .
Orthotropy.
The
quartic
for
T is now
AT 4 + BT 2 + C = 0 Where
2}
(19)
complex
for
A
= H66H22
B
= H22(HII - pv 2) + H66(H66 - pv 2) - (HI2 + H66
C
= ( H l l - pv 2) (H66 - pv 2)
T 2 = - B +_~]B 2 - 4AC
2A
Thus
I f B = 0 f o r PVl 2 >/ H66 , t h e n t h e r o o t s o f T a r e
v a l u e s of V l e s s t h a n t h i s .
Therefore,
if the constants are such that
H22HII - H I 2 2 - 2HI2H66 > (H22 + H66)H66
(20)
we c a n t h e n take pv 2 = H66 a s o u r l i m i t i n g ve10city a n d s u b stituting in the various
formulae
f o r the c o n s t a n t s we o b t a i n
H22Hll
T 2 = +iK,
T1 = 0
where
K2 =
- (H12 + H66) 2 - H66H22
H66H22
and
A x =0,
Ay=l
"~
By = (HI2 +H66)iK, B x = H22K 2
A
= 0;
A I .= H66;
B
= _K2H22H66
(21)
B I = ik(H22Hll - H I : - H66H22 - H66H12
8V
As we a r e c o n s i d e r i n g the s i g n of ~ ,
we n e e d to c o n s i d e r the
C. A tktnson
55
1
coefficient
of r'~ as this is the largest
part
we need to consider
the sign of the real part
"
o f x = 0.
of Cy.
Hence,
V
R(Cy) = 2~riAlBH66{P(HI2 + H66)iK - H66iK[H22K2 + (HI2 + H66)]}
_
Hence,
- V t
2zrA1B P.H66H22
.
KS
= +ve.
~V - ( - v e ) x r - 2z - , -**, 7 --* O.
~t
Thus the crack would close rapidly under these conditions; so
these last two examples give us s o m e indication of the applicability of the criterion that only elliptical equations of motion
have physical significance in crack propagation problems.
RI~SUlvlE-La theorie est dgveloppge pour la propagation d' uue fracture plane et semi-infinie,
duns un matgriaux allotropique, off la direction de propagation a lieu dans un plan de symmetric. La quantit¢ de travail produit par les forces appliquges en maintenant une vitesse
constante, est calcul~e. Ce travail est dissip¢ sous forme d'energie de surface de la fracture
propagge. C e c i donne/une relation entre la force appliqude, l'euergie de surface et la vitesse de propagation en rant que fonction de la direction. Une telle relation pourrait permettre la prCvisiou des directions pr~f~rentielles des fractures duns les matdriaux.
ZUSAMMENFASSUNG - Die Theorie fuer die Fortpflanzung eines halbunendlichen ebenen Bruches
in einem aeolotropischen Material wird entwickelt, wobei die Fortpflanzungsrinhtung in einer
Symmetrieebene liegt. Die zur Aufrechterhaltung einer konstanten Bruchgeschwindigkeit erforderliche Arbeitsleistung der angewendeten Kraefte wird errechnet. Diese Arbeit dissipiert
in Oberflaecheneuergie des sich ausdehnenden Bruches. Dadurch wird eine Beziehung zwischen
den angewandten Kraeften, der Oberflaechenenergie uud der Fortpflanzungsgeschwindigkeit in
Abhaengigkeit yon tier Bruchrichtung erhalteu. Diese Beziehung koennte Einsicht in die bevorzugten Bruchrichtungen in einem aeolotropischen Material verleiheu.