Trifurcation of a crack due to plane SH-waves in an infinite elastic medium

Engineering Fracture Mechanics Vol. 52, No. 2, pp. 205-213, 1995

Pergamon

0013-7944(95)00021-6

Copyright (~) 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0013-7944/95 $9.50+0.00

T R I F U R C A T I O N OF A C R A C K D U E TO P L A N E SH-WAVES IN AN I N F I N I T E ELASTIC M E D I U M A. N. DAS Department of Mathematics, North Bengal University, Darjeeling, West Bengal, 734 430, India A~traet--The dynamic anti-plane problem of trifurcation of a semi-infinite crack due to incidence of two linearly varying plane SH-waves with non-parallel wave fronts in an infinite elastic medium has been considered. The semi-infinite crack is assumed to trifurcate when the plane waves intersect the crack tip. The problem has been solved using the self-similar technique, which is based on the observation that certain field variables show dynamic similarities. The results include the expressions for shear stress in the planes of the cracks and the stress intensity factors at the crack tips. Numerical calculations have been carried out to show the variations of stress intensity factors at the crack tips with the angle of skew for different values of the crack tip velocity and angle of incidence.

I. INTRODUCTION SEVERAL INVESTIGATIONSon symmetric or non-symmetric extension of a crack in its own plane in an infinite elastic medium have been carried out until now, but when the extension of the crack occurs under an arbitrary angle with its own plane (a primary crack may bifurcate) the study becomes more relevant. Solutions for dynamic crack bifurcation in anti-plane strain for two special cases were solved by Burgers and Dempsey [1]. Corrected results for mode III kinking of a crack under an arbitrary angle were given by Dempsey et aL [2]. Numerical approaches for the study of dynamic propagation of a kinked or bifurcated crack in anti-plane strain and also dynamic kinking of a crack in plane strain have been given by Burgers [3, 4]. Achenbach et aL [5] have developed a method based on the superposition principle to derive approximate expressions for the elastodynamic stress intensity factors of the kinked crack. Recently, Das [6, 7] has solved the problems of tearing of a half-plane and extension of cracks in an infinite elastic medium due to the plane SH-waves. To the best knowledge of the author, the problem of trifurcation of a semi-infinite crack in an infinite elastic medium has not been solved so far. In this paper, the dynamic anti-plane problem of trifurcation of a semi-infinite crack due to the incidence of two linearly varying plane SH-waves with non-parallel wave fronts in an infinite elastic medium has been considered. The semi-infinite crack is assumed to trifurcate when the plane waves intersect the crack tip. For constant crack tip velocities, the shear stress and particle velocity are self-similar which allow Chaplygin's transformation to reduce the problem to the solution of the Laplace equation in a semi-infinite strip containing a slit. The Schwarz-Christoffel transformation is employed to map the semi-infinite strip on a half-space. Expressions for shear stress in the planes of the cracks and stress intensity factors in the vicinity of the crack tips have been derived. Finally, numerical results for stress intensity factors have been presented to show their variations with the angle of skew for different values of the angle of incidence and crack tip velocity.

2. STATEMENT OF THE P R O B L E M Let two identical plane waves defined by co"

Wi~c= T - - - z + H ( z ± ) , referred to the coordinate system (r, 0, z) where z± = t + r cos(O -T-Oo)/C, 0 <~Oo <~~/2 2o5

(l)

206

A . N . DAS

and H ( ) is the Heaviside step function, strike the tip of a stationary semi-infinite crack at t = 0 and cause the crack to trifurcate in such a way that two branches emanate symmetrically with velocity v ( < c ) under an angle krr with the remaining one which extends along the plane of the semi-infinite crack with velocity u ( < c ) . Thus, at time t > 0, crack tips are defined by r = vt, 0 = + k n and r = ut, 0 = 0. The expanding cracks, circular wave front associated with its motion and plane wave fronts are shown in Fig. 1. The shear stress component tr°: corresponding to incident waves is as°: = a[sin(O - Oo)H(z+) - s i n ( O + Oo)J(z_ )].

(2)

Superposing the fields due to the incident waves and scattered waves we see that the conditions on the crack faces due to the scattered wave are 0=+n;

r>0:tr0:=-2asin00H(t-rcos00/c) 0 <<,r < vt : tr0== 2tr sin 00 cos k n

0 = + krr;

0 = 0;

0 ~< r < ut : tr0~ = 2a sin 00.

(3a-c)

The shear traction given by eq. (3a) generates the plane waves with constant particle velocity, i.e. of magnitude -T-2ca/#. Since stresses and velocities are continuous across the cylindrical wave front the conditions for the particle velocity are

(a)

...-

/

-...\

/

-\o

(b)

° /

Y

Fig. 1. Geometry and coordinate system.

Trifurcation of a crack due to plane SH-waves n--00<0
207

r = ct : ~b = -- 2crr /l~

- Tr < O < - rt + Oo,

r = ct : w = 2c(r /t~

-rt+00<0
(4a-c)

r=ct:w=O.

The problem is obviously anti-symmetric about the x-axis with respect to particle velocity, so only a half-plane need be considered. In polar coordinates (r, 0), the two-dimensional anti-plane wave motions are governed by 1 (32w- 1 (32w (or J 4 r2 (302 c2 (3t2

(5)

1 8 (r(oW~

r (or \

,

where w ( r , O, t) is the displacement in the z-direction and c = x / I t / P is the velocity of transverse waves. Absence of any characteristic length in the geometrical configuration of the problem and the boundary conditions (3) and (4) suggest that the particle velocity ~bis self-similar, implying thereby that it depends on r/t, 0 rather than on r, 0, t separately. Introducing the variable s = r / t it is found that w(s, 0) satisfies the equation s2(1 - s2/c 2) ~ s 2 + s(1 - 2s2/c 2) G + ~

= 0.

(6)

Within the half-circular region ABEMDCPA (Fig. 1) the boundary conditions on w ( s , O ) are

O=rt,

s~c

7t--00<0 <~,

~=0

s=c:w--

0<0
s=c:~=0

0=0,

u<<.s<<.c:~=O

0=0,

0
O=klr

2err

++_G 0 ~ s < v : ~ = 0 .

(7a-f)

For s < c, the Chaplygin's transformation fl = cosh '(c/s)

(8)

reduces eq. (6) to the Laplace equation (oj~2 "]- ~

= 0

(9)

and maps the interior of the half-circular region [0 ~< 0 ~<~z, s ~(~),

~ = ~ +i~.

An appropriate transformation is U du iu, /7---75 ~ = c 0 f~, (u +~c)(U--~M)X/1 --U ~ +

(10)

208

A.N. DAS

0'

M

'IT

,n.-00 -B klr

M

D~

C iP

C

Fig. 2. The

IP

O-fl plane.

where Co is an arbitrary complex constant. The ~-plane is shown in Fig. 3. The transformation given by eq. (10) implies that the points E, A and D are mapped onto ( = 1, - 1 and 0, respectively. Equation (10) may be integrated to yield

=

~c

~M+ ~c

~

[in{x/(1 -- ~2)(1 -- (2) + (~c + 1} --In(( +¢c)]

~M

~

- (2) _ (~M + 1} -- ln(( - ~M)] + in.

[ln{~/(1 - ~ ) ( 1

(11)

gM + ~c Considering the change in imaginary parts in M and C, we obtain ~M

CO

= 1- k

(12)

and ~c

-

-

~M + ~c x/1

Co -

¢2

(13)

~k~

respectively. Thus the result (11) becomes with the aid of (12) and (13) = k[ln{x/( 1 _ ¢2)( 1 _ (2) + (~c + 1} - ln(( + ~c)] + (1 -- k)[ln{x/(1 -- ~2)(1 -- (2)--(~ M+ l} -- In(( - ~M)] + in.

(14)

Comparing the coordinates of point D in the v-plane and (-plane we obtain

),D=COsh-'C +ikn = k In 1 + ~ / 1 - ~ v

¢c

F(1--k)ln l + x / 1 - ~ 2 ~

~-ikn.

(15)

Comparison of the coordinates of point B in the v-plane and (-plane results in the relation ksin -II+¢c~B

A

-I

P

C

"~r -~c

(1-k)sin -l~B~M-l=n/2-00.

D

M

E

B

0

~M

I

~n

Fig. 3. The ~-r/ plane.

(16)

Trifurcation of a crack due to plane SH-waves

209

The mappings of point P yield cosh_~ c = k In u

1 -- ¢p¢c + .~/(1 -- ¢2)(1 -- ¢2)

1 + eP¢~a + ~/(1 -- ¢~a)(1 -- ¢~)

+ (1 - k)ln

~P - ~c

~p + ~M

(17) Equations (12), (13), (15), (16) and (17) can be used to solve for Co, eM, ¢c, ~B, eP. The boundary conditions given by eqs (7a-f) turn into the following conditions in the l-plane ~/=0,

-~<~<-¢p:w=0

r/=0,

0}0 --¢p < ~ < l:-0-~q = 0

r/=0,

2c~r I~<~<¢B:}0=----

q=0,

CB<¢ < ~ : } 0 = 0 .

(18a-d)

Following the technique [2] used in determining the relation between small distances from the crack tip in the physical plane and (-plane, it can be easily shown that for small values of r - vt in the crack plane 0 = kn for which Ill <<1

(2 __

2

[1 -- /)2/C2]-I/2 (r - vt)/vt,

COO

(19)

where COo= k x/1¢2

/ L + (l - k) x / ~ TMeM.

~C2

(20)

Equation (19) has been derived by expanding eq. (14) and maintaining terms of 0((2). The terms 0(() in the expansion are f o u n d to vanish with the aid o f the results (12) and (13). Further, eq. (19) suggests that ( = it/,

r/= [ 2 (1 - v2/c2) -I/2 -c r - vt ] I/2 . l)

ct

(21)

..]

Again expanding eq. (14) up to the order 0(( + ep) for I( + eel<< 1, we obtain for small values ofr-utin the p l a n e 0 = 0 ( + ep = - - - 1 (1 - u2/c2) - 1 / 2r - ,- ut COl

(22)

ul

with

(D, = ~ 1

2

,/1-¢I,]

k N/l~p_--¢c , / , ~-5c i i - k) ~P -~-~M" "

(23)

Next, if we take w = Re F(() then in view of the conditions given by eq. (18) it is found convenient to work with F ' ( O . Accordingly, we consider

F'(() = F~(¢) + F~(() + F~(l),

(24)

where F~ (() =

A

(25)

( l - eB)x/(1 - l ) ( l + ep)

1 r ~ ( ( ) = x / ( l _ l ) ( l + ~e)

B + ~5 ~-

(26)

210

A.N. DAS

and D

F~(()

x/(1 _ ( ) ( ( + ~a)3"

(27)

I n t e g r a t i n g eq. (25) with respect to ( a n d using the c o n d i t i o n that w possesses a j u m p d i s c o n t i n u i t y at ~ = ~B as seen f r o m eqs (18c, d) we find that A = 2 ca x/(¢B + CP)(~a - 1). re#

(28)

I n t e g r a t i n g eqs (26) a n d (27) we o b t a i n

F3 ( ( ) =

2D

1 +~p

1- ~

(29a, b)

Since the term involving the l o g a r i t h m gives rise to a l o g a r i t h m i c singularity at ( = 0 which is n o t acceptable, we require B = (1 - ~p)C/2~p.

(30)

T h e s h e a r stress at r > vt, 0 = k n can be o b t a i n e d using the relation

__0wo0 = - Im[F'(~) ~1 as zo: = - - /.t I m fr' F ' ( ~ ) d~ dt. r /,.

(31)

F o r all values o f k, ( can n o t be expressed in terms o f t explicitly. Hence, for k =A0 the i n t e g r a t i o n (31) i s to be c a r r i e d o u t numerically. In o r d e r to extract the singular term we c h a n g e the i n t e g r a t i o n o f F ~ ( ( ) over t in eq. (31) to an i n t e g r a t i o n over ~ as follows

/2 =

- - ~ Im ~ sinh fl(~)F'2(~) d(, C ,JI

where F is the c o r r e s p o n d i n g c o n t o u r in the ~-plane. Table 1 v/c = 0.2

k

St

0.0 0.1 0.2 0.3 0.4 0.5

- 14.857 --3.775 --2.145 --1.171 --0.312 --0.472

$2 3.631 3.982 4.623 5.319 6.040 6.702

0.0 0. I 0.2 0.3 0.4 0.5

-- 12.351 --2.983 -- 1.813 --0.992 --0.109 0.812

3.238 3.899 4.262 4.862 5.692 6.195

u/c = 0.6

0.0 0.1 0.2 0.3 0.4 0.5

--0.813 0.112 1.018 2.138 2.893 3.623

3.127 3.691 4.023 4.687 5.314 6.007

u/c = 0.8

00 = 90°.

u/c = 0.4

(32)

Trifurcation of a crack due to plane SH-waves

211

Table 2 k

SI

S2

vlc = 0.3

0.0 0.1 0.2 0.3 0.4 0.5

- 10.283 -2.781 - 1.783 - 1.021 -0.249 0.853

3.628 4.262 5.328 6.297 7.206 7.911

u/c = 0.4

0.0 0.1 0.2 0.3 0.4 0.5

-9.887 -2.527 - 1.232 -0.329 0.516 1.272

3.466 3.880 4.538 5.347 6.265 7.213

u/c = 0.6

0.0 0.1 0.2 0.3 0.4 0.5

--0.414 0.156 1.109 2.383 3.152 3.919

3.423 3.778 4.308 5.020 5.925 6.950

ulc = 0.8

Oo= 90°. Integrating eq. (32) by parts and then changing the variable in the integration over F; (() and F2(() to the variable s we obtain from eq. (31) *o~1 v =

__( #

sr'l'/2Im F 2(~)

1 -

(33)

-- #I,

where (34)

I = 13 + CI4 + D I s ,

with /3 -- 2 ~

N/(~a + ~v)(~a - 1) Im

s -2 d~

ds

d~ (~ _ ¢ . ) J ( i 14 = ~p-' Im 3,f" ~-~x/(( + ~p)(l - ~) sex/1 ds • __

- ~)(~ + ep) $2/¢

2

(35) (36)

and

/s = I m

s-2 d~ ds d'~ W ( 1 -- ~)(~ + ¢p)3"

(37)

Similarly, it can be easily shown that the shear stress at r > ut, 0 = 0 is Zo:l.=

-7~

( sq.2 1-

7~)

ImF3(()-#I'

(38)

where 1'=/3+

CI6+D17,

(39)

with /6 = Im

f[

1 + ~ p ) [ ( l - ¢ p ) / 2 ¢ p ~ + 1/~2]ds d-~ ~/(1 - ()(~

's_2d~

(40)

and

l, = 1 + ~---~ Im

~/ ¢ + ¢~ ~ / I - ~ /~ "

(41)

212

A.N. DAS

The stress intensity factors Nt and N2, at the crack tip defined by r = vt, 0 = k n and r = ut, 0 = 0, respectively, are obtained using eqs (21), (22), (29a, b), (33) and (40) as N, = r~,,,Ltx / 2 ~ ( r - vt) Zo:L,, = - L ~(°° vc

Nz= Lt x/2~(r-ut)zo=[.

l-

#C c,~-~p

I-~,]

L uc

]--~pj

2#D~t.

(42)

(43)

F r o m the asymptotic analysis o f the deformation field a b o u t a dynamically extending crack tip, we see that izo~ --* z as r --* vt - 0 on 0 = kn, then the regular term is z0~ as r --* vt + O, 0 = k n should also be equal to z. So, we have from eqs (3b, c), (33) and (40)

131,=.+

C141s=~, + D l s ) . . . . --

:31,=,+

2(7 - - - sin 00 cos krc # 2~

CI~I,=~+ DI~I~=,=- - -

sin 00.

(44)

(45)

P

Equations (44) anti (45) give the values o f C and D.

3. N U M E R I C A L R E S U L T S A N D D I S C U S S I O N In this section numerical calculations for the dimensionless stress intensity factors SL and $2, where S~ = NI/trx/~t and $2 = N : / o x / / - ~ , have been carried out. The variations o f stress intensity factors with k for different values o f v/c, u / c and 00 have been presented in tabular forms. F o r 00 = 90 °, the variations o f stress intensity factors are shown in Tables 1 and 2. F r o m Tables 1 and 2 we see that the values o f stress intensity factors increase with the increase in the values o f k and an increase in the values o f u / c decreases the values o f stress intensity factors at the edge r = ut, 0 = 0, but increases the values o f stress intensity factors at the edge r = vt, 0 = +_ kn. Further, it is noted from these tables that the values o f stress intensity factors at both the edges increase with the increase in the values o f v/c. In Tables 3 and 4, we have presented the variations o f stress intensity factors with k, u / c and v / c for 00 = 60 °. The variations o f stress intensity factors with varying crack tip speeds are now complicated as c o m p a r e d to the results given in Tables 1 and 2. Table 3 v/c = 0.2

k

SL

$2

0.0 0.1 0.2 0.3 0.4 0.5

-- 14.991 --4.064 --2.519 - 1.630 --0.851 --0.149

3.482 3.934 4.640 5.418 6.196 6.880

u/c

0.0 0.1 0.2 0.3 0.4 0.5

-- 12.431 --3.001 -- 1.927 - 1.028 --0.291 0.215

3.350 3.786 4.209 4.818 5.620 6.357

u/c = 0.6

0.0 0.1 0.2 0.3 0.4 0.5

-- 1.002 0.223 1.314 2.411 3.205 4.013

3.242 3.542 3.978 4.524 5.206 5.923

u/c = 0.8

0o =

~°.

=

0.4

Trifurcation of a crack due to plane SH-waves

213

Table 4 k

Si

S~

v/c = 0.3

0.0 0.I 0.2 0.3 0.4 0.5

-9.824 -2.842 - 1.948 - 1.273 -0.568 0.081

3.548 4.187 5.360 6.420 7.372 8.085

u/c = 0.4

0.0 0.1 0.2 0.3 0.4 0.5

-8.868 - 2.646 - 1.530 --0.702 0.076 0.773

3.267 3.695 4.391 5.207 6.083 6.907

u/c = 0.6

- 1.987 0.558 0.024 0.655 1.358 2.065

3.164 3.488 4.007 4.669 5.472 6.351

u/c = 0.8

0.0 0.1 0.2 0.3 0.4 0.5

-

00 = 60 ~'. 4. C O N C L U S I O N S T h e m a t h e m a t i c a l a n a l y s i s p r e s e n t e d in this p a p e r is a p o s t - t r i f u r c a t i o n a n a l y s i s o f a s e m i - i n f i n i t e c r a c k d u e to t w o p l a n e S H - w a v e s . U n d e r w h i c h c o n d i t i o n s a c r a c k m a y t r i f u r c a t e h a v e n o t b e e n d i s c u s s e d here, b u t it is k n o w n t h a t t h e r e are several f a c t o r s w h i c h c o n t r i b u t e to c r a c k c u r v i n g a n d b r a n c h i n g . O n e f a c t o r , o f c o u r s e , is b a s e d u p o n the c r i t e r i o n t h a t a c r a c k m a y p r o p a g a t e in the d i r e c t i o n o f the m a x i m u m s h e a r i n g stress o r in the d i r e c t i o n s o f p r e - e x i s t i n g flaws. H e n c e , t r i f u r c a t i o n o f a s e m i - i n f i n i t e c r a c k is i m p o s s i b l e if there are p r e - e x i s t i n g flaws in the d i r e c t i o n s 0 = +kr~ a n d the m a x i m u m s h e a r i n g stress o c c u r s a l o n g the line o f t h e s e m i - i n f i n i t e crack.

REFERENCES

[1] P. Burgers and J. P. Dempsey, Two analytical solutions for dynamic crack bifurcation in anti-plane strain. J. appl. Mech. 49, 366 (1982). [2] J. P. Dempsey, M. K. Kuo and J. D. Achenbach, Mode III crack kinking under stress wave loading. Wave Motion 4, 181 (1982). [3] P. Burgers, Dynamic propagation of a kinked or bifurcated crack in anti-plane strain. J. appl. Mech. 49, 371 (1982). [4] P. Burgers, Dynamic kinking of a crack in plane strain. Int. J. Solids Structures 19, 735 (1983). [5] J. D. Achenbach, M. K. Kuo and J. P. Dempsey, Mode III and mixed mode I-II crack kinking under stress wave loading. Int. J. Solids Structures 20, 395 (1984). [6] A. N. Das, Tearing of an elastic half-space. Engng Fracture Mech. 49, I (1994). [7] A. N. Das, Extension of cracks in an infinite elastic medium due to plane SH-waves. Engng Fracture Mech. 49, 243 (1994). (Received 31 May 1994)

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