On a Moving Griffith Crack at the Interface of two Bonded Dissimilar Orthotropic Half-Planes

SHORT COMMUNICATION

ZAMM � Z. Angew. Math. Mech. 81 (2001) 4, 281––287

Das, S.; Debnath, L.

On a Moving Griffith Crack at the Interfaceof two Bonded Dissimilar Orthotropic Half-Planes

The plane strain problem of determining strain energy release rate and crack energy for an interfacial moving Griffith cracksituated at the interface of two bonded dissimilar orthotropic half-planes is considered. The problem is reduced to solution ofa pair of simultaneous singular integral equations which are finally solved by using Jacobi polynomials. Expressions forstrain energy release rate and crack energy are obtained for some particular cases and the results are presented graphically.

Key words: interfacial crack, orthotropic medium, Mach number, stress intensity factor, strain energy release rate, crackenergy, integral equation

MSC (2000): 74A10, 74A50, 45E05

1. Introduction

Anisotropy and flaw at the interface of two bonded materials are of great importance in designing engineering structuresand machines. Due to practical importance, there is a need for a study of stress field in the presence of a crack at theinterface of two dissimilar anisotropic bonded materials. Several authors including Erdogan and Gupta [5, 6], Dhali-wal et al. [4], Rice and Sih [13], Kadioglu and Erdogan [10],Wang and Choi [14], Lowengrub and Sneddon [11],Das and Patra [1, 2], Erdogan andWu [7], and He et al. [8] have considered problems of interfacial Griffith cracks onthe bonded materials. In spite of these studies, the corresponding problems dealing with a moving Griffith crack at theinterface of two bonded orthotropic half-planes have received much less attention. So this paper deals with such a problem.

The main purpose of this work is to study the plane strain problem of determining strain energy release rate andcrack energy for an interfacial moving Griffith crack situated at the interface of two bonded dissimilar orthotropic half-planes. The problem is then reduced to the solution of a pair of simultaneous singular integral equations with theCauchy kernel which have ultimately been solved by using the Jacobi polynomials. Expressions for the strain energyrelease rate and crack energy are obtained for some particular cases and results are presented graphically.

2. Formulation of the problem

We consider the plane elastodynamic problem in orthotropic half-plane 1 ð�1 < X <1; 0 � Y <1Þ bonded to adissimilar orthotropic half-plane 2 ð�1 < X <1; �1 < Y � 0Þ with a moving Griffith crack of finite length situatedat the interface of the two materials. The principal axes of the materials coincide with the Cartesian coordinate axes.As in Yoffe model [15], it is assumed that the cracks are propagating with constant velocity c and without change inlength along the positive X-axis. In what follows and in the sequel the quantities with superscripts i ¼ 1; 2 refer tothose for the half-planes 1 and 2, respectively.

Under the assumption of plane strain in an orthotropic medium, equations of motions for the displacement fieldsare (see Piva and Viola [12], De and Patra [3], Itou [9])

CðiÞ11

@2uðiÞ

@X2þ C

ðiÞ66

@2uðiÞ

@Y 2þ ðCðiÞ

12 þ CðiÞ66 Þ

@2vðiÞ

@X @Y¼ qðiÞ

@2uðiÞ

@t2; ð2:1Þ

CðiÞ22

@2vðiÞ

@Y 2þ C

ðiÞ66

@2vðiÞ

@X2þ ðCðiÞ

12 þ CðiÞ66 Þ

@2uðiÞ

@X @Y¼ qðiÞ

@2vðiÞ

@t2; ð2:2Þ

where i ¼ 1; 2, qðiÞ and CðiÞjk are respective densities and elastic constants of the half-planes 1 and 2. Applying the

Galilean transformation x ¼ X � ct, y ¼ Y , t ¼ t, eqs. (2.1)––(2.2) become independent of t and reduce to

@2uðiÞ

@x2þ 2bðiÞ @

2vðiÞ

@x @yþ aðiÞ @

2uðiÞ

@y2¼ 0 ; ð2:3Þ

@2vðiÞ

@x2þ 2b

ðiÞ1

@2uðiÞ

@x @yþ a

ðiÞ1

@2vðiÞ

@y2¼ 0 ; ð2:4Þ

Short Communication 281

where uðiÞðx; yÞ ¼ uðiÞðX;Y; tÞ; vðiÞðx; yÞ ¼ vðiÞðX; Y; tÞ, and

aðiÞ ¼ CðiÞ66

CðiÞ11 ð1�M

ðiÞ21 Þ

; aðiÞ1 ¼ C

ðiÞ22

CðiÞ66 ð1�M

ðiÞ22 Þ

; ð2:5a; bÞ

2bðiÞ ¼ CðiÞ12 þ C

ðiÞ66

CðiÞ11 ð1�M

ðiÞ21 Þ

; 2bðiÞ1 ¼ C

ðiÞ12 þ C

ðiÞ66

CðiÞ66 ð1�M

ðiÞ22 Þ

; i ¼ 1; 2 ; ð2:6a; bÞ

MðiÞj ¼ c

nðiÞj

; j ¼ 1; 2 ; ð2:7Þ

nðiÞ1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC

ðiÞ11 =q

ðiÞq

; nðiÞ2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC

ðiÞ66 =q

ðiÞq

: ð2:8a; bÞ

The Mach numbers MðiÞj ði; j ¼ 1; 2Þ are less than unity for a subsonic propagation. The stress-displacement relations

are

sðiÞxx ¼ C

ðiÞ11

@uðiÞ

@xþ C

ðiÞ12

@vðiÞ

@y; ð2:9Þ

sðiÞyy ¼ C

ðiÞ22

@vðiÞ

@yþ C

ðiÞ12

@uðiÞ

@x; ð2:10Þ

sðiÞxy ¼ C

ðiÞ66

@uðiÞ

@yþ @vðiÞ

@x

� �; i ¼ 1; 2 : ð2:11Þ

Other stress components being zero as displacements are assumed to be in xy-plane and are independent of z. It isassumed that the crack located at jxj � 1; y ¼ 0, is opened by internal tractions p1ðxÞ and p2ðxÞ on the crack face.The boundary conditions at the interface y ¼ 0 are

sð1Þyy ðx; 0Þ ¼ sð2Þ

yy ðx; 0Þ ¼ �p1ðxÞ ; jxj � 1 ; ð2:12Þ

sð1Þxy ðx; 0Þ ¼ sð2Þ

xy ðx; 0Þ ¼ �p2ðxÞ ; jxj � 1 ; ð2:13Þ

uð1Þðx; 0Þ ¼ uð2Þðx; 0Þ ; jxj > 1 ; ð2:14Þ

vð1Þðx; 0Þ ¼ vð2Þðx; 0Þ ; jxj > 1 ; ð2:15Þ

sð1Þyy ðx; 0Þ ¼ sð2Þ

yy ðx; 0Þ ; jxj > 1 ; ð2:16Þ

sð1Þxy ðx; 0Þ ¼ sð2Þ

xy ðx; 0Þ ; jxj > 1 : ð2:17Þ

In addition, all components of stress and displacement fields vanish at far distances from the crack.

3. Solution of the problem

The displacement and stress components are given by

uðiÞðx; yÞ ¼ 1ffiffiffiffiffiffi2p

pð1

�1

x�1½AðiÞ1 ðxÞ expfð�1Þi g

ðiÞ1 xyg þ A

ðiÞ2 ðxÞ expfð�1Þi g

ðiÞ2 xyg� e�ixx dx ; ð3:1Þ

vðiÞðx; yÞ ¼ ð�1Þiþ1ffiffiffiffiffiffi2p

pð1

�1

ix�1½AðiÞ1 ðxÞ kðiÞ1 ðxÞ kðiÞ1 expfð�1Þi g

ðiÞ1 xyg þA

ðiÞ2 ðxÞ kðiÞ2 expfð�1Þi g

ðiÞ2 xyg� e�ixx dx ;

ð3:2Þ

sðiÞyy ðx; yÞ ¼ � 1ffiffiffiffiffiffi

2pp

ð1

�1

i½ðCðiÞ12 þ C

ðiÞ22 k

ðiÞ1 g

ðiÞ1 ÞAðiÞ

1 ðxÞ expfð�iÞi gðiÞ1 xyg

þ ðCðiÞ12 þ C

ðiÞ22 k

ðiÞ2 g

ðiÞ2 ÞAðiÞ

2 ðxÞ expfð�1Þi gðiÞ2 xyg� e�ixx dx ; ð3:3Þ

sðiÞxyðx; yÞ ¼

ð�1Þiffiffiffiffiffiffi2p

pð1

�1

½CðiÞ66 ðg

ðiÞi � k

ðiÞ1 ÞAðiÞ

1 ðxÞ expfð�1Þi gðiÞ1 xyg

þ CðiÞ66 ðg

ðiÞ2 � k

ðiÞ2 ÞAðiÞ

2 ðxÞ expfð�1Þi gðiÞ2 xyg� e�ixx dx ; ð3:4Þ

282 ZAMM � Z. Angew. Math. Mech. 81 (2001) 4

where gðiÞ1 , g

ðiÞ2 ð<g

ðiÞ1 Þ are the positive roots of the equation

aðiÞ2 g4 � 2a

ðiÞ1 g2 þ 1 ¼ 0 ; ð3:5Þ

and

KðiÞ1 ¼ aðiÞg

ðiÞ21 � 1

2gðiÞ1 bðiÞ ; K

ðiÞ2 ¼ aðiÞg

ðiÞ22 � 1

2gðiÞ2 bðiÞ ;

2aðiÞ1 ¼ aðiÞ þ a

ðiÞ1 � 4bðiÞb

ðiÞ1 ; and a

ðiÞ2 ¼ aðiÞa

ðiÞ1 :

Boundary conditions (2.12) and (2.16) yield

hð1Þ1 A

ð1Þ1 ðxÞ þ h

ð1Þ2 ðxÞ ¼ h

ð2Þ1 ðxÞ þ h

ð2Þ2 A

ð2Þ2 ðxÞ ; ð3:6Þ

where

hðiÞj ¼ C

ðiÞ12 þ C

ðiÞ22K

ðiÞj g

ðiÞj ; i; j ¼ 1; 2 :

Again boundary conditions (2.13) and (2.17) give

mð1Þ1 A

ð1Þ1 ðxÞ þ m

ð1Þ2 A

ð1Þ2 ðxÞ ¼ m

ð2Þ1 A

ð2Þ1 ðxÞ þ m

ð2Þ2 A

ð2Þ2 ðxÞ ; ð3:7Þ

where

mðiÞj ¼ C

ðiÞ66 ðg

ðiÞj � k

ðiÞj Þ ; i; j ¼ 1; 2 :

If we restrict our attention to the anti-symmetric part of the problem in which p1ðxÞ ¼ �p1ð�xÞ, p2ðxÞ ¼ p2ð�xÞ,the boundary conditions (2.12)––(2.15) in conjunction with eqs. (3.6) and (3.7) lead to the following conditions:

ffiffiffiffiffi2

p

r ð1

0

½hð1Þ1 A

ð1Þ1 ðxÞ þ h

ð1Þ2 A

ð1Þ2 ðxÞ� sin xx dx ¼ p1ðxÞ ; 0 � x � 1 ; ð3:8Þ

ffiffiffiffiffi2

p

r ð1

0

½mð1Þ1 A

ð1Þ1 ðxÞ þ m

ð1Þ2 A

ð1Þ2 ðxÞ� cos xx dx ¼ p2ðxÞ ; 0 � x � 1 ; ð3:9Þ

ffiffiffiffiffi2

p

r ð1

0

½L1Að1Þ1 ðxÞ þ L2A

ð1Þ2 ðxÞ� sin xx dx ¼ 0 ; x � 1 ; ð3:10Þ

ffiffiffiffiffi2

p

r ð1

0

½M1Að1Þ1 ðxÞ þM2A

ð1Þ2 ðxÞ� cos xx dx ¼ 0 ; x � 1 ; ð3:11Þ

where

Li ¼ 1þ hð1Þi ðmð2Þ

i � mð2Þ2 Þ þ m

ð1Þi ðmð2Þ

1 � hð2Þ2 Þ

mð2Þ2 h

ð2Þ1 � m

ð2Þ1 h

ð2Þ2

;

Mi ¼ ki þhð1Þi ðkð2Þ1 m

ð2Þ2 � k

ð2Þ2 m

ð2Þ1 Þ þ m

ð1Þi ðkð2Þ1 h

ð2Þ2 � k

ð2Þ2 h

ð2Þ1 Þ

mð2Þ2 h

ð2Þ1 � m

ð2Þ1 h

ð2Þ2

:

Following the work of Erdogan and Gupta [6], we set

ffiffiffiffiffi2

p

r ð1

0

½L1Að1Þ1 ðxÞ þ L2A

ð1Þ2 ðxÞ� sin xx dx ¼ 1

x

ð1

0

f1ðtÞ cos xt dt ; ð3:12Þ

ffiffiffiffi2

p

r ð1

0

½M1Að1Þ1 ðxÞ þM2A

ð1Þ2 ðxÞ� cos xx dx ¼ 1

x

ð1

0

f2ðtÞ sin xt dt ; ð3:13Þ

where f1ðtÞ and f2ðtÞ are even and odd functions of t, respectively. Then eqs. (3.10) and (3.11) under the conditions

Ð1�1fiðtÞ dt ¼ 0 ; i ¼ 1; 2 ; ð3:14Þ


are identically satisfied. Eqs. (3.8) and (3.9), after some algebra, become

a1f1ðxÞ þ1

pb1

ð1

�1

f2ðtÞ dtt� x

¼ � 1

pp1ðxÞ ; ð3:15aÞ

c1f2ðxÞ �1

pd1

ð1

�1

f1ðtÞ dtt� x

¼ � 1

pp2ðxÞ ; d < x < 1 ; ð3:15bÞ

where

a1 ¼ � hð1Þ1 M2 � h

ð1Þ2 M1

L1M2 � L2M1;

1

b1¼ � h

ð1Þ1 L2 � h

ð1Þ2 L1

L1M2 � L2M1; ð3:16aÞ

c1 ¼mð1Þ1 L2 � m

ð1Þ2 L1

L1M2 � L2M1;

1

d1¼ m

ð1Þ1 M2 � m

ð1Þ2 M1

L1M2 � L2M1: ð3:16bÞ

As a1, b1, c1, and d1 depend on the material constants and the velocity of propagation c, the signs of these quantitiesmay be of any combination. Varying c such that the Mach numbers remain less than unity, if the signs of thesequantities are all positive, then eqs. (3.15a, b) can be expressed as

fkðxÞ þ1

piefk

ð1

�1

fkðtÞ dtt� x

¼ �gkðxÞ ; �1 < x < 1 ; ð3:17Þ

where

fkðxÞ ¼ffiffiffiffiffiffiffiffiffia1b1

pf1ðxÞ þ ifk

ffiffiffiffiffiffiffiffiffic1d1

pf2ðxÞ ;

gkðxÞ ¼1

p½

ffiffiffiffiffiffiffiffiffiffiffib1=a1

pp1ðxÞ þ ifk

ffiffiffiffiffiffiffiffiffiffiffid1=c1

pp2ðxÞ� ; k ¼ 1; 2 ;

f1 ¼ 1 ; f1 ¼ �1 ; e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1b1c1d1

p:

When a1, c1, and d1 are positive and b1 ¼ �b2 < 0, eqs. (3.15a, b) can be put into the form

fkðxÞ þ1

pefk

ð1

�1

fkðtÞt� x

dt ¼ �gkðxÞ ; �1 < x < 1 ; ð3:18Þ

where

fkðxÞ ¼ffiffiffiffiffiffiffiffiffia1b2

pf1ðxÞ þ fk

ffiffiffiffiffiffiffiffiffic1d1

pf2ðxÞ ;

gkðxÞ ¼1

p½


pp1ðxÞ þ fk


pp2ðxÞ� ; k ¼ 1; 2 :

f1 ¼ �1 ; f1 ¼ 1 ; e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1b1c1d1

p:

For other combination of signs of a1, b1, c1, and d1, eqs. (3.15a, b) can similarly be handled.

4. Solution of the integral equations

The solution of the integral equations in (3.17) may be assumed as

fkðxÞ ¼ wkðxÞP1n¼0

CknPðak; bkÞn ðxÞ ð4:1Þ

where

wkðxÞ ¼ ð1� xÞak ð1þ xÞbk ; ak ¼ �12 þ iwk ; bk ¼ �1

2 � iwk ; wk ¼ wfk ; k ¼ 1; 2 ;

w ¼ 1

2pln

1þ e

1� e

; and Ckn are unknown constants:

By virtue of (3.14), we have

Ck0 ¼ 0 ; k ¼ 1; 2 : ð4:2Þ


Using the result

1

pi

ð1

�1

wkðtÞ P ðak; bkÞn ðtÞ dt

t� x¼ �efkwkðxÞ P ðak; bkÞ

n ðxÞ þffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p

2iP

ð�ak;�bkÞn�1 ðxÞ ; �1 < x < 1 ;

ð1� efkÞ ½wkðxÞ P ðak; bkÞn ðxÞ �G1

knðxÞ� ; jxj > 1 ;

8><>:

where G1kn is the principal part of wkðxÞ P ðak; bkÞ

n ðxÞ at infinity, the integral equation (3.17) with the aid of (4.1) givesffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p

2iefk

P1n¼1

CknPð�ak;�bkÞn�1 ðxÞ ¼ �gkðxÞ ; k ¼ 1; 2 : ð4:3Þ

Multiplying both sides of (4.3) by w�1k P

ð�ak;�bkÞj ðxÞ and integrating with respect to x from �1 to þ1 and using ortho-

gonality relation the values of the unknowns Ckj (k ¼ 1; 2, j ¼ 0; 1; 2; . . .) are given by

Ckjþ1 ¼ � iefkgkjffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p ðjþ 1Þ! Gðjþ 2ÞGðj� ak þ 1Þ Gðj� bk þ 1Þ ; ð4:4Þ

where

gkj ¼Ð1�1gkðxÞ w�1

k ðxÞ P ð�ak; bkÞj ðxÞ dx :

The stress intensity factors near the crack tip x ¼ 1 may be calculated asffiffiffiffiffiffiffiffiffiffiffib1=a1

pKI þ ifk


pKII ¼ lim

x!1þðx� 1Þ�ak ðxþ 1Þ�bk ½


psð1Þyy ðx; 0Þ þ ifk


psð1Þxy ðx; 0Þ�

¼ � ipffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p

efk

P1n¼1

CknPðak; bkÞn ð1Þ : ð4:5Þ

To find the energy dU available for fracture at a crack tip x ¼ 1 for a crack extension da, one may consider, withoutloss of generality the “fixed grip” condition under which dU can be calculated from the ‘crack closure energy’ asfollows:

dU ¼ 12

Ð1þda1

fsð1Þyy ðx; 0Þ ½vð1Þðx� da; 0Þ � vð2Þðx� da; 0Þ� þ sð1Þ

xy ðx; 0Þ ½uð1Þðx� da; 0Þ � uð2Þðx� da; 0Þ�g dx : ð4:6Þ

In a closed neighborhood of the crack tip x ¼ 1 the displacement derivatives may be expressed as

fkðxÞ ’ ð1� xÞak 2bkP1n¼1

CknPðak; bkÞn ð1Þ : ð4:7Þ

Now,

K ¼ffiffiffiffiffiffiffiffiffiffiffib1=a1

pKI þ irk


pKII � � ip

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p

2erk

P1n¼1

CknPðak; bkÞn ð1Þ :

Therefore,

fkðxÞ ’2ierk

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p 2bkð1� xÞak K ;

sðxÞ ¼ffiffiffiffiffiffib1a1

rsð1Þyy ðx; 0Þ þ irk

ffiffiffiffiffiffid1c1

rsð1Þxy ðx; 0Þ � 2bkðx� 1Þak 2bk ; ð4:8Þ

V ðxÞ ¼ffiffiffiffiffiffiffiffiffia1b1

p½uð1Þðx; 0Þ � uð2Þðx; 0Þ� þ irk

ffiffiffiffiffiffiffiffiffid1c1

p½V ð1Þðx; 0Þ � V ð2Þðx; 0Þ�

� 21þbke

pirkffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p ð1� xÞ1þak

1þ akK : ð4:9Þ

By substituting (4.8) and (4.9) into (4.6), we obtain

dU ¼ 1

2

ð1þda

1

AssðxÞ V ðx� daÞ dx ¼ e

2da K AKK : ð4:10Þ

The strain energy release rate is given by

G ¼ dU

da¼ e

2K AKK : ð4:11Þ


The expression for crack energy is given by

W ¼ �Ð1�1

p1ðxÞ ½vð1Þðx; 0Þ � vð2Þðx; 0Þ� dx : ð4:12Þ

When a1; c1; d1 > 0 and b1 ¼ �b2 < 0, the corresponding system of algebraic equations for the determination of Ckn isffiffiffiffiffiffiffiffiffiffiffiffiffi1þ e2

p

2efk

P1n¼1

CknPð�ak;�bkÞn�1 ðxÞ ¼ �gkðxÞ ; k ¼ 1; 2 ; ð4:13Þ

with the following modified values of

ak ¼ � 1

2� fk

ptan�1 e ; bk ¼ � 1

2þ fk

ptan�1 e ; k ¼ 1; 2 ;

w ¼ i

ptan�1 e and e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1b2c1d1

p:

The stress intensity factors are calculated as


pKI þ fk


pKII ¼ �p

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ e2

p

efk

P1n¼1

CknPðak; bkÞn ð1Þ ð4:14Þ

and the strain energy release rate and the crack energy can be calculated from eqs. (4.11) and (4.12).

5. Numerical results and discussion

We consider the particular case of the problem when p1ðxÞ ¼ p and p2ðxÞ ¼ 0, p being a constant. Then

Ck1 ¼� 2ipfkeffiffiffiffiffiffiffiffiffiffiffiffiffi

1� e2p


p; when b1 > 0 ;

� 2pfkeffiffiffiffiffiffiffiffiffiffiffiffiffi1þ e2

pffiffiffiffiffiffiffiffiffiffiffib2=a1

p; when b1 ¼ �b2 < 0 ;

8>>><>>>:

and

Ckj ¼ 0 ; j 6¼ 1 :

The stress intensity factors are then calculated as

KI ¼ �p

KII ¼� p

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib1c1=a1d1

pln

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1b1c1d1

p

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1b1c1d1

p

; when b1 > 0 ;

� 2p

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2c1=a1d1

ptan�1 ð


pÞ ; when b1 ¼ �b2 < 0 :

8>>><>>>:

ð4:15Þ

The strain energy release rate is calculated as

G

p2¼


p

2

b1a1

K2I þ

d1c1K2II

� �; when b1 > 0 ;


p

2

b2a1

K2I �

d1c1K2II

� �; when b1 ¼ �b2 < 0 :

8>>><>>>:

ð4:16Þ

The crack energy is calculated as

W ¼

p

2

b21p2ffiffiffiffiffiffiffiffiffiffiffiffiffi

1� e2p ð1þ 4w2Þ sech pw ; when b1 > 0 ;

p

2b2p

2 1� 4

p2ðtan�1 eÞ2

� �; when b1 ¼ �b2 < 0 :

8>>><>>>:

As an illustration, graphical plots of strain energy release rate G=p2ð Þ and crack energy W=p2ð Þ with crack speed c fora-uranium and Beryllium composite are presented through Figs. 1––4 for subsonic propagation. It is observed fromFig. 1 and Fig. 3 that both the G=p2ð Þ and W=p2ð Þ continuously increase in magnitude up to the crack speed c ¼ 0:575and then these have an expected oscillatory nature for 0:575 < c � 0:624 as there is a change of propagation phasefrom the subsonic to supersonic. These phenomena have been depicted in Fig. 2 and Fig. 4, which are the zoomedportion of Figs. 1 and 3 corresponding to 0:575 < c � 0:624.


References

1 Das, S.; Patra, B.: Moving Griffith crack at the interface of two dissimilar orthotropic half planes. Engin. Fracture Mech. 54(1996), 523––531.

2 Das, S.; Patra, B.: Stress intensity factors for an interfacial crack of a pair of bonded dissimilar orthotropic layers. Internat. J.Computers Structures 69 (1998), 472––492.

3 De, J.; Patra, B.: Moving Griffith crack in an orthotropic strip. Internat. J. Engin. Sci. 28 (1990), 809––819.4 Dhaliwal, R. S.; Saxena, H. S.; Rokne, J. G.: Elasticity: Mathematical methods and applications. In: The I. N. Sneddon 70thbirthday volume. Ellis Horwood Ltd. 1990, pp. 59––86.

5 Erdogan, F.; Gupta, G.: Layered composites with an interface flaw. Internat. J. Solids Structures 7 (1971), 1089––1107.6 Erdogan, F.; Gupta, G.: In: Sih, G. C. (ed.): Methods of analysis and solutions of crack problems. 1973, pp. 368ff.7 Erdogan, F.; Wu, B.: Interface crack problems in layered orthotropic materials. J. Mech. Phys. Solids 41 (1993), 889––917.8 He, W.; Dhaliwal, R. S.; Saxena, H. S.: Griffith crack at the interface of two orthotropic elastic layers. Engin. Fract. Mech.

41 (1992), 13––25.9 Itou, S.: Dynamic stress intensity factors around two coplanar Griffith cracks in an orthotropic layer sandwiched between twoelastic half planes. Engin. Fract. Mech. 34 (1989), 1085––1095.

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12 Piva, A.; Viola, E.: Crack propagation in an orthotropic medium. Engin. Fract. Mech. 29 (1988), 535––548.13 Rice, J. R.; Sih, G. C.: Plane problems of cracks in dissimilar media. J. Appl. Mech. 32 (1965), 418––432.14 Wang, S. S.; Choi, I.: The interface crack between dissimilar anisotropic composite materials. J. Appl. Mech. 50 (1983),

169––178.15 Yoffe, E. F.: The moving Griffith crack. Philos. Mag. 42 (1951), 739.

Received July 28, 1999, revised February 29, 2000, accepted March 10, 2000

Addresses: Prof. S. Das, Department of Mathematics, B.E. Poddar Institute of Management and Technology, Poddar Vihar, 137 ––V.I.P. Road Calcutta 700 052, West Bengal, India; Prof. L. Debnath, Department of Mathematics, University of CentralFlorida, Orlando, FL 32816, USA


Fig. 1. Plot of ðG=p2Þ against c Fig. 2. Plot of ðG=p2Þ against c

Fig. 3. Plot of ðW=p2Þ against c Fig. 4. Plot of ðW=p2Þ against c

Documents

On a Moving Griffith Crack at the Interface of two Bonded Dissimilar Orthotropic Half-Planes