A griffith crack at the interface of two bonded dissimilar orthotropic elastic half-planes

E~g~eeri~g Frveture technics Printed

Vol. 37, No. 4,

~~3-7~/~

pp.8 17-824, 1990

$3.00 + 0.00

PergamonPresspk.

in GreatBritain.

A GRIFFITH CRACK AT THE INTERFACE OF TWO BONDED DISSIMILAR ORTHOTROPIC ELASTIC HALF-PLANES RANJIT S. DHALIWAL and HIRDESHWAR

S. SAXENA

Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada, T2N lN4 Abstrae&-The problem considered is that of dete~ining the stress intensity factors when a Griffith crack is located at the interface of two bonded dissimilar orthotropic elastic half-planes. A Fourier transform technique is employed to reduce the mixed boundary value problem to solving a set of simultaneous singular integral equations of the second kind. By the use of Jacobi polynomials, these equations are further reduced to a system of simultaneous algebraic equations. It is shown that both the stress intensity factors K, and Kz occur and also that they depend on elastic properties of the materials when the crack is opened by an applied normal stress, which is contrary to the results obtained by Zhang.

1. INTRODUCTION THE THEORY of crack problems in two-dimensional elasticity was first developed by Griffith[l] in 1921. Sneddon and Elliot[2] considered the problem of dist~bution of stresses in the vicinity of a crack as a mixed boundary value problem. Rice and Sih[3], Erdogan and Gupta[4], Lowengrub and Sneddon[S], Sih[6] and others have discussed the problems of interface crack in isotropic elasticity. Dhaliwal[7], Parhi and Atsumi[8] and Parihar and Lalitha[9] considered the crack problems in anisotropic elasticity. Recently Zhang[lO] discussed the problem of a central crack at the interface of two dissimilar orthotropic elastic media in modes I and II. Due to some fundamental errors in his method~lO], his conclusions do not agree with those obtained earlier by other authors. In this paper, we consider the problem[lO], to present the correct method, of determining the stress intensity factors K1 and K2 when a Griffith crack 1x1g 1, y = 0, is located at the interface of two bonded dissimilar orthotropic elastic half-planes. Fourier transform method is used to obtain solution of the basic equations. Lowengrub and Sneddon’s technique[5] is employed to reduce the system of dual integral equations to a set of sim~taneous singular integral equations which are further transformed to simultaneous algebraic equations by using Jacobi polynomials[4, 1I]. Contact stresses on the interface for /xl> 1 and stress intensity factors are then obtained analytically.

2. BASIC EQUA~ONS Under the assumption of plane strain in an orthotropic elastic medium when the rectangular coordinate axes coincides with the principal material axes, we have u = u(x, y),

v = u(x, y),

where u, u and w are displacement components stress-displacement relations are given by

au

flxx = Cl1z

au

+ Cl2 ;j;;’

au ~YY’%

z

817

w = 0,

(1)

in x-, y- and z-directions

av + S2dy’

respectively.

av au

axy = GiIj

( > ;ji; + ay

,

The

where cV (i,j = x, y) and c0 (i,j = 1,2,6) are respectively the stress components and the elastic moduli of the orthotropic material. Equations of equilibrium in terms of displacements, in the absence of body forces, may be expressed as follows:

-

e11~+ct$5~+(e,,+c,)

axay

8Y’

2

a2u

a2V

% ;

+

c22 -

a2u

+

3Y2

h2

-t

cd

-

ax ay

-0, (3) = 0.

~

The Fourier transforms uf eqs (3) are given by

(4) where

G, Y) = -

1 2x

J-

s m

_

m

~Cw~eiCXdx, W,y) =-

1

d-2z

s m

u fx,y )ertx dx,

--OS

(5)

and L> = d[dy. Let ctiI and c$ be the elastic moduli for the upper (Y >, 0) and the lower Cy < 0) orthotropic media respectively. Eqs (4) yield the following solution:

where

and Ai( I$({) (i = 1,2) are the unknowns to be determined. When there is symmetry about the line x = 0, the Fourier inverse transforms of eqs (6) lead to the following solution for u(x, Y) and @G y):

u(x, y) =

2e6z6 + B2es2b) sin cx d& y CO.

V(X?Y)’ Js

2 0e (a2Azep2tY+ y2B2ed2cY} R cos TX d<, i

(9)

Griffith crack at the interface of two half-planes

819

substituting eqs (8) and (9) into eqs (2), we get

a,,@, Y) =

2 om Js ?c

(a,lAle-81tY+

cos 5x d&

yI,B,e-'ltY){

Y >o,

a,,(& y) =

0,(&Y)

(10)

sin 5x d&

* (a,2Ale-B1CY + y,2B,e-61@)<

eszcy+ y21B2e6zeY)rcos 5x d&

=

m (az2 A, ehty+

1

Js n 0

yz2B2es2’J’)
where amI= c12m+ c22ma,8,V am2= ~~~0% - a,), Yml= Clzm+ C22mYm4n, 3. DERIVATION

Ym2 = %m&L

OF THE SINGULAR

(m = 1,2).

(12)

- ym),

INTEGRAL

EQUATIONS

We consider the problem of determining stress intensity factors in the neighborhood of the central Griffith crack Ix) 6 1, y = 0 located at the interface of the two bonded dissimilar orthotropic half-planes (y > 0 and y < 0). It is assumed that the upper and lower faces of the crack are each subjected to prescribed tractions p,(x) and p2(x). The boundary conditions at the common interface may be stated as follows: ~,,(x,o)=P,(x),

u(x, o-) = u(x, o+),

~xyw)=P2(~),

1x1 G 1,

u(x, o-) = V(X,o+),

1x1> 1,

e,,(x, 0-) = a,,(& O+), cxy(x, 0-) = ex,(x, Of), The stresses and the displacements are assumed to be zero as J(x’ to modes I and II respectively when (i) p&d

= -P(X),

(13) (14)

1x1> 0.

(15)

+ y2)+ cc, The problem reduces

p2(x) = 0,

(16)

P2@)

(17)

and (ii) pI (4 = 0,

= 7(x).

Applying boundary conditions (15), eqs (10) and (11) yield A2(5)=~13,A,(r)+a~~B,(e),

&(r)=

-[a,lA,(r)+a~2BI(r)l,

Uf9

where +I =

(ally22 + a12y2JlR

a4l = (a11a22+ a12a2JlD,

a32 = (~~~~22+

Y~~Y~~)/&

a42 =

a21~12)lQ

(a22yll

+

D = a2!y22- a22Y21.

(19) I

Substituting from eqs (8) and (9) into boundary conditions (14), leads to the equations

s

m[(A2 + BJ - (A, + B,)] sin
0

m Ka2A2 s 0

+

y2B2) - (a, AI + Y14)lcos

[XI>

5~ dt = 0,

1,

(20)

820

RANJIT S. D~ALIWAL

and ~I~ESHWAR

S. SAXENA

respectively, and using eqs (18) we may reduce these in turn to m K%, - c(~,- l)A, + (~32s0

0~42-

sin {X d< = 0,

I)&]

/XI>

?( a2a31-

y2C41+a,)Al

1.

(21)

V$lcosb d5 =%

+(tf2~--y~a~~+~,

s0

Applying boundary conditions (13), we obtain, from eqs (lo), the following relations: mta,,A,+~,,B,)L:cos5~d5= s0

%a,(~), J Ix/<

z (a,2A, +~,~h)t J0

sintx

d5=

-

1.

(22)

:~2(x), J

;

If we now express A, (5) and B,(S) in terms of two new functions cbi(T) (i = 1,2) through the relations A,(<) = 5-‘E(a2aj2 -YzC142+Y,)6),(5)+(a32-a42-

l)~*(~)l~~*,

B, (0 = 5 -%a2a1, - Yza4,+ a,M,(5) + (a3, - a4, -

~M~)l/~*,

(23)

where D* = -(aza3, - yzcr,l + g,)(aS2 - a42- 1) +&a,,

- y2a4, + y,)(a,, - a4, - 1),

(24)

we may reduce eqs (22) and (21) to the set of simultaneous dual integral equations m~,,~,(5)+11,2~2(t;)lcosexd9= s0

;P,(x),

(xl< 1,

(25)

~pdx),

/x/c 1,

(26)

\i

mI~l~l(g)fILU~2(r)lsintxdg= s0

dom t-' 4,(t)

sin 5x dr = 0,

1x1> 1,

(27)

I+l,

(28)

s

~~~-,~2(~)cos~~d~=O, s

h

=

[alIta2a32-

y2a42+

Y,)- Ylltw31 -

y2a4t +

yll t% - a41-

1)1/D*,

adIP*,

k2 =

La,,ta32-

p21=

Fa12ta2a32 - Y2a42+ YJ + y12ta2a3, - y2a4, -+ aJl/D*~ [-a12(a32 - a42 - 1)+ Y12ta3, - aql - 1)1/D*.

1122=

a42-

1)-

(29)

:

To reduce the system of dual integral eqs (25)-(28) to a set of singular integral equations of second kind we follow the method of Lowengrub and Sneddon[S]. If we differentiate both sides of eqs (27) and (28), cbi(<) (i = 1,2) must satisfy the following equivalent set of simultaneous dual integral equations: m~,,~,(S)+~,~~~(~)1cos5x sa “[~2,$,(4))+ s0

d5=

~~4~~(5)lsintx dt= xqM4)costxdt=0,

J

J

;P,(x),

1x1~ 1,

~PAx), /X/-C 1, I++

(30)

(31) (32)

821

Griffith crack at the interface of two half-planes

m&(5)sin
Ixl>l.

(33)

0

Let us now set d, (5) cos Cx d5 =

Il/,o(xWU

-

XL 1

-2 m A(5) sin e-x d5 = $20(x)Wl -x), rc 0

Js

(34)

J

where H( ) denotes the Heaviside unit function, then from ref. [5], we have

where e,(x) and &(x) are respectively the even extension of tile(x) and odd extension of tizo(x) to [- 1, 11. Using relations (34) and (35), we find that eqs (32) and (33) are identically satisfied whereas eqs (30) and (31) are equivalent to the following pair of simultaneous integral eqs:

PlI~I(x) +: Pz*,iW) - $ The system of singularintegral

s1,$$ ;,gdf

dt =p,(x),

1x1< 1,

=p2(x),

1x1< 1.

(36)

s

eqs (36) and (37) may be further simplified to the following form:

s

’ MfNf _ t-_x aL(x)+& k

=gk(X), 1x1 < 1, (k = 17%

(38)

I

where a=&,

b, = 1, bZ= -1,

v, =P,,/P,~,

4. SOLUTION OF SINGULAR INTEGRAL EQUATIONS The solution of the system of singular integral eqs (38) may be expressed in the form[4, 111: ck(x) = wk(x) f C,,P$‘Bk,(x), n=O

(k = 1, 2),

(40)

where ck,, and P$ *&) are respectively the unknown constants and the Jacobi polynomials, and wk(x) is the weight function defined by wk(x) = (1 - x)““(l + .)” ak =

-i+

iwk,

flk= -f - iOk,

wk =

bkO,

(41)

.

It is assumed that &(x) (k = 1,2) are H-continuous. Since the unknown &(x) (k = 1,2) and therefore I(lk(x) and [k(x) (k = 1,2) are “flux type” quantities, they have integrable singularities

822

RANJIT S. DHALIWAL

and HIRDESHWAR

S. SAXENA

at the end points + 1. Thus the index for both eqs in (38) will be K,,,= -(a, conditions (14) or (27) and (28) are identically satisfied if

_, $,(n) dx = 0, S’

S’

&(I) dx = 0,

+ &,) = 1. Continuity

(k = 1,2) which implies (k = I, 2).

(42)

I

Substituting from eqs (40) into eqs (42), we get C,=O,

(k=1,2).

(43)

We shall, now, make use of the following relations:

1 7

HI

s ’

.I

[2(x - l)‘*(x + l)8*PF,flk’(x)-

G,-,(n)],

Ix/ > 1,

where G;(x) is the principal part of +(I) Pfi.@*) (x) at infinity. Substituting from eqs (40) into eqs (39) and using eqns (43) and (44) we obtain Lx

c C&‘:‘~‘“)(x)

= &&(x),

n=,

Ix/ c 1 k = 1,2.

(45)

Multiplying both sides of eqs (45) by w;‘(x)P~~;:.-~~’ (n), by truncating the series at the N”’ term, integrating in (- 1, 1) and using the orthogonality relation [12] 0, wx(r)Pr*,4*)(X)P~,6*1(X)dn

n#m;n,m=O,1,2 ,.... 2% +& + 1 ~I-(rz+a~+l)r(n+p,+l) epi,si,= n!T(n +a,+&+ 2n + bk + & + I

=

.r -1 1

1) n =M,

we find the following algebraic equations for the determination

’

(46)

of Cb:

The contact stresses along the interface for 1x1z 1 are obtained as follows:

J;I a?,(x. 0)+ ibkJ;; a,,(x, 0) = iJ(1

r(li

1,

-a2)

a 1 C*.[Z(n - I)““(x + l~Pp”.fl~)(x)

ii

n=,

2b

- G,“.(x)],

(k = 1,2).

(48)

Defining the stress intensity factors by [4]

we find

(50)

823

Griffith crack at the interface of two half-planes

If v1 > 0 and v2 > 0, then c*(x) = c](x). 5. PARTICULAR

CASES

When the deformations in the two half-planes are due to the application of a symmetrically distributed pressure to the faces of the crack, we have = -p(x),

p,(x)

P*(x) = 0.

(51)

In this case, eqs (39) and (47) yield JG

g,(x) = g2(x) = -zP(X), 2i,.&b,

c,, = -

’

w

(52) k = 1,2,

p(x)w;‘(x)P+-bk’(x)dx,

s -1 n = 1 to N.

(53)

Substituting from eqs (53) into eqs (50), we obtain K, = “$ ki, pjPt*@k)(l)~~, p(x)w;l~x)Pj~“:,-~~‘(x)dx, (54) 1,

p(x)w;‘(~)P~~~~~-~~)(x)

dx.

s

If the faces of the crack are under the action of a constant applied pressure, i.e. p,(x) = -p.

(a constant),

pz(x) = 0,

(55)

the constants C,, and the stress intensity factors K, and K, take the following simplified forms: - 2ibk,,&

c/f,=

Yl2JV

-

C,,=O;

n>2,

k=1,2,

(56)

a2JPD’

K,=p,,

K2=2poo

(57)

The values of K, and K2 in eqs (57) are similar to those obtained by Erdogan and Gupta[4] in isotropic elasticity. The effect of the materials to be orthotropic is also seen in eq (57). In this case, the contact stresses along the interface for 1x1> 1, noting that G,, = 1 (k = 1,2), are given by ~IJ

YY

(x ’ O)+ibi$o

XY

(x0)-p 3 -

0

~(~(x-~)~~(x+~)~~P~,~&)(x)-I]

=po+2 G [(x -

1)+(x + l)fl”(x + 2iOb/J - 11, 1x1> 1,

(58)

which may be further expressed as /XI> 1, 121 P22 -

0) = Po(x2 cos - 1)-“2(2w sin ~yy(X, 0) = Po[(x2 - 1)-‘/2(x 00 + 2wcos sin00 00)- -x l] , 00) J( PllPl2 >

cyk

(59)

7i

where x+1 x-l.

8=ln (

>

Expressions for contact stresses in eqs (58) and (59) are similar to those obtained respectively by Erdogan and Gupta[4] and Lowengrub and Sneddon[S]. In a similar manner, the results can be obtained for the case when the faces of the crack are subjected to shear stress only by taking p,(x) = 0 and p2(x) = -‘c(x).

824

RANJIT S. DHALIWAL and HIRDESHWAR

S. SAXENA

6. CONCLUDING REMARKS The problem considered here has recently been discussed by Zhang[lO]. Due to some fundamental errors, he obtained too simple dual integral equations to lead to the right conclusion. We present here some discussion on Zhang’s paper[lO]. (i) The basic equations for an orthotropic elastic medium, and the statement of the problem alongwith the boundary conditions have been stated correctly in ref. [lo]. (ii) Although the total number of unknown functions to be involved in the solution of basic equations for the displacement components U, and uYfor the two half-spaces should be 4 as taken in ref. [IO], the method of choosing the solution (eqs 28 and 29 in Section 3 and eqs 51 and 52 in Section 4) is not correct for the following reasons: The solution, in ref. [lo], for U, and U, which satisfy equations of equilibrium and conditions at infinity for each half-space should involve only two instead of four unknown functions, and that the unknown functions chosen for the upper half-space y > 0 should be different from those for the lower half-space y < 0. Consequently, the expressions obtained for the stresses in eqs (30)-(33) in Section 3, and eqs (53)<56) in Section 4[10] are incorrect. (iii) The boundary conditions {(23), (24)) and {(25), (26)) in ref. [lo] are valid only for 0 < 1x1< a and 1x1> a respectively. Equations (35)-(37) in Section 3 and eqs (57x59) in Section 4 ref. [lo] can not be obtained unless one assumes these boundary conditions (23x26) to be valid for all x 2 0. (iv) Due to these errors, Zhang[lO] obtained only one pair of dual integral equations instead of two pairs of simultaneous dual integral equations, and consequently arrived to the following incorrect conclusions: (i) K, and K,, do not both appear in each mode, (ii) the stress intensity factors do not depend on elastic properties of the materials, and (iii) although it has not been stated but can be deduced from eqs (28), (35) and (36) in Section 3 and from eqs (51), (57) and (58) in section 4 ref. [lo] that a,,(~, 0) = 0 and u,(x, O+) = u,(x, O-) for all x satisfying 1x12 0. In Section 5 of this paper, it has been shown that in general K, and K2 both appear when the crack is subjected to normal stress only, i.e. p,(x) = -p(x), p?(x) = 0, and that they depend on elastic properties of the two materials. The shear stress B,,,(x, 0) is continuous but not identically equal to zero at the interface. These results are in agreement with those obtained earlier by other authors. Similar results for K, and K2 may be proved valid when the faces of the crack are subjected to shear stress only by taking pi(x) = 0, p*(x) = -r(x). REFERENCES A. 221, 163 (1921). I. N. Sneddon and H. A. Elliot, The opening of a Griffith crack under internal pressure. Q. appl. Math. 4,229 (1946). ::; J. R. Rice and G. C. Sih, Plane problems of cracks in dissimilar media. J. appl. Mech. 32, 418 (1965). F. Erdogan and G. D. Gupta, Layered composites with an interface flaw. Znf. J. So/i& Structures 7, 1089-I 107 (1971). t;j; M. Lowengrub and I. N. Sneddon, The stress field near a Griffith crack at the interface of two bonded dissimilar elastic half-planes. Znr. J. Engtzg Sci. 11, 1025-1034 (1973). G. C. Sih, Mechanics of Fracture-6 Cracks in composite materials. Noordhoff, The Netherlands (1981). :; R. S. Dhaliwal, Two coplanar cracks in an infinitely long orthotropic elastic strip. Util. Math. 4, 115-128 (1973). 181H. Parhi and A. Stsumi, The distribution of stress in a transversely-isotropic cylinder containing a penny-shaped crack.

111A. A. Griffith, The phenomena of rupture and flow in solids. Phil. Trans.

Znt. J. Engng Sci. 13, 675-668 (1976).

[91 K. S. Parihar and S. Lalitha, Cracks located on a single line in an orthotropic elastic medium in which body forces are acting. Znr. J. Engng Sci. 25, 735-754 (1987).

PO1 X. S. Zhang, A central crack at the interface between two different orthotropic media for the mode I and mode II. Engng Fructure Mech. 33, 327-333 (1989).

VII F. Erdogan and G. D. Gupta, Method of Analysis and Solutions of Crack Problems (Edited by G. C. Sib), Noordhoff, The Netherlands, p. 368 Erdtlyi (1973).

iI21 A. Erdblyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1953). (Received 23 October 1989)