J integral for delaminated composite laminates

Composites Science and Technology 47 (1993) 185-192

J I N T E G R A L FOR D E L A M I N A T E D COMPOSITE LAMINATES L. J. Lee & D. W. Tu Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan (Received 15 July 1991; revised version received 19 April 1992; accepted 30 June 1992)

Abstract An integral, J, is derived for edge-delaminated composite laminates subjected to pure bending and/or uniform axial extension. This J integral for a quasithree-dimensional edge-delamination problem subjected to uniform extension has the same form as Rice's J integral for a two-dimensional crack problem. However, there is an additional area integral involved in the derived J for delamination cracking of laminates under pure bending. It can be shown that the derived J integral has the physical meaning of total energy release rate for a crack extension. In addition, J is pathindependent. Total energy release rates are calculated according to this J integral along with a quasi-threedimensional finite element analysis. Numerical examples are shown for [O/-O]s, [O/O]s and [0/0]~ laminates with edge delaminations and the results are compared.

behavior. Wang 1'2 analytically obtained the mixedmode stress intensity factors and associated strain energy release rates for a delamination crack based on the anisotropic laminate elasticity theory along with an eigenfunction expansion method. Later, many researchers 3-7 calculated the energy release rates for an edge-delamination crack of a composite laminate under uniform axial extension. They used a virtual crack closure technique incorporated in a quasi-threedimensional (Q3D) finite element analysis. In order to calculate the energy release rate for Q3D edgedelamination problems by using the Rice's J integral which is applicable to two-dimensional problems, Aoki and Kondo4'5 used a transformed Q3D finite element analysis which was based on a concept of superposition proposed by Whitcomb and Raju. 8 The integration path to evaluate J should be away from the delamination crack-front and free edge in order to satisfy the plane stress assumption. Stress intensity factors and strain energy release rates due to crack growth are separated into three fracture modes, I, II, and III, in the theory of linear elastic fracture mechanics (LEFM) for homogeneous materials. For an interlaminar crack, these modes are coupled inherently as a consequence of material discontinuity. Manoharan and Sun 6 and Raju et al. 7 found that the energy release rates are not well defined for each individual GI, Gu, and Gi11 unlike the total strain energy release rate, G. In addition to the aforementioned stress intensity factors and energy release rates, the well known J integral proposed by Eshelby9 and Rice 1° for homogeneous materials is also considered to be an effective fracture mechanics parameter. In the development of LEFM, J is equivalent to the energy release rate, G. Later, many authors H-~3 extended the J integral for cracks in anisotropic solids. Miyamoto and Kikuchi ~4 suggested Jk integrals for mixed-mode crack problems. Their ./1 integral is identical to the J integral and both J~ and J2 can provide two independent pieces of information on the crack tip for inhomogeneous materials. Chu and Hong ~5 derived the analytical relations between J~, J2 and KI, Ku.

Keywords: edge delamination, J integral, path independency, energy release rate, composite laminates, energy momentum tensor 1 INTRODUCTION

It is known that edge delaminations or interracial cracks in a composite laminate could result in the degradation of its mechanical properties, e.g. stiffness and strength, and a premature failure could occur during its use in service. Hence, the delamination problem has been one of the major considerations for composites design. Recently, much research effort has been devoted to this area in order to try to obtain a better understanding of the interracial cracking mechanism. As a result of geometric and material discontinuities, a delamination crack problem is extremely complex. There are many parameters, e.g. stress intensity factors, total energy release rate, J integral, etc., which may be used to characterize crack tip

Composites Science and Technology 0266-3538/93/$06.00 © 1993 Elsevier Science Publishers Ltd. 185

L. J. Lee, D. W. Tu

186

One of the most fascinating points about using the J integral as a fracture parameter is that the J integral provides a possible means of determining an energy release rate for cases where plasticity effects are not negligible. The objective of this paper is to derive a J integral analytically for an edge-delamination crack of composite laminates. Under the assumption of Q3D displacement field, the derived J integral for a delamination crack in a laminate subjected to uniform axial extension has the same form as Rice's J integral. However, there is an additional term which involves an area integral in the derived J for the case of pure bending. The presently proposed J integral has the physical meaning of total energy release rate for a delamination crack and preserves the property of path-independence. The Q3D finite element method is employed to calculate the value of J for interracial cracks in laminates subjected to uniform axial extension and pure bending. The results are compared with the total energy release rates, G. Effects of crack length, fiber orientation on the J integral are also investigated.

3

CRACK FRONT

\

(a)

"

\

s1

S2

CRACK SURFACE

INTE F A C E

Fi "--...~ S

--Y

(b) 2 MATHEMATICAL FORMULATION Consider an elastic body containing a singularity P within it, as shown in Fig. 1. Eshelby 9 stated that the total energy change, OH, as the singular point P is displaced by an arbitrary amount, 0r/k, is equal to the work done by an effective force, Fk, acting on P, i.e. -Fkbr/k=bFl. The effective force is given by integrating the normal component of Eshelby's energy momentum tensor, 9 Pkp over a closed surface, So, which contains P. Thus:

OH = fs Pkjnj dS

Fk = --6r/k

k = 1, 2, 3,

F q

(1)

Fig. 2. Through thickness crack between two parallel surfaces. (a) Crack in a homogeneous material; (b) crack between two dissimilar materials.

The energy momentum tensor, Pkj, is expressed as:

P~,i = WOkj -- (YijUi.k

where W is the strain energy density, oq is the stress tensor, ui is the displacement vector, and 6kj is the Dirac delta function. As for the problem of a crack in an elastic solid, the crack tip can be treated as a singular point. In this situation, Fk is considered to be a crack driving force for the crack tip propagates a unit distance in the direction of 6r/k. Consider a thin slice of elastic material with thickness B and a through-thickness crack as shown in Fig. 2(a). This thin slice is cut from a general through-thickness crack problem. Let (x, y, z) denote the Cartesian coordinates with the origin being at the crack tip. The x axis is tangent to the crack, the y axis is lying on the crack surface and normal to the crack-front, and the z axis is normal to the crack surface. There is only one component, Fy, which remains for a crack growing in the y direction. Thus, the subscript y can be dropped hereafter without ambiguity. The J integral is defined as: J=-~-

Fig. 1. Singularity shifting within an elastic solid.

(2)

1 ~ fs,+s:+s (W0iy - oiiUi.y)n, dS

(3)

J has the meaning of force per unit length on a crack tip which propagates a unit distance in the direction

J integral for delaminated composite laminates along the crack surface. In eqn (3), $1, $2 and $3 are the side, front and back surfaces of the thin slice, respectively; and $1 + Sz + $3 totals the surface surrounding the crack tip. If the thickness B of the slice is reduced so as to be infinitely small, the J integral becomes: =,im B--*O

Ills

S' : area S": area S~ : area S~':area

enclosed enclosed enclosed enclosed

by by by by

187

re+ rf+ rg+ rh ra+ rb 1-1+rc+ rd+ rj rc+ i-d

Interface i+$2+S3

=fr(Wdz-oiin, ui,~dr)-f

(oi~ul,y),xdS

(4)

where F denotes any contour embracing the crack tip. In the derivation of the above equation, the crack surfaces are assumed to be traction-free and the relationship ny dF = dz has been imposed. The J integral given by eqn (4) is the crack driving force or the energy release rate for a crack extension in homogeneous materials. In order to employ eqn (4) for delamination problems which contain a crack between two dissimilar materials (two different plies), J has to be modified in the following form:

,Y Fig. 3. Two different integration paths for the J integral. following. Let J' and J" be two integrals corresponding to two different integration paths as shown in Fig. 3. The J ' and J" can be expressed as follows:

J' = fr.+r,+rg+rh (W6jy - aqUi,y)nj dr

J = fr(W dz - aqnjUi.y dF) - fs(a~U~,y),~ dS -[fr (W dz-aijnjui,ydF)- fs (O~ui,y).xdS]

-- fS' (OixUi'y)'x as (5)

- [fr~+r~+r~+r (W6'y - °qui,y)n' dr where F is the contour surrounding the crack tip as shown in Fig. 2(b); S is the area enclosed by F and the crack surfaces; F~ is a contour surrounding the interface of the laminate within the contour F; and SI is the area enclosed by F1. The reason that the last two terms are added in eqn (5) will be explained using Eshelby's theory 9 as follows. The first two terms in eqn (5) are just the same as eqn (4) which is the total force acting on the crack tip (defect) due to a crack extension in a homogeneous material along the y direction. However, for the case of composite laminates, the first two terms represent an integral over a closed surface including the crack tip as well as an inhomogeneity (interface boundary). This means that the total force is acting on the crack tip and the inhomogeneity, but the force acting on the inhomogeneity contributes nothing to the crack extension and it has to be eliminated from the total force. In Fig. 2(b), the integration contour contains only one interface boundary. Had the integration contour contained several interface boundaries, all contributions from these interfaces would have had to be eliminated from the integral.

3 PATH-INDEPENDENCE

OF THE

J

INTEGRAL That the J integral defined by eqn (5) has the path-independent characteristic will be shown in the

(6)

- fs (O,,,u,.y).x dS] J" - ( --

(W~/. - o,/u,,.)n/ dr - fs,, (O,xUi.) . dS

JUa+Ft,

'

'

- [fr~+r (WOjy - °ijui.y)njdF- L (°ixUi.y).x dS ] (7) where Fa, Fb . . . . . Fj represent for integration paths; and S', S", S~, and S~' are the areas enclosed by the F as shown in the figure. In eqns (6) and (7), integration over the crack surface, i.e. Fg and Fh, have been added to the integral. This added integration contributes nothing to J since it is traction-free and dz = 0 on the crack surfaces. Consider the difference of the two integrals:

j, j,, f aF

(W6iy - oqUl.y)ni d r

n+Fe-Fi--F

+]

a

(w6j~ - oqu~,,3nj dr +r~-r~-r~

- [fs,_s (°~ui.y).x dS - ;s~_si (°ixui.y).x dS ] (8) The first two terms vanish, since they are integrated over a closed path within materials I and II (Fig. 3),

L. J. Lee, D. W. Tu

188

respectively. The last two terms in the bracket represent the integration over the hatched area where the discontinuity over the phase boundary has been eliminated. Using Gauss's theorem, the area integral in the bracket can be transformed to a line integral exactly the same as the first two terms, and once again it vanishes. Consequently: J' =J"

(9)

sections which are perpendicular to the x axis and away from the ends will deform in the same manner. For this reason, only an arbitrary x = constant plane needs to be investigated. Furthermore, as a result of symmetry, only one-quarter of this plane is analyzed, as shown in Fig. 4(b). Thus, a Q3D displacement field is derived, following Lekhnitskii: 16

u(x, y, z) = e,,x + kxxz - (J(y, z)

The J integral is therefore path-independent.

v(x, y, z) = ~-~xz + (/(y, z) 4 J INTEGRAL FOR DELAMINATION CRACKS IN COMPOSITE LAMINATES

(10)

kxx2 - ~--~xy + I,~'(y, z) w(x, y, z) = ---f

Consider a symmetric composite laminate as shown in Fig. 4. Assume that the laminate is long in the x direction and it is subjected to a uniform axial extension e0 a n d / o r bending curvature k~. Delamination cracks at both edges are located symmetrically about the midplane of the laminate. Since the composite laminate is long in the axial direction, the end effects are negligible in the region away from the ends. Consequently, all the cross-

where u, v, and w are the displacements in the x, y, and z directions, respectively, U, I7, and W are unknown displacement functions of y and z; and kx and kxy are bending and induced twisting curvatures, respectively, and are given by: kx

=

-

02w OX2 ; kxy -

32w Ox Oy

(11)

Based on the Q3D displacement field, eqn (10), the J integral can be reduced to: e o

J = fv(W dz - oi]n,Ui.y) dF + ~-~ fs r~z dS + £, o,jnju,.y dF - - ~ £ ~xz dS

(12)

where the integration on W dz in F~ has been cancelled, for dz is zero along F1 ( G is parallel to the y axis). In addition, the stress and displacement across the interface should satisfy the equilibrium and continuity conditions, i.e.:

- Y

0.(1) = (a)

(2);

Oz

u 0 ) = u(2);

7.(1)= _(2). r(') = r~ ~ ~yz t~yz , ~xz

v ° ) = v(2);

w ~ ) = w ~2)

(13) (14)

where superscripts 1 and 2 denote for materials 1 and 2, respectively. Furthermore, in integration over F~, one can choose a path along the interface such that the area enclosed by F~ tends to zero. Thus the J integral for the edge delamination crack due to uniform bending can be reduced to the following form:

7 02 LAYER / /

01 LAYERI / - -~y

(b) Fig. 4. Edge delamination of a composite laminate. (a) Crack geometry; (b) integration path surrounding the crack tip.

J = fv(W dz - o,,nju,.,) dF +-k-~ ~sG~ dS

(15)

The twisting curvature k~y appears in eqn (12) and eqn (15) is an induced quantity for a generally symmetric laminate under pure bending. This was explained by Salamon.27 A brief discussion about this is given as follows. Owing to the effect of bending/twisting coupling in a symmetric composite

J integral for delaminated composite laminates laminate, the constitutive equations for the plate are:

mx = O11kx + O12ky + Ol6kxy My = D12kx + D22ky + D26k~y Mxy =

(16)

where D 0 are the bending stiffnesses; and ky is the bending curvature in the yz plane and can be expressed from eqn (10) as:

ky -

a2~, r

~y2

ay2

(17)

For the problem of pure bending about the x axis, both ky and kxy are also induced and can be solved in terms of the prescribed kx by setting My - - M x y = 0 in eqn (16): D16D26 - DIED66

ky- 022066_026 DIED26 - D16DE2

kx

(18)

kx

The value of ky is in general non-zero, but it is incorporated into the unknown function 17¢(y, z) as seen from eqn (17) and does not appear in eqn (10). However, the induced k~y should be included in order to guarantee pure bending. For laminates subjected to uniform axial extension only, the J integral is further reduced to:

J=

fr (Wdz -

oijnjUi.y dF)

(19)

It is interesting to note that the J integral for the Q3D edge-delamination problem has the same form as Rice's J integral for a two-dimensional crack in homogeneous materials. It should be pointed out that the presently derived J involves the integration of 3D stresses, strains and displacements, i.e. all six components of stress and strain and three components of displacement are involved. While only twodimensional quantities are involved in Rice's J integral. 5 NUMERICAL

RESULTS

AND

high-modulus unidirectional graphite/epoxy prepreg with the following ply properties:l'2 E1 = 137.9 GPa (20-0 x 106 psi); E2 = E3 = 14-48 GPa (2.1 x 106 psi)

+ D26ky + D6 kxy

~2 W

189

DISCUSSION

The J integral along an arbitrary contour around the crack tip defined herein is the change in total potential energy for a crack extension. J is therefore equal to the energy release rate, G. In order to demonstrate the J integral approach, the problems of edge delamination at the 01/02 interface of a four-layer [0~/02]s composite laminate subjected to uniform axial extension or pure bending were studied. The geometry of the delaminated laminate is shown in Fig. 4. It has a width of 2 b = 4 0 . 6 4 c m (16in.), a ply thickness of h -- 2.54 cm (1 in.), and the crack length a - - 2 . 5 4 c m (1 in.). The laminate is made of

G12 = G23 = G31 = 5 . 8 6

G P a (0.85 x 106 psi)

V12 ~- 1/23 ~--- V13 ~- 0"21

As a consequence of the symmetry, only one quadrant of an arbitrary laminate cross-section at x - - c o n s t a n t is analyzed. An eight-noded, quadrilateral, isoparametric element with three degrees of freedom per node was employed in the finite element analysis. Total of 120 elements were used to discretize one quadrant of the cross-section. A finite element method based on the Q3D displacement assumptions given by eqn (10) is used to analyze the stresses and strains of the cross-section. The selected quantities are then substituted into eqn (15) or (19) to obtain the energy release rate along a specific path. 5.1 Uniform axial e x t e n s i o n case The energy release rates of angle-ply laminates [ 0 / - 0 ] s calculated by the J integral method are compared with those obtained by Wang 2 for 0 -- 15 °, 30°, 45 °, 60 °, and 75 ° . The comparisons are listed in Table 1. The difference between these two results is less than 1% which indicates that the J integral approach presented here is effective in computing total energy release rates for the Q3D edgedelamination crack problems. Edge delamination at the 0 / 0 interfaces of [0/0]s laminates is also investigated. Figure 5 shows the variation of the J integral with respect to the fiber orientation in a 0 ply. This curve has a shape similar to that for [+ 0]~ angle-ply laminates presented in Ref. 2. From Fig. 5, it is noted that the energy release rate J has a maximum value for a delaminated [0/0]~ laminate with 0 ~ 14° while for [+0]s laminates the maximum value occurs at 0 ~ 18° as obtained in Ref. 2. The latter has a much higher maximum value of J (1433J/m z) than the former (232J/m2). It is also shown in Fig. 5 that there is a minimum J for a [0/0]s laminate with 0 ~ 6 3 °. This p h e n o m e n o n could be Table 1. Comparison of J with total strain energy release rate G for a [ 0 / - 0 ] , composite laminate

0

J (Jim 2)

G (j/m2) a

Difference (%)

15° 30° 45° 60° 75°

1415.00 707-00 100-40 2-42 0.64

1419.00 709.00 100-40 2-42 0-63

0-28 0-28 0.00 0.00 1.59

° Ref. 2.

190

L. J. Lee, D. W. Tu

6.0-

250"0t

\

50

200.0

eo = 0.1%, a = h

4.0

\ \

~ 5.0 -~

[o/75]~

2,0-

= ~.

/ -

50"0V 0 ] S 1.0 ~'~ 0.0

~t . . . .

~ ....

0

15

~ .... 30

, ....

~ ....

45

O ((:leg.)

60

, , , ',T - ~ . 75

= = :

~ 90

0.0

0.2

0.4

0.6

0.8

1.0

o/b

Fig. 5. J versus 0 for [0/0], laminates due to uniaxial extension.

Fig. 7. J versus crack length for [0/0], laminates due to

explained by the fact 18 that the difference of coefficients of mutual influence between neighboring plies has a maximum value and, at the same time, the Poisson ratio mismatch is relatively high between 0 / 0 plies with 0 ~ 1 4 °. On the other hand, both the Poisson mismatch and shear coupling effects between 0 / 0 plies are low for 0 -~ 63 °. Figures 6 and 7 illustrate the J values versus the crack length for [0/0]s laminates. From a careful examination of these results, it is found that stable crack growth would occur under monotonically increasing load as the crack length, a, is greater than one or two ply thickness of the laminates studied. This phenomenon was also observed by Wang 2 for [+0]~ angle-ply laminates.

5.2 Pure bending case Since there is very little literature concerning the calculation of energy release rates for delamination cracks due to uniform bending, the energy release rate obtained by the J integral method in this paper is compared with G which is calculated from the

250.0

-

[0/15]s

y 200.0

\

-

eo =0.1%, a = h 150.0 -

100.0 -

/ 50.0-

{o/,ls]~

I "- = 0.0

0.2

0.4

definition, i.e. - ( % - ~ a ," Here, W denotes the strain energy density function and the subscript u denotes the quantity under fixed displacement condition. Table 2 shows the energy release rates calculated by these two approaches for various edge-delaminated laminates under uniform bending curvature kx = 0.04/m. From these results, it can be seen that the differences in the energy release rates obtained by these two methods do not exceed 2%. We may conclude that the J integral, eqn (15), does represent the energy release rate for delamination cracking of a composite laminate due to pure bending. Effects of fiber orientation for edge delaminated laminates [+0],, [0/0], and [0/0]~ are illustrated in Fig. 8. This shows that angle-ply laminates [+ 0]~ have the highest J value of the three laminates for 0 less Table 2. Comparison of J with - ( 0 W / ~ ) , for various laminates subjected to pure bend~£

[o/3o] s

v

uniaxial extension for 0 = 60°, 75°, and 90°.

0.6

0.8

1.0

q/b

Fig. 6. J versus crack length for [0/O], laminates due to uniaxial extension for 0 = 15°, 30°, and 45°.

Laminate

J (J/m 2)

- ( aW / aa ). (J/m 2)

Difference (%)

[o/90]s [90/0], [15/-151s [30/-30]s [45/-45]s [60/-60], [75/-75]s

8.940 2.190 282.800 251.500 55.700 1.530 0.304

9.010 2-210 285-200 253.700 56-300 1-550 0-303

0.77 0-90 0-84 0.87 1-06 1-29 0-33

J integral for delaminated composite laminates I0.0

320.0 -

k.

28O.O-

-

191

[o/9o]s

o=h

=0.04

8.02.,40.0-

200.0

k~

-

=

6.0-

"E ~

160.0 -

v -) 4.0-

120.0-

[9o/o]~ 80.0

=

-

2.0.

=

/¢~r=

40,0-

0.0

0.015

30

45

e

60

75

90

. . . .

0

i

0.2

. . . .

i 0.4

. . . .

i

. . . .

0.6

i

0.8

. . . .

i 1

(deg.)

Fig. 8. J versus fiber orientation for different laminates subjected to pure bending.

Fig. 10. J versus crack length for cross-ply laminates

than 64°. The reason for this could be as in the uniaxial extension case. TM The [+0], laminates have a much higher shear coupling effect between + 0 and - 0 layers (for 0-< 640) than that between 0/0 layers. Although there is Poisson ratio mismatch between 0/0 layers, it is not a dominant factor in inducing interlaminar stresses. In fact, the delamination crack due to pure bending is dominated by the tearing mode. For 0 greater than 640, the shear coupling effect for a [+0], laminate is the lowest of the three laminates investigated here and hence has the smallest J. The effects of crack length on the J integral for [+O], ( 0 = 15°, 30°, 45 °, etc.), [0/90],, and [90/0], laminates are illustrated in Figs 9 and 10, respectively.

It was found that stable cracking would occur when the crack length, a, is greater than half a ply thickness.

500.0

k~ = 0.04 400.0.

300.0.

v

[3o/-3o1~

2oo.o-

subjected to pure bending.

6 CONCLUSIONS An integral J has been derived for edge-delaminated composite laminates. The J integral for delamination crack of laminates under uniaxial extension has the same form as Rice's J integral for a two-dimensional crack problem. However, J involves the integration of 3D stresses, strains, and displacements. On the other hand, the J integral for the case of pure bending involves an area integral in addition to the line integral. The J integral derived here has been proved to have the physical meaning of total energy release rate for delamination crack extension of the laminate. It is also shown, theoretically, that J is pathindependent. Numerical studies are carried out for computing the energy release rates by the J integral method for a symmetrical laminate with edge delaminations subjected to axial extension and pure bending. The results are compared with those in the existing literature. The J integral derived in this paper is an effective parameter for Q3D delamination crack problems. REFERENCES

100.0 -

0.0

. . . . . . . . . . . . . . . . . . . . 0

0.2

0,4

0.6

0.8

i

1

Fig. 9. J versus crack length for angle-ply laminates subjected to pure bending.

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L. J. Lee, D. W. Tu

4. Aoki, T. & Kondo, K., Free-edge delamination of anisotropic composite laminates---(I) Theoretical approach. Trans. Jpn Soc. Aeronautical & Space Sci., 37 (1989) 29-38 (in Japanese). 5. Aoki, T. & Kondo, K., A simple method for calculating energy release rates due to free-edge delamination in laminates. Trans. Jpn Soc. Aeronautical & Space Sci., 37 (1989) 193-201 (in Japanese). 6. Manoharan, M. G. & Sun, C. T., Strain energy release rates of an interracial crack between two anisotropic solids under uniform axial strain. Comp. Sci. & Technol., 39 (1990) 99-116. 7. Raju, I. S., Crews, J. H., Jr & Aminpour, M. A., Convergence of strain energy release rate components for edge-delaminated composite laminates. Engg. Fract. Mech., 3ll (1988) 383-96. 8. Whitcomb, J. D. & Raju, I. S., Superposition method for analysis of free-edge stresses. J. Comp. Mater., 17 (1983) 492-507. 9. Eshelby, J. D., The continuum theory of lattice defects. In Prog. Solid State Physics, Vol. 3, ed. F. Seitz & D. Turnbull. Academic Press, New York, 1956, pp. 79-144. 10. Rice, J. R., A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech., 35 (1968) 379-86. 11. Wang, S. S., Yau, J. F. & Corten, H. T., A

12. 13. 14.

15.

16. 17. 18.

mixed-mode crack analysis of rectilinear anisotropic solids using conservative laws of elasticity. Int. J. Fract., 16 (1980) 247-59. Wu, K., Representations of stress intensity factors by path-independent integrals. J. Appl. Mech., 56 (1989) 780-5. Tsamasphyros, G., Path-independent integrals in anisotropic media. Int. J. Fract., 40 (1989) 203-19. Miyamoto, H. & Kikuchi, M., Evaluation of Jk integrals for a crack in two-phase materials. In Numerical Methods in Fracture Mechanics, ed. D. R. J. Owen & A. R. Luxmoore. University of Swansea, UK, 1980, pp. 359-70. Chu, S. J. & Hong, C. S., Application of the Jk integral to mixed mode crack problems for anisotropic composite laminates. Engg. Fract. Mech., 35 (1990) 1093-103. Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, Inc., San Francisco, CA, 1963. Salamon, N. J., Interlaminar stresses in a layered composite laminate in bending. Fibre Sci. & Technol., 2 (1978) 305-17. Herakovich, C. T., On the relationship between engineering properties and delamination of composite materials. J. Comp. Mater., 15 (1981) 336-48.