Mode II stress intensity factors for interlaminar cracked composites under arbitrary shear crack surface loading

Pergamon Engineering Fracture Mechanics Vol. 59, No. 4, pp. 509-520, 1998 © 1998 Elsevier Science Ltd, All fights reservmt

Printed in Great Britain PII: 80013-7944(97)00136-7 0013-7944/98 $19.00 + 0.00

M O D E II S T R E S S I N T E N S I T Y F A C T O R S F O R

I N T E R L A M I N A R C R A C K E D C O M P O S I T E S U N D E R

A R B I T R A R Y S H E A R C R A C K S U R F A C E L O A D I N G

SUNG HO KIMt

Agency for Defense Development, Taejeon, Korea

KANG YONG LEE

Department of Mechanical Engineering, Yonsei University, Seoul, Korea

MOON BOK PARK

Department of Mechanical Engineering, Yonsei University, Seoul, Korea

Abstract--A model is constructed to evaluate the mode II stress intensity factors (SIFs) for composites with an interlaminar crack subjected to an arbitrary shear crack surface loading. A mixed boundary value problem is formulated by Fourier integral transform method and a Fredholm integral equation of the first kind is derived. The integral equation is solved numerically. The mode II SIFs are evaluated for various ratios of shear moduli of successive layers and ratios of crack length to layer thickness. Fur- thermore, the terms as well as the shape of crack surface polynomial loading and the number of layers affect the variation of SIFs. © 1998 Elsevier Science Ltd. All rights reserved

a

q(x) 6j hj

n

p2k ,p2k- 1

tj I l l

t f2

tm0 lm W(~m), W(~t) VLF V~F /*j v/ rj

(T yy(.h,t3 xyO), r3 xx(l)

uO~,vo~ Xm,Otl

NOMENCLATURE

half crack length arbitrary crack surface loading Fourier transformation of Airy stress function for the jth layer distance from the crack to the exterior interface of the flh layer Bessel function of the first kind of order 0 and 1, respectively the number of layers influence coefficients for symmetric and anti-symmetric loading parts, respectively the order of each term of polynomial loading fiber layer thickness the first fiber layer thickness the second fiber layer thickness cracked resin layer thickness sound resin layer thickness integration weights for Gaussian and Laguerre integrations, respectively local fiber volume fraction global fiber volume fraction shear modulus for the jth layer Poisson's ratio for the jth layer shear modulus ratio (-/~j/}tj + 1) Fourier transformation variable stress components for the flh layer displacement components for the jth layer integration abscissas for Gaussian and Laguerre integrations, respectively unknown coefficients.

1. INTRODUCTION

THE DETERMINATION of the stress intensity factors (SIFs) for the layered material with a crack is important in fracture analysis. Gecit and Erdogan[1], Hilton and Sih [2], Lin and Keer[3] and Fowser and Chou [4] considered the layered material subjected to a uniform loading. Also, Hilton and Sih [5] considered the cracked layer bonded between two half-planes and derived

tAuthor to whom correspondence should be addressed. Present address: Samsung Motors Inc, Technology Center, San 50, Kongse-Ri, Kihung-Eup, Yongln-City, Kyungld-Do, Korea 449 900.

509

510 S.H. KIM et al

SIFs by integral transform method. Delale and Erdogan [5] obtained mode I and mode III SIFs for two homogeneous elastic half planes bonded through a nonhomogeneous layer with a central crack. Kim et al. [6, 7] treated the cracks for three bonded elastic layers under in-plane, anti- plane shear and uniform tension loadings, respectively. For nonuniformly stressed center crack problems, Isida[8, 9] considered the cracks embedded on the finite and infinite plates by Laurent expansion of the complex stress potential. Chen [10] obtained the SIFs for the anti-symmetric crack surface loading by integral transform method. Chen and Chang [11] obtained the mode I and II SIFs for an infinite plate by efficient finite element alternating method. Fan and Keer[12] evaluated the mode I and II SIFs for the strip subjected to an arbitrary distributed traction on the infinite crack. Recently, Lee et al.[13] considered the cracked three layered material subjected to linearly varying normal crack surface traction and evaluated the mode I SIF. Also, Kim et aL [14] and Lee et al[15] studied the cracked multi-layered material subjected to anti-symmetric normal crack surface traction.

In this study, the model is a composite with an interlaminar crack subjected to a plane- strain type arbitrary shear crack surface loading. By following a linear elasticity, a Fredholm integral equation of the first kind is derived using Fourier integral transform method and the mode II SIF is evaluated numerically. The mode II SIF is analyzed for each term of crack surface polynomial loading as well as the effects of the number of layers, shear modulus ratio between the layers, Poisson's ratio, crack length to layer thickness ratio and the thickness of

/ '~ll ~ ]'llbl

y

nth layer

/.13, W3

~2,/ '12

3rd layer I

2nd layer I

1st layer [

J21, Vl ~ - - . ~ - - 4-

q ( x ) [ t i

P2, v2 2a _ [ t2

/ ' /3 , W3

/a , , v . nth layer

I t3 I

t . 1

• h2 ha

hi _ !

h i

h2 ha

h

h

X

Fig. 1. Configuration of the model.

Stress intensity factors for cracked composites 511

each layer and the mode II SIFs for the E-glass-epoxy composites and the hybrid composites are evaluated as possibly applicable examples.

2. PROBLEM FORMULATION

Consider the plane-strain type layered material with a central crack subjected to an arbitrary shear surface loading as illustrated in Fig. 1. It is assumed that the crack is parallel to the layer interface and each elastically dissimilar isotropic layer is perfectly bonded. Using Fourier integral transformation, stress and displacement components can be formulated for symmetric and anti-symmetric loadings, respectively, in the form:

where

1 [oz ~ G j [ sin ~x ]d~ ~xx0~ = ~ j 0 --~-y2 i.cos ~ x j

¢ryy~) ~ f o ~ "2~ [" sin ~x ]d~ = ¢ ~J L cos ~x

i f /0oj[ cos x, axyO~ = ~ ¢--~'y sin cxJ d~

(1 + vj) f~F(1 02Gj uq~ = ,, rcEj Jo k - vJ)--ff72

v0, (13vj ) ~ [(1 a3Gj = - o

+ pj~2Gj][-G°S~X] sin ~x Jd~/~

+ (vj - 2)~: ~2/] [ :~¢¢x ]d~/~ 2 , (1)

axyO)

uo)

ayy(l)

axy0)

ayyO)

uo)

vo)

a xy(n)

ff yy(n)

: O'yy(j+l ) (--OO < X < CO, y = hj)(j = 1 . . . . . n - 1)

= u(/+O ( - o o < x < oo, y = hj)(j = 1 . . . . . n - 1)

m-Vq+l) ( - o o < x < o o , y = h j ) ( j = 1 . . . . . . n - l ) = 0 ( - o o < x < oo, y = hn)

= 0 ( - o ~ < x < oo, y = h.). (3)

The arbitrary shear crack surface loading q(x) is defined as:

q(x) = Z . , , p ~ a ] = 2_~P2k~=] + ~_~P2k-l~-~] (4) #=0 k=0 k=l

where p ~ and p ~ _ ~ are the given influence coefficients for the symmetric and anti-symmetric loading parts, respectively.

For simplicity of the formulation, the arbitrary crack surface loading q(x) is separated into symmetric and anti-symmetric loading parts without loss of generality. By applying the boundary conditions to eq. (1), the following pair of dual integral equations can be obtained for each

Gj = [Aj(O + ~yBj(Olcosh(~y) + [Cj(O + ~yDj(O]s inh(~y) ( /= 1, 2, 3 . . . . . n). (2)

The subscript j = 1 represents the first layer where a crack is located. It is assumed that the thickness of the layers is identical, vj and Ej are Poisson's ratio and Young's modulus for the jth layer, respectively. The unknown coefficients Aj(O, Bj(O, Cj(O and Dj(O are to be determined.

Because of the symmetry of the configuration for the considered model as shown in Fig. 1 in relation to the crack axis, it needs to examine the upper half plane for solving the stated problems. The boundary conditions are as follows:

= - q ( x ) ( - a < x < a, y = O)

= 0 ( I x l > a , y = O )

= 0 (-oo < x < oQ, y = 0 )

= ffxyq+l) (--00 < X < 00, y = hj)(j = 1 . . . . . n - 1)

512

loading part ,

< / e q n > where

S. H. KIM et al

COS ~X

oo COS ~X fo CF(OM(O[sinCx] d¢=nF p~(~)2k l L_p~k_,(~)~k_ l j (I x I< a), (6)

Bl(~) + CI(~) can be defined as follows,

where

¢{a~(~) + G(O} F(~) = , (7)

M(~)

M(~) - - ~DI (¢), (8)

A 1 (O = 0. (9)

T = Q-IR, (10)

T = [B1, Cl, A2, B2, C2, D2 . . . . . An, Bn, Cn, Dn] T,

R = [R1, R2, R3, R4 . . . . . R4n-2] T

(11)

(12)

where

K(z, a)= ~ ~ ot[F(~/a)- ]d= L Jl (~"c)al t~o') j

f2k(a) -- (2k - 1)!! 0.2k+1/2 (2k)!!

fEk-l(tr) = (2k - l)!! o.2k_112 (2k),

(2k - 1)l! -- (2k - 1) x (2k - 3) . . . . . 3 x 1 (2k)!! -- (2k) x (2k - 2) . . . . . 4 x 2

t - - az, s - - aa, a - - C a , x - - a Y e

qb2k(l ) ~- 7t~V/-~p2kO2k('C), ~OZk_l(/) ~- ~t~r~P2k_l~IJ2k_l('Q. (16)

RI = -Zl t anhz l , R2 = - z l - t anhz l , R3 = -2 (1 - Vl) - zl t anhz l

R4 = - (1 - 2v l ) tanhzl + Zl, R5 = R6 . . . . . R4n-2 = 0(zl -- ~hl), (13)

and 0 is defined in Appendix A. By following the method o f Copson[16], M(O is defined for symmetric and anti-symmetric

loading problems, respectively, in the form:

M ( ~ ) = fao[ C~2k(t)Jo(~t)ld" (14) 1~2k_ l(t)Jl (¢t) J - t '

where J0 and J~ are the Bessel function o f the first kind o f order 0 and 1, respectively, whereas q~2k(t) and ~bZk- 1(0 are to be determined. By eq. (14), eq. (5) is satisfied automatical ly and eq. (6) is reduced to the form of a Fredholm integral equat ion of the second kind as follows:

[ ~2k(a) ] l (15) + = [ 1 L I2k- l tZ)J --f2k-I (0) J


To solve the Fredholm integral equation, the Gaussian-Laguerre integration technique is uti- lized. At zm = am(m = 1 to Na), ~2k(Z,,,) and ~2k- 1(Tin) can be determined numerically as follows:

r ( e = l t o N G ) . = 6me + K(zm, Ze)i. r2k_l[Zm)_ ] = Lf2k-l(Ze) (17)

u, !1 [ ] K(T~, %) = ~ E °~t[F(°~da) - ' ~ L JI (O~l'~m)Jl (O~l'~e) J W(~Xl), (18)

1=5

where ~rae is Kronecker's delta, Zm and ctt are the abscissas for Gaussian and Laguerre integrations, respectively and Ze is the collocation point. W(~t) and W(zm) are the weight factors for Laguerre and Gaussian, respectively, with NI and Nc being the number of integration points. The mode II SIFs for both sides of crack tips are calculated in the forms:

KIIR = lim v/2na(2 - l)o'xy(1)(x, 0) ~...~ 1 +

KIIL = lim v/--2na(2 + 1)axy(l)(Yc, 0), (19) ~---,- l -

where KnR and KI1 L represent the mode II SIFs for the right and left crack tips, respectively. After some integral manipulations for axy(l)(X, 0), the mode II SIFs are obtained in the forms:

KnR = ~ - a q~o(1)po + ~--~{q~(1)PEk -- ~ - 1 (1)PEk-l} k=l

KIL = q r ~ O0(1)po + {tl)2k(1)P2k + O'~2k-l(1)P2k-1} • k=l

(20)

3. NUMERICAL RESULTS AND DISCUSSION

For the homogeneous material, the kernel K(z,a) in eq. (15) approaches zero because F(ct/ a) = 1. Therefore, the mode II SIFs are obtained explicitly as follows:

KIXR = ,/ Ipo

K.L = 4 Ipo

-~(2k - 1)!! + k=~ (2k)!!

-~ (2k - 1)!! + k=l (2k)!!

{P2k -- P2k-I }] =

- - {P2k + P 2 k - 1 } ] =

1 _ + p2 .-

( , ' ' ) 0 + ~ P l + ~ P 2 + " " V/-~. (21)

The results coincide with Ficher[17] and Sih and Chen[18]. For simplicity of the analysis in the three layered material case, it is assumed that each

layer is composed of equal thickness (q = t2 = t3 = to) and the shear modulus of each layer is gradually increasing or decreasing. In earlier studies, Kim et al. [6] showed that Poisson's ratio does not significantly influence SIF, and, therefore, Poisson's ratio for each layer is assumed as 0.3. Figure 2 shows the dimensionless mode II SIFs for each term (fl) of polynomial loading in the right crack tip, < tl > K~{)l(pl~vr~), with variations o f crack length to layer thickness ratio (a/to) and shear modulus ratio. The dimensionless mode II SIF increases as a/to increases. The dimensionless mode II SIFs for layered material with F~ = F 2 = 2 . 0 are higher than those obtained with F1---F2=0.5. Also, the dimensionless SIFs reduce significantly as discussed by Isida [9] and Lee et al. [13, 15] as the order (fl) of each term of polynomial loading increases when the even and odd terms of the polynomials are considered separately. The convergencies of the solutions are examined in Table 1 and the accuracy needed in the engineering viewpoint is required.

514 S . H . KIM et al

$

] .5

1.2

0.9

0.6

0.3

0.0

." ..... r~ =G =1.0

, " . . . . . . . . r ~ = r 2 =O.S ?

B--1 B--2

2""

B=3 B=4 [

I I l 0 2 4 6 8 10

a/to Fig. 2. Dimensionless mode II stress intensity factor for three layered case.

Next, a multi-layer case is considered. For simplicity of the problem, it is assumed that each layer is composed of identical thickness (tl = t2 . . . . . tn = to) and the shear moduli of the layers are gradually increasing or decreasing. As shown in Fig. 3, the dimensionless mode II SIF for each term of polynomial loading increases as a/to and shear modulus ratio increase and fl decreases, as discussed in the three-layer case. When the layer is relatively thin (alto = 4.0), the dimensionless mode II SIF decreases drastically as the number of layers (n) increases.

For the E-glass-epoxy composites case shown in Fig. 4, the shear moduli of E-glass fiber and epoxy are 4.4× 10 6 psi and 0.19x 10 6 psi with Poisson's ratios being 0.2 and 0.35,

Table 1. Converging behavior for dimensionless mode II SIFs for various numbers of Laguerre and Gaussian points

Dimensionless SIFs (I~n)/p#v/-~)

/'1 = / '2 =0.5, FI = F2 = 2.0, -Fi = r2 = 0.5, r l = / ' 2 = 2 . 0 , a/to = O. 1, a/to = 4.0, a/to = 4.0, a/to = 4.0,

NL No f l = 0 f l = 0 r = 1 r = 1

I0 50 0.998249 1.643582 0.531460 0.628197 I00 0.998248 1.643582 0.531459 0.628196 II0 0.998248 1.643582 0.531459 0.628196

20 50 0.999811 1.647560 0.534724 0.624701 lO0 0.999810 1.647559 0.534722 0.624701 II0 0.999810 1.647559 0.534722 0.624700

30 50 0.999687 1.648907 0.535079 0.624194 100 0.999686 1.648906 0.535077 0.624193 110 0.999686 1.648906 0.535077 0.624193

40 50 0.999479 1.648950 0.535088 0.624180 100 0.999479 1.648947 0.535086 0.624179 l l 0 0.998479 1.648947 0.535086 0.624179


1.3

1.1

0.9

0 .7

0 .5

0 .3

(a)

w ~ m . t

~ B = O - - - - - - B = I

- = 2 - 3 n = 4 ~ 5

I I I 1 2 5 4

r0

(a) a/to = 1.0

2.4 (b) ~ B = O

------B=I

2.0 n=2 ___------

/ 1.2

0 . 8 ~

0.4 ~ - . ~=

0.0 I 1 2 3 4

r'o

(b) a/to =4.0

Fig. 3. Dimensionless mode II stress intensity factor for multi-layer case: (a) alto = 1.0; and (b) a~ to = 4.0.

516 S.H. KIM et al

/ r a i l l/ca

Y

It~, P~ [

p/, v/ 2a

tin'

t j

l/2t.o

1/2t~

t j

tm

X--

Fig. 4. Geometry of monolithic composites.

respectively[19]. The local fiber volume fraction near the cracked epoxy layer region, VLF, is calculated as[20]:

tf - - . (22)

VLF -- tmO + tf Also, the global fiber volume fraction far from the cracked epoxy layer region, VCF, is calculated as:

tf V~F = - - (23) tm+ tf

In the numerical analysis, VLF to be equal to 0.2, 0.4, 0.6 and 0.8, whereas VGF takes the values 0.3, 0.5 and 0.8 and a/t~ varies from 0.1 to 4.0. Figure 5 shows the dimensionless mode II SIF for each term (p = 0 and 1) of polynomial loading in case of various local and global fiber volume fractions. The dimensionless mode II SIFs depend more on the local fiber volume fraction (VLF) than the global fiber volume fraction (VGF). For the lower local fiber volume fractions, relatively larger dimensionless mode II SIFs are produced. The global fiber volume fraction (VcF), however, contributes significantly to the dimensionless mode II SIFs if the cracked epoxy layer gets thinner (a/tmo > 2) and the local fiber volume fraction is low enough (VLF = 0.2). It is noted that the dimensionless mode II SIFs for the linear crack surface loading case (fl = 1) produces slightly dissimilar patterns, compared with those for the uniform crack surface loading case (fl = 0).

Finally, the hybrid composites case shown in Fig. 6 is considered. The local fiber volume fraction near the cracked epoxy layer region, VLF, is calculated as:

tfl = . (24)

VLF tmO +/fl


1 . 3 5

k • 1 . 2 0 cQ.

:~ 1.05

0 .90

0 .75

0 .60

0.45

0 . 3 0

V w = 0.3

- - - - - - Vc~ =0.5 . . . . . - " "

. . . . . . . Vcp=0.8 . . . . . . . .

. 5 . - - - - ~ _ " _ ~ ~ - ~ ~--0 Vn~- = 0.2

" ~ " m ~ ~ - . , ~ . B . , . e . .

~=1 . ~

V ~ = 0 . 2 V,~-=0.4 Vu~=0.6 V ~ = 0 . 8 I I I 1 2 3 4

a/t~

Fig. 5. Dimensionless mode II stress intensity factor for monolithic composites.

Y

p~, v~

[~m o Vm

p~ ,v~

PlL, VII

Pro, Vm

pl~, v12

___1___ 2=

Fig. 6. Geometry of hybrid composites.

t=

11/2t~

I tjl I I t,, I I t~I

;c

518 S.H. KIM et al

$

1.35

1.20

1.05

0.90

0 .75

0.60

0.45

E-glass/epoxy/Graphite/epoxy E-glass/epoxy Gral~hite,/epoxy/E- glas s/epoxy Gral~hite,/elx~xy

I - - /

/

~=0 VLF ffi 0.2 ~ - _ 1

V~ =0.8 " - ~=1

Vze ffi 0.2 Vw = 0.5 V~ = 0.8 t I 2 3 4

I 0.30 0 1

a/ t,o

Fig. 7. Comparison of dimensionless mode lI stress intensity factors between monolithic and hybrid composites.

Also, the global fiber volume fraction far from the cracked epoxy layer region, VGF, is calculated as:

tfl + tf2 = . (25)

VOF 2tin + tfl + tf2

In the numerical analysis, the shear modulus of graphite fiber is assumed to be 33.3 x 106 psi with Poisson's ratio being 0.2 [19]. Figure 7 shows the dimensionless mode II SIF for each term of polynomial loading in case of various composites. The graphite-epoxy composites yields the least mode II SIFs. When the local fiber volume fraction is low (VLF=0.2), the mode II SIF value for E-glass-epoxy-graphite-epoxy composites is similar with that for graphite-epoxy-E- glass-epoxy composites. However, when the local fiber volume fraction is relatively high (VLF = 0.8), the mode II SIF value for graphite-epoxy-E-glass-epoxy composites is much smal- ler than that for E-glass-epoxy-graphite-epoxy composites.

4. CONCLUSIONS

The mode II stress intensity factors (SIFs) for a central crack subjected to an arbitrary shear loading in the multi-layered material composed of elastically dissimilar layers are analyzed by Fourier integral transform method. The following conclusions are obtained from the numerical analysis:

1. The dimensionless SIFs for the half space and finite number of layers agree with the previous results.

2. For multi-layered materials, the dimensionless SIFs for each term of polynomial loading increases as crack length to layer thickness ratio and shear modulus ratio increase, and the order of each term of polynomial loading and the number of layer decrease.


3. For monolithic and hybrid composites, the dimensionless SIFs for each term of polynomial loading depends mainly on the local fiber volume fractions. As the local fiber volume fraction increases, SIFs increase.

REFERENCES

1. Gecit, M. R. and Erdogan, F., ASME Journal of Engineering Materials and Technology, 1978, 100, 2-9. 2. Hilton, P. D. and Sih, G. C., International Journal of Solids and Structures, 1971, 7, 913-930. 3. Lin, W. and Keer, L. M., ASME Journal of Applied Mechanics, 1989, 56, 63--69. 4. Fowser, S. W. and Chou, T. W., ASME Journal of Applied Mechanics, 1991, 58, 464-472. 5. Hilton, P. D. and Sih, G. C., A sandwiched layer of dissimilar material weakened by crack like imperfections. In

Proceedings of the Fifth South Eastern Conference on the Theoretical and Applied Mechanics, 5, ed. Rogers, G. L., Kranc, S. C. and Henneke, E. G., 1970, pp. 123-149.

6. Kim, S. H., Oh, J. H. and Ong, J. W., The Korean Society of Mechanical Engineers, 1991, 15(5), 1611-1619. 7. Kim, S. H., Oh, J. H. and Ong, J. W., The Korean Society of Mechanical Engineers, 1992, 16(5), 838-849. 8. Isida, M., International Journal of Fracture, 1971, 7(3), 301-316. 9. Isida, M., Fracture Mechanics and Strength of Materials 1976, 2.

10. Chen, Y. Z., International Journal of Fracture, 1989, 41, R29-R34. 11. Chen, H. and Chang, C. S., Engineering Fracture Mechanics, 1989, 34(4), 921-934. 12. Fan, H. and Keer, L. M., ASME Journal of Applied Mechanics, 1990, 61, 250-255. 13. Lee, K. Y., Park, M. B. and Kim, S. H., The Korean Society of Mechanical Engineers, 1994a, 18(6), 1382-1387. 14. Kim, S. H., Lee, K. Y. and Park, M. B. (to appear), Mode I stress intensity factors for layered materials under anti-

symmetric loadings. Engineering Fracture Mechanics. 15. Lee, K. Y., Kim, S. H. and Park, M. B. (to appear) Stress intensity factors for multi-layered material under anti-

symmetric polynomial loading. The Korean Society of Mechanical Engineers. 16. Copson, E. T., Proceedings of the Glasgow Mathematical Association, 1961, 5, 19-24. 17. Ficher, W. B., Compendium of Stress Intensity Factors. Hillingdon Press, 1976. 18. Sih, G. C. and Chen, E. P., Cracks in Composite Materials (VI: Mechanics of Fracture). Martinus Nijhoff, The

Hague, 1981. 19. Schwartz, M. M., Composite Materials Handbook, 2nd edn. McGraw-Hill, New York, 1992. 20. Bechel, V. T. and Kaw, A. K., International Journal of Solids and Structures, 1994, 31(15), 2053-2070. 21. Tada, H., Paris, P. and Irwin, G., The Stress Analysis of Cracks Handbook. Del Research, Hellertown, PA, 1973.

(Received 8 November 1996, in finalform 27 September 1997, accepted 28 September 1997)

APPENDIX A

The components for matrix Q in eq. (10) are defined as follows:

QII =Zl Qx2 = tanhzl

Q21 = 1 + Zl tanhzl

Q22 = 1

Q31 = 2(1 - vl)tanhzl + zl

Q32 = tanhzl

Qal = -z l tanhzl + (1 - 2vl)

Q42 = -1 , (A1)

520 S . H . K I M e t a l

Q(4j-3X4j-I) = --1

Q(4j-3X4~ = - z j

Q(4j-3)~4j+l) = - t a n h zj

Q(4j-3x4j+2) = - z j t a n h zj

Q(4j -2x4 j - I ) = - t a n h z j

Qt4j-2)(4j) = - z j t a n h z j - 1

Q(4j-2)(4j+ l) - - - 1

Q(4j-2X4j+2) = - t a n h zj - zj

Q(4j-I)(4j-I) = - - F j

Q(4j-I)(4j) = - F j [ - 2 ( 1 - vj+l ) t a n h zj + zj]

Q(4j-I)(4j+]) = - F j t a n h z j

Q(4j-i)(4j+2) = - F j [ 2 ( 1 - vj ) + zj t a n h z j ]

Qt4j~t4j-l) = Fj t a n h z j

Q(4jX4.i) = r j [ - I + 2Vj+l -~ Zj t a n h z j ]

Qt4jX4j+I) = r j

Q¢4j×4i+2) = Fj[z j - (1 - 2v j ) t anhz j ]

Q~4j+zx4j-l) = 1

Q(4j+])(4j) = --Zj+I

Q(4j+IX4j+I) : tanhzj+] Q(4j+I)(4j+2) : Zj+l t anh zj+]

Qt4j+E)(4j-J) = t a n h z j + l

Q(4j+2X4j) = 1 + Zj+j t a n h z j + l

Q(4j+2X4j-I) = 1

Q(4j+2)(4j+2) = t anhz j+ ] + z j+ t ( j = 1 . . . . . n - 1),

Q(4j+3)(4j-l) = 1

Q(4j+3x4j) = 2(1 - Vj+ l ) t anhz j+] + zj+l

Q(4j+3)(4j+l) - - t anhz j+l

Q(4j+3x4j+2) = zj+l t anhz j+ l + 2(1 - vj+t)

Q(4j+4X4j-l) = - t a n h z j + l

Q(4j+4X4j) : - Z j + l t anhz j+ l + 1 - 2vy+]

Q(4j+4X4j+ I) = - 1

Q(4j+4X4j+2) = t a n h Zj+l(1 - 2vj+]) - z j+ l ( j = 1 . . . . . n - 2),

w h e r e F ~ # y / # j + t, z F ~ h j (1" = 1 . . . . . n - 1). Al l o t h e r c o m p o n e n t s a re zero .

(A2)

(A3)

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Mode II stress intensity factors for interlaminar cracked composites under arbitrary shear crack surface loading