Composites Science and Technology 46 (1993) 157-166

REFINED TWO-DIMENSIONAL ANALYSIS OF CROSS-PLY LAMINATES WITH TRANSVERSE CRACKS BASED ON THE

A S S U M E D CRACK OPENING D E F O R M A T I O N

J. H. Lee & C. S. Hong* Department of Aerospace Engineering, Korea Advanced Institute of Science & Technology, 373-1 Kusung-Dong, Yusong-Ku,

Taejon, 305-701, South Korea

(Received 15 July 1991; revised version received 6 November 1991; accepted 31 January 1992)

Abstract A refined two-dimensional analysis method, taking into account the crack opening deformation, is proposed for the evaluation of stress distributions and the prediction of stiffness reduction in transverse cracked cross-ply laminates. The interlaminar stresses which play an important role in laminate failure are evaluated by using the concept of interface layer. A series expansion of the displacements is employed and the thermal residual stresses and Poisson effects in the laminate are taken into consideration in the formulation. The stress distributions and the stiffness reductions due to transverse cracks are compared with finite element results and experimental data, respectively. The prop- osed method represents well the characteristics of the stress distributions. The predicted laminate stiffness reductions are in excellent agreement with experimental data for various materials. The proposed analysis can be applied as a basis for the prediction of the induced delamination onset by using appropriate failure criteria.

Keywords: transverse crack, interlaminar shear layer, shear-lag method, interlaminar stress, delamination stiffness reduction

1 INTRODUCTION

The predominant damage mechanism of laminated composites in the initiation stage is the formation of transverse cracks. Although the transverse cracks do not lead immediately to the final fracture of the laminate, they result in a redistribution of layer stresses which can affect the stiffness reduction of laminated composites and interlaminar stresses: these,

* To whom all correspondence should be addressed.

Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd.

157

in turn, can induce the delamination. 1 The key to the analytical treatment of the transverse cracking problem is, of course, a stress analysis of laminates with transverse cracks. The development of a satisfactory analysis method for cross-ply laminates with transverse cracks has attracted a substantial number of investigators. ~-13 However, the previous methods were developed to focus on the prediction of the stiffness reduction. It is well known that the delamination induced by the matrix cracks is the major damage mechanism of the laminate fracture. The first step for the prediction of the induced delamination onset is the evaluation of the interlami- nar stress near the transverse crack.

In general, the existing methods may be grouped into two categories, viz. the crack model approach and the continuum damage model approach. The first approach is the method of evaluating the stress distributions in a unit cell which contains a single crack or two cracks on the basis of uniform spacing of transverse cracks at a certain limit, called the characteristic damage state (CDS). The stress distributions around a transverse crack are determined by various methods such as shear lag, 1~ variational, 7 and approximate elasticity. 8'9 The second approach 1°-13 is the method of evaluating the locally averaged constitutive properties from a transverse cracked local volume so that the effect of the transverse cracks are reflected in the constitutive equation, i.e., the homogeneous degenerated layers are substituted for the transverse cracked layers. In this approach, only the in-plane stress components are obtained because the stresses are determined from the damaged constitutive equations and classical laminate theory. Therefore, the stress distributions and the interlami- nar stresses which play an important role in laminate failure can't be determined by this method.

Many investigators have used the shear lag method for its simplicity. Lim and Hong proposed a simple modified shear lag analysis which uses the concept of

158 J. H. Lee, C. S. Hong

the interlaminar shear layer. 3"4 Han and Hahn proposed a shear lag method with a second-order polynomial assumed for the crack opening displacement. 5 In their analysis, the traction free conditions are not satisfied on the transverse crack surfaces. Lee and Daniel proposed a simplified shear lag method which assumed a linear shear stress distribution through the thickness. 6 The shear-stress- free condition on the crack surface is not satisfied in their analysis. Also, the distribution of interlaminar normal stress contradicts the self-equilibrium condi- tion. Generally the shear lag method is simple and gives the reasonable prediction of stiffness reductions. However, the shear and through-thickness deforma- tion due to opening displacement of the transverse crack are not considered in the shear lag method. Moreover, the shear lag methods are one-dimensional analyses and the complete stress distributions cannot be determined.

Hashin used a variational method with admissible stress systems on the basis of the principle of minimum complementary energy. 7 The stress distribu- tions were obtained and were in qualitative agreement with numerical results. However, the normal stress gradient through the thickness is neglected. Aboudi proposed an approximate elasticity method with a second-order expansion of displacements in the Legendre polynomials. 9 In the variational method and approximate elasticity method, the stress distributions can be evaluated and the predicted stiffness reduction is found to be in good agreement with experimental data. However, the interlaminar shear stress at the crack tip cannot be determined and only the estimate of stress distributions can be obtained.

In this paper, we propose a refined two-dimensional analysis method for the evaluation of the stress distributions and the analytical prediction of stiffness reduction in transverse cracked cross-ply laminates. A series expansion of the displacements, considering the symmetry or anti-symmetry about the mid-plane, is employed together with equilibrium equations, con- tinuity of tractions at the interface, and traction-free conditions on the outer surface of the laminate. The concept of interface layer which is the extended assumption of the interlaminar shear layer is applied for the evaluation of the interlaminar stresses. The thermal residual stresses and Poisson effects are taken into consideration in the formulation. Numerical results are presented for the first expansion case of the proposed analysis method. The stress distribuions are compared with finite element results. The predictions of the stiffness reduction due to transverse cracks are also compared with those of previous analyses and experimental data. T M The proposed analysis can be applied as a basis for the prediction of the induced delamination onset by using appropriate failure criteria. '-~

Gx,AT ~ . ~ : T Z - r Z - r : - ] " : ~-'-'~

:::::::::::::::::::::::::::::::::::::::::::::: -,~_ ~ . \ / I

Z

O-x, AT

b I 2d

Layer (1)

oo + _ __ ~/_ i - go%m 7 - ~ - - -

On "~. \ ~ Interface layer

I- 2L --'H " " Layer (2)

Fig. 1. Transverse cracked cross-ply laminate and analytical model.

2 A N A L Y S I S O F T R A N S V E R S E C R A C K

2.1 Genera l f ormula t ion Consider the cross-ply laminated composites subjected to mechanical and thermal loading and a unit cell of this laminate containing two cracks as shown in Fig. 1. The two outside 0 ° layers are of equal thickness, b, the two interface layers are of thickness do, the central 90 ° layer is of thickness, 2d, and the crack spacing is 2L. The interface layer is the extended concept of the interlaminar shear layer where only the shear deformation is taken into consideration, i.e., the shear and normal deformation of the interface layer is included for the evaluation of interlaminar shear and normal stress, respectively.

Each lamina is considered to be transversely isotropic so that its stress/strain relationships in material principal axes 16 are given by

o~ Ic , : c:: c~J o o o I / ~ / o~ I c , : c2~ c:~ o o o I , J ~ t " ~ = l o o o c4,, o , , l l Y ~ [ (1)

V ' I L°o o o o c~6 o I / r ,~ I t f ,~J o o o o c ~ tY,~)

where the C u denote the stiffnesses and C44- (C22-C23)/2. For simplicity of formulation, C44 is used instead of (C22- C23)/2. The relationship of the C u to material properties is given in Ref. 16.

The displacements are expressed as the summation of a linear damage-free component and a perturbed component due to the opening of transverse cracks. The perturbed displacement fields in the vicinity of a transverse crack are expanded in terms of polynomials in the z direction; the coefficients of these polynomials are function of x only. Symmetry or anti-symmetry

Refined 2-D analysis of cross-ply laminates with transverse cracks 159

about the x axis is taken into consideration in the expressions for layer 1 (90 ° layer), and only the portion z -> 0 is considered for layer 2 (0 ° layer). From now on script 1 indicates the 90 ° layers and 2 indicates the 0 ° layers. The expressions for the displacement fields are as follows.

Ul(X , Z) ~- ~.xo X "~ e l i ( X ) i=0

v l (y )= eyoY (Z'~ 2/+1

WI(X, Z ) = ~'zo (,)z "~- lPli(X ) \ ~ ] i=0

U2(X , Z) ~" exoX "~ E ~2i(x) i=0

v2(y) = eyoy

w2(x, z ) = (2) +d(e~lo ) (2) 2. ( z ) i e~oZ - ~ o ) + ~ ~ , ( x ) i=0 k¢~/

(2)

Displacements u, v, and w denote the components in the x, y, and z directions, respectively. The arbitrary functions qh , ~P,i, q~z/ and ~P2i a re to be determined later. As seen in the expressions for displacements, we assume that v, taking into account Poisson effects in the laminate, is not disturbed by the transverse cracks. The subscripts 'o ' denote the linear components. The linear components which are determined by classical laminate theory (CLT) 16 are given by

e~o = eM + e~o, eyo -..~- exMo+eyo \ A22

1 e(')--- (C23exo + C,2t~yo), (3) zo C22

1 e(2) = -- - - ( Clzex ° .~_ C23Syo) zo C22

where the superscript M and T denote the mechanical and thermal strain components, respectively. The A u denote the laminate extensional stiffnesses. The components of the strains are given by

e(xi)(x, z ) = aul(x, z) e(/) _--dvi(y ) OX ' dy '

E~n(x, z)= awAx' z) Oz

(J) _ _ + Y= (X, Z)= aU/(X, Z) aW:(X, Z) az ax

(4)

where the subscript j represents variables associated with the 90 ° or 0 ° layer, i.e., j = 1 refers to a 90 ° layer and j = 2 is for a 0 ° layer.

The interface layer is represented by equivalent shear spring and equivalent linear springs in the x and z direction, respectively. Thus, the interlaminar shear stress, v, and the interlaminar normal stress, a, in the

interface layer are given by

r(x) = ~ [u~(x, z ) - u,(x, Z)lz=~ (5) 1 7

O(X) = ~ [w2(x, z ) - Wl(X, z)L= ~

where the Go and E0 denote the shear and elastic moduli of the interface layer, respectively.

The equilibrium equations of two-dimensional linear elasticity are

~,,~'(x, z) Ox

(J) arx~ (x, z) Ox

(J) 0r~z (x, z) + = 0

az

aO~z'(X, z) t- = 0

Oz

(6)

The equilibrium conditions at the interface layer are

~(~ , z) (~) = ~z (x, z) = r(x), a~O(x, z) = oT)(x, z) = o(x) at z = d (7)

The stress-free conditions on the outer surfaces of the laminate are

CxZ)(x, z) = a(z2)(x, z) = 0 at z = d + b (8)

Equations (6) are multipled by (z/d) k and integrated by parts through-thickness for each layer. During integration the equilibrium conditions at the interface layer (eqn (7)) and stress-free conditions on the outer surfaces (eqn (8)) are imposed. This process yields

dN~J)(x) k v ~ Q , ( x ) - ( - l ~ r ( x ) = O (9) dx d

dV~J)(x) kQ(J)--l(X) - - ( -1)Ja (x) = 0 (10)

dx d

The value of k in eqns (9) and (10) are the same as the power of (z/d) k used in the expressions of u and w, respectively, That is, for j = 1 (90 ° layer), k =0 , 2 . . . . . 2n in eqn (9) and k = 1, 3 . . . . . 2n + 1 in eqn (10). For j = 2 (0 ° layer), k = 0, 1, 2 . . . . . 2n in eqns (9) and (10).

In eqns (9) and (10), N, Q, and V are resultant forces given by

{W )(x)] ~,f o(2(x, ~)) Q ~ D ( x ) ~ : f { o ~ ' ) ( x , z ) } ( d ) k d z (11)

v~'(x)J "', l~(2(~, z ) J

where f o r j = l , ol i=0, f l i=d and f o r j = 2 , o~ i = d , /3;=d+b.

Substituting eqns (1), (2) and (4) into eqn (11) and using CLT, the resultant forces are obtained in terms of the unknown functions and is written as follows.

d a°) ~ [C222i + d U~'(x) = T - ~ vxo + i=0 "+ 1 #)li(X)

2 i + 1 ] + C23 2i + k + 1 ~u(x)

160 J. H. Lee, C. S. Hong

Q~')(x) = C23 dpu(X) ~=o 2i + k + 1

2 i + 1 ] + C222i + k + 1 ~li(X)

W ) ( x ) =

d~,+l ~r,2 , ~ [ d ~ i + , + l Nl2)(x) = ~ ~ x o q'- Cll i=0 i + k + l

i¢~i+, ~P2i (X) ] -F C12

2 n I dt~i+k+l QlZ)(x) = E C,2

i=o i + k + 1

n [ 2i d ] C44 ~] ~ q~,~(x) + ~P'li(X) (12)

i=0 L Z I " I - K 2 i + k + 2

- - ( ~ 2 i ( X )

i O i + k , , , ] - - + w, x)j

2, [ibi+k dOi+g+l , ] = E ep ,(x) + - - W,(x)

~=o1_i + k i + k + l

where 6 m = (1 + (b/d)) m - 1 and a prime denotes differentiation with respect to x. O~/o ), j = 1, 2 denote the summation of mechanical and thermal stress components of the undamaged cross-ply laminate in the x-direction.

Substituting eqns (5), (12) into eqns (9) and (10) yields the following governing differential equations whose solution describes the approximate elastic field around a transverse crack.

[ d C23 (2i + 1) - C44k , C22 ~)~i(x) '~ lP li(X)

~=0 ~'~[ 2 i + k + l 2 i + k + l

- d 2i + k + 1 l- ~li(X) At- Z i=0E ~2i(x) = 0

(13)

2 . [ d6i+k+~ ,, + (C12i - C 6 6 k ) 6 i + k , E c,, i=o i + k + l i + k

(C66ik6i+k-lGd~) ] Gox -q"

~ [ d 2C44i - - C23 k , c4, ) + ¢ . (x )

i=0 2 i+ k + 2 2i + k

(2i + 1)k Eo] Eo E ~- lPli(X ) At- ~/2i(X)

2i + k do/ do i=o

= 0

(14)

= 0

(15)

- d6i+k+ I (C66i - - C l 2 k ) 6 i + k ,

i=0 [C66i + k + 1 ~p'~i(x) + 7+-k ~2 i ( x )

(16)

For eqn (13), k=O, 2 , . . . ,2n; for eqn (15), k = l , 3 . . . . . 2n + 1 and for eqns (14) and (16), k =0, 1, 2 , . . . , 2n.

The governing equations are a system of coupled, second-order differential equations. This system is

similar in nature to those obtained for coupled oscillation problems. These equations should be supplemented with boundary conditions, including the stress free surface at the transverse crack and the applied displacement, at x = 0, 2L which are given by

z ) = o, r=(x, z) =0, (2) rx~ (,~, z) = O, u2(x, z) = e~ox (17)

2.2 First expansion ease For the numerical evaluation, the governing eqns (13)-(16) are expanded for the first expansion case, n = 1, for which the governing equations are as follows:

d e d [A] ~ {(I)} + [B] ~ {(I)} + [C]{(I)} = 0 (18)

where {O} = {~10~ll~20~21~22~10~)ll~320~J211P22} T. The components of [A], [B] and [C] are shown in

the Appendix. The general solution of the governing equations is obtained by the well-known state vector method. 17 The solutions take the form of sums of characteristic solutions multiplied by arbitrary con- stants. The application of boundary conditions (17) determines the arbitrary constants. From the solu- tions, the displacement and stress fields are obtained.

2.3 Laminate st i theas refluetion due to transverse cracks Multiple transverse cracking with additional applied stress in laminated composites has been known to cause a progressive laminate stiffness reduction. The stiffness reduction due to multiple transverse cracking in the 90 ° layer is derived.

Because of overall equilibrium, the average axial stress of a damaged cross-ply laminate with a crack spacing of 2L can be calculated at arbitrary x. For the convenience of calculation, the average axial stress, Ox, is evaluated at the transverse crack surface (x =o).

_ ! ( (rx b + d)d (~(x2)(0, z) dz (18)

From the stress/strain relationships for the un- damaged and the damaged cross-ply laminate for a constant applied strain, the normalized laminate stiffness reduction due to the multiple transverse cracking is defined by

Ex ox = - - (19)

Ego axo

where Ego and Ex denote the Young's moduli of the undamaged laminate and the damaged laminate with crack spacing 2L in the x direction, respectively, o,,o denotes the axial stress of the undamaged laminate for the constant applied strain.

Refined 2-D analysis of cross-ply laminates with transverse cracks 161

In eqn (18), the average axial stress of damaged cross-ply laminate is the summation of the mechanical and residual stress. The residual stress released due to the multiple cracking in one loading-unloading E~(GPa) 127.8 procedure, is small compared to the applied stress. E2(GPa) 9.4 Therefore, the effect of the residual stress is GI2 (GPa) 4.2 neglected. (;23 (GPa) 3.1

~/12 0"28 h (mm) b 0-12 (0-05)

3 EXPERIMENTS

An experimental program was performed to investi- gate the stiffness reduction due to multiple transverse cracking. One of the first difficulties to be overcome is that the laminate stiffness changes are small since the axial behavior of laminates is so dominated by the 0 ° layers. In order to increase the stiffness reduction due to multiple cracking, the transverse cracked layer (90 ° layer) can be made thick. It is noted that for thicker 90 ° layers there are more curved or partial curved cracks which is not the assumed shape in the present analysis that straight cracks. 14 Therefore, prepreg of which one ply thickness is 0.05 mm is used to fabricate cross-ply laminates.

Two laminates, [0/905/0]T and [ 0 / 9 0 1 0 / 0 ] T , w e r e

tested in uniaxial tension. The graphite/epoxy prepreg used P3051-F05 was manufactured by Torayca. The laminates were fabricated in a panel autoclave according to the manufacturer 's recommended curing cycle. Specimens of dimensions 180mm by 20mm were obtained from the laminates. Abrasive papers were used for tabs, as in 90 ° tensile tests, instead of the conventional glass/epoxy tabbing material. In order to follow the progression of the transverse cracking, edge replication was used. The free edges of specimens were polished with 1 #m diameter alumina particles to enhance the detail observed in edge replicas.

Mechanical testing was conducted in an Instron 1350 universal testing system. All tests were conducted under monotonic loading at a constant strain rate of 0-0025min -1. The axial strain was measured with a 3 cm strain gauge (Kyowa KFG-30- 120-CI-23). The load/strain curve was continuously plotted during the test. The specimens were unloaded as soon as a pop-in was detected in the load/strain curve. After the edge replica was obtained under an about 2000N tensile load, the specimens were loaded. This loading-unloading procedure was repeated. The edge replicas were subjected to microscopic examina- tion to measure the number of cracks in the gauge section. The stiffness was measured only on the unloading portion of the stress/strain curves.

4 RESULTS AND DISCUSSION

Material properties of typical graphite/epoxy and glass/epoxy composites used in the present analysis

Table 1. Material properties of used composite systems

P3051-F12 (F05) AS4/350214 E-Glass/epoxy L7

144.78 41.7 9-58 13.0 4.79 3.4 3.15 a 4.58 0-31 0"3 0.127 0-203

a Calculated from assumed value of v23 = 0.52. b One-ply thickness.

Table 2. Interface layer properties of used composite systems

P3051-F12 (F05) AS4/3502 TM E-Glass/epoxy 3'4

E o (GPa) 3.45 3.45 1-78 Go (GPa) 1.28 1.33 0.684 do (mm) 0.012 (0.005) 0-0127 0.0203

are given in Table 1. Since the shear modulus of AS4/3502 graphite/epoxy in the 2-3 plane, which is necessary for numerical evaluation, is not known, it is assumed that (;23 is 3-15GPa, which implies that Poisson's ratio, v2a, is 0-52. Experimental data for the interface layer properties are not generally available owing to experimental difficulties. Values similar to those of Refs 1, 3, 7 and 14 for the interface layer properties are used in the present analysis as shown in Table 2. The proposed method is first checked with finite element results by comparing the stress distributions. A finite element code 'ANSYS' was used to determine the stress field in a P3051-F12 [02/90]s laminate where a half-crack spacing, L, is 8d. Due to symmetry, only one quadrant of the section is modeled by using 8-node isoparametric quadrilateral plane strain elements. Finite element model consists of 1260 elements with 3927 nodes as shown in Fig. 2. Only the mechanical loading (1% of tensile strain) was applied.

C r a c k 1 Surface(

/ I> / l,lUllllllil ll I I I I I ) l l I

I I I I I I I I I I ILH'I i I I J I I I I I I l > n i l l l l l l l J ~ l l l [ l l l l l l l l I I I I I

i l l l l l J 1 1 1 1 l l [ l l l l l [ l l I I I I I I I I I ~ 14 I I l > - - ' - - ~ . ' . ' . ' . ' . : : : [ t l l t ~ ' . ', . . . . . I I I I I I 1 I I + ~XO

Crock :;:;i, . . . . ' ' ' Sor,oce~ = : : :~ ! ! ! i i i i i i ! ! i i i ~ i i ! ! ! ! ' : i i i [ ,--i= ' d

Fig. 2. Finite element model and boundary conditions.

162 J. H. Lee, C. S. Hong

N

1.0

0.8

0.6

0.4

0.2

0.0 -20

102/90]S L=8d ~I do 6xMo=0.01 AT=0.0 I ~ t ~ .

I x 2L

~ ~dd--S/x/d=O.016

1/. I - : ""nt

,~1 ~ t .t._ n ' - " i, I I 0 20 40 60 80 1 O0 120

a

1.o

0.8

0.6

0.4

0.2

o.o - lOO-75 -50 -25

P3051-F12 [02/90]s h=Sd ~ xMo =0.01 AT=0.0

0 0 Q • •

•l'x/diO'0161i]

o

l q x 2L

/ x/d=0.979 / x/d=2.0

Present oAl* FEM

I I I

25 50 75 100 125

O-(X 1) (MPa) -(1) (ePa) Z

Fig. 3. Through-thickness distribution of axial stress in the 90 ° layer for P3051-F12 [02/90]= graphite/epoxy laminate.

The stress distributions through the thickness are described as polynomials in the z direction owing to the displacement assumption. To confirm the validity of the displacement assumption, a comparison was made between the through-thickness stress distribu- tions in the cracked 90 ° layer predicted by the present and the finite element method as shown in Figs 3-5. The through-thickness distribution of the axial stress, ax, in the cracked 90 ° layer is shown in Fig. 3. The

P.I_F12 ..o.o :il L=Sd - - (T _ 6xo=O'01 I- ,,I x 2L

1.0 ~ 6

0.8 L "/ ; ;4 ,/d:o5o2

j E

~ . /~ / ' ~ le : FEM

o.o -10 0 10 20 30 40 50 60 70 80 90 100

T(1) (MPa) xz Fig. 4. Through-thickness distribution of shear stress in the

90 ° layer for P3051-F12 [02/90]s graphite/epoxy laminate.

Fig. 5. Through-thickness distribution of lateral normal stress in the 90 ° layer for P3051-F12 [02/90]= graphite/epoxy

laminate.

variation of the axial stress through the thickness is remarkable near the transverse crack. It should be noted that the axial stress has been assumed constant through the thickness in the shear lag and variational method. This assumption is reasonable for the prediction of transverse cracking onset or multiplica- tion since only the far field stresses are the significant ones. 5 However, for the prediction of the stiffness reduction, the situation is changed, i.e., the stresses around the transverse crack are important. The distribution by the present method shows the slight compression region near the transverse crack surface. This phenomenon occurred because the present method evaluates the stresses from the assumed displacement fields.

The through-thickness distribution of the shear stress, rxz, in the cracked 90 ° layers is shown in Fig. 4. The shear stress which takes charge of the stress transfer from the transverse cracked (90 ° ) layer to the adjacent (0 °) layer, becomes smaller apart from near the transverse crack. It is noted that the shear stress has been assumed to have a linear distribution through the thickness in the previous methods. T M

Figure 5 shows the through-thickness distribution of the lateral normal stress, o=, in the cracked 90 ° layer. The lateral normal stress induced by the shear stress for moment equilibrium changes dramatically from compression to tension near the transverse crack. As expected, the distributions determined by the present method do not describe the singularity near the crack tip. However the present results are in good

Refined 2-D analysis of cross-ply laminates with transverse cracks 163

1 2o I [ - - : Present I lOO t - - - : FEM

8 0 P3051 -F12

6°l! x%=O.OlAT=O.O

4 0 z / d = 0 . 6 2 9 b O. t

20 d . . . . . . fro =

2L x

o

I I I - 2 0 i i ~ i = o 1 2 3 4 5 6 7

x/d

20 ~ ~ z / d = 0 . 8 9

~. - 2 0 i lL . .> z/o=o629 :o2/ o1

-40 qo =°°1

- 6 0 == =

- - : Present - 8 0

- 1 0 0 L ~ 0 1 2 3 4 5 6 7 8

x/d

Fig. 6. Distribution of axial stress along the x axis in the 90 ° layer for P3051-F12 [02/90], graphite/epoxy laminate.

agreement with the finite e lement analysis results in most regions.

In Figs 6-8 , the stress distributions along the x axis determined by the present and the finite e lement method are compared. Figure 6 shows the distribution of the axial stress in the cracked 90 ° layer. The distribution by the present method shows a slight compressive value near the crack tip as ment ioned above. Since the present method evaluates the stresses considering Poisson effects of the laminate and the finite element analysis evaluates the stresses under plane strain condition, the far field stresses given by the present and the finite e lement methods are different.

The distributions of the shear stress and the lateral normal stress in the cracked 90 ° layer are shown in Figs 7 and 8, respectively. The shear stress is zero at the crack surface, then increases to a maximum value near the crack surface. The shear stress decreases to

Fig. 8. Distribution of lateral normal stress along the x axis in the 90 ° layer for P3051-F12 [02/90], graphite/epoxy

laminate.

zero at the mid-point between two adjacent cracks by virtue of anti-symmetry. The lateral normal stress shows a maximum compressive value at the crack surfaces, then changes to tension for the self- equilibrium and converges to the far field value. The results obtained by the present method represent well the characteristics of the distribution of the stresses.

The distributions of interlaminar stresses are shown in Figs 9 and 10. The interlaminar shear stress shows a maximum value at the crack surface, then decreases to zero at the mid-point between cracks due because of anti-symmetry. The interlaminar normal stress shows a maximum tensile value at the crack surfaces, then changes to compression for the self-equilibrium and converges to the far field stress. As shown in Figs 9 and 10, there are considerable differences in the magnitude of interlaminar stresses very close to the crack tip. However the delamination onset is usually

80

70 - - : Present

- - - : FEM P3051-F12

6 0 [ 0 2 / 9 0 ] s L=Sd

5 0 ' ~ z / d = O . 8 9 E:xUo=O.01 ZXT=O.O

x N 40 b tc h v z/o=o.629 °ii ;o:-)

0 , I I 0 1 2 3 4 5 6 7 8

x / d

Fig. 7. Distribution of shear stress along the x axis in the 90 ° layer for P3051-F12 [02/90], graphite/epoxy laminate.

120

100

P3051 -F12

~ . 8 0 [ 0 2 / 9 0 ] s L=Sd E~ £L / Present ~;xUo=O.01 &T=O.O

6o k / . z T N FEM b| i .

, o

2O

0 i I I ~ I 0 1 2 3 4 5 6 7 8

x / d

Fig. 9. Distribution of interlaminar shear stress along the interface for P3051-F12 [02/90]s graphite/epoxy laminate.

164 J. H. Lee, C. S. Hong

300

250

200

n 150

io N 1oo

P3051-F12 [02 /90 ] s L=8d

~:x% =0.01 &T=0.0

I, 2L I x

FEM

50 ~ n t

- 5 0 i i i i J i i o 1 2 3 4 5 6 7 8

x/d

Fig. l@. Distribution of interlaminar normal stress along the interface for P3051-F12 [02/90], graphite/epoxy laminate.

predicted by the average stress concept, ~5 and the values at the crack tip are not important.

For the application of the present method, the stiffness reductions of the laminates due to multiple cracking are compared with those of previous analyses and experimental data. The stiffness of the damaged laminates is normalized with respect to the stiffness of the undamaged laminates and plotted as a function of the transverse crack density, i.e. the number of cracks per unit length. Figure 11 shows the stiffness reduction of a [0/903]~ E-glass/epoxy laminate with various thicknesses of interface layer. Generally, the thickness of the interface layer is assumed one-tenth of the thickness of one ply on the basis of experimental observation. ~ In order to see the thickness effect of the interface layer on the stiffness reduction, several cases in the range d/lO to d/2000 are considered. It is

noted that the case for do=d~2000 implies the condition of nearly perfect bonding. As shown in Fig. 11, the agreement between predicted and experimen- tal results is excellent when do--d/lO. In previous analyses, except for the variational method, 6 the prediction was higher than the experimental data for this case. This is explained by the fact that the previous analysis methods without consideration of shear and through-thickness deformation were stiff modeling for E-glass/epoxy. In Fig. 11, the limiting value of normalized stiffness for the case of do = d/lO as the crack density tends to infinity was calculated numerically to compare with the ply discount result based on the classical laminate theory. When the transverse crack density is larger than 10 cracks/mm, the predicted value tends to the limit. The difference between the above two cases is caused by the assumption that the lateral displacement of a cracked laminate is exactly the same as that for an undamaged laminate subjected to the same applied strain (see eqn (2)). However, the limiting value of the present method is slightly lower (4%) than the ply discount result.

The stiffness reductions of AS4/3502 cross-ply laminates are shown in Fig. 12. The predicted values are slightly lower than the experimental data. However, the general characteristic of the predicted curve suggests more similarity with experimental data than that obtained by a previous shear lag method. 3 Fig. 13 shows the comparison of the present prediction of the stiffness reduction due to transverse cracks with the present experimental data for P3051-F05 graphite/epoxy cross-ply laminates. It is found that the normalized stiffness reduction is more pronounced with increasing thickness of 90 ° layer and the agreement between the present analysis and experimental data is excellent.

o x

k l J

x 1.0 I , I

03/3 03 e 0.8 c

03 0.6

© N 0 . 4

E 0.2 O

Z

E-Glass/Epoxy [0 /903 ] s

/;:::;:::o 007,;o0

~ - - . ~i Ym i~ii~°U~tu e

( d o = d / l O )

- - : Present

- - - : S h e e r - l a g 3

• : Exper iment 1

I I 0.2 0.4

0.0 i J 0.0 0.6 0.8 1.0

Tronsverse c rack dens i t y ( c r a c k s / m m )

Fig. 11. Normalized stiffness reduction due to transverse cracks in [0/903], E-glass/epoxy laminate.

o x

Ld 1.00 x

LJ 0.98 03 03 0.96

~ 0.94 4--'

m 0.92

N 0.90

E 0.88 o 0.86 Z

AS4/3502 Grophite/Epoxy

\ • .. " _ _ _ _ [ 0 2 / 9 0 2 ] ,

L 0 / 9 0 3 ] s " Present

- - - Sheer - l og 3

I A I t Exper iment 14 <l I I I I

0.0 0.5 1.0 1,5 2.0 2.5 5.0

Tronsverse c rock dens i t y ( c r o c k s / r a m )

Fig. 12. Normalized stiffness reduction due to transverse cracks in AS4/3502 graphite/epoxy cross-ply laminates.

Refined 2-D analysis of cross-ply laminates with transverse cracks 165

o I P3051-F05 Grophite/Epoxy L~ 1 . 0 0 ~ ~x 0.98 i,i

0.96 ~) [0/905/0] T ~_ 0.94

0.92

-~ 0.90

.N 0.88 - -

-6 E 0 . 8 6 • • Experiment [0 /9010/013. 0 z 0.84

I J I 0.0 0.2 0.4 0.6 0.8 1.0

Tronsverse crock dens i ty ( c r o c k s / m m )

Presen ~.

Fig. 13. Normalized stiffness reduction due to transverse cracks in [0/90m/0]T P3051-F05 graphite/epoxy laminates

(m = 5, 10).

5 CONCLUSIONS

A refined two-dimensional analysis method is proposed for the evaluation of the stress distributions and the analytical prediction of stiffness reduction in transverse cracked cross-ply laminates. The stress distributions and the stiffness reduction due to transverse cracks are compared with finite element analysis result and experimental data, respectively. From the present result, the following conclusions can be drawn:

(1) The proposed method represents well the characteristics of the stress distributions in transverse cracked cross-ply laminates.

(2) The stress gradient through-thickness is very pronounced near the transverse crack because of the crack opening deformation.

(3) The interlaminar stresses which can induce the delamination have significant values at the transverse crack tip.

(4) The analytical predictions of the laminate stiffness reduction as a function of transverse crack density are in excellent agreement with experimental data for various materials.

ACKNOWLEDMENT

The authors wish to thank the Korea Agency for Defense Development for the support of this study.

REFERENCES

1. Highsmith, A. L. & Reifsnider, K. L., Stiffness- reduction mechanisms in composite laminates. In Damage in Composite Materials, ASTM STP, 775, 1982 pp. 103-17.

2. Laws, N. & Dvorak, G. J., Progressive transverse

cracking in composite laminates. J. Comp. Mater., 22 (1988) 900-16.

3. Lim, S. G. & Hong, C. S., Effect of transverse cracks on the thermomechanical properties of cross-ply laminated composites. Comp. Sci. Technol., 34 (1989) 145-62.

4. Lim, S. G. & Hong, C. S., Prediction of transverse cracking and stiffness reduction in cross-ply laminated composites. J. Comp. Mater., 23 (1989) 695-713.

5. Han, Y. M. & Hahn, H. T., Ply cracking and property degradations of symmetric balanced laminates under general in-plane loading. Comp. Sci. Technol., 35 (1989) 377-97.

6. Lee, J. W. & Daniel, I. M., Progressive transverse cracking of crossply composite laminates. J. Comp. Mater., 24 (1990) 1225-43.

7. Hashin, Z., Analysis of stiffness reduction of cracked cross-ply laminates. Eng. Fract. Mech., 25 (1986) 771-8.

8. Ohira, H., Analysis of the stress distribution in the cross-ply composite. Proceedings of ICCM V, San Diego, 1985, pp. 1115-24.

9. Aboudi, J., Stiffness reduction of cracked solids. Eng. Fract. Mech., 26 (1987) 637-50.

10. Dvorak, G. J., Laws, N. & Hejazi, M., Analysis of progressive matrix cracking in composite laminates I. Thermoelastic properties of a ply with cracks. J. Comp. Mater., 19 (1985) 216--34.

11. Talreja, R., Transverse cracking and stiffness reduction in composite laminates. J. Comp. Mater., 19 (1985) 355-75.

12. Allen, D. H., Harris, C. E. & Groves, S. E., A thermochemical constitutive theory for elastic compos- ites with distributed damage--I. Theoretical develop- ment. Int. J. Solids Struct., 23 (1987) 1301-18.

13. Allen, D. H., Harris, C. E. & Groves, S. E., A thermomechanical constitutive theory for elastic com- posites with distributed damage--II. Application to matrix cracking in laminated composites. Int. J. Solids Struct., 23 (1987) 1319-38.

14. Groves, S. E., Harris, C. E., Highsmith, A. L., Allen, D. H. & Norvell, R. G., An experimental and analytical treatment of matrix cracking in cross-ply laminates. Exp. Mech., 27 (1987) 73-9.

15. Kim, R. Y. & Soni, S. R., Experimental and analytical studies on the onset of delamination in laminated composites. J. Comp. Mater., 18 (1984) 70-80.

16. Tsai, S. W., Composites Design--1986, Think Compos- ites, Dayton, Ohio, USA, 1986.

17. Meirovitch, L., Analytical Methods in Vibrations, Macmillan, New York, 1967.

APPENDIX: THE COMPONENTS OF MATRIX IN GOVERNING EQUATION FOR n = 1 CASE

[A] is a symmetric matrix, i.e., [A] = [A] T = [aij] and non-zero components are given by

all = dC22

dC22 322 -- 5

a33 ----- bell

dC22 a12 = 3

dCll dCn a34 = T 62 a35 = T 63

166 J. H. Lee, C. S. Hong

a44 = a35

dCn a45 ~ ~ 4

4

dCn 6s a55- 5 "

a66 dC44 dC44

a67 = 3 5

dC44 a77 ----

7

a88=bC66 a 8 9 - 2 2 a s m - 3 3

a99 = a81o dC66 6

a91o -- ~ - 4

dC66 6s aamo - 5 "

where 6,. = (1 + b/d),. - 1

(A.1)

[B] is a anti-symmetric matrix, i.e., [B] = - [ B ] r = [bq] and non-zero components are given by

b16 = b17 = C23

C23 - 2C44 3 C 2 3 - 2C44 b26 - b27 -

3 5

bCl2 b 3 9 - b310 = C12(~2

d

bC66 b49 - - C12 - - b48 - d 2 6 6 6 (~2

2 C 1 2 -- C66 b41o

3

(A.2)

63

bs8 = -C6662 b 5 9 - Cl2 - 2C66 . . ~ 3

3

C12 - C66 bs|o - - (~4

2

[C] is a symmetric matrix, i.e., [C] = [C] r = [cij] and non-zero components are given by

Go GO Cl l ~ C12 -- do C13 = c14 ~ c15 = d o

4C44 Go G, c22 - 3d do c23 = c24 = c25 =

Go C33 ~ C34 = C35 - -

do

bC66 Go C6662 Go c44 = d 2 do c45 - d do

4C66 63 Go C55 - -

3d do

C22 E o Eo = - - C66 = C67 = d do c68 = C69 = C610 do

9C22 Eo Eo C77 ~ C78 ~ C79 = C710 ~ - -

5d do do

Eo C88 = C89 = C810 do

bC22 Eo C22 62 Eo c99 - d2 do c9m = d do

4C22 b3 Eo Cl010 - -

3d do

(A.3)