90n]s composite laminates. Part 2: Development of transverse ply cracks

90n]s composite laminates. Part 2: Development of transverse ply cracks

Analysis of multiple matrix cracking, in [ ± 0.m/90n] s compos=te lam=nates Part 2: Development of transverse ply cracks J. ZHANG*, J. FAN" and C. SOU...

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Analysis of multiple matrix cracking, in [ ± 0.m/90n] s compos=te lam=nates Part 2: Development of transverse ply cracks J. ZHANG*, J. FAN" and C. SOUTISt ('Chongqing University, People's Republic of China/tUniversity of Leicester, UK) Received November 1991; accepted in revised form 28 January 1992 In this paper the progressive transverse ply cracking in [ ± 0m/90n]s composite laminates is investigated theoretically. A general and simple expression for the energy release rate due to transverse matrix cracking is obtained using the potential energy approach in classical fracture mechanics and the assumption of a through-the-thickness flaw; thermal residual stresses are taken into account. The laminate resistance to crack multiplication is examined for both uniform and nonuniform crack spacing by substituting the measured applied stress/crack density data into corresponding energy release rate expressions. The resistance curve concept is employed to predict crack growth with increasing applied load. The results indicate that the assumption of uniform cracking and a resistance curve concept is a convenient and acceptable method for modelling crack initiation and multiplication in composite laminates. Key words: composite materials; cross-ply laminates; transverse ply cracking; energy release rate; resistance curve; crack density; crack spacing parameter, modelling

The damage growth in composite laminates prior to catastrophic failure and its effects on stiffness properties have received considerable attention in the literature Hg. Much of this effort has been directed towards predicting the development of matrix cracking in cross-ply laminates and over the years a number of analytical and numerical models have been proposed ~,3,6~. However, very few models consider the crack multiplication. Nuismer and Tan 9 used the energy balance principle to predict the progressive ply cracking in [+ 0m/90,]s laminates. An approximate elasticity theory solution was employed to estimate the energy released when a crack formed. Additional cracks were assumed to form when the energy released was equal to the energy absorbed. Fukunaga e t al. 8 examined the transverse crack multiplication using a statistical strength analysis. A shear-lag analysis was carried out to determine stress distributions. It was assumed that a new crack occurred midway between any two adjacent cracks at 50% failure probability. Laws and Dvorak v investigated progressive cracking using a probability density function. The stress necessary to cause additional cracking was assumed to be an integral

of the critical stress multiplied by the probability density function over the distance between two neighbouring cracks. This value was evaluated numerically. The stress distributions were determined by using a shear-lag analysis. Several models based on fracture mechanics concepts have been proposed which assume the growth of an inherent flaw or defect to be the mechanism of ply cracking '°-t3. Han and Hahn 13analysed the initiation and multiplication of transverse cracking in symmetric crossply laminates and developed a method based on the through-the-thickness flaw concept proposed by Hahn and Johannesson ~4. A resistance curve was proposed in order to characterize the resistance to the multiplication of transverse cracks. This is analogous to the R-curve used in linear elastic fracture mechanics which represents the increasing fracture resistance associated with crack growth in homogeneous and isotropic materials. The Rcurve concept was also used with some success for the analysis of multiple matrix cracking of [ q- 0m/90,]s laminates by Fan and Zhang Is. The authors developed the equivalent constraint model (ECM)which predicts the

0010-4361 •92•050291-06 © 1992 Butterworth-Heinemann Ltd COMPOSITES. VOLUME 23. NUMBER 5. SEPTEMBER 1992

299

Using the classical laminate theory, the laminate stress vector is written as:

L

_

State O:oi(k) =0

Fig. 1

State 1: ~i = O,~i=°(k)~ °{k}~O State 2: ~i,gi,C 6

Schematic diagram showing the laminate deformation states

reduction in stiffness properties due to transverse ply cracks and also the initiation and growth of matrix cracking with increasing mechanical load. An improved two-dimensional (2-D) shear-lag analysis15,16was used to determine the stress distributions in the cracked laminates. In the present paper, the energy release rate due to transverse ply cracking, incorporating residual thermal stresses, is derived by using the ECM modeP 5. The R-curve behaviour of cracked balanced symmetric laminates is presented and a discussion on whether or not resistance curves represent material behaviour independent of the laminate stacking sequence is given. Finally, the effect of non-uniform crack spacing on the resistance curve is quantitatively analysed.

1

(1 + X)(o~i) + 2 Z~P)

where Z is the thickness ratio of layer 1 over layer 2. Combining Equations (2)-(4) together results in the constitutive relation of the cracked laminate, i.e., ~i = Qij(Cj -- CP)

C-ip =

,

-- Sij (h I 9-

In this section we derive the energy release rate due to multiple matrix cracking, incorporating residual stress effects.

Energy release rate

Consider a [ + 0m/90n]slaminate element with a finite gauge length of 2l and width of w containing transverse ply cracks. The potential energy (PE) of the element is:

G=-

o,

,9)

Rearranging Equations (5)-(8) and substituting into Equation (9) we derive an expression for the energy release rate due to matrix cracking (see Appendix):

kth layer and laminate, respectively.

- h2

" -lSjm~l~m 9- 2 S i ~ ,

(~0~2) + gjp) + (E~=) + g,p )(~j=) + ~ )1

In order to include thermal stresses in the laminate, we introduce a fictitious configuration state '0', by debonding fictitiously the interlaminar surfaces in the laminate at state T and state '2'. At state '0' all lamina stresses are zero and the temperature is the same as that of state '1' and state '2'. The thermal strain of state ' I' is expressed as:

(2)

where ~k) and E°!k) are the total residual thermal stress and strain vectors of the/cth ply, respectively. Hence, the average constitutive equations of a damaged lamina are: (3)

where ~(I) and ~(i2) are the total stress vectors of the constraining layers 4- 0m and the 90* ply, respectively Q¢~) is the stiffness of the constraining layer (ply 1), and 6(i~) is the reduced in-plane stiffness of the 90 ° ply (ply 2); Q(~ has been derived in References 15 and 16 by expressing the in situ damage effective function (IDEF), Aij, as a function of crack density.

COMPOSITES. SEPTEMBER 1992

(8)

The energy release rate is equal to the first partial derivative of the potential energy with respect to crack surface area, A, under fixed applied laminate stresses, i.e.,

G(~,D a) =

where U~) and Ei represent the total strain vectors of the

300

(7)

k=l

(1)

1, 2)

2(h~ + h2)w21~(~

2

Let a laminate be subjected to in-plane loading under isothermal conditions, and be deformed and cracked progressively from state '1' (laminate stress-free state) to state '2', Fig. 1. According to the definition of macrostrain of damaged lamina introduced in Part 1 of this w o r k 16, w e have:

~(ik) = Q(k) (~j 9- ~ k ) ) (k =

(6)

where Qij and Sij are the in-plane stiffness and compliance matrices of the damaged laminate, respectively. It is observed that the laminate acquires a permanent strain gP due to the interaction effect of damage and residual stresses.

U = Vz ~. 2/W2hk~'(~, + C-0~k))cCj 9- ~-j 9- ~ k ) )

Constitutive relationship of the cracked laminate

~ik) = 5~ik) ~0(~) (k = 1, 2)

±

h2) k = I hk ,o(k) ~ jl cO~k) ~-~Im ~0(k)

where U is the total strain energy stored in the laminate element. Using the constitutive relation of the lamina, Equation (3), we get:

EXPRESSION .OF THE ENERGY RELEASE RATE INCLUDING RESIDUAL STRESSES

( k = 1,2)

(5)

with

PE = U -

~)=~,

(4)

(10a)

with D" = 2h2Cd

(10b)

The parameter Da represents the total crack surface area per unit length and width of laminate; C d is the average crack density. The second and third terms of the righthand side are caused by residual stresses and their interaction with damage. If the residual stresses are zero, Equation (10) is reduced to Equation (42) of Reference 15. The above expression of the energy release rate is general but quite simple to use. Knowing the in-plane stiffness matrix of the cracked lamina for a given crack density, the energy release rate can be derived from Equation (10), and the tedious integration operation found in existing fracture models t,6,7,13,17is completely avoided. R-CURVE BEHAVIOUR

Experimental work by various investigators JJ.~shas shown that multiple transverse ply cracking in cross-ply

laminates subjected to tensile loading is a stable fracture process. It has been observed that the resistance of the composite to transverse crack multiplication increases with increasing crack density. In this section we propose to use a resistance curve, analogous to the R-curve concept of classical fracture mechanics, as a measure of the composite resistance to initiation and growth. The criterion for stable growth of transverse ply cracking was introduced in Reference 6 and 15 as: G(~, D~) = GR(D")

is the laminate resistance to multiple transverse ply cracking. Hahn, Han and Kim ~9plotted GR against crack density Cd (not D9 for three composite systems and concluded that the R-curve is dependent on the 90° ply thickness but independent of the stiffness of the constraining layer. In previous work 15we found by curve fitting a simple mathematical expression for GR, i.e., G=c + G0(1 - e RD)

,,oo~-

•,,-•"

.~g 200~"%

(l la)

w h e r e GR

GR =

5°°I

Idealized uniform crack array F o r a u n i f o r m m a t r i x crack array, the in situ damage

effective functions, A22 and A66 were derived in References 15 and 16, and are given by (D a = D for uniform crack spacing): Az2 = 1 - ~b, + ~b2Dtanh(~,,/D)

0, + q~3Dtanh(k,/D)

(12a)

A66 = 1 - F~ + F2D tanh(kz/D) FI + F3D tanh(X2/D)

(12b)

The laminate resistance to transverse ply cracking is calculated by using Equations (10), (1 la) and (12) and Equation (1) presented in Part 116, and substituting measured laminate stress/crack density data into the lefthand side of Equation (1 la). Here, the indirect experimental resistance curves for T300/934 [0/90"]s and [ 4- 25/90j]~ laminates are estimated. The lamina properties of the T300/934 (Toray 300 fibres in an epoxy resin 934) composite material areS: E, = 144.8 (GPa) E2 = 11.38 (GPa) GI2 = 6.48 (GPa) G23 = 3.45 (GPa) v12=0.3 ~l = 0.36pc°C -~ ~ 2 = 28.8p~:°C-' A T = - 125°C t = 0.132mm In Fig. 2 the crack resistance G R is plotted against the accumulated damage ( O a = 2h2Cd) for [0/90.], and [ 4- 25/90i]s laminates. The solid and dashed lines are plotted by curve fitting of the resistance data using Equation (1 lb). When matrix cracking starts the resistance increases rapidly but becomes stable with further cracking. The critical energy release rate G~c and the R-curve for the [0/902]s laminate are different from those of the [0/90], lay-up, Fig. 2, suggesting that the parameters Glc, Go and R used in our fracture model are not independent of stacking sequence. Similar observations were made by Hahn et al?9; they concluded that the slope of the resistance curve increases as the transverse ply thickness increases. We believe that this is due to the different failure mechanism observed in crossply laminates with thicker transverse plies. Highsmith and

-~ o-fi---~-



O n = 1] [] n 2j'[o/90n] s IO0 •

JJ = iJ2][25/-25/90j]s

• i 0 - - - Eq (11b) (CIC=190J m-2, G0=125J m-2,R=6.5) - - - Eq (11b) (GIc=228J m-2, G0=178J m-2,R=6.2) 0.~

I 0.I

I 0.2

I 0.3

I 0.4

I 0.5

0.6

Damage

(1 lb)

where G~c is the critical energy release rate for damage nucleation, that Go and R are laminate constants. In the following section, we plot GR against damage, D ~.

[]q

Fig. 2

Resistance curve of T300/934 [O/90n]s and [25/-25/90j],

Reifsnider2 found delamination between the 0 ° and 90 ° plies and extensive crack branching in cross-ply laminates with a high number of consecutive 90° layers (n > 2). Groves and co-workers 2° in their experimental work observed that the damage in [0/90Z]s and [0/903]s laminate consists of three distinctly different types of transverse crack straight cracks, partial angled cracks and curved cracks - each having its own unique growth mechanism. A straight crack extends completely through the 90° ply, grows very suddenly and probably emanates from a preexisting microscopic flaw. This behaviour is a form of 'brittle' fracture and can be described by fracture mechanics concepts. It is widely accepted that the density of straight cracks is driven by shear lag between the transverse plies and the constraining layer. A curved crack initially forms as a partial angled crack extending from the 00/90 ° interface oriented, according to the observations of Groves et al. 2°, at an angle less than 90 ° to the laminate plane. The growth of such a crack is much slower than that of a straight crack, suggesting a 'ductile' fracture mode and that a different approach of analysis might be required. Fig. 2 also shows the resistance curves for T300/ 934 [+ 25/90j]s laminates. Whenj = 2 the R-curve coincides with that of the [0/90], laminate. F o r j > 2 the resistance curve is the same as that of the [0/902], lay-up. This suggests that the transverse ply thickness and the thickness ratio of 90 ° plies to constraining 4-0 layers are the important parameters controlling the ply cracking resistance. The resistance curve is a useful concept to describe transverse ply crack progression in various composite laminates, Figs 3 and 4. In Fig. 3 we plot the crack density as a function of applied stress for the [4-25/902], and [0/90]s laminates. For both lay-ups the ratio (thickness of the constraining layer over the 90 ° ply thickness) is equal to unity and the values of G~c, Go and R are the same. They are found by fitting GRID data for the [0/90]s laminate with Equation (1 lb), and are equal to: G~c = 190 (J m -2)

Go = 125 (J m -E)

R = 6.5

From Fig. 3 it can be seen that the agreement between the theoretical values and the experiment data is acceptable. In Fig. 4 the crack initiation and growth for [4- 25/904],

COMPOSITES.

SEPTEMBER 1992

301

20

/

15

r g

2S*p

O Experiment for [01901s (Wang11) O Experiment for [251-251902]s (Crossman 8 Wang18) ~Calculation (GIc= 190J m-2, G0= 125J m-2,R=6.5)

10

,, 0o..0 h.~

~

CR

L

O Fig. 5

,( oo...o... I

TZ

2

CM

CM+ICp

Gauge length, 2/

1

1

=-1

Configuration of a cracked laminate with non-uniform crack

O

Non-uniform matrix cracking In Part 1 of this work ~6, we derived expressions for the in A66, for a finite gauge length, 2/, containing arbitrary M transverse ply cracks with non-uniform crack spacing 2s~, 2s2 ..... 2SM. To model crack propagation, the remaining problems to answer are where the next new crack forms and how much additional energy or applied load is required for its formation.

situ damage effective functions, A22 and 200

400

600

800

1000

Laminate stress (MPa} Fig. 3

Crack density vs. laminate stress for T3001934 [0190], and

[25/-251902], 16

O Experiment. for [01902] s (Wang 11) [] Experiment for [251-251904]s (Crossman & Wang18) Calculation (GIC=228J m-2 G0=178J m-2,R= 6.2)

o

12

o

j

i 100

o/°

200

. , ~ t OO 300

I 400

I 500

600

Fig. 4 Crack density vs. laminate stress for T300/934 [0/902], and [25/-25/904],

and [0/902], laminates are presented. These lay-ups have a high number of consecutive 90" plies and Z = 1/2. The values of the G~c, Go and R parameters are found from the experimental resistance/damage data of the [0/90Z]s laminate and are equal t o : G 0 = 1 7 8 ( J m -2)

R = 6.2

Again the agreement between the theory and the experiment is good. It can be seen that the resistance curve concept is useful to describe laminate crack development and the parameters G~c, Go and R are constant as long as the ratio X remains the same. So far in our analysis of crack propagation we have ignored delamination between the 0/90 ° plies and assumed that the transverse ply cracks are straight, grow through the 90 ° ply thickness and that the crack spacing is uniform. In the following section we examine the effect of non-uniform crack spacing on the R-curve. The effect of local delamination will be considered in future work.

302

M

s* = 21

(13a)

sj = 2l

(13b)

J=l M+I J=l

sj = s j

( J = 1,2 . . . . . M)

(13c)

SM+I +So=S*

Laminate stress {MPa)

G l c = 2 2 8 ( J m -2)

Let M transverse cracks pre-exist and a new crack form between crack Cp and its left-hand adjacent crack CR under fixed laminate stresses, Fig. 5. The crack spacings before and after formation of the new crack satisfy the following conditions:

COMPOSITES. SEPTEMBER 1992

(13d)

where the asterisk (*) represents the parameters corresponding to the configuration before the formation of a new crack. We then introduce an off-eentre crack spacing parameter for the newly formed M + I crack as: eM+l =

1 - s~---o-p, 0

~ eM+l <

1

(14)

So, 2sp = s*(l - eM + ,)

(15a)

2SM + l = SO(1 + eM + t)

(15b)

The average crack density before and after the formation of a new crack is:

C] = M/21

(16a)

Ca = (M + 1)2/

(16b)

It can be seen that there are two parameters controlling the exact location of a new crack. One is number 'P', which indicates that the new crack forms between crack C o and its left-hand neighbouring crack. Another is the off-centre cracking factor eM + ~which determines the exact location between the two cracks at which the new crack forms; the parameters P and eM + ~are arbitrary. However, it seems that the probability of the formation of a new crack between two pre-existing cracks with the largest crack spacing is the biggest. So we can specify P by assuming that the new crack develops between two cracks

500 Uniform, thick laminate 4001

('q

E



• • ~ I/~_ ~

300

•_

-__

.t.

__

_-_o-o =o o ~ U n i f o r m , thin = 0.2 laminate

e

~'~



"-.~

" e=0.6

c 200

GR=GIc

tn

O & 0 O • 41J ---

100

0

n = n = n = n = n = n = Best Best

0.0

I, u n i f o r m I, e = 0.2 I, e = 0.6 2, u n i f o r m 2, e = 0.2 2, e = 0.6 fit fit

crack

array

crack

array

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

,

0.6

Damage Fig. 6 Energy release rate vs. damage with uniform and nonuniform crack spacing for T 3 0 0 / 9 3 4 [ 0 / 9 0 , ] , (calculations based on the experimental stress/crack density d a t a " )

'°°f 7

E

Uniform, thick laminate

I

,

---P

3OO

~J"

~

e=0 6 ......Uniform, thin

# ~ 'w; ' - - , .



~

. ~

"

• laminate

200

c>o

oc 10(

0

A O II • 41. -----

n = 2, u n i ~ r m crack a r r a y n=2, e=0.2 n=2, e=0.6 n = 4, u n i ~ r m c r a c k a r r a y n = 4, e = 0.2 n = 4, e = 0.6 Best fit Best fit ]

0.0

O.l

1

I

0.3

0.2

I 0.4

I 0.5

0.

Fig. 7 Energy release rate vs. damage with uniform and nonuniform crack spacing for T 3 0 0 / 9 3 4 [ ± 2 5 / 9 0 . ] , (calculations based on the experimental stress/crack density data la)

with the largest crack spacing. For the off-centre crack spacing factor, we chose: el = e 2 . . . . . = e M + , . . . . . = e

(17)

The first derivative of the reduced in-plane stiffness matrix with respect to damage, see Equation (10), can be approximately written as: dQ(2) 0( 2)- Q!~,* 0D a AD a (18a) with D" -

D ~* =

h2/l

In practice, the off-centre crack spacing parameter e changes with the amount of crack density and depends on the laminate stacking sequence. It cannot be measured from experiments and therefore the true energy release rate cannot be obtained. However, our analysis shows qualitatively and quantitatively that the crack interaction affects substantially the energy release rate. Other failure modes such as delamination should be addressed.

CONCLUDING REMARKS

Damage

AD ~ =

the energy release rate due to multiple matrix cracking as a function of damage (D" = 2h2Cd) for uniform and nonuniform crack spacing (e = 0.2, 0.6) for T300/934 [0/90,], and [ + 25/90n], laminates. The stress/crack density data used to evaluate the energy release rate are taken from Wang's experiments "As. It is interesting to see that the effect of crack interaction (due to non-uniform crack spacing) on energy release rate is small when damage is less than 0.2 (average crack spacing is about five times the thickness of the 90 ° lamina), regardless of the number of the consecutive transverse plies and the orientation angle of the constraining layer. However, the resistance curve of the [0/90z], and [ ± 25/904]s laminates is still above the Rcurve of laminates with a thin 90° ply, suggesting that there is another failure mechanism dominating the resistance behaviour at the initial stages of damage. As damage accumulates, the energy release rate of thick laminates varies considerably with the parameter e, while for thin laminates the variation is small. This implies that the effect of crack interaction on energy release rate is significant for laminates with high number of consecutive transverse plies (n > 2). When e = 0.6, crack interaction is severe and the R-curve comes closer to that of thin laminates.

(18b)

where Q(~j)* and Q(2)~are the reduced in-plane stiffness matrices of the 90* ply before and after formation of a new crack, respectively. For a given off-centre crack spacing parameter e, the energy release rate is calculated by combining Equations (10) and (18), and substituting measured crack density Co and corresponding laminate stresses. Figs 6 and 7 show

A simple expression for the energy release rate as a result of transverse ply cracking is obtained; thermal residual stresses are included in the expression. The calculated energy release rate is then used to predict the crack growth as a function of applied load. A resistance curve for crack propagation is proposed as a means of characterizing the laminate resistance to transverse matrix cracking. The R-curves are independent of the type of laminate lay-up as long as the ratio of thickness of the 90° ply to that of the constraining layers remains constant. The effect of non-uniform crack spacing on the resistance curve is significant for laminates with a high number of consecutive 90 ° plies. Further work is required in order to examine the effect of other failure mechanisms such as local delaminations upon the resistance curve.

ACKNOWLEDGEMENTS This work was supported by the British Council and by Chongqing University in China. The authors would like to express their appreciation to Drs P.A. Smith and L. Boniface from Surrey University, UK, for helpful discussions.

REFERENCES Bailey, J.E., Curtis, P.T. aml Parvizi, A. 'On transverse cracking and longitudinal splitting behaviour of glass and carbon fibre reinforced epoxy cross-ply laminates and the effect of Poisson and

COMPOSITES

. SEPTEMBER

1992

303

2

3 4 5 6 7 8

9 10 11 12 13 14 15 16

17

18 19

thermally generated strain' Proc Roy Soc London A 366 (1979) pp 599-623 Highsmith, A.L. and Reifst~ler, K.L. 'Stiffness reduction mechanism in composite laminates' Damage in Composite Materials, A S T M STP 775 (Amercian Society for Testing and Materials, 1982) pp 103-117 Ogln, S.L. and Smith, P.A. 'Fast fracture and fatigue growth of transverse ply cracks in composite laminates' Scr Metal119 (1985) pp 77%784 Smith, P.A. and Wood, J.R. 'Poisson's ratio as a damage parameter in the static tensile loading of simple crossply laminates' Composites Sci & Techno138 (1990) pp 85-93 Raggs, D.L. 'Prediction of tensile matrix failure in composite laminates' J Composite Mater 22 (1985) pp 2%50 l-hm,Y.M., Hahn, H.T. and Croman, R.B. 'A simplified analysis of transverse ply cracking in cross-ply laminates' Composites Sci & Techno131 (1988) pp 165-177 Laws, N. and Dvorak, G.J. 'Progressive transverse cracking in composite laminates' J Composite Mater 22 (1988) pp 901~915

A U T H O R S

J. Zhang, a visiting scholar at the University of Leicester, UK, and J. Fan are with the Department of Engineering Mechanics, Chongqing University, Chongqing, 630044, People's Republic of China. Correspondence should be addressed to C. Soutis at the Department of Engineering, University of Leicester, University Road, Leicester LEI 7RH, UK.

Appendix

Fukunaga, H., Chon, T.W., Peters, P.W.M. and Schulte, K.

'Probabilistic failure strength analysis of graphite/epoxy cross-ply laminates' J Composite Mater 18 (1984) pp 339-356 Tan, S.C. and Nuismer, R.J. 'A theory for progressive matrix cracking in composite laminates' J Composite Mater 23 (1989) pp 1029-1047 Wang, A.S.D. and Cromman, F.W. 'Initiation and growth of transverse cracks and edge delamination in composite laminates 1: an energy method' J Composite Mater 14 (1980) pp 71-87 Wang, A.S.D. 'Fracture mechanics of sublaminate cracks in composite materials' Composites Technol Rev 6 (1984) pp 45-62 Wang, A.S.D., Chou, P.C. and Lei, S.C. 'A stochastic model for the growth of matrix cracks in composite laminates' J Composite Mater 18 (1984) pp 239-254 Han, Y.M. and Hahn, H.T. 'Ply cracking and property degradations of symmetric balanced laminates under general in-plane loading' Composites Sci & Techno135 (1989) pp 377-397 Hahn, H.T. and Johannesson, T. 'Fracture of unidirectional composites: theory and applications' in Mech of Composite Mater AMD 58 (ASME, 1983) pp 13:%142 Fan, J. and Zhang, J. 'In-situ damage evolution and micro/macro transition for laminated composites' submitted to Composites Sci & Technol Zhang, J., Fan, J. and Soutis, C. 'Analysis of multiple matrix cracking in [+ 0~/90,]~ composite laminates. Part 1: In-plane stiffness properties' Composites 23 No 5 (September 1992) pp 291298 Boniface, L., Ogin, S.L., and Smith, P.A. 'Strain energy release rates and the fatigue growth of matrix cracks in model arrays in composite laminates' Proc Roy Soc London A 432 (1991) pp 42~ 444 Crossman, F.W. and Wang, A.S.D. 'The dependence of transverse cracks and delamination on ply thickness in graphite-epoxy laminates' Damage in Composite Materials op. cit. pp 118 139 Hahn, H.T., Han, Y.M. and Kim, R.Y. 'Resistance curves for ply cracking in composite laminates' Proc 33rd lnt SAMPE Syrup (1988) pp 1101-1108

20 Groves, S.E., Harris, C.E., Highsmith, A.L., Allen, D.H. and Norvell,

304

R.G. 'An experimental and analytical treatment of matrix cracking in cross-ply laminates' Exptl Mech 130 (1988) pp 73-79

COMPOSITES.

SEPTEMBER

1992

DERI VA TION OF ENERG Y RELEASE RATE

Differentiating Equation (8) with respect to area A under fixed laminate stresses ~, we get: ,3/'1(2)

~U_ 12lw 2 h ; ~ i

+ E%2,)(~j + ~2))

2

+ ~,, 21 w 2hk Q~)(~, + ~k)~Egi

k= I

i ~

(19)

Substituting Equations (5) and (3) into the first and second terms of the right-hand side of Equations (19), respectively, gives:

OUoA--

l

,qf)(2) _

">~'~:fO -- , + ~2) + ~ ) 2l w 2hfd~{Si,o

X (Sjm~m + ~ 2 ) + ~p)

+ 2l w 2(h, + h2)~j c~i

~A

(20)

Differentiating Equation (7) under fixed laminate stresses and then combining with Equations (20) and (9), results in: ~t3(.2)

--

--

~[SilSjmo-iCYm

h2 2l + 2S,~,~ j + ~ ) + (U~) + ~)(~0~2) + c~)]

G(~,A)

=

--

-

t~o(2)

(21)

Noting that A = w 2h2 2l Cd = W21 D" we finally obtain Equation (10) (see main text).

(22)