‘ ,!. t SEMI-ANNUAL REPORT {’”/ ~ 19 

ANALYSIS OF INTERLAMINAR FRACTURE IN COMPOSITES 

UNDER COMBINED LOADING 

NASA GRANT NAG-1-637 

GEORGIA TECH PROJECT E16-654 

lNaSA-a-185055) ANBLYSIS OF INTERLAaINAB N89-24674 FRACTURE IN COHPOSITES UNDER COMBINED 

LOADING Semiannual Report, 12 Auq. 1988 - 11 

Feb. 1989 [Georgia Inst. of Tech.) 31 p Unclas 

CSCL 20R G3/39 0212668 

v 

PRINCIPAL INVESTIGATOR 

ERIANAARMANIOS 
ovenriew 

This report covers the research work performed under Grant 

NAG-1-637 for the period ending February 28, 1989. A detailed description of the fracture analysis of transverse crack tip delaminations 

is presented in the following sections. This work was performed 

during the first six months of the grant period and has been accepted 

for presentation at the 30th Structures, Structural Dynamics and 

Materials Conference (Mobile, Alabama, April 1989). The following 

sections are adapted from the aforementioned paper. 

Abstract 

Delamination is a predominant failure mode in continuous fiber 

reinforced laminated composite structures. One type of delamination 

is the transverse crack tip delamination which originates at the tip of 

transverse matrix cracks. An analytical model based on the sublaminate approach and fracture mechanics is developed in this paper to 

study the growth of such delaminations. Plane strain conditions are 

assumed and estimates are provided for the total strain energy release 

rate as well as the mode I and mode I1 contributions. The energy release rate estimates are used in combination with a simple failure law 

to predict critical delamination growth strains and stresses. These 

predictions are compared with experimental data on T300/934 

Graphite Epoxy [f25/90& laminates in the range n=.5 to 8. A good 

agreement is demonstrated for the range of n where the experimental 

observations indicate transverse crack tip delamination to be the predominant failure mode. 

1 
Introduc tion 

Fiber reinforced composites are now being used in a wide variety 

of engineering structures. The concept of directional strength and 

stiffness has been, for the most part, understood sufficiently to enable 

efficient load bearing designs. One of the current major issues in 

composite structures is the understanding and prediction of damage 

modes and failure mechanisms. A thorough knowledge of the failure 

mechanisms is bound to lead to the design of efficient and durable 

structures. Failures in these materials often initiate in the form of matrix cracks or delaminations. Matrix cracks refer to intralaminar failures whereas delaminations refer to interlaminar failures. 

Matrix cracks usually occur within laminates where the fibers 

run at an angle to the primary load direction. Hence, such matrix 

cracks are also called transverse cracks. Based on the location and direction of growth, two distinct types of delamination can be discerned. 

These two types are called edge delamination and local or transverse 

crack tip delamination. Edge delaminations initiate at the load free 

edges of the laminate whereas local delaminations start from a transverse matrix crack. In many cases, both types occur concurrently with 

varying levels of interaction. It has been observed in simple tension 

tests of uniform rectangular cross section specimen (Edge 

Delamination tests) that delaminations initiate along the load free 

edges and propagate normal to the load direction. Transverse matrix 

cracks running parallel to the fibers have also been observed in off axis 

plies such as 90" plies. Such transverse cracks terminate where the 

ply orientation changes. Delaminations can also originate at the interfaces where transverse cracks terminate. These delaminations, called 

2 
transverse crack tip delaminations or local delaminations, grow normal to the transverse crack from which they originate. In the case of 

90" plies, the growth direction is parallel to the load. 

The growth process of edge delaminations and local delaminations is often modelled using a fracture mechanics approach leading to 

the calculation of a strain energy release rate. This is because the 

strain energy release rate can correlate delamination behavior from 

different loading conditions and can account for geometric dependencies. The strain energy release rate associated with a particular 

growth configuration is a measure of the driving force behind that failure mode. In combination with appropriate failure criteria, the strain 

energy release rate provides a means of predicting the failure loads of . 

the structure. 

Several methods are available in the literature for analyzing edge 

delaminations. These include finite element modelling l-3, complex 

variable stress potential approach4, simple classical laminate theory 

based techniques and higher order laminate theory including shear 

deformation&. Finite element models provide accurate solutions but 

involve intensive computational effort. Classical laminate theory (CLT) 

provides simple closed form solutions and is thus well suited for preliminary design evaluation. However, CLT provides only the total energy release rate, and thus, in a mixed mode situation, there is insufficient information to completely assess the delamination growth tendency. A higher order laminate theory including shear deformations 

has the ability to provide the individual contributions of the three fracture modes while retaining the simplicity of a closed form solution. A 

3 
shear deformation model is available for edge delamination and has 

been shown to agree well with finite element predictionss. 

Crossman and Wang7 have tested T300/934 Graphite epoxy 

[f25/90n]s specimens in simple tension and reported a range of behavior including transverse cracking, edge delamination and local delamination. O'Brien8 has presented classical laminate theory solutions 

for these specimen, demonstrating reasonable agreement in the case 

of edge delamination but with some discrepancies in the local delamination predictions. A finite element model combining edge and local 

delaminations has been proposed by Laws. His predictions, however, 

do not fully explain the dependency of the critical strain on the number of 90° plies. 

In this paper, a shear deformation model is developed for the 

analysis of local delaminations originating from transverse cracks in 

90° plies located in and around the specimen midplane. Plane strain 

conditions are assumed and thickness strain is neglected. Delaminations are assumed to grow from both ends of the transverse crack tip. 

The transverse crack is treated as a free boundary and the delamination is considered to be the crack whose growth behavior is to be 

modelled. The sublaminate approachlo. 11 is used to model different 

regions of the specimen. The resulting boundary value problem is 

solved to obtain the interlaminar stresses, total strain energy release 

rate and energy release rate components. Critical local delamination 

growth loads are predicted for [-+25/9On]s specimens. 

Analvtical Model 

The formulation is based on the sublaminate approach detailed 

in Ref. 10. A longitudinal section illustrating the geometry of a generic 

4 
configuration is shown in fig. 1. The central region is assumed to be 

made of 90° plies with an isolated transverse crack in the middle. 

Delaminations are assumed to grow from both ends of the transverse 

crack, and towards both ends as shown. From symmetry considerations, only one quarter of the configuration is modelled. The modelled 

portion is divided into four sublaminates as shown In fig. 2. The top 

surface (sublaminates 1 and 4) is stress free. In order to skpllfy the 

analysis, plane strain conditions are assumed and the thickness strain 

(eZ) is set to zero. The consequence of this, combined with the fact 

that the w displacement is zero along the center line, is that w is zero 

in sublaminates 1, 2 and 3. Also, this approximation does not allow for 

the enforcement of boundary conditions on the shear stress resultants, 

leading to incorrect estimates of the interlaminar normal stresses. 

The interlaminar shear stresses, however, are not affected by this assumption6.10. These assumptions lead to considerable simplifications 

in the analysis. In spite of the simplifications, reliable energy release 

rate components can be estimated based on the interlaminar shear 

stress distributions6.10. 

- 

A generic sublaminate is shown in fig. 3 along with the notations 

and sign conventions. The peel and interlaminar shear stresses are 

denoted by P and T, respectively, with t and b subscripts for the top 

and bottom surfaces, respectively. The axial stress resultant, shear 

stress resultant and bending moment resultant are denoted by N, Q 

and M, respectively. A summary of the governing equations is presented in the following paragraphs for convenience. These equations 

are derived for a generic sublaminate using the principle of virtual 

work in Ref.12. 

5 
The x and z displacements within the sublaminate are assumed 

to be of the form 

u(x,z) = u(x)+zp(z) (1) 

w(x,z) = W(X) (2) 

Here U represents the axial midplane stretching and W is the transverse displacement. The shear deformation is recognized through the 

rotation p. The origin of the coordinate axes for the sublaminates is 

taken at the delamination tip as shown in fig. 4. The equilibrium 

equations take the form 

N,x+Tt-Tb = O (3) 

Q,x+Pt-Pb = 0 (4) 

(5) 

h 

M,x-Q+Z(Tt+Tb) = 0 

where h is the thickness of the sublaminate. The constitutive relations in terms of the force and moment resultants are 

N = A1IU9x+B1lp,x (6) 

9 = A55(P+W*d (7) 

M = B11U*x+D11P*x (8) 

where Ail, Bq and Dg are the classical laminate theory axial, coupling 

and bending stiffnesses, respectively. The boundary variables to be 

prescribed at the sublaminate edges are 

N or U 

M or p 

Q orW 

Additionally, at the interfaces between sublaminates, reciprocal 

traction and displacement matching boundary conditions have to 

specified. 

6 
polution Procedure 

A detailed solution is provided in the Appendix. A summary is 

provided in this section for convenience. The variables are subscripted 

to indicate the sublaminate in which they occur. The solutions in 

sublaminates 1 and 2 are coupled by the reciprocal interlaminar 

stresses denoted T1 and P1 and by displacement continuity at the 

common interface. Assuming exponential solutions for the axial force 

and bending moment resultants leads to an eigenvalue problem involving the exponential parameter s. The eigenvalues turn out to be 0 and 

two nonzero values (say s1 and s2) occurring in positive and negative 

pairs. Since the response decays from the delamination (crack) tip, 

only the exponentially decaying terms are considered in the solutions. 

The following boundary conditions from the ends of the modelled region are enforced. 

N2(O) = 0 (9) 

94(4 = 0 (10) 

P4(d = 0 (1 1) 

N1+N2 = Applied Load (12) 

Further. the following displacement 

applied. 

u1 (2G-2) = u2 (x,?) 

matching conditions are 

It should be noted that a P2 and p3 matching condition cannot be 

applied at this level of modeling since it would amount to specifying 

both W and Q6.12; Consequently, there is a displacement discontinuity 

7 
at the delamination tip. The effect of this will be discussed subsequently. To eliminate rigid body displacements, U1 is set to zero at 

the left end. The following solutions can then be obtained for the resultants in sublaminates 1 and 2. 

N1 = alesi+a2e%+EA11(1) (17) 

N2 = -aleSiX-a2e%+&A11(2) (18) 

M1 = alklesi+a2k2e%x (19) 

M2 = al~esi+a2~es2x (20) 

The interlaminar shear and peel stresses between sublaminates 1 and 

2 can be obtained from equilibrium as 

T1=als lesi+a2s2e%x 

In the above solutions, the ki parameters are dependent on the 

eigenvalues and the stiffness of sublaminates 1 and 2, the aj parameters depend on the ki parameters and the crack length a, and E is defined as 

P 1 &=- 

2b A1 1( 1)+A1 l(2) 

where P is the uniform axial force applied on the specimen and b is 

the specimen width. Expressions for the eigenvalues and the aj and ki 

parameters are provided in the Appendix. 

Proceeding on to sublaminates 3 and 4, the following solutions 

can be written. 

N3 = 0 

M3 = 'p1 sinh(a3x)+'p2 cosh(a3x) 

where 

8 
and 

N4 = (A11(1)+A11(2)) 

a = alkl+azkz (3 0) 

The corresponding displacement solutions are provided in the 

Appendix. The compliance of the specimen can be evaluated as 

The total energy release rate, GT, per crack is then given by 

P2 dC 

*=2b da 

Use of the previously described solutions leads to the following expression. 

1 +I1-12) Or= (G-All(l)+All(2) 

P2 1 

(33) 

where the quantities I1 and 12 contain exponential terms dependent 

on the delamination length. Using the virtual crack closure technique, 

from the relative displacements in the cracked portion and the interlaminar stresses ahead of the crack tip, the mode I and mode I1 energy release rate contributions can be obtained. The mode I11 energy 

release rate is zero from the assumption of plane strain. The mode I1 

energy release rate is given by 

where 6 is the virtual crack step size. The result of the limiting process is zero if there is no singularity in the stress fieldlo. So, the limit 

is usually taken as the crack step size d tends to a small value, say A, 

based on the decay length or the length required to capture the 

9 
essential features of the stress and displacement fields near the crack 

tip. The decay length is dependent on the eigen values s1 and sa. In 

this study, the value of A has been set to 

since it reasonably fuifills the criterion given above. In a similar fashion, the mode I energy release rate can be obtained based on the normal stress (P) and the w displacements near the crack front. The 

normal (peel) stress estimate is inaccurate due to the absence of 

thickness strain. Hence, an alternate approach was used to estimate 

GI, the mode I energy release rate. The total energy release rate for 

this problem is made up entirely of GI and GII (GIII=O). From an estimate of and GII, an estimate for GI can be obtained simply as 

GI = *-GI1 (36) 

The critical load for a given specimen can then be evaluated 

based on an appropriate fracture law. This is illustrated in the following section. 

Results and Discuss ion 

The solutions derived in the previous section have been used to 

model the behavior of [+25/90& T300/934 Graphite Epoxy specimen 

for n values of .5,1,2,3,4,6,and 8. These correspond to the specimen 

tested by Crossman and Wang7. The specimen width and length were 

fixed at .0381m and .015m, respectively, as in the tests and the applied uniform axial stress was 1OOMPa. The solutions were generated 

using a simple computer program based on the closed form expressions for the interlaminar stress and energy release rates. 

10 
An example of the total energy release rate variation with the 

crack length is presented in fig. 5. The asymptotic value of %is denoted by -0 in the figure. It can be observed that after a certain 

crack length, the GT is independent of the crack length. On the basis 

of curves like the one shown in fig. 5, the crack length was fixed at 10 

ply thicknesses for the remainder of the study. The dependence of 

the mode I1 contribution of the energy release rate on crack length (a) 

is depicted in fig. 6. Typical interlaminar shear and normal stress 

profiles are presented in figs. 7 and 8, respectively. The corresponding energy release rates have also been calculated and are presented in 

Table I and fig. 9. 

In order to evaluate the critical loads, an appropriate mixed 

mode fracture law has to be applied, based on the calculated energy 

release components. Since the calculated mode split shows only a 

small variation with n, the simple Griffith criterion %=GTC has been 

used to scale the stresses to obtain the critical delamination growth 

stress (oC) and strain (E,) values. The critical energy release rate 

was chosen as 415 J/m2 to obtain the critical stresses and strains 

listed in Table I. This value of ec is larger than GI, to account for the 

presence of mode I1 and the fact that for the material system under 

consideration, GII~ isabout four times GI,. The critical strains are 

plotted against n, the number of 90° plies in fig. 10. The experimental 

results of Ref. 7 and the predictions of Refs. 8 and 9 are also presented in the figure for comparison. The predictions of the model developed in this paper are represented by the solid line while the experimental results are shown as filled squares. The classical laminate 

theory and finite element critical strain predictions of Refs. 8 and 9 

11 
are represented by triangles with a connecting line and a dotted line 

respectively. The CLT based model agrees well with the shear deformation model in terms of the total energy release rate. However, the 

CLT based model does not provide information on the mode split and 

thus, the value of GC(=G1,) used leads to bias in the predictions. 

In the experiments, the local delamination phenomenon was observed as the predominant failure mode only for the n=4, 6 and 8 

specimens. The shear deformation model presented in this paper 

provides good agreement with the experimental data in this range. 

For n<4, edge delamination either in the mid plane or in the 25/90 

interface was observed in the tests. Hence, the predictions of the local delamination models in this region are not of consequence as long 

as they do not predict critical loads lower than those predicted by 

edge delamination models. Thus, it can be seen that the shear deformation model predicts the observed behavior with reasonable accuracy 

and can be used in conjunction with an appropriate edge delamination 

model to predict critical loads accurately for the complete range of n 

values. The edge delamination model presented in Refs. 6 and 12 can 

be used for this purpose. However, a separate model is required to 

account for the midplane (Mode I) edge delamination behavior. The 

development of such a model is described in Ref. 13. 

Conclusions 

A shear deformation model has been developed to analyze local 

delaminations growing from transverse cracks in 90" plies located 

around the mid plane of symmetric laminates. The total energy release rate calculations yield the same results as in the case of CLT 

based models. The predictions of the .shear deformation model agree 

12 
reasonably with critical strain experimental data from [k25/90n]s 

T300/934 Graphite Epoxy laminates. The predicted behavior is such 

that, in combination with an edge delamination model, the critical 

loads can be predicted accurately in the range of n from .5 to 8. 

ficknowledfle ments 

The authors gratefully acknowledge the financial support provided by NASA under grant NAG-1-637 for performing the research 

reported in this paper. The authors also wish to thank Mr. A. Badir for 

help in verifylng the analytical model. 

References 

Wilkins, D.J., Eisemann, J.R., Camin, R.A., Margolis, W.S. and 

Benson, R.A., "Characterizing Delamination Growth in GraphiteEpoxy," in Damage in Composite Materials, ASTM STP 775, K.L. 

Reifsnider, Ed., pp. 168-183 (1982). 

O'Brien, T.K., "Mixed-Mode Strain Energy Release Rate Effects on 

Edge Delamination of Composites," in Effects of Defects in 

Composite Materials, ASTM STP 836, pp. 125-142 (1984). 

Wang, S.S. and Choi, I., 'The Mechanics of Delamination in Fiber 

Reinforced Composite Materials. Part I1 - Delamination Behavior 

and Fracture Mechanics Parameters," NASA CR- 172270 ( 1983). 

Wang, S.S., "Edge Delamination in Angle Ply Composite 

Laminates," Proceedings of the 22nd AIAA/ASME/ASCE/AHS 

Structures, Structural Dynamics and Materials Conference, Atlanta, Georgia, 6-8 April, 1981, pp. 473-484. 

O'Brien, T.K., "Characterization of Delamination Onset and 

Growth in a Composite Laminate," in Damage in Composite 

13 
Materials, ASTM STP 775, K.L. Reifsnider, Ed., pp. 140-167 

( 1982). 

Armanios, E.A., and Rehfield, L.W., "Interlaminar Analysis of 

Laminated Composites using a Sublaminate Approach," Proceedings of the 27th AIAA/ASME/ASCE/AHS Structures, Structural 

Dynamics and Materials Conference, San Antonio, Texas, 19-21 

May, 1986, Part 1, pp. 442-452. AIAA Paper 86-0969CP. 

Crossman, F.W., and Wang, A.S.D.,'The Dependence of Transverse 

Cracking and Delamination on Ply Thickness in Graphite/Epoxy 

Laminates," in Damage in Composite Materials, ASTM STP 775. 

K.L. Reifsnider, Ed., pp. 118-139 (1982). 

O'Brien, T.K., "Analysis of Local Delaminations and Their 

Influence on Composite Laminate Behavior," in Delamination and 

Debonding of Materials, ASTM STP 876, Johnson, W.S., Ed., pp. 

282-297 (1985). 

Law, G.E., "A Mixed Mode Fracture Analysis of (+25/90n)s 

Graphite/Epoxy Composite Laminates," in Effects of Defects in 

Composite Materials, ASTM STP 836, pp. 143-160 (1984). 

[lo] Armanios, E.A., "New Methods of Sublaminate Analysis for 

Composite Structures and Applications to Fracture Processes," 

Ph.D. Thesis, Georgia Institute of Technology (1984). 

[ 111 Armanios, E.A., Rehfield, L.W., and Reddy. A.D., "Design Analysis 

and Testing for Mixed-Mode and Mode I1 Interlaminar Fracture 

of Composites," in Composite Materials: Testing and Design 

(Seventh Conference), ASTM STP 893. J.M.Whitney, Ed., pp. 

232-255 (1 986). 

14 
[ 121 Armanios, E.A., and Rehfleld, L.W., "Sublaminate Analysis of 

Interlaminar Fracture in Composites: Part I - Analytical Model." 

submitted for publication in the Journal of Composites 

Technology and Research (July, 1988). 

[ 131 Armanios, E.A., Badir, A., and Sriram, P., "Sublaminate Analysis of 

Mode I Edge Delamination in Laminated Composites," 30th 

AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and 

Materials Conference, Mobile, Alabama (April 1989). 

ADDendix 

@Uamhate Analvs is for Local De lamhations 

A generic sublaminate is shown in fig. 3 along with the notations 

and sign conventions. The interlaminar normal (peel) and shear 

stresses are denoted by P and T respectively with the t and b subscripts for the top and bottom surfaces respectively. The axial force 

resultant, shear force resultant and bending moment resultant are denoted by N, Q and M respectively. Plane strain conditions are assumed 

to prevail in the x-z plane and the thickness strain &zz is neglected. 

These assumptions lead to considerable simplification in the analysis. 

The displacements in the x and z directions are assumed to be of the 

form 

u = U(x)+zb(x) (A. 1) 

w = W(x) (A. 2) 

Here U represents the axial stretching and W is the transverse 

(thickness direction) displacement. This formulation recognizes 

shear deformation through the rotation p. The equilibrium equations 

take the form 

N,,+Tt-Tb = 0 (A. 3) 

15 
where h is the thickness of the sublaminate. The constitutive equations in terms of the force and moment resultants are 

where A,B and D are the classical laminate theory axial, coupling and 

bending stiffnesses defined in the customary manner as 

h/2 

-h/2 

(Aij,Bij,Dij) = lCij(1 ,z,z2)dz 

Here, the Cp are the material moduli. For the case of plane strain in 

the x-z plane, the Cs are defined as follows. 

The boundary quantities to be prescribed at the sublaminate 

edges are 

N or U 

M or f3 

Q orW 

Further, at the interfaces between sublaminates, reciprocity of tractions and continuity of displacements have to be enforced. 

The four sublaminates along with the loads acting on each are 

shown in fig. 4. Setting PI and T1 as shown automatically satisfies the 

traction matching boundary condition at the 1-2 interface. From 

symmetry, we get w=O and zero shear stress along the bottom faces of 

16 
I. 

sublaminates 2 an( 3. This leads to w=O in sublaminates 1, 2 and 3. 

Thus, W has been prescribed in these sublaminates and the vertical 

shear force resultant Q cannot be prescribed on these sublaminates. 

Consequently, the calculated peel stress distribution .will not be correct. In addition, at the 2-3 interface, the ps cannot be matched, 

since in these sublaminates, specifying p is equivalent to specifylng Q 

(through Eq. A.7). Inspite of these simplifications, reliable energy release rate components can be estimated based on the interlaminar 

shear stress distributions. The mode I contribution can then be evaluated using the total energy release rate, which is not affected significantly by these simplifications. 

For the (&25/90n)s laminates under consideration, B11 is zero in 

all the four sublaminates. For sublaminates 1 and 2, the equilibrium 

equations and constitutive relationships can be written as 

N1,x-Tl = 0 (A. 10) 

N2,x+T1 = 0 (A. 1 1) 

Q1.x-R = 0 (A. 12) 

Q2,x+P1-P2 = 0 (A. 13) 

(A. 14) 

(A. 15) 

N1 = All(1)Ul.x (A. 16) 

N2 = A1 1(2)U2,X (A. 17) 

91 = &5(1)p1 (A. 18) 

Q2 = A55(2)p2 (A. 19) 

M1 = Dll(1)Pl.x (A.20) 

M2 = D11(2)P2.X (A.2 1) 

hl 

h2 

Ml,x+2Tl-Q1 = 0 

M2,x+5T1-Q2 = 0 

17 
The subscripts in parentheses refer to the sublaminates to which the 

stiffness coefficients correspond. Eqs. A.14, A.15 and A.12 can be 

rewritten in a modified form as 

. 

Matching the u displacement along the 1-2 interface implies 

or 

(A.22) 

(A.23) 

(A.24) 

(A.25) 

Combining the equations to eliminate the displacement and interlaminar stress terms leads to the following system of homogeneous 

coupled ordinary differential equations. 

Nl,X+N2,X = 0 (A.26) 

-- N1 hlMl N2 h2M2 =o 

All(1) 2Dll(l) - All(2) - 2D11(2) 

The solution is assumed of the form 

(A.27) 

(A.28) 

(A.29) 

(A.30) 

Substitution of this solution into Eqs. A.26-A.29 leads to an eigenvalue 

problem with the following characteristic equation. 

18 
s[B~s~+B~s~+B~] = 0 (A.3 1) 

where the B's involve the stiffness and thickness parameters A, D and 

h. For the material system and ply stacking sequence considered, 

B22>4BlB3. Hence, the roots can be written as 

(A.32) 

Only the zero and positive roots of eq. A.32 are considered as they give 

solutions decaying exponentially from the crack tip. Then, the axial 

force and moment resultants can be written as 

N1 = alesix+ a2eS2X+ a1 (A.33) 

(A.34) 

(A.35) - 

(A.36) 

The k parameters in the above solutions involve the eigenvalues 

and the stiffness coefficients (A,D). For example, we have the definition for kl as 

(A. 3 7) 

Using the equilibrium Eqs. A. 10, A. 12 and A. 14 along with the 

applied axial force P and specimen width b. the axial force resultants 

and interlaminar stresses can be written as 

(A.38) 

(A. 39) 

(A.40) 

19 
(A.4 1) 

The constitutive equations are used to write down the displacement solutions. The arbitrary constants associated with the displacements and rotations are determined from the matching conditions between sublaminates l and 2 and the end conditions. Proceeding to 

sublaminate 3, the governing equations are 

N3.x = 0 (A.42) 

Q33+ p3 = 0 (A.43) 

M3.x- 93 = 0 (A.44) 

(A.45) 

Q3 = A55(2)P3 (A.46) 

M3 = Dll(2) P3.x (A.47) 

Matching U at the 2-3 interface and applying the boundary conN3 = A1 1 (2)u3 ,x 

dition at the free end, N3(a) = 0, gives 

N3 = 0 (A. 48) 

A+ a3 (A.49) a1 

SlAll(2) - SZAll(2) u3 = U2(0) = - 

In order to solve for the 

are combined to yield 

M3 = q1 

where 03 is defined by 

bending moment, Eqs. A.44, A.46 and A.47 

sinh 03 x + q2 cosh 03 x (A.50) 

(A. 5 1) 

Since the p matching conditon cannot be used at the 2-3 interface, 

the (remaining) boundary conditions are 

M3(a) = 0 (A. 52) 

MdO) = M2(0) (A.53) 

The 9s can be solved using the boundary conditions A.52 and A.53 as 

20 
(A. 54) 

(A.55) 

The solution for sublaminate 3 can be completed by writing the expressions for 93, p3 and P3 based on the M3 solution. 

The equilibrium equations for sublaminate 4 are 

N4.x = 0 (A. 56) 

94.x = 0 (A.57) 

M43- 94 = 0 (A. 5 8) 

The constitutive relations take the form 

N4 = A1 1( 1)u4,x (A. 59) 

94 = &5( 1)(p4+w4d (A.60) 

M4 = D11( $4.~ (A.61) - 

P Using Eq. A.56 with the boundary condition N4(a)= yields 

P N4 =- 2b (A. 62) 

Similarly, using Eq. A.57 with Q4(a)=O results in 

Q4=O (A. 63) 

Matching M1 and Q at the 1-4 interface and using Eq. A.58 

gives 

M4 = alkl + a2k2 (A.64) 

The U4 displacement is obtained by integrating Eq. A.59 and 

using the displacement matching boundary condition 

a1 ") u4=-(-x+-+- 1P +a3 

All(1) 2b s1 s2 

Similarly, integrating Eq. A.64 and setting p4(a) 

U4(0) = um. 

(A.65) 

to zero gives the 

solution for p4. Using the solutions for 94 and p4 and the boundary 

condition w4(0)=0 in Eq. A63 yields the solution for W4. 

21 
In order to determine al, a2 and ag, the following boundary 

conditions are used. 

P 

Nl(0) = 2b 

Pl(0) = P4(0) 

Ul(-L+a) = 0 

It is convenient to define the following parameters. 

ed = 03 - €I1 + (04 - 02)a 

The nominal (far field) strain is given by 

(A.66) 

(A.67) 

(A.68) 

(A.69) 

(A. 70) 

(A. 7 1) P 1 &=- 2b A1 1( 1)+ A1 l(2) 

The a parameters are obtained as 

03+84a (A. 72) 

ed a1 =All(Z)E 

(A. 73) 

(A. 74) 

The specimen compliance C is defined as the ratio of specimen 

extension to applied load. This is obtained as 

2Wa) c= p 

22 
(A.75) 

The total energy release rate associated with the crack 

(delamination) growth under a constant load P is given by 

P2 dC *=-- 2b da (A.76) 

Substituting the compliance expression from Eq. A.75 in Eq. 

A.76 yields the following expression for the total energy release rate. 

1 +I 1-12 

P2 1 1 

Or= (G-A11(1)+A11(2) (A.77) 

with 

A1 l(2) 

= All(l)+A11(2) All(1) 

1 (A. 8 0) 

The individual fracture mode contributions to the energy release 

rate can be calculated using the virtual crack closure method, based on 

the interlaminar stresses and displacements in the vicinity of the 

crack tip. From the assumed plane strain condition, the mode I11 contribution is zero (GIII=O). The mode I1 energy release rate, GII, is calculated using the virtual crack closure technique while GI is evaluated 

using 

(A.8 1) 

GII is calculated from the interlaminar shear stress and relative 

GI = Gr - GI1 

sliding displacement as 

(A.82) 

23 
In the absence of a singularity in the stress field, the limiting 

process leads to the trivial result GII=O. Hence, the limit is calculated 

as 6 tends to some finite value, say A. The value of A is chosen depending on the decay length associated with the problem i.e. the 

length within which the presence of the crack significantly alters the 

specimen response in comparison with the corresponding far field 

values. The decay length in this problem is dependent on the eigenvalues s1 and s2. The following value of A has been chosen in order to 

reasonably fulfil the decay length criterion. 

(A.83) 

The relative sliding displacement Au is based only on the difference U4-U3 so that the kinematic condition of zero relative displacement at the crack tip is fulfilled. This also simplifies the calculations. 

If the true value of Au (based on u4-113) is used, the p mismatch at the 

3-4 interface leads to a kinematically inadmissible displacement discontinuity at the interface. This discontinuity causes a non trivial limiting value GII as 6+0. But this value is an artifact of the modeling assumptions and cannot be used as the true value of GII. The mode I1 

energy release rate component, using Au=U4-U3, is obtained as 

I3 GI1 = (A.84) 

where the parameter 13 depends on A11(1), A11(2), SI, s2, al, a2, A and 

the specimen nominal strain E. 

24 
* 

number of 

90" plies 

112 

1 

2 

3 

4 

6 

8 

Table I Summary of Results 

GT 

'J/m2 

2.404 

6.752 

22.849 

51.049 

93.603 

228.871 

440.065 

GII/GT 

0.276 

0.275 

0.267 

0.261 

0.256 

0.250 

0.247 

0, 

MPa 

1313.9 

784.0 

426.2 

285.1 

210.6 

134.7 

97.1 

EC 

% 

1.6747 

1.1685 

0.8058 

0.6427 

0.5444 

0.4264 

0.3555 

c- 

#\ ,\ 

Symmetry 

Plane 

DP 

Delamination 

Transverse 

Crack 

Fig. 1 Specimen Cross Section 
free free 

P12 

Symmetry Shear stress=O Shear stress=O 

Plane w=o w=o 

Fig. 2 Modelled Region and Sublaminate Scheme 

Fig. 3 Generic Sublaminate 
q, 

M2 

tz 

\ 

Q4 Q4 

- 

L-a 

Z 

Q3 t c - 

a 

Fig. 4 Sublaminate Forces and Coordinate Systems 
1.05 

24.5 - 

24.0 

1.04 

1 1 1 1 1 

1.03 

1.02 

1 

cs 

0.99 

0.98 

0.97 

1 

/ n=8 

3 5 7 

Crack Length (Ply Thickness) 

Fig. 5 Total Energy Release Rate Variation 

9 

with Crack Size 

27.5 

27.0 

7 26.5 

26.0 

0=25.5 

2 

\ 

n=8 

Crack.Length (Ply Thickness) 

Fig. 6 Mode I1 Energy Release Rate Dependence on Crack Size 
50 I I 

a 

Y 

e -n 40 

0.2 4 6 8 10 12 14 

Distance from Crack Tip (Ply Thickness) 

Fig. 7 Interlaminar Shear Stress Distribution 

-3 I I I I I I I I I I I I I I 

0 2 4 6 8 10 12 14 

Distance from Crack Tip (Ply Thickness) 

8 Interlaminar Normal Stress (Peel Stress) Distribution 
0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

0 

450 

400 

350 

hl 300 E 

7, 250 

\ 

- 

a + (F *0° 

150 

100 

50 

0 

0.5 1 2 3 4 6 a 

Number of 9Cf'piies 

Fig. 9 Energy Release Rate Comparison 

I Experimental 

Present Model 

Ref. 9 

A Ref. 8 

- 

---- 

2 4 6 

Number of 98 plies 

8 

Fig. 10 Critical Delamination Growth Strain Variation