Damage evolution and constitutive behavior of advanced composites

Composite Structures 16 (1990) 341-357

Damage Evolution and Constitutive Behavior of Advanced Composites

G. Frantziskonis

Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona 85721. USA

&

S. E Joshi

Aerospace Engineering Department, University of Texas at Arlington, Arlington, Texas 76019, USA

ABSTRA CT

The aim of this paper is to present a mechanics based theory that accounts for damage evolution in advanced composites. Single-ply behavior, under various loading conditions, is studied in detail herein. It is shown that the proposed theory can capture the essential deformational characteristics of single composite material plies, subjected to external load. Emphasis is placed on the properties of the theory and its capability to predict the observed behavior; further, the present approach is compared to classical analysis procedures. The paper extends previous work by the first author; it considers loading path effects, relation of damage growth to apparent Poisson ratio, strain based distinction between tension and compression, failure criteria under different loading conditions, and thermodynamic restrictions.

1 INTRODUCTION

Composite material structural components suffer from microcrack initiation and growth when loaded. Such growth affects considerably the overall behavior of composite materials, and, can be termed as damage evolution. At critical load, microcrack growth (i.e. damage evolution) becomes unstable resulting in the failure of the structure (specimen).

341 Composite Structures 0263-8223/90/S03.50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain

342 G. Frantziskonis, S. P. Joshi

In the early stages of constitutive modeling work, the material was theorized as a continuum without any cracks or discontinuities. However, in recent years it has been realized that a slightly damaged material does not necessarily lead to the end of its life, and it is important to understand the behavior of materials containing acceptable levels of damage in order to predict the remaining life. In addition, it is important to understand the effects of damage evolution on the f'mal unstable state of a structure subjected to critical loading.

A large body of information on the subject of damage evolution in composites has been collected from nondestructive testing such as ultra- sonic C-scan, simple and stereo X-ray radiography, light and electron microscopy and replication. Also, destructive techniques such as sec- tioning and de-ply for microscopic observations are used. All the above testing techniques provide important information, in a qualitative sense, that show the multiplicity and complexity of cracking in composite materials. Here, it is not intended to comprehensively review the experimental information on damage evolution. Such reviews can be found in Refs 1-3. In general, the following stages of damage evolution have been recognized, under fatigue loading. The first stage consists of multiple matrix cracking up to a saturation point termed as 'the characteristic damage state'. During the second phase of damage development, secondary crack growth, fiber matrix debonding, fiber microbuckling and delaminations are observed. In the third stage, the rate of damage growth increases rapidly resulting in unstable collapse. Comprehensive discussions of the above damage modes can be found in many recent papers, e.g. Stinchcomb. 3 An essential conclusion is that damage growth affects considerably and in a very complicated manner the deformational and unstable failure characteristics of composite materials.

There is a need for a realistic theoretical description of damage guides to a theory that comprises the essential features of the known damage mechanisms. Damage must be so defined that the effect of a growing microcrack network is taken into account. Further, the theory should be able to predict the loading conditions (level) under which damage growth becomes unstable resulting in failure.

A number of investigators have considered the effect of damage development in composite material modeling. Tsai 4 considered various schemes in order to discount ply properties in the presence of damage. The general (classical) laminated analysis is employed for the analysis of laminated structural components. Schapery 5 utilized fracture mechanics concepts for life prediction. The theory has been utilized for quasi- isotropic random particulate composites such as solid rocket propellant. Laws et al. 6 obtained analytical solutions for effective moduli of elastic

Damage evohttion in advanced composites 343

bodies (composites) with distributed cracks. The cracks were aligned in predetermined directions, and the overall crack surface area had a first- order effect on the stiffness. Talreja 7 employed vector fields where the direction and magnitude of each vector provided a characterization of damage in fatigue, or any other loading mode. Each set of vector fields introduced represented a specific damage mode. Allen e t al. ~'~ charac- terized damage by a set of second-order tensor valued internal state variables representing locally averaged measures of specific damage modes. Various such modes were considered such as parallel and transverse to the fiber direction, interlaminar crack modes, etc. Certain applications 9 illustrated the capabilities of the theory. The first author of this paper, Frantziskonis'°-'2 considered material behavior as composed of two fractions, the first one being intact (free of cracks) and the second one incorporating the effects of microcrack growth. Further, the mechanics of free edge effects and instability criteria, due to damage and/or edge delamination, were introduced in Refs 11 and 12.

This paper presents further developments and addresses important issues on the approach introduced in Ref. 10. These issues include effects of loading path, failure criteria under different loading directions, damage growth effects on the overall Poisson's ratio, strain based distinction between tension and compression, and thermodynamic restrictions. In Section 2, the problem is formulated and the final constitutive equations, including damage progression, are derived. Section 3 addresses important issues related to damage growth in single plies. Finally, in Section 4, back prediction of commonly performed tests is presented; the theory is verified with respect to Graphite-Epoxy/T300 test results and also the results are compared with classical analysis.

2 FORMULATION

As mentioned previously, the material behavior is decomposed in two fractions, namely intact (undamaged) or u-part and relieved (damaged) or d-part. Let us consider a small volume element A V of the material. This volume consists of A V u or the intact volume, and of A V d or the damaged volume. The first fraction represents topical behavior and the material in it is intact in the sense that no microcracks are embodied. Due to inhomogeneities of the material at the microlevel, weak regions are developed, resulting in crack initiation and subsequent growth, thus the material suffers structural changes. The effects of such structural changes are not fully understood, and as a result, a theory that considers, in detail, all such changes is not yet available. Here, a theory is presented


that can provide an effective tool in describing the average influence of structural changes phenomenologically. In doing so, we consider the effect of an isolated or coupled fracture site in the sense that an influence zone is around it (as shown schematically in Fig. 1). The above considerations suggest use of the theory of mixtures, or theory of interacting continua. As a consequence, the following is applicable: t2" ~3

Oij=(1 -- r)o~+ ro~ (1)

where o,~ and o,~ are the stress tensors of the u-part and d-part, respectively, oii is the average Cauchy stress tensor and r is a volume ratio such that r--AVJAV. The general theory of mixtures ~a is simplified considerably if there is no diffusion among the components. There is no diffusion between u-part and d-part, thus the strains in the two components are equal j4" ~5 or

ij= (2)

Let o,~. ~ and a,~ f (index notation holds for the subscripts only) be the stress tensors in the matrix and in the fibers of the d-part, respectively. In a material element of a ply, damage progresses, due to the enforced deformation, and damage influence zones evolve. Such influence zones portray the mechanical influence of a system of microcracks, thus no shear stiffness is attributed to the d-part, or

S g = 0 (3)

where S~ is the deviator tensor of o,~. Fibers are assumed to be linear and elastic up to failure such that in the principal material direction of the ply tr~ =Efe~, and E~ is the Young's modulus of the fibers. Further, the spherical part of tr,~. is related to the spherical part of e~/, due to statistical isotropy of damage in the matrix of the d-part,

o~k = Aekk (4)

where A is bulk modulus for the d-part. The above considerations allow us to establish the constitutive

relations for the d-part as ~°.~ d ~ d

a i r -- C ijkle kl (5)

For plane stress conditions eqn (5) reduces to (in the principal material coordinate system)

O"121 0 Q6d6 tel2j

Damage evolution in advanced composites 345

Fig. 1.

Ti

sso • ° ~ . ~. o, ~ ,e°°~ i~'

aR

Stress relieved zone

Schematic of damaged body and stress relieved zones. A, Potential crack extension sites; B, unlikely zone for crack extension.

where Q~,=E~d(1 -v ,2v2 , ) , Qd2=vlzE~i/(1--vt~v,. ,) , Qd,_=E~,./ (1--vlzv21), Q66=G~2 a n d E d, vl2, G~z are the Young's modulus, Poisson's ratio, and shear modulus of the d-part, respectively, and

E~t = 2Ee + ( 1 - 2 ) A ( 1 - v,2) (7)

E~2= A(1 - v2,) (8)

where ~ = A f/A which represents the ratio of the area of fibers to total area in a plane transverse to the fibers. From eqn (3), it follows that G~2-- 0. Then the constitutive matrix in eqn (6) is a function of ~., El, A and v~2. Further details on these parameters are given subsequently. The undamaged fraction is considered linear and elastic

o~ = C~.k,e k, (9)

where C0k t is the usual orthotropic constitutive tensor; ~6 for plane stress conditions it is a function of E n , E22, VI2 , and GI2, these constants being the initial (tangent) elasticity ones. From eqns (1), (2), (5) and (9) the overall constitutive relations are established as (the irreversible nature of damage implies a rate formulation)

Oij= Li/kt~kt-- ['7"i] (10)

where

Li jk t=(1- r) " d (11) Ci]kl + rCqk I


and T,~ is the second-order damage stress rate tensor

:Pij = ? (o~- o,~) (12)

and an overdot indicates time derivative. Important properties of the constitutive eqns (10) with respect to degradation, damage induced anisotropy (due to/'ii), failure are discussed later.

For loading rate independent behavior, based on the physical inter- pretation of r, L its control on the constitutive equations and on the consideration that when ply failure occurs r reaches a critical value rcr, the following law is proposed, for plane stress conditions

=Agl l + Bg:_, + Cgl_, (13)

in the principal material coordinate system. A, B and C are functions of total strain, eij. The failure criterion is written as

rcr=a( ," , __ e, ,)- + a,I e'i, e';. I+ b(ee._2) 2 + c(e~2) z (14)

where re, is the critical value of r such that r = r , at ply failure. Note that the above failure criterion 'reminds" us of the classical Tsai-HiU failure theory. A superscript f denotes strain at failure. Assuming that at pre- failure an equation similar to eqn (14) holds as

r =a(ell)2+ al I el i e221 + b(e_,_,)2 + c(e12)2 (1 5)

then from eqns (13) and (15), we obtain

A=2ael l -a l le2"- } B=2be22-at[e~l (16)

C =2ce,,_

Constants a, b, c and at can be obtained from a series of commonly performed tests on plies. The procedure to obtain these damage related constants and their physical interpretations is presented in Section 3.

2.1 Poisson's ratio effects

Consider an angle ply subjected to uniaxial load (tensile or compressive) in the fiber direction. The condition that ¢ryy--0 and a ,y--0 yield (from eqns (1), (6)and (9))

t :%=0 = Hl2exx + H22e~. ~. + H2sexy (17)

Ctxy= 0 = Hlaexx + H23 ey7 + H33gxy (18)

Damage evoh~tion in advanced composites 347

where " l l _ _ - U _ _ H i i - Q i j r(Oij 0_~) i , j = 1 , 2 , 3 (19)

where an overbar denotes reduced stiffness matrix components. From eqns (17) and (18), we obtain

n_,3Hi3 - H33H)2 e>.>.= H,.2H33- H_~3 e.r x (20)

/-/_,3 Ht 2 -HI3H:2 ex>, = H,.2H33_H~.3 txx (21)

Since r increases during deformation, it is obvious from eqn (20) that the apparent Poisson's ratio is not, in general, constant. In the spatially orthotropic case, the expression for Poisson's ratio is given by

Q~'2 - r(Q~2 - Q~_,) e>.~. = Q~ QL o _ egg

,, - r( Q22 ,_) (22)

Typical variations of the apparent Poisson's ratio is discussed later.

2.2 Induced anisotropy and thermodynamic considerations

The damage approach introduced herein is based on the possibility of a coexistence and an interaction of two types of states: damaged, charac- terizing the nucleation and evolution of microstructures in the form of microcracks; and undamaged or 'normal' accommodating such microstructures to come about. For continuously distributed microstructures, we can introduce continuous fields such as r. It is important to note that although r is a scalar quantity, a tensor Tij (eqn (12)) is introduced in the formulation to describe damage evolution. This tensor is responsible for induced anisotropy, an obvious property of cracked media, and is clearly the result of damage evolution. As an example, if the material was initially isotropic then a sufficient condition that eqns (10) are not isotropic in terms of Oij and go is that / '~ is not isotropic thus it should be equal to k6~j, k being a finite real number. For illustration we consider a volumetric increment (rate) of strain and from eqns (10) it can be seen that the corresponding increment (rate) of stress is not volumetric, due to the presence of T~i, although the relation between the total stress and total strain, as expressed through eqns (1), (5) and (9) is isotropic. Here, it is not unreasonable to assume isotropy since these relations refer to the


initial stress-free state which is isotropic; a discussion on such issues is given by Biot. t7 Thus although the relationship between total average stress and total strain is assumed to be isotropic, a damage-induced incremental anisotropy is present. Its presence is, of course, necessary since microcracks follow damage-induced preferred orientations.

From the above discussion it is clear that unloading behavior should be anisotropic though the present model gives isotropic (orthotropic) unloading behavior (increasing r means loading in the present formulation). This is a simplification of the reality. Such cases related to slow cyclic loading will be presented in a forthcoming paper.

The material (ply) is initially orthotropic in the case of composites. Here, a sufficient condition that eqns (10) are not orthotropic, but generally anisotropic is that at least one of the shear components of 7"ij is non-zero, in the ply coordinate axis. If all shear components of Tii are zero then eqns (10) remain orthotropic. Cases of simple loading and the relevant anisotropy are discussed in Refs 10-12.

The thermodynamic constraints on constitutive relations are imposed by the second law of thermodynamics (Marsden and Hughes). ~8 Mathe- matically, it is expressed by the entropy production inequality. The relation between free energy and internal energy is E = # + NO, where E is the internal energy, # is the free energy, N is the generalized entropy force associated with a generalized entropy displacement variable. The second term characterizes the increment of entropy due to increment of damage. N and O are conjugate variables. The formulation in this paper (which is based on physical principles in the theory of mixtures) can be interpreted in the following manner from the thermodynamic point of view.

O is represented by a scalar valued function r. A scalar valued function signifies that the distribution of damage volume embedded in the intact volume is isotropic (statistically homogeneous). It should be noted that isotropic distribution of damage and anisotropic constitutive behavior of a damaged continuum are two separate issues. For example, consider a unit cube of a unidirectional fiber-reinforced composite; the fiber fraction of the unit cube remains constant with respect to the orientation of the unit cube (isotropic) but the constitutive behavior is not isotropic.

The generalized entropy force may be interpreted as forces associated with damage variable r. This generalized entropy force is a tensor in our case and represents stress difference, the term . _ a (Oij Oij ) in eqn (12), due to damage accumulation. The damage associated constitutive behavior (anisotropy) is brought about by the generalized entropy force variable rather than by the generalized entropy displacement variable.


3 PARAMETER DETERMINATION

Here, a step-by-step procedure for determination of the constants involved in the theory is presented. It is a modification of the procedure presented in Ref. 10. One significant modification is the strain based distinction between tension and compression. Consider a uniaxial tension (compression) test in the fiber direction. The initial slope of the stress-strain curve provides the Young's modulus E') (El). The stresses and strains at failure provide o~'~ (a~i¢~) and the corresponding strain e'it~ (e]¢~), respectively. Similarly, from a unaxial test transverse to the fiber direction, E~2 (E~:), of_,', (a~2)and e'i', (etiCt)can be obtained. From a shear test on a ply, the initial tangent modulus on the stress-strain curve provides G~2, while from the conditions of failure, o't: and etj_, are obtained. Also, from the shear test at failure the following holds

cr]2 =(1 - re,) G~2tfll: (23)

and from the damage evolution law (eqn (15)),

rcr = C(E~2) 2 (24)

Equation (23) provides the value of rcr and its subsequent substitution in eqn (24) provides the value of c. Next, from the failure stress-strain data of the uniaxial tension (compression) tests in the fiber and transverse to the fiber directions, the values of a, b are obtained. It is to be noted that a, b are different for tension and compression, denoted as a,, b, and G, b~, respectively. Constant a~ is expressed as ~!(aiai) , where i--t, c, and j -- t, c, and i depends on the sign of strain in the fiber direction, while j depends on the sign of strain in the direction transverse to the fibers. From the uniaxial test data the following set of nonlinear equations are obtained

I fc x2 -ft 2 fc - f t r¢,=a¢ie,,) +bt(e22) +~](a,a,)e,,e=] -fc ft re,--a¢(gf~,): + bt(e~2): + ,~(Ga,)e,, e : : [

r_ / -re ~2 ft =re r . =at(e~,)' +ocie=) + ~(acat)e , te=( (25) t t fc ~2 -ft fc rcr----at(gf~,) 2 +Oc(ezz) +,/(a~a,)e,,e=J

In the above equations, an overbar indicates the value of strain in the direction transverse to the loading direction. In obtaining these strains the apparent Poisson's ratios at failure (experimental values) are used, which depict the effect of damage evolution on the apparent Poisson's ratios. Equations (25) can be solved by introducing a simple iterative procedure for determination of a t, ac, bt, b c. Constant A assumes different values


A , a n d A c f o r t e n s i o n a n d c o m p r e s s i o n , r e s p e c t i v e l y , a n d c a n be d e t e r -

m i n e d b y wr i t i ng the s t r e s s - s t r a i n r e l a t i o n s h i p at f a i l u r e f o r un iax ia l

s t r e ss t r a n s v e r s e to the f ibe r s as

022 = [( 1 - r) E22 + rA( 1 - vl2)] E22 (26)

F o r T 3 0 0 G r a p h i t e - E p o x y , t he c o n s t a n t s s h o w n in T a b l e 1 a r e o b t a i n e d .

T h e e las t i c i ty a n d fa i lu re r e l a t e d c o n s t a n t s a r e t a k e n f r o m Ref . 19. F r o m the va lue s in T a b l e 1, t he d a m a g e r e l a t e d c o n s t a n t s a r e t h o s e

s h o w n in T a b l e 2.

TABLE 1 Material Constants for T300 Graphite-Epoxy

E'~ = 27 125 ksi E~ = 22 979 ksi

Et, = 1542 ksi E~; = 2898 ksi

Gi2 = i000 ksi vt~_, = v~_, (initial)=0"28

o~'1 = 216 ksi o~cl = 216 ksi

or;, = 5.78 ksi 0~¢, = 34 ksi

o~1_, = 9.33 ksi

e]'l = 0.008 rill = 0.0094

e ~_, = 0.0037 e~q = 0-016

~ z = 0 . 0 1 4 6

v~l _, (failure) = 0-28 v~., (failure) = 0.38

( 1 psi = 6.89 kPa)

TABLE 2 Damage Related Constants for T300 Graphite-

Epoxy

r . -- 0"36

c = 1668-8

at = 5355.24

bt = 26 284-1

At = 1"567 x 10 3 ksi

a c -- 61-22

b c = 1395.77

A¢--- 0-7887 x 10 3 ksi

Note that all damage related constants, except A, are dimensionless.

Damage evolution in advanced composites 3 51

10000

A

. . . . Experiment

1 I I I I I I t t I I I (I a ) 0O05 0010 (>015

S t r a i n

. . . . E x p e r i m e n t - - Theory

oOO2 0 0 O 4 0 0 O 6 0OO8 Strain

(b)

i 0-010

3

x

. . . . T h e o r y - - E x p e r i m e n t

¢ )

O.OO2 0004 0006 0008 0010 S t r a i n

60O0

4 O O O

2OOO

. . . . Experiment /

0001 0 0 0 2 0 0 0 3 0 004 S t r a i n

4 O O O O

A

. . . . E x p e r i m e n t

J i i i I i i i I I I , t , ~ , 0 0 0 8 0<)10 0.015 0020

S t r a i n

Fig. 2. Comparison between predicted and experimentally observed stress-strain response of T300 Graphite-Epoxy: (a) pure shear; (b) tension in fiber direction; (c) compression in the fiber direction; (d) tension transverse to the fibers; (e) compression

transverse to the fibers.

4 RESULTS A N D DISCUSSION

In predicting single-ply force-deformation behavior, the constitutive equations described previously must be integrated. For this purpose, a computer code has been developed; it predicts stress-strain response of angle plies subjected to a complex sequence of external loads.


Fig. 3.

0-4

0-2

(,3i c,)/ ;~b) cc!," / ~ ) , /

I / / , , ' / / / ; : / / / / / i , . , /

0 0 0 0 5 0.010 0.015 Strain

I I

OO20

Evolution of parameter r with respect to strain for the tests presented in Fig. 2.

Fig. 4.

0.40

0

,~ o .~5

E

._~ ?.

0.30 C

L

Q. <

0.25 0

I , I , I i .... I . t 0 0 0 2 0 0 0 4 0 0 0 6 0 0 0 8 0010

Strain

Increase of the apparent Poisson's ratio due to damage progression, with respect to strain for uniaxial compression test in the fiber direction.

The capability of the theory to predict the constitutive behavior up to final unstable failure is tested first with respect to the commonly performed experiments on single plies (laminae). Figure 2 shows comparison of the predicted and experimentally observed t9 stress-strain response for T300 Graphite-Epoxy. Stress-strain response in the fiber direction is basically linear. However, in the direction transverse to the fibers, the response is linear in tension and nonlinear in compression. Further, the shear response is highly nonlinear. As can be seen from Fig. 2, the model predicts these characteristic single-ply behaviors accurately. Figure 3 shows the evolution of the damage related parameter r with

Damage evohltion in advanced composites 353

30000

2OOO0

~, 10OO0

0

- - Tension . . . . Compression

. ... .....

(a)

, I , I , I , 0.002 0-004 0 0 0 6 0.008

Strain

20000

== 10900

_ _ Tension .... Compression

.,,°"

0.002 0-004 0-006 Strain

. ° . . . . .

t z (b) 0.008 0010

20000

1ooo0

Fig. 5.

/ -

Tension . / - . - C o m p r e s s i o n / ' /

/ / *

/ " / "

/ "

/ /

(c) t t t I t I I I I I t t I I I t I I I I I I I l

0 0 0 2 5 0 - 0 5 0 0 . 0 7 5 0 . 1 0 0 0 . 1 2 5

S t r a i n x 1 0 "1

Prediction of angle ply behavior: (a) 30*; (b) 45°; (c) 60 °.


Fig. 6.

0.4

Tension / ,.'

t. 0"2 ,,,..,"

0 0'002 0"004 0006 0008 0"010 Strain

V a r i a t i o n o f parameter r with respect to strain for 45 ° angle pry.

Fig. 7.

106

._•105

::2 = 104 tl_

103[ I 0

%

4

I I I I I I 5O

Ply angle

1 100

Comparison of present and Tsai-Hill failure criteria for T300 Graphite-Epoxy. I, Present (C); 2, Tsai-Hill (C); 3, Tsai-Hill (T); 4, Present (T).

respect to strain for the tests presented in Fig. 2(a). It is interesting to note that the rate of damage evolution increases continuously as strain increases. Figure 4 shows the increase of the apparent Poisson's ratio, due to damage progression, with respect to strain for uniaxial compression test in the fiber direction. For tensile loading, however, the apparent


Poisson's ratio remains unchanged in the present formulation (see Tables I and 2).

The present theory is used to predict the constitutive behavior of angle plies. Representative results for fiber orientations of 30 °, 45 ° and 60* are shown in Figs 5(a), (b) and (c), respectively. Both tension and compression responses are presented up to failure. Figure 6 shows the corresponding growth of r with respect to strain for the 45* single ply.

The present theory should be able to predict the failure stresses and strains accurately for different angle plies. It is well known that the Tsai-Hill failure theory compares well with experimental observation. In absence of experimental results for failure of T300 Graphite-Epoxy angle plies, the failure criterion presented herein is compared with the Tsai-Hill theory in Fig. 7, where T and C denote tension and compression, respectively. Variation of failure strain (in loading direction) for

Fig. 9.

0.020

E _ _ Tension . . . . Compression

: o-o,ot-

0.005

I- , I , I , I , I , 0 20 40 60 80

Ply angle

Fig. 8.

100

Failure strains for angle plies of T300 Graphite-Epoxy.

0"75

...... Tension ,,.,.. . . . . . . . . . . . , , , - - - - Compression

• 0 . 2 5 :"'" """'"""'"",. I

0 2O 4O 60 80 100 Ply angle

Variation of apparent Poisson's ratio at failure with respect to fiber orientation.

356 G. Frantziskonis. S. P. Joshi

angle pries is shown in Fig. 8. The lowest failure strain in tension and compression occurs at about 10 ° fiber orientation. The failure strain values in tension remain relatively constant for pries with angle orientation between 10 ° and 90 °, while they vary in the case of compression. Variation of the apparent Poisson's ratio at failure with respect to fiber orientation is shown in Fig. 9. This variation is characteristically similar to the Poisson's ratio variation for undamaged angle plies.

5 CONCLUSIONS

A theory is presented which is capable of incorporating damage evolution in the constitutive equations. Damage evolution is 'responsible' for nonlinear behavior, induced anisotropy, and final unstable failure. The constants involved in the theory may be easily obtained from the common tests performed on single plies (laminae). The theoretical model is verified by back-predicting such tests. In addition, the angle-ply behavior is predicted and discussed; although not directly compared with experimental observations, the basic characteristics of angle ply behavior are captured by the theory. 211

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2. Schulte, K. & Stinchcomb, W. W., Damage development near the edges of a composite specimen during quasi-static and fatigue loading. Comp. Tech. Review, 6 (1984) 3-9.

3. Stinchcomb, W. W., Nondestructive evaluation of damage accumulation processes in composite laminates. Comp. Science and Tech., 25 (1986) 103-18.

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Damage evohttion in advanced composites 357

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Damage evolution and constitutive behavior of advanced composites