A Continuum Damage Model for Composite Laminates-Part I Constitutive Model

Mechanics of Materials 39 (2007) 897–908

www.elsevier.com/locate/mechmat

A continuum damage model for composite laminates:Part I – Constitutive model

P. Maimı a, P.P. Camanho b,*, J.A. Mayugo a, C.G. Davila c

a AMADE, Escola Politecnica Superior, Universitat de Girona, Girona, Spainb DEMEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

c NASA Langley Research Center, Hampton, VA 23681, USA

Received 21 March 2006; received in revised form 9 December 2006

Abstract

A continuum damage model for the prediction of the onset and evolution of intralaminar failure mechanisms and thecollapse of structures manufactured in fiber-reinforced plastic laminates is proposed. The failure mechanisms occurring inthe longitudinal and transverse directions of a ply are represented by a set of scalar damage variables. Crack closure effectsunder load reversal are taken into account by using damage variables that are established as a function of the sign of thecomponents of the stress tensor. Damage activation functions based on the LaRC04 failure criteria are used to predict thedifferent failure mechanisms occurring at the ply level.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Fracture mechanics; Continuum damage mechanics; Composite materials

1. Introduction

The methodology for designing high-perfor-mance structures of composite materials is stillevolving. The complexity of the response of com-posite materials and the difficulty in predictingstructural modes of failure result in the need for awell-planned test program. The recommended prac-tice to mitigate the technological risks associatedwith such materials is to substantiate the perfor-mance and durability of the design in a sequence

0167-6636/$ - see front matter � 2007 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2007.03.005

* Corresponding author. Tel.: +351 225081753; fax: +351225081315.

E-mail address: [email protected] (P.P. Camanho).

of steps known as the building block approach(BBA) (MIL-HDBK-17-3F, 2002). The BBAensures that cost and performance objectives aremet by testing greater numbers of smaller lessexpensive specimens, assessing technology risksearly in the program, and building on the knowl-edge acquired at a given level of structural complex-ity before progressing to a level of more complexity.

Achieving substantiation of structural perfor-mance by testing alone can be prohibitively expen-sive because of the number of specimens andcomponents required to characterize all loadingconditions. BBA programs can achieve significantcost reductions by seeking a synergy between testingand analysis. The more the development relies onanalysis, the less expensive it becomes.

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mailto:[email protected]

898 P. Maimı et al. / Mechanics of Materials 39 (2007) 897–908

The use of advanced analytical or numericalmodels for the prediction of the mechanical behav-ior of composite structures can replace some ofthe mechanical tests and can significantly reducethe cost of designing with composites while provid-ing to the engineers the information necessary toachieve an optimized design.

Strength-based failure criteria are commonlyused to predict failure in composite materials. Alarge number of continuum-based criteria have beenderived to relate stresses and experimental measuresof material strength to the onset of failure (Sodenet al., 1998; Davila et al., 2005; Pinho et al., 2004).Failure criteria predict the onset of the differentdamage mechanisms occurring in composites and,depending on the material, the geometry and theloading conditions, may also predict final structuralcollapse.

For composite structures that can accumulatedamage before structural collapse, the use of failurecriteria is not sufficient to predict ultimate failure.Simplified models, such as the ply discount method,can be used to predict ultimate failure, but they can-not represent with satisfactory accuracy the quasi-brittle failure of laminates that results from theaccumulation of several failure mechanisms.

The study of the non-linear response of quasi-brittle materials due to the accumulation of damageis important because the rate and direction of dam-age propagation defines the damage tolerance of astructure and its eventual collapse. To model thephenomena of damage propagation, non-linear con-stitutive models defined in the context of themechanics of continuum mediums have been devel-oped and implemented in finite elements codes inrecent years. The formalism of the thermodynamicsof irreversible processes is a rigorous frameworkfrom which the constitutive models can bedeveloped.

The simplest way to describe damage is using asingle scalar damage variable, as proposed byKachanov (1958). Damage can be interpreted asthe creation of microcavities, and the damage vari-ables as a measure of the effective surface densityof the microdefects. Such a mechanical interpreta-tion of damage assumes that the loads are resistedonly by the undamaged ligaments in the material.The stresses ð~rÞ in the ligaments, referred to as effec-tive stresses, continue to increase until all ligamentsare severed and the material has failed.

The tensorial representation of damage is a for-mal and general procedure to represent the direc-

tionality of microcracks, which can take anydirection in a medium depending on the load his-tory, geometry, boundary conditions, and materialproperties. After Kachanov’s pioneering work, sev-eral damage models have been developed thatdescribe damage as a second order tensor (Chab-oche, 1995; Carol et al., 2001a,b; Luccioni andOller, 2003) or as a fourth order tensor (Ortiz,1985; Simo and Ju, 1987a,b). Second order tensorsdescribe an initially isotropic material as an ortho-tropic one when damage evolves, whereas fourthorder tensor models can remove all material symme-tries and provide a more general procedure to simu-late damage (Cauvin and Testa, 1999).

The application of continuum damage models inorthotropic or transversely isotropic materials, suchas fiber-reinforced plastics (FRP), results in addi-tional difficulties. The nature and morphology ofthe material induces some preferred directions forcrack growth, i.e., crack orientations are not onlyinduced by the loads, geometry and boundary con-ditions, but also by the morphology of the material.The interface between fiber and matrix is weakerthan the surrounding material and interfacial deb-onding is normally the first damage mechanism tooccur. Furthermore, residual thermal stresses occurin the composite plies due to different coefficients ofthermal expansion of the fiber and matrix (micro-mechanical residual thermal stresses) and due tothe different coefficients of thermal expansion inthe longitudinal (fiber) and transverse (matrix)directions (macromechanical residual thermalstresses).

Multiscale models and mesomodeling are twoapproaches used to evaluate the elastic and inelasticresponse of a material. Using homogenization laws,multiscale models define relations between a meso-scale, normally the scale of the finite elements,where material is considered homogeneous, andthe microscale, which is the scale of the fiber andmatrix constituents. The constitutive models aredefined at the microscale, and the strain and stressfields at the microscale and the mesoscale are relatedvia transformation field tensors (Chaboche et al.,2001; Fish and Yu, 2001; Fish et al., 1999; Voyiadjisand Deliktas, 2000; Car et al., 2002), or solved usingfinite elements (Car et al., 2002; Oller et al., 2005).To reduce the amount of computations that needto be performed, periodicity of the material isinvoked.

Mesomodeling is an alternative way to definedamage models for composite materials that is more

P. Maimı et al. / Mechanics of Materials 39 (2007) 897–908 899

appropriate for large scale computations. Meso-models treat the composite lamina (Ladevezeet al., 2000; Barbero and Lonetti, 2001; Barberoand Devivo, 2001; Matzenmiller et al., 1995; Olleret al., 1995) or sub-laminate (Williams et al., 2003)as a homogeneous material. When diffuse damagelocalizes in a narrow band and becomes a macro-crack, the response is dominated by the crack tipand its ability to dissipate energy. On the otherhand, the material morphology, which is the mainbasis of homogenization techniques, loses impor-tance due to the loss of periodicity. Therefore, instructures exhibiting stable crack propagation, i.e.,when the macrocrack length does not increase underconstant load, mesomodeling is more appropriate topredict the structural collapse than multiscalemodels.

The main objective of the present paper is todevelop a continuum damage model able to repre-sent the quasi-brittle fracture of laminated compos-ite structures, from the onset of non-critical failuremechanisms up to final structural collapse.

The majority of the material properties requiredby the present model can be measured using stan-dard test methods. Most of these properties can beobtained from ply-based test methods. The use ofply properties, rather than laminate properties, isan advantage because it avoids the need to test lam-inates every time the lay-up or stacking sequence ismodified.

The proposed constitutive model accounts forcrack closure effects under load reversal, an impor-tant phenomenon in cases where a composite struc-ture is subjected to multiaxial loading.

One important issue regarding the numericalmodeling of damage is that the convergence of thesolution through successive mesh refinement mustbe ensured. The objectivity of the numerical modelis ensured by adjusting the energy dissipated by eachfailure mechanism using a characteristic elementlength (Bazant and Oh, 1983). The constitutivemodel proposed herein can be integrated explicitly,making it computationally efficient and, therefore,suitable to be used in large scale computations.

This paper is organized as follows: a briefdescription of the failure mechanisms occurring inlaminated composites is presented. Based on themechanisms identified, a new constitutive modelwhich relates the failure mechanisms with a set ofinternal variables is proposed. An accompanyingpaper (Maimı et al., in press) presents the detailsof the computational implementation of the model

and the comparison between the model predictionsand experimental data.

2. Mechanisms of damage and fracture in laminated

composites

The model proposed in this work predictsintralaminar failure mechanisms in laminatedcomposites: matrix cracking and fiber fracture.Intralaminar failure mechanisms trigger structuralcollapse of unidirectional laminates almost immedi-ately. In matrix-dominated failure modes, laminatecollapse occurs as soon as a matrix crack is created.Failure of unidirectional laminates loaded in thelongitudinal (fiber) direction results from the accu-mulation of fiber fractures (Rosen, 1964).

However, multidirectional laminates can sustainincreasing amounts of intralaminar failure mecha-nisms before structural collapse occurs. Consideringthe multidirectional laminate as a representativevolume element, the intralaminar failure mecha-nisms can be regarded as damage mechanisms, i.e.distributed microcracks in a laminate whose tangentstiffness tensor is positive definite. When the tangentstiffness tensor of the laminate ceases to be positivedefinite, a macrocrack is formed and structural col-lapse ensues.

The main characteristics of the intralaminar fail-ure mechanisms occurring in laminated compositesare briefly described in the following sections. Thesecharacteristics are the basis for the definition of thefailure criteria and damage variables used in thiswork.

2.1. Longitudinal failure

In fiber-reinforced composites, the largest por-tion of the loads is resisted by the fibers. When thesefail under either tension or compression, the internalloads must redistribute to other areas of the struc-ture, and may cause a structural collapse.

In composites with high fiber volume fractionand those where the strain to failure of the resinmatrix is higher than the one of the reinforcing fiber,such as carbon–epoxy composites, longitudinal fail-ures start by isolated fiber fractures in weak zones.The localized fractures increase the normal andinterfacial shear stresses in the adjoining fibers,and the local stress concentrations promote matrixcracking, fiber matrix debonding and, for ductilematrices, conical shear failures (Daniel and Ishai,


1994). When increasing the load further, additionalfiber fractures occur, leading to final collapse.

Failure under longitudinal tensile loading occursin both constituents, and fracture occurs along aplane whose normal is parallel to the fiber direction.A simple non-interacting failure criterion based onmaximum stress or maximum strain along the longi-tudinal direction can usually provide an accuratemeasure of longitudinal tensile failure (Davilaet al., 2005).

Compressive failure of aligned fiber compositesoccurs from the collapse of the fibers as a result ofshear kinking and damage of the supporting matrix(Fleck and Liu, 2001; Schultheisz and Waas, 1996).The analysis of the kinking phenomena is usuallybased on the assumption of a local initial fiber mis-alignment. Fiber misalignment causes shear stressesbetween fibers that rotate the fibers, which in turnincreases the shear stress and leads to instability.

2.2. Transverse failure

Failure in the transverse direction encompassesboth matrix cracking and fiber–matrix debonding.Under the presence of transverse tensile stressesand in-plane shear stresses, the combined effect ofsmall defects present in a ply, such as small fiber–resin debonds, resin-rich regions, and resin voids,trigger a transverse crack that extends through-the-thickness of the ply. The transverse cracks areformed without disturbing the fibers: they occur atthe fiber–resin interface and in the resin.

When an unidirectional laminate is loadedin shear, a non-linear shear stress-shear strainresponse is observed before the laminate fails bythrough-the-thickness transverse matrix cracking.This non-linearity results from the visco-plastic

1

3

2

3a) Longitudinal tensile fracture b)

c) Transverse fracture with α=0º d)

Fig. 1. Ply fracture planes co

behavior of the matrix, and from the nucleationand growth of microcracks. The non-linear shearresponse which occurs before transverse matrixcracking is not taken into account in the modelproposed here, and will be the subject of futureresearch. According to Chang et al. (1984), shearnon-linear effects should be taken into account inthe strength prediction of cross-ply and angle-plylaminates.

A fundamental issue that needs to be consideredin the analysis models is the effect of ply thicknesson the ply strength, usually called the ‘in situ effect’.As shown in Parvizi et al. (1978) and Chang andChen’s (1987) experiments, the constraints imposedby the neighboring plies of different fiber orienta-tions cause an apparent increase in the tensile andshear strengths of a ply compared to those of anunconstrained ply. The in situ effect is a determinis-tic size effect that can be represented using fracturemechanics models of plies containing defects (Dav-ila et al., 2005; Camanho et al., 2006; Dvorak andLaws, 1987).

Experimental results have shown that moderatevalues of transverse compression have a beneficialeffect on the strength of a ply (Soden et al., 1998):when the in-plane shear stress is large comparedto the transverse compressive stress, the fractureplane is perpendicular to the mid-plane of the ply.However, increasing the compressive transversestress causes a change in the angle of the fractureplane. Normally, for carbon–epoxy and glass–epoxycomposites loaded in pure transverse compression,the fracture plane is at an angle (fracture angle,a0) of 53� ± 3� with respect to the thickness direc-tion (Puck and Schurmann, 1998). Therefore,matrix cracking does not occur in the plane of themaximum transverse shear stress (45�).

1

2

3

3Longitudinal compressive fracture

Transverse fracture with α=53º

nsidered in the model.


Following the observations outlined above, fourpossible fracture planes are considered in the modelproposed here, as schematically represented inFig. 1.

The purpose of present work is to propose amodel for the calculation of the initiation and prop-agation of intralaminar failure mechanisms. Conse-quently, delamination is not considered here.

3. Constitutive damage model

The thermodynamics of irreversible processes is ageneral framework that can be used to formulateconstitutive equations. It is a logical frameworkfor incorporating observations and experimentalresults and a set of rules for avoiding incompatibil-ities. In this section, we present a constitutive dam-age model for laminated composites that has itsfoundation in irreversible thermodynamics, and thatuses the LaRC04 criteria as damage activationfunctions.

3.1. Complementary free energy and damage

operator

To establish a constitutive law, it is possible todefine a scalar function corresponding to the com-plementary free energy density in the material. Thisfunction must be positive definite, and it must bezero at the origin with respect to the free variables(the stresses) (Malvern, 1969). The proposed defini-tion for the ply complementary free energy densityis

G ¼ r211

2ð1� d1ÞE1

þ r222

2ð1� d2ÞE2

� m12

E1

r11r22

þ r212

2ð1� d6ÞG12

þ ða11r11 þ a22r22ÞDT

þ ðb11r11 þ b22r22ÞDM ð1Þ

where E1, E2, m12 and G12 are the in-plane elasticorthotropic properties of a unidirectional lamina.The subscript 1 denotes the longitudinal (fiber)direction, and 2 denotes the transverse (matrix)direction. a11 and a22 are the coefficients of thermalexpansion in the longitudinal and transverse direc-tions, respectively. b11 and b22 are the coefficientsof hygroscopic expansion in the longitudinal andtransverse directions, respectively. DT and DM arethe differences of temperature and moisture contentwith respect to the corresponding reference values.The stress tensor r corresponds to the average stress

tensor over a representative volume that is assumedto be much larger than the diameter of a fiber.

The set of scalar damage variables d1,d2, and d6

ensure that the composite ply maintains the originalplanes of material symmetry, regardless of the dam-age state of the material. The damage variable d1 isassociated with longitudinal (fiber) failure, whereasd2 is the damage variable associated with transversematrix cracking and d6 is a damage variable influ-enced by longitudinal and transverse cracks.

To ensure the thermodynamically irreversibilityof the damage process, the rate of change of thecomplementary free energy _G minus the externallysupplied work to the solid at constant strains,_r : e, must not be negative:

_G� _r : e P 0 ð2Þ

This inequality corresponds to the positiveness ofthe dissipated energy and has to be fulfilled by anyconstitutive model (Malvern, 1969). Expanding theinequality in terms of the stress tensor and damagevariables gives:

oGor� e

� �: _rþ oG

od1

_d1 þoGod2

_d2 þoGod6

_d6 P 0 ð3Þ

Since the stresses are variables that can vary freely,the expression in the parenthesis must be equal tozero to ensure positive dissipation of mechanical en-ergy. Therefore, the strain tensor is equal to thederivative of the complementary free energy densitywith respect to the stress tensor:

e ¼ oGor¼ H : rþ aDT þ bDM ð4Þ

The lamina compliance tensor can be represented inVoigt notation as:

H ¼ o2G

or2¼

1

ð1� d1ÞE1

� m21

E2

0

� m12

E1

1

ð1� d2ÞE2

0

0 01

ð1� d6ÞG12

266666664

377777775ð5Þ

The closure of transverse cracks under load reversal,also known as the unilateral effect, is taken into ac-count by defining four damage variables associatedwith longitudinal and transverse damage. To deter-mine the active damage variables, it is necessary todefine the longitudinal and transverse damagemodes as follows:


d1 ¼ d1þhr11ijr11j

þ d1�h�r11ijr11j

d2 ¼ d2þhr22ijr22j

þ d2�h�r22ijr22j

ð6Þ

where hxi is the McCauley operator defined ashxi := (x + jxj)/2.

The present model tracks damage caused by ten-sion loads (d+) separately from damage caused bycompression loads (d�). Depending on the sign ofthe corresponding normal stress, a damage modecan be either active or passive.

The model also assumes that the shear damagevariable, d6, is not affected by the closure effect.Shear damage is caused mainly by transverse cracksand these do not close under shear stresses (r12).Transverse cracks are influenced by transverse stres-ses (r22) producing the closure of cracks and a fric-tion retention whereas longitudinal cracks producethe same effect under longitudinal stresses (r11)(Chaboche and Maire, 2002). The effect of frictionis neglected in the present model.

3.2. Damage activation functions

The determination of the domain of elasticresponse under complex stress states is an essentialcomponent of an accurate damage model. Based onthe previously described mechanisms of crack gen-eration in advanced composites, a strain space isconsidered where the material is linear elastic. Inthe present model, it is assumed that the elasticdomain is enclosed by four surfaces, each of themaccounting for one failure mechanism: longitudinaland transverse fracture under tension and compres-sion. Those surfaces are formulated by the damageactivation functions based on the LaRC03 andLaRC04 failure criteria. The LaRC03-04 failurecriteria have been shown to represent accuratelythe physical process of failure onset in laminatedcomposites. The LaRC04 criteria represent an evo-lution of the LaRC03 criteria: some criteria such asthe one for fiber kinking, are more accurate inLaRC04. However, the improvement in accuracyis associated with a significant increase in the com-putational effort. The present damage model uses acombination of both sets of criteria to achieve acompromise between accuracy and computationalefficiency. The full details of the derivation and val-idation of the LaRC04 failure criteria are presentedin references (Davila et al., 2005; Pinho et al.,2004).

The four damage activation functions, FN, asso-ciated with failure mechanisms in the longitudinal(N = 1+, 1�) and transverse (N = 2+, 2�) direc-tions, shown in Fig. 1, are defined as

F 1þ ¼ /1þ � r1þ 6 0; F 1� ¼ /1� � r1� 6 0

F 2þ ¼ /2þ � r2þ 6 0; F 2� ¼ /2� � r2� 6 0ð7Þ

where the loading functions /N (N = 1+, 1�, 2+,2�) depend on the strain tensor and material con-stants (elastic and strength properties). The elasticdomain thresholds rN (N = 1+, 1�, 2+, 2�) takean initial value of 1 when the material is undam-aged, and they increase with damage. The elastic do-main thresholds are internal variables of theconstitutive model, and are related to the damagevariables dM (M = 1+, 1�, 2+, 2�, 6) by the dam-age evolution laws. The elastic domain threshold de-fines the level of elastic strains that can be attainedbefore the accumulation of additional damage.

3.2.1. Longitudinal tensile fractureThe LaRC04 criterion for fiber tensile failure,

Fig. 1a, is a non-interacting maximum allowablestrain criterion defined as

/1þ ¼E1

X T

e11 ¼~r11 � m12~r22

X T

ð8Þ

where the effective stress tensor ~r is computed as~r ¼ H�1

0 : e. H0 is the undamaged compliance tensorobtained from Eq. (5) using d1 = d2 = d6 = 0.

3.2.2. Longitudinal compressive fracture

The LaRC03 failure criterion for longitudinalcompressive fracture, Fig. 1b, postulates that a kinkband is triggered by the onset of damage in the sup-porting matrix. Under this circumstance, the fiberslose lateral support and fail under the effect of lon-gitudinal compressive stresses. The initial fiber mis-alignment and the rotation of the fibers as afunction of the applied stress state are the parame-ters used in the damage activation function.

The damage activation function used to predictdamage under longitudinal compression ð~r11 < 0Þand in-plane shear (fiber kinking) is established asa function of the components of the stress tensor~rðmÞ in a coordinate system (m) representing the fibermisalignment:

/1� ¼hj ~rm

12 j þgL~rm22i

SL

ð9Þ

where the longitudinal friction coefficient can beapproximated as (Davila et al., 2005).


gL � � SL cosð2a0ÞY C cos2 a0

ð10Þ

The components of the effective stress tensor in thecoordinate system associated with the rotation ofthe fibers are calculated as

~rm22 ¼ ~r11 sin2 uC þ ~r22 cos2 uC � 2j~r12j sin uC cos uC

~rm12 ¼ ð~r22 � ~r11Þ sin uC cos uC þ j~r12jðcos2 uC � sin2 uCÞ

ð11Þ

where the absolute value of the shear stress is takenbecause the misalignment angle can be positive ornegative.

The misalignment angle (uC) is determined usingstandard shear and longitudinal compressionstrengths, SL and XC (Davila et al., 2005):

uC ¼ arctan

1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4 SL

X Cþ gL

� �SL

X C

r

2 SL

X Cþ gL

� �0BB@

1CCA ð12Þ

It should be noted that the LaRC03 failure crite-rion, derived to predict the onset of damage in lam-inated composites, calculates the misalignmentangle as a function of the applied stress. TheLaRC03 failure criterion is modified here assuminga constant misalignment angle, corresponding to therotation of the fibers at failure under pure longitudi-nal compression. This modification of LaRC03 fail-ure criterion ensures that /1� is a monotonicincreasing function under any state of proportionalloading.

It should be pointed out that two criteria are usedin LaRC03 for fiber kinking: Eq. (9) for ~rm

22 6 0 anda second equation for ~rm

22 P 0. However, the omis-sion of the second equation results in a minor lossof accuracy because the equation is the same asEq. (13.a) with the stresses transformed into themisaligned coordinate frame.

3.2.3. Transverse fracture perpendicular to the

laminate mid-plane (a0 = 0�)

Transverse matrix cracks perpendicular to themid-plane of the ply, i.e. with a0 = 0� (Fig. 1c),are created by a combination of in-plane shear stres-ses and transverse tensile stresses, or in-plane shearstresses and small transverse compressive stresses.These conditions are represented by the followingLaRC04 failure criterion:

/2þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� gÞ ~r22

Y Tþ g ~r22

Y T

� �2

þ ~r12

SL

� �2r

if ~r22 P 0

1SLhj ~r12 j þgL~r22i if ~r22 < 0

8><>:

ð13Þwhere g is the fracture toughness ratio defined as:g ¼ GIc

GIIc.

3.2.4. Transverse fracture with a0 = 53�The LaRC04 matrix failure criterion for high

transverse compressive stresses, Fig. 1d, consists ofa quadratic interaction between the effective shearstresses acting on the fracture plane:

/2� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~sT

eff

ST

� �2

þ ~sLeff

SL

� �2s

if ~r22 < 0 ð14Þ

where the effective stresses ~sTeff and ~sL

eff are computedas (Pinho et al., 2004)

~sTeff ¼ h�~r22 cosða0Þ½sinða0Þ � gT cosða0Þ cosðhÞ�i

~sLeff ¼ hcosða0Þ½j~r12j þ gL~r22 cosða0Þ sinðhÞ�i

ð15ÞThe sliding angle h is calculated as (Pinho et al.,2004)

h ¼ arctan�j~r12j

~r22 sinða0Þ

� �ð16Þ

The transverse shear strength and transverse frictioncoefficient can be approximated as

ST ¼ Y C cosða0Þ sinða0Þ þcosða0Þ

tanð2a0Þ

� �

gT ¼ �1

tanð2a0Þ

ð17Þ

The fracture angle a0 is approximately 53� in uniax-ial compression. With increasing amounts of in-plane shear, the fracture angle diminishes up toabout 40� and then abruptly switches to 0�. To findthe correct angle of fracture, a maximization of theLaRC03-04 failure criteria as a function of a shouldbe performed. However, in order to improve thecomputational efficiency of the present model, it isassumed that the fracture angle can only take oneof two discrete values: 0� or 53�.

The elastic domain in the ~r11, ~r22, ~r12 space rep-resented by the LaRC04 failure criteria is shown inFig. 2.

3.3. Dissipation

The rate of energy dissipation per unit volumeresulting from the evolution of damage is given by

Fig. 2. Elastic domain in the ~r11, ~r22, ~r12 space.


N ¼ oGod1

_d1 þoGod2

_d2 þoGod6

_d6

¼ Y 1_d1 þ Y 2

_d2 þ Y 6_d6 P 0 ð18Þ

The form of the complementary free energy definedin Eq. (1) ensures that the thermodynamic forces(YM) conjugated to their respective damage vari-ables (dM) are always positive:

Y 1 ¼oGod1

¼ r211

2ð1� d1Þ2E1

P 0

Y 2 ¼oGod2

¼ r222

2ð1� d2Þ2E2

P 0

Y 6 ¼oGod6

¼ r212

2ð1� d6Þ2G12

P 0

ð19Þ

Therefore, the condition of positive evolution ofdamage variables ð _dM P 0Þ is a sufficient condition

Fig. 3. Evolution of elastic dom

for the fulfillment of the second law ofthermodynamics.

It is important to note that the proposed modeldoes not generate spurious energy dissipation, i.e.,the loss or gain of mechanical energy, under crackclosure or opening. At load reversal, the time deriv-ative of the damage variable is non-zero ð _dM 6¼ 0Þ.Considering Eq. (18), the thermodynamical forces,YM, associated with the damage variables, dM, mustbe zero to avoid spurious energy dissipation at loadreversal (Carol and Willam, 1996). This condition istrivially satisfied in the present model (Eq. (19)).

Damage evolution without energy dissipation isphysically inadmissible. Therefore, it is necessaryto avoid damage evolution when the correspondingconjugated thermodynamic force is zero. Considerthe load history represented in Fig. 3: the materialis loaded in transverse tension and shear to t1 andthen loaded to t2. At time t2, the damage variabled2 evolves because the corresponding damage acti-vation function is activated. However, the corre-sponding thermodynamic force is zero (r22 = 0,Y2 = 0).

This non-physical response is avoided by modify-ing the longitudinal and transverse damage activa-tion functions. The transverse damage activationfunction is modified using the following equation:

/2� ¼ min

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~sT

eff

ST

� �2

þ ~sLeff

SL

� �2s

;~r22

X

8<:

9=; ð20Þ

where the constant X is equal to ~r22 when/2� = /2+ (Fig. 3).

The longitudinal damage activation function ismodified by taking into account that under sheardominated loads, matrix cracking is the first form

ain in the ~r22–~r12 space.


of damage to occur. After matrix cracking, thetransverse and shear stresses are zero and the fibermisalignment angle u tends to p/4. Under these cir-cumstances, the kink band criteria, Eq. (9), reads

/1� ¼ minhj ~rm

12 j þgL~rm22i

SL

;gL � 1

2SL

~r11

ð21Þ

3.4. Damage evolution

The evolution of the damage threshold values rN

is expressed by the Kuhn–Tucker conditions:

_rN P 0; F N 6 0; _rN F N ¼ 0 ð22ÞNeglecting viscous effects, the damage activationfunctions, Eqs. (7), always have to be non-positive.While the damage activation function FN is nega-tive, the material response is elastic. When the strainstate activates a criterion, FN = 0, it is necessary toevaluate the gradient _/N . If the gradient is not posi-tive, the state is one of unloading or neutral loading.If the gradient _/N is positive, there is damage evolu-tion, and the consistency condition has to besatisfied:

_F N ¼ _/N � _rN ¼ 0 ð23ÞTwo important characteristics of the model pro-posed here are that the damage threshold valuesare a function of the damage variables, and thatthe loading functions depend on the strain tensor.Under these conditions, it is possible to integratethe constitutive model explicitly (Simo and Ju,1987a,b).

In the definition of the constitutive model, it isnecessary to represent the relation between activeand inactive elastic domains. The evolution of anactive elastic domain is defined by the consistencycondition, i.e., it is defined with respect to the corre-sponding damage activation function. However, it isalso necessary to specify how the inactive elasticdomain evolves if other damage modes are active.It is assumed that the longitudinal and transversedomains are not coupled. On the other hand, com-pression damage is coupled with tension damage, asexplained in the next section.

3.4.1. Transverse loading

As described previously, transverse damage inthe form of matrix cracks can have different orienta-tions as a result of tension, shear, or compression-dominated loads. Under load reversal, transversecracks, which are perpendicular to the ply mid-

plane, do not affect the compression response: theelastic domain and the compressive damage variable(d2�) are unaffected by r2+.

On the other hand, matrix cracks at a fractureangle a0 = 53� caused by high compressive transversestresses have the same effect as cracks perpendicularto the mid-plane (a = 0�) when the load is reversedfrom compression to tension. Therefore, the evolu-tion of the transverse tensile elastic domain threshold(r2+) is governed by both damage mechanisms.

Based on the above considerations, the evolutionof the elastic domain in the transverse direction canbe represented by the following equations:

Tension loading: _r2þ ¼ _/2þ and _r2� ¼ 0

Compression loading: _r2� ¼ _/2� and

_r2þ ¼_/2� if r2þ 6 r2�

0 if r2þ > r2�

(

The integration of the previous expressions results in

r2þ ¼ max 1;maxs¼0;tf/s

2�g;maxs¼0;tf/s

2þg

r2� ¼ max 1;maxs¼0;tf/s

2�g ð24Þ

3.4.2. Longitudinal loading

Under longitudinal tensile stresses, the fractureplane is perpendicular to the fiber direction. Whenreversing the load, the cracks close and can stilltransfer load. However, the broken and misalignedfibers do not carry any additional load. Therefore,the compressive stiffness is influenced by longitudi-nal damage. However, the elastic domain isassumed to remain unchanged.

Under longitudinal compression, damaged mate-rial consisting of broken fibers and matrix cracksforms a kink band, and there is not a unique orien-tation for the damage planes. When the loads arereversed, the cracks generated in compression openand the elastic domain threshold increases.

Therefore, the evolution of damage thresholdsfor longitudinal damage are defined as

Tension loading: _r1þ ¼ _/1þ and _r1� ¼ 0

Compression loading: _r1� ¼ _/1� and

_r1þ ¼_/1� if r1þ 6 r1�

0 if r1þ > r1�

(

ð25Þ


The integration of the previous expressions results in

r1þ ¼ max 1;maxs¼0;tf/s

1þg;maxs¼0;tf/s

1�g

r1� ¼ max 1;maxs¼0;tf/s

1�g ð26Þ

3.5. Damage evolution laws

The internal variables rN define the threshold ofthe elastic domains, and are related to the damagestate of each lamina, i.e., the damage variablesdepend on the values of the internal variables. Inorder to fully define the constitutive model, it is nec-essary to define the relation between the internalvariables and the damage variables.

When the material is undamaged the internalvariables rN take the initial value of 1, anddN(rN = 1) = 0. Eqs. (24) and (26) define the evolu-tion of the internal variables ensuring that _rN P 0.As shown in Eqs. (18) and (19), the condition forpositive dissipation is satisfied if _dN P 0. The condi-tion for positive dissipation is automatically fulfilledif the damage evolution law satisfies the conditionodN/orN P 0. When the material is completely dam-aged, a fracture plane is created, the strains arelocalized in a plane in which rN!1 and the relatedcomponents of the stiffness tensor are zero,dN(rN!1) = 1.

Matrix cracks are related to the internal variablesr2+ and r2�. The internal variable r2� accounts forcompressive damage only, whereas r2+ accountsfor both, compressive and tensile, failure mecha-nisms. Therefore, for tensile transverse stresses dam-age is a function of d2+(r2+) because both types ofcracks (a = 0� and a = 53�) are open. Under trans-verse compressive loads, the damage is only influ-enced by the inclined cracks, d2�(r2�).

Kink bands are related to the internal variabler1�. The internal variable r1+ accounts for kinkbands and fiber tensile fracture. For tensile longitu-dinal stresses, the material loses stiffness as a resultof both damage modes because the cracks open.Therefore, the damage variable can be expressedas d1+(r1+).

When a lamina which is fully damaged in tension(d1+ = 1) is subjected to load reversal and the crackcloses, some of the original stiffness is recoveredbecause the tractions can be transmitted throughthe crack faces. However, the broken fibers losetheir alignment. Assuming that the fibers do notcarry any load, which can be considered as a limit

case, the compressive stiffness can be approximatedby the rule of mixtures applied for components inparallel as: (1 � d1�)E1 = VmEm. The generalizationof the above arguments for an intermediate damagestate can be expressed as d1� ¼ A�1 d1þ, with

A�1 � bV f Ef

V mEm þ V fEf

� bE1 � E2

E1

ð27Þ

where Ef and Em are the fiber and matrix Youngmodulus, Vf and Vm the corresponding volume frac-tions, and b is an adjustment parameter between 0and 1. If b = 1, the stiffness recovery is due onlyto matrix closure, and if b = 0, the stiffness recoveryis total and it is assumed that broken fibers do notlose alignment under compressive loads, and the ini-tial stiffness is recovered.

The damage law for longitudinal compressioncan be expressed as a combination of the failuremechanisms caused by tension loads, d1+(r1+), andthe failure mechanisms generated under compres-sion, d1�(r1�):

d1� ¼ 1� ½1� d�1�ðr1�Þ�½1� A�1 d1þðr1þÞ� ð28Þ

Longitudinal failure (d1±) is not influenced bymatrix cracking (r2±) as shown in experimentalresults carried by Carlsson and Pipes (1987) and inmicromechanical models (Dvorak et al., 1985; Lawset al., 1983; Nuismer and Tan, 1988; Tan andNuismer, 1989) of cracked composites. Therefore,the longitudinal stiffness is not function of trans-verse matrix cracks.

The shear stiffness is reduced as a result of longi-tudinal and transverse cracks. Under these circum-stances, the damage variable d6 is given by

d6 ¼ 1� ½1� d�6ðr2þÞ�ð1� d1þÞ ð29Þ

The damage evolution laws used force strain-softening as soon as one damage activation criterionis satisfied. Softening constitutive equations mayresult in physically inadmissable responses: thedamage is localized in a plane and fracture occurswithout energy dissipation. The numerical imple-mentation of softening constitutive equations usingthe finite element method may result in mesh-depen-dent results because the energy dissipated is a func-tion of the element size.

The solution normally used to ensure the correctcomputation of the energy dissipated regardless ofthe refinement of the mesh is to adjust the damageevolution laws using a characteristic dimension ofthe finite element. The definition of the damage evo-lution laws is therefore related to the computational


model, and it is described in the accompanyingpaper (Maimı et al., in press).

4. Conclusions

A new constitutive model for the prediction ofthe onset and growth of intralaminar failure mech-anisms in composite laminates under plane stresswas proposed. The model is based on four possibleply fracture planes related to fiber tensile fracture,fiber kinking, and matrix cracking with fractureplanes oriented at 0� and 53� with respect to theply thickness direction.

The onset of the intralaminar failure mechanismsrelated to the four fracture planes is predicted usinga simplification of the LaRC04 failure criteria. Theeffect of ply thickness on the onset of transversematrix cracking is taken into account by replacingthe unidirectional strengths by the ‘in situ’ strengthsin the failure criteria.

The unilaterality of damage is taken into accountby the ability of the model to represent complexload histories, including tension–compression loadreversals. The computational implementation ofthe constitutive model and its validation aredescribed in part two of this work (Maimı et al.,in press).

Acknowledgements

The research visits of the first author at the Uni-versity of Porto, Portugal, were funded by theUniversity of Girona (BE-UdG-2004) and by theSpanish government under ‘‘Acciones IntegradasHispano-Portuguesas’’ (HIP2004-0031). The firstand third authors also acknowledge funding fromthe Spanish government through DGICYT underthe contract MAT 2003-09768-C03-01.

The financial support of the Portuguese Founda-tion for Science and Technology (FCT) under theproject PDCTE/50354/EME/2003 is acknowledgedby the second author.

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A Continuum Damage Model for Composite Laminates-Part I Constitutive Model

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