Composites Science and Technology 72 (2012) 269–277

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Composites Science and Technology

journal homepage: www.elsevier .com/ locate/compsci tech

A generic approach to constitutive modelling of composite delaminationunder mixed-mode loading conditions

Irene Guiamatsia ⇑, Giang D. NguyenThe University of Sydney, School of Civil Engineering, Sydney, NSW 2006, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 August 2011Received in revised form 27 October 2011Accepted 11 November 2011Available online 25 November 2011

Keywords:A. Polymer–matrix compositesB. DelaminationB. Plastic deformationC. Damage mechanicsCohesive model

0266-3538/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compscitech.2011.11.012

⇑ Corresponding author. Tel.: +61 2 9351 2134; faxE-mail addresses: irene.guiamatsia@sydney.edu.au

ca (I. Guiamatsia).

A generic approach to constitutive modelling of composite delamination under mixed mode loading con-ditions is developed. The proposed approach is thermodynamically consistent and takes into account twomajor dissipative mechanisms in composite delamination: debonding (creation of new surfaces) andplastic/frictional deformation (plastic deformation of resin and/or friction between crack surfaces). Thecoupling between these two mechanisms, experimentally observed at the macro scales through the stiff-ness reduction and permanent crack openings, is usually not considered in depth in many cohesive mod-els in the literature. All model parameters are shown to be identifiable and measurable from experiments.The model prediction of mixed-mode delamination is in good agreement with benchmarked mixed-modebending experiments. It is further shown that accounting for all major dissipative mechanisms in themodelling of delamination is the key to the accurate prediction of both resistance and damage of theinterface.

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1. Introduction with fibre surface treatment. Not only is the plastic/frictional

In fibre-reinforced polymer composites, the interfacial regionbetween plies of different orientations is particularly vulnerableto damage, due to the increased likelihood of voids and defects.Because of the geometrical constraint of stiff fibres in adjacentplies, delaminating cracks are constrained to evolve within the re-sin-rich interface which typically has very little resistance to frac-ture growth. Consequently, this mode of failure can easily lead tocatastrophic loss of the structural integrity of a laminated compos-ite component. Therefore, it is crucial, in the modelling of delami-nation, to accurately predict both the resistance and damageinduced delamination of the interfacial region.

Experimental evidence [1,2] shows that the nature of delamina-tion, including the associated energy loss, is strongly dependent onthe fracture properties of the polymeric resin. Thermoset epoxies,preferred in the early days due to their superior manufacturingand in-service properties, are gradually being replaced by thermo-plastic or particulate-enhanced resins, which offer higher fracturetoughness thanks to substantial plastic deformation. Large perma-nent crack openings, a testimony of frictional/plastic dissipation atthe crack faces, have been witnessed in experiments presented byFan [3] on mixed-mode bending (MMB) tests and Rikards et al. [4]on double-cantilever beam (DCB) tests on composite laminates

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dissipation concomitant to the actual damage process, as notedby Carlsson et al. [5] for the End-Notched Flexure (ENF) test, butit may also exist a posteriori, should the newly created crack sur-faces happen to come into contact. The need to consider the irre-versible permanent deformation provides a rationale for thedevelopment of an improved interface constitutive model that iscapable of handling both (i) mixed-mode loading conditions, and(ii) the coupling between debonding and plastic dissipation or fric-tion that is witnessed experimentally.

In the literature, cohesive zone models have been widely usedfor the prediction of interfacial behaviour of laminated composites[10–19]. The combination of both strength and fracture criteria inthese models give them advantages over models based on linearelastic fracture mechanics or virtual crack closure technique (e.g.[6,7]). The issue of mode interaction under mixed-mode loadinghas been treated with such strategies as a fully coupled interfacestiffness matrix [12] or equivalent mixed-mode interface separa-tion [13]. Furthermore, to handle the variable mode ratio thatoccurs in realistic loading situations, Turon et al. [14] proposed astrategy akin to switching between constitutive laws that are com-puted, a priori, under conditions of fixed-mode mixity. Althoughhaving gained in popularity to the extent of being considered as‘‘standard models’’ for composite delamination, most existingcohesive models in the literature are based on damage theoryand do not take into account the plastic/frictional dissipation. Thistreatment is appropriate only if this dissipative mechanism is neg-ligible compared to the release of energy due to the creation of newsurface areas, which is not always the case. It has been recently

270 I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277

shown [8] that overlooking the plastic/frictional dissipation, evenin pure mode I, leads to inaccurate prediction of the damage state,even when the structure’s resistance can be well predicted. It istherefore rational to question the capability of damage-basedcohesive models in predicting interfacial damage.

In the context of delamination modelling, there are not manycohesive models possessing coupling behaviour between damageand plasticity. In fact, Schipperen and De Borst [30] suggested, bycomparing a pure damage model to two coupled damage/plasticitymodels (isotropic and orthotropic hardening), that pure damagemodels are computationally more efficient and just as accurate interms of predicting the extent of delamination. It will be shownthat, because friction and fracture are both dissipative mechanismscontributing to the interface damage process, a numerical modelfor delamination can only have true predictive capabilities if thosephysical mechanisms are adequately accounted for. For instance,Tvergaard and Hutchinson [32] proposed a traction separationlaw where the post-peak regime consisted of a region of pure plas-tic yielding followed by softening. More recently, Kolluri [22] fol-lowed the same principle by introducing a parameter, called theplastic limit, defining the boundary between pure plastic dissipa-tion and pure damage dissipation. Su et al. [31] presented a formu-lation with separate yield functions for the normal and shearfailure modes and a mixed-mode coupling parameter linking onlythe two plastic dissipations but resulting in a non-smooth yieldsurface that could cause numerical difficulties. The reader is alsoreferred to Scheider [21] for a more extensive review of the litera-ture on cohesive models. Existing coupled damage/plasticity mod-els primarily aim to reproduce a particular form of the tractionversus separation response that may be observed experimentally.Hence, the most common strategies are either to make use of phe-nomenological functions to fit the interface constitutive response,or enhancing classical plasticity models with stiffness reductionto capture the unloading response. Although these approaches(e.g. in [22]) are able to successfully produce the observed perma-nent deformation and stiffness reduction, the coupling parameteris usually not easy to identify, let alone calibrate. This is becausethe connection between the dissipative mechanisms representedby damage and plasticity remains unclear in those coupled models.

In this study, the development of cohesive zone models isapproached from a different angle underpinned by physicallybased concepts and fundamentals. The formulation of a new classof cohesive models is put in a thermodynamic framework featuringstrong coupling between different dissipative mechanisms [9]. Theresulting constitutive models are thermodynamically consistentand possess contributions from damage and plastic/frictionalmechanisms. Emphasis is put on establishing links between thecoupled dissipative processes and a measurable quantity: the ratiobetween the damage dissipation and the total fracture energy. Themode interaction is adequately dealt with, thanks to the use of anexplicitly defined dissipation potential. The proposed approach al-lows different experimentally observed strength and fracture crite-ria to be incorporated to guide the model predictive capability inmixed mode loading conditions.

The paper is organised as follows. First a general framework forthe development of cohesive models featuring coupling betweennormal and shear modes, as well as between damage and plasticity,is presented. A cohesive model is derived from the general formula-tion using typical strength and fracture criteria in the literature. Itsbehaviour is then assessed in pure and variable mode loading condi-tions. The parameter identification shows that it is able to calibrateall parameters from experiments. This model was implemented inthe finite element package ABAQUS/Explicit [23] and used for theprediction of failure in mixed-mode delamination benchmarksavailable in the literature. The importance of accounting for all ma-jor dissipative mechanisms in the constitutive modelling and its

consequence in the correct prediction of both resistance and im-pact-induced delamination is addressed in the simulation of an im-pact test.

2. A thermo-mechanical formulation

The thermodynamic formulation is based on earlier develop-ments [8,9], where further details can be found. The followingexpression for the Helmholtz energy potential is proposed:

W ¼ 12ð1� DÞ Tn ue

n

� �2 þ Ts ues

� �2h i

þ 12

DTn �uen

� �2 ð1Þ

where D is a scalar variable representing the interface damage; u isthe interfacial separation, connected to the elastic ue

n

� �and perma-

nent upnð Þ parts through the incremental relationship: du = due + dup;

Tn and Ts are the initial normal and shear stiffnesses of the interface,respectively. The Macauley brackets h�i are introduced in the normaldirection in order to prevent interpenetration of the crack faces. Theinterface normal (tn) and shear (ts) tractions, and damage energy (v)are derived from the energy potential as:

tn ¼@W@ue

n¼ ð1� DÞTnue

n � DTn �uen

� �ð2Þ

ts ¼@W@ue

s¼ ð1� DÞTsue

s ð3Þ

v ¼ � @W@D¼ vn þ vs ð4Þ

It can be seen from Eq. (2) that sgnðtnÞ ¼ sgn uen

� �. The damage

energies associated with mode I (vn) and II (vs) in Eq. (4) can beexpressed in terms of elastic strains or stresses and damage as:

vn ¼12

Tn uen

� �2 � �uen

� �2� �

¼t2

n

2ð1�DÞ2Tn; tn > 0

0; tn 6 0

(ð5Þ

vs ¼12

Ts ues

� �2 ¼ t2s

2ð1� DÞ2Ts

ð6Þ

For a strong coupling between damage and plasticity, the dissi-pation potential is taken of the form:

U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2

D þ /2np þ /2

sp

qð7Þ

where the contributions to the dissipation potential from damage/D and plasticity in normal (/np) and shear (/sp) modes are respec-tively assumed of the forms:

/D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFðk;DÞv

pdD

cosa; /np ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFðk;DÞ

ptndup

n

sinaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� kÞv

p ;

/sp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFðk;DÞ

ptsdup

s

sinaffiffiffiffiffiffikv

p ð8Þ

In the above expressions, the mixed-mode ratio k is defined asthe ratio between the mode II dissipation and the total dissipation;F(k,D) is a function controlling the damage evolution and a is aparameter governing the coupling between damage and plasticity.The forms and roles of F(k,D) and a will be discussed in the nextsections. The stresses and damage energy are given as derivativesof the explicitly defined dissipation potential:

v ¼ @U@dD

¼ @U@/D

@/D

@dD¼ /Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/2D þ /2

np þ /2sp

q @/D

@dDð9Þ

tn ¼@U@dup

n¼ @U@/np

@/np

@dupn¼

/npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2

D þ /2np þ /2

sp

q @/np

@dupn

ð10Þ

I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277 271

ts ¼@U@dup

s¼ @U@/sp

@/sp

@dupn¼

/spffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2

D þ /2np þ /2

sp

q @/sp

@dups

ð11Þ

The Legendre transformation of the dissipation potential, using(9)–(11), results in the following yield function in generalisedstress space:

y� ¼ v@/D@dD

!2

þ tn@/np

@dupn

0@

1A

2

þ ts@/sp

@dups

0@

1A

2

� 1 6 0 ð12Þ

As a property of this transformation, the evolution rules forinternal variables, damage and permanent deformations, are de-rived from the above yield function (dk is the plasticity-typemultiplier):

dD ¼ dk@y�

@v ¼ 2dkv

@/D@dD

� �2 ¼ 2dkcos2a

Fðk;DÞ ð13Þ

dupn ¼ dk

@y�

@tn¼ 2dk

tn

@/np

@dupn

� �2 ¼ 2ð1� kÞdkvsin2a

Fðk;DÞtnð14Þ

dups ¼ dk

@y�

@ts¼ 2dk

ts

@/sp

@dups

� �2 ¼ 2kdkvsin2a

Fðk;DÞtsð15Þ

Substituting (8) into (12) and rearranging the resulting expres-sion yield:

y ¼ vFðk;DÞ � 1 6 0 ð16Þ

3. Dissipative properties and parameter identification

The dissipative properties of the proposed formulation in bothpure and mixed mode conditions are directly obtained from theabove formulation, without having to integrate the constitutiveequations:

U ¼ UD þUnp þUsp ¼ vdDþ tndupn þ tsdup

s ð17Þ

where UD, Unp and Usp are the (rates of) dissipations due to damage,and plastic/frictional deformations in the normal and shear direc-tions, respectively. Using the flow rules (13)–(15) and yield condi-tion (16), the total dissipation can be expressed in terms ofdamage as:

U ¼ UD þUnp þUsp ¼Fðk;DÞdD

cos2að18Þ

where the individual dissipations are

UD ¼ Fðk;DÞdD; Unp ¼ ð1� kÞtan2aFðk;DÞdD;

Usp ¼ ktan2aFðk;DÞdD:

It is readily seen that the coupling parameter cos2a representsthe ratio between the dissipation due to damage and the total dis-sipation; similar to our previous study on constitutive modellingfor quasi-brittle materials [8]. Using (4)–(6), the total damage dis-sipation UD = vdD can be divided into contributions from mode I(UnD = vndD) and mode II (UsD = vsdD). Therefore the dissipationrates in modes I and II are, respectively:

Un ¼ UnD þUnp ¼ vn þ ð1� kÞvtan2a

dD ð19Þ

Us ¼ UsD þUsp ¼ vs þ kvtan2a

dD ð20Þ

The mode ratio can then be expressed in terms of stresses(assuming that tn > 0):

k ¼ Usp

Unp þUsp¼ UsD

UnD þUsD¼ Us

Un þUs¼

t2s

Ts

� �t2

nTn

� �þ t2

sTs

� � ð21Þ

Substituting vn = (1 � k)v and vs = kv into (19), (20) and using(16), the normal and shear total dissipations are obtained as afunction of the damage variable and mode mixity:

Un ¼ ð1� kÞ Fðk;DÞdDcos2a

; Us ¼ kFðk;DÞdD

cos2að22Þ

3.1. Pure mode I/II loading condition

Under pure mode I (k = 0) or II (k = 1) conditions, the stress–strain relationship becomes:

tr ¼ ð1� DÞTruer ð23Þ

with yield function and evolution rules:

y ¼ vr

FrðDÞ� 1 ¼ 1

2t2

r

ð1� DÞ2FrðDÞTr

� 1 6 0 ð24Þ

dD ¼ 2dkcos2ar

FrðDÞ; dup

r ¼ 2dksin2ar

trð25Þ

In the above, r stands for either n or s, and also either I or II; it isassumed that F(k,D) = Fr(D) and cos2a = cos2 ar in pure mode I/IIloading condition. The following form of function Fr(D) is adopted,which produces a linear softening law in the absence of plasticdissipation:

FrðDÞ ¼12

1þ T0r

Tr

� �2

1� Dþ T0r

Tr

� �2 ð26Þ

where Sr is the interface strength and T0r a parameter dependent on

the fracture energy Grc of the material in pure mode condition. It iseasy to show, using Eqs. (23)–(25) above, that T0

r is in fact the soften-ing modulus in the case of pure damage model with linear softening,e.g. cos2ar = 1. For a general coupled damage plasticity model the dis-sipative properties, Eq. (22), yield the total fracture energy:

Grc ¼R 1

0 ðFrðDÞdDÞcos2ar

ð27Þ

Substituting the form of Fr(D) into (27) and then integrating theobtained expression lead to:

T0r

Tr¼ S2

r

2GrcTrcos2ar � S2r

� � ð28Þ

The ratio between damage dissipation and total dissipation inpure mode, cos2ar, can be calibrated from an experiment with suf-ficient number of unloading–reloading cycles [20], Fig. 1. Theeffects of coupling damage with plasticity on the model responseare illustrated in Fig. 1. For a fixed fracture energy Grc, the modelbehaviour transforms from linear to nonlinear softening if the cou-pling parameter cos2ar goes from 1 to 0 (properties used are:Tr = 7500 MPa/m, Grc = 1.8 Nm/m2 and Sr = 70 MPa). Accordingly,the unloading curves show increasing permanent deformation. Inprinciple, it is possible to keep the softening law unchangedregardless of the variation of cos2ar, as has been done in [8].However, it will be seen later in this paper that changes in thesoftening law, while keeping Grc fixed (Fig. 1), in fact have very lit-tle effects on the structural response, as far as the load carryingcapacity versus the crack opening displacement is concerned. Onthe other hand, the damage, as an indicator of the delaminationprocess, is significantly different if the model behaviour goes from

Fig. 1. Left: Extraction of cos2a. Right: Effect of cos2a in pure mode loading.

Fig. 2. Loading surface compared with the quadratic [25] and BK [14] strengthinitiation criteria.

272 I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277

pure damage to coupled damage-plasticity. This effect cannot beneglected in both the constitutive (unloading curves in Fig. 1)and structural responses (e.g. damage spreading, in impactsimulation).

3.2. Mixed mode loading condition

In principle, there are several ways to couple the above puremode responses following the strength and fracture criteriaobserved experimentally, provided that the following constraintsare met:

Fðk;DÞjk¼0 ¼ FnðDÞ; Fðk;DÞjk¼1 ¼ FsðDÞ ð29Þcos2ajk¼0 ¼ cos2aI; cos2ajk¼1 ¼ cos2aII ð30Þ

Given the dissipation properties of the formulation, cos2a can bedetermined directly as a function of the mode mixity k from mixedmode experiments with sufficient number of unloading–reloadingcycles. Since data from such mixed mode experiments are usuallynot available (yet), a linear scaling law for cos2a based on the (pre-sumably known) dissipation ratios cos2aI and cos2aII in pure modescan be used as an approximation:

cos2a ¼ ð1� kÞcos2aI þ kcos2aII ð31Þ

In a similar way, the following form of F(k,D) is proposed, whichis sufficiently flexible to accommodate various combinations ofstrength and fracture criteria:

Fðk; DÞ ¼ ð1� kÞcFnðDÞ þ kbFsðDÞ ð32Þ

The two parameters b and c are function of mode mixity k andcan be determined from the experimentally observed fracture andstrength criteria. Assuming a constant mode ratio and using (18),(22) and (32), the modal energy dissipations can be found as afunction of b and c as:

GIðb; cÞ ¼ ð1� kÞGðk;b; cÞ; GIIðb; cÞ ¼ kGðk;b; cÞ ð33Þ

where the total energy dissipation is given by:

Gðk;b; cÞ ¼Z 1

0

Fðk;DÞdDcos2a

� �

¼ 1cos2a

ð1� kÞccos2aIGIc þ kbcos2aIIGIIc

h ið34Þ

Hence a fracture propagation criterion maybe imposed bysubstituting Eq. (33) into the expression of the mixed-mode frac-ture locus, to yield an equation in terms of b and c:

gðGIðb; cÞ;GIIðb; cÞÞ ð35Þ

With respect to damage onset, a strength initiation criterion f(tn,ts), e.g. the quadratic interaction proposed in [25], may be used toobtain the second equation for b and c:

f ðtnðb; cÞ; tsðb; cÞÞ ¼ 0 ð36Þ

where the tractions tn(b,c) and ts(b,c) are obtained from Eqs. (16)and (21) as:

v ¼ FðDÞjD¼0; k ¼ vs=v

) ½tnðb; cÞ�2 ¼ 2ð1� kÞTn½ð1� kÞcFnð0Þ þ kbFsð0Þ�½tsðb; cÞ�2 ¼ 2kTs½ð1� kÞcFnð0Þ þ kbFsð0Þ�

ð37Þ

Eqs. (35) and (36) form a system of two nonlinear equations,where the two unknowns are the model parameters b and c, whichcan be solved at every solution increment (Appendix A).

4. Model behaviour

As mentioned, the model has its own strength criterion given bythe yield function. This is obtained by substituting Eqs. (5) and (6)into the yield function (16):

12Fðk;DÞð1�DÞ2

t2n

Tnþ t2

sTs

h i� 1 ¼ 0; tn > 0

12Fðk;DÞð1�DÞ2

t2s

Ts� 1 ¼ 0; tn 6 0

8><>: ð38Þ

This loading surface in stress space, obtained from the abovecalibration procedure with the Benzeggagh–Kenane (BK) fracturecriterion [26](cf. Table 2) and shown in Fig. 2, ‘‘shrinks’’ towardsthe origin as the value of the damage variable increases. This isconsistent with the physically observed mixed-mode interfaceresponse. The limiting case (D = 0) represents, in fact, the initiationlocus, which is traditionally imposed onto classical delaminationmodels, but is naturally obtained here as a function of the damagecontrolling function. Comparison with other initiation criteriafound in the literature is also provided in Fig. 2.

To illustrate the response of this constitutive model, three load-ing scenarios with variable mode-mixity, Fig. 3, are simulatedusing the interface properties also utilised in Appendix A. The BKfailure locus (with g = 2.) is imposed as a model constraint throughcoefficients. It can be seen from Fig. 3 that the mixed-mode failurelocus imposed through parameters b and c is indeed reproduced by

Fig. 3. Displacements (left) and failure locus (right) for three loading scenarios.

Fig. 4. Shear (top) and normal (bottom) tractions (N) against opening displacements (m). Pure damage results are shown with a continuous line while the coupled damage/plasticity model’s are shown with a dashed line.

I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277 273

the model, regardless of the damage/friction coupling, for the sce-nario with constant mode ratio, S1. This is not the case, however,for variable mode loading as illustrated with scenarios S2 and S3;such a result is not unexpected given that mixed-mode fracturecriterion are typically obtained from constant mode ratio experi-ments. For each scenario, the response for a pure damage modeland that of a coupled model (cos2a = 0.5) are compared, in Fig. 4,in terms of the evolution of interface tractions against openingdisplacements.

The main differences are the variable tangent stiffness duringsoftening and the permanent opening displacement of the coupledmodel; the compressive loading (normal mode), which appears inS2, has the effect of preventing interpenetration of the crack faces.Finally, the response corresponding to scenario S3 emphasises howthe strength in both modes is affected by damage evolution.

y ¼ y t ; t ;D ð42Þ

@D @tn @D @ts @D

5. Numerical implementation

This model was implemented as a user-defined interface ele-ment in the explicit finite element code ABAQUS/explicit [23].The topology of such elements, which can be of zero or finitethickness, is presented in several papers [14,15,19]. Only thestress-return algorithm is described here.

Step 1: Stress predictor For a typical displacement-controlledloading, the predictor (upper prefix p) for elastic separa-tions and tractions are obtained as:

ptþDttn ¼

ttn þ ð1� DÞTnDun; ttn > 0

ttn þ TnDun; ttn 6 0

ð39Þ

ptþDtts ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiptþDtt

2s1 þ

ptþDtt

2s2

q; p

tþDttsi

¼ ttsi þ ð1� DÞTsDusi; i ¼ 1;2 ð40Þ

Step 2: Mode ratio and parameter identification The mixed-mode ratio is written in terms of the tractions:

k ¼ptþDtts� �2

=Ts

ptþDttn� �2

=Tn þ ptþDtts� �2

=Ts

ð41Þ

Based on the mode ratio k, the parameters b and c are calculatedas described in Appendix A.Step 3: Internal state update The yield function based on predic-

tor stresses is evaluated:

p p p� �

tþDt n tþDt s

If the loading condition is violated, i.e. py > 0, then the damagevariable and permanent displacement jumps are updated. Droppingthe prefixes, the sought yield point is determined by solving:

y ¼ pyþ @y@tn

@tn

@upndup

n þ@y@ts

@ts

@upsdup

s

þ @y þ @y @tn þ @y @ts� �

dD ð43Þ

274 I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277

Using the flow rules (13)–(15), the plasticity-type multiplier isobtained: p

TableMater

Lay

E11

E22

G1

dk¼� y

2 @y@Dþ

@y@tn

@tn@D þ

@y@ts

@ts@D

� �cos2aFðDÞ þ

@y@tn

@tn@up

n

vð1�kÞsin2atnFðDÞ þ

@y@ts

@ts@up

s

vksin2atsFðDÞ

h ið44Þ

All the derivatives required for the calculation of dk are given inAppendix B and the increments of damage and permanent openingsare computed using the flow rules given by Eqs. (13)–(15). In threedimensions, the increments in the two shear directions are found byusing the chain rule:

� � � �

dup

si ¼ dk@y@tsi¼ dk

@y@ts

@ts

@tsi¼ dup

stsi

ts; i ¼ 1;2 ð45Þ

Step 4: Stress update Finally, the stress is updated according to:

1ia

e

2

tþDttn ¼ptþDttn � dD

ptþDt

tn

1�t D

� �� ð1�tDÞTndup

n; tn > 0ptþDttn ; tn 6 0

8<: ð46Þ

tþDttsi ¼ ptþDttsi � dD

ptþDttsi

1�tD

� �� ð1�tDÞTsdup

si; i ¼ 1;2 ð47Þ

6. Numerical validation

6.1. Mixed-mode delamination benchmarks

The model was utilised in pure damage mode (cos2a = 1) for thesimulation of mixed-mode delamination experiments performedby Reeders and Crews [27] on AS4/PEEK unidirectional laminates,and presented in Camanho et al. [19], to which reference is madehere for the details of material properties and specimen geometry.There are three mixed-mode bending (MMB) tests with moderatios k = 0.2; 0.5; 0.8, as well as a double cantilever beam (DCB)and End-Notched Flexure (ENF) tests corresponding, respectively,to k = 0 and k = 1. A three dimensional model with mesh size of0.5 mm (lengthwise) by 1 mm (width) was utilised for the nonpre-cracked section of the beam. The full experimental setup was

Fig. 5. Left: FE model of the mixed-mode bending test (k = 20%). Right: Predict

l properties used in the analysis of the GFRP composite [4].

r Inte

40.7 GPa G23 3.98 GPa Sn

7.4 GPa m 0.3 Tn

4.2 GPa GIc

modelled, including the loading bar (Fig. 5). The prediction isobtained in terms of reaction versus displacement of the loadingpoint for all the specimens, except for the DCB, where the total sep-aration (twice the displacement) of the beam’s arms is considered.There is a good agreement between the prediction and experimen-tal results, as per Fig. 5. Additionally, the response of this modeland the variable mode delamination proposed by Turon et al.[14]are in such excellent agreement that they cannot be distinguishedfrom one another (results presented here for the model by Turonet al. [14] were obtained by utilising their formulation with ourFE model, in order to remove discrepancies associated with theFE idealisation).

We now consider, for validation of the proposed coupled damageand plasticity constitutive model, a DCB experiment realised on uni-directional surface-treated glass fibre/epoxy specimens, by Rikardset al. [4]. The specimen dimensions are 100 mm (length) � 3 mm(thickness) � 25 mm (width) with a total pre-crack length of mm(31 mm of inserted film and 2.5 mm additional pre-crack). Thematerial properties and parameters used for the analysis are sum-marised in Table 1.

Several unloading/reloading curves are provided in Ref. [4],therefore, it is straightforward to calculate, using the area methoddescribed in Section 3.1, the ratio of the portion of fracture energydue to damage, cos2aI = 0.7. Realistic material properties are usedin the simulation, for both the interface and the composite layers.Note the range of values of the fracture toughness, which neverreaches a plateau; this is rather common for many glass fibre UDlaminates, as reported in [24]. This behaviour is also imposed inthe simulation, in the form of a variation of the toughness propertyas a function of the crack mouth opening displacement (CMOD), inorder to enable the prediction. Three rows of first-order, two-dimensional, four-node plane strain elements are utilised per beamarm; the prediction of the specimen response for several loading–unloading cycles is in excellent agreement with both experimentalresults and analytical predictions with simple beam theory (BT), asshown in Fig. 6. It is worth mentioning that there would be nonoticeable difference between a pure damage model and the cou-pled model, in the force versus displacement plot during mono-tonic loading. Upon unloading, however, the distinct permanentopening displacement provides an understanding of the coupled

ion of current model against experimental data [27] and Turon et al. [14].

rface

58 MPa cos2aI 0.70.75E14 Pa/m thickness 40 lm244 � 600 Nm/m2

Fig. 6. Left: Prediction of the response of a DCB specimen. Right: Delaminated region, pure versus coupled model.

Fig. 7. 5.1 J impact simulation. From left to right: FE model, impact force history and delaminated area.

I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277 275

dissipative processes. This is further illustrated in Fig. 6, where pre-dicted delaminated regions from pure and coupled models arecompared at two loading steps. The final crack length is recorded,for the numerical simulation and experiments, as 116.5 mm and117.1 mm, respectively.

6.2. Impact-induced delamination

The potential of the model in handling both variable mode load-ing and frictional dissipation is now demonstrated through the sim-ulation of an impact experiment proposed by Aymerich et al. [29],which the reader is referred to for detailed dimensions and materialproperties for the 5.1 J impact of a [03,903]s cross-ply graphite/epoxylaminate. Generic ABAQUS continuum shell elements, with Hashinintra-laminar damage capability [23] (Strength: S11t = S11c =2400 MPa, S22t = 100 MPa, S22c = S12 = 300 MPa; Energy: GIt = GIc =

Fig. 8. 1.15 J impact simulation. Left: Damaged area. Middle: Dam

40 Nm/m2, GIIt = GIIc = 1 Nm/m2), are utilised for the plies (Fig. 7),while the 0/90 (bottom) and 90/0 (top) interfaces are modelled with8-node interface element using the proposed coupled damage/fric-tion model. The mesh size in the impacted region is approximately0.5 mm � 0.5 mm; artificially high values of the mode II interfacetoughness are utilised to mimic the apparent increase in the inter-face toughness that is experimentally observed at the upper inter-face under compression.

Comparisons are made between our new model in pure damagemode (cos2a = 1) and in coupled damage/friction mode (cos2a = 0.3),in order to model the difference between composite laminates withbrittle or ductile matrix. In principle, the damaged areas (D > 0) pre-dicted with both models should be more or less the same, given thatthe amounts of energy absorbed, linked to the impact force historiesare identical. However, the damage distribution or spread, will differdue to the fact that part of the impact energy in the simulation using

age profile along dashed line. Right: Impact force versus time.

Fig. 9. From left to right (fquad, gquad), (fquad, glin), and (fBK, gBK).

Table 2Identifying b(k) = c(k) based on imposed propagation criterion.

Criterion b = c Fracture locus Initiation locus

gquad

gBK

276 I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277

the coupled model has been dissipated through the plastic/frictionalmechanism. Indeed, the coupled damage-plasticity model predicts asmaller delaminated area (D = 1) than the pure damage one.

To further illustrate the argument, the treated glass fibre/epoxyfrom Section 6.1, which is a more ductile material system, is now con-sidered. The imposed initial velocity on the 2.3 kg impactor is1000 mm/s, for a total impact energy of 1.15 J and no intra-laminardamage model is used in this simulation. Three hypothetical valuesof the coupling parameter, 0.01, 0.1 and 1.0, are considered. Again, de-spite identical impact forces, the predicted damaged areas (Fig. 8) areclearly different. This infers a real potential for overestimating thedelamination damage if the contribution of frictional dissipation isneglected for materials (resins) of substantially higher ductility.

7. Conclusion

A thermodynamically-consistent approach with a strong cou-pling between damage and plasticity was proposed for the analysisof mixed-mode delamination in composite laminates. The formula-tion is capable of accurately predicting the initiation and propagationof mixed-mode delamination. Two significant contributions are:

1. The proposed coupling between the energy dissipation resultingfrom the creation of new surfaces (damage) and the plastic/fric-tional dissipation is directly calibrated from experimentalunloading–reloading curves. The proposed coupled damage-plasticity model offers an accurate prediction of the permanentcrack opening observed experimentally.

2. The handling of mode-mixity is performed through a definitionof a mixed-mode damage controlling function that can bedirectly calibrated with constant mode ratio experiments. Such

a function is used to obtain excellent predictions of classicalmixed-mode delamination benchmarks. Additionally, the treat-ment of variable mode loading is accomplished seamlessly byintroducing derivatives of the model’s parameters (b and c)with respect to the mode ratio k.

Because tougher and more ductile resins, such as PEEK, arebeing considered for higher performance future composite lami-nates, it will no longer be possible to neglect the energy dissipatedthrough plastic deformation and/or friction. For interfaces wherethe plastic dissipation cannot be neglected, a pure damage modelcalibrated with the total measured fracture energy would over-predict damage, besides the inability to capture the permanentdeformation, as shown with the example of a dynamic impactevent. This is clearly detrimental for the accurate assessment ofthe vulnerability of a composite structure.

Detailed comparisons of the results by our coupled model andexperiments cannot be presented due to lack of experimental dataon unloading paths. Experiments are under investigation to extractrealistic values of cos2a for the analysis of PEEK-based laminates.Finally, the treatment of the compressive regime, which is simplyhandled here via the use of a penalty stiffness to prevent interpen-etration, is currently being improved in order to incorporate adependence of frictional effects on compressive normal tractions.

Acknowledgements

The authors wish to thank A/Prof Itai Einav for insightful discus-sions. This research was supported by the Australian ResearchCouncil (project number DP1093485). Support from the University

I. Guiamatsia, G.D. Nguyen / Composites Science and Technology 72 (2012) 269–277 277

of Sydney Postdoctoral Fellowship scheme to the second author isalso gratefully acknowledged.

Appendix A. Model calibration based on mixed modeexperiments

The parameters b and c are determined by finding the solutionto the system of Eqs. (35) and (36). An interface with the followingproperties is considered (Tn = Ts = 6000 MPa/m; Sn = 60 MPa;Ss = 90 MPa; GIc = 1.0 N m/m2; GIIc = 3.0 N m/m2); frictional dissipa-tion is neglected. The initiation criteria selected are the quadratic(fquad) and BK (fBK) stress interaction given by:

fquadðtn; tsÞ ¼ ðtn=SnÞ2 þ ðts=SsÞ2 � 1 ¼ 0 ð48Þ

fBKðtn; tsÞ ¼ ðtnÞ2 þ ðtsÞ2 � S2n � S2

s � S2n

� �kg ð49Þ

Two types of failure surfaces with widespread use are the Ben-zeggagh–Kenane criterion (gBK) [26], and the linear/quadraticpower-law criteria (glin/gquad) [28]:

gBKðGI;GIIÞ ¼ GI þ GII � ðGIIc � GIcÞ½GII=ðGI þ GIIÞ�g ð50Þglin=quadðGI;GIIÞ ¼ ðGI=GIcÞh þ ðGII=GIIcÞh � 1; hlin ¼ 1; hquad ¼ 2 ð51Þ

First, both initiation and fracture criteria are imposed onto themodel, yielding results shown in Fig. 9. Then, a more realistic sce-nario, where only the fracture criterion is known, is solved. Theadditional constraint b(k) = c(k) is imposed in order to obtain aunique solution (Table 2).

Interestingly, the combination that fails to find solutions for band c, for k > 0.5, is the rather common [quadratic stress interac-tion/ quadratic failure locus]. This may indicate that the model isstill not flexible enough or, alternatively, that such combinationsof criteria are not realistic. In order to gain physically-based knowl-edge with respect to admissible combinations of initiation and fi-nal failure criteria, an experimental investigation of the stressstate at the initiation of mixed-mode delamination would beneeded, but such a study is beyond the scope of the present paper.

The strength and fracture locus presented in Table 2 are themodel’s prediction; as expected, the latter coincides with the im-posed criterion. The results presented in Section 4, as well as theyield surface in stress space presented in Fig. 2, are based on theassumptions of b(k) = c(k) and the B-K final fracture criterion.

Appendix B. Useful derivatives

The derivatives shown here are required for the computation ofthe plastic multiplier, dk, Eq. (44). The index r is used as a genericnotation for either the normal (n) or shear (s) mode.

Mode ratio

@k@tn¼ �2tn

Tn

ðtsÞ2=Ts

ðtnÞ2=Tn þ ðtsÞ2=Ts

h i2 ;

@k@ts¼ 2ts

Ts

ðtnÞ2=Tn

ðtnÞ2=Tn þ ðtsÞ2=Ts

h i2 ;

Tractions and dissipative force

@tr

@D¼ �tr

ð1� DÞ ;@tr

@upr¼ �ð1� DÞTr ð53Þ

v@tr¼ tr

ð1� DÞ2Tr

;@v@D¼ 1

ð1� DÞ3t2

n

Tnþ t2

s

Ts

� �ð54Þ

Damage function

@F@tr¼ @k@tr

@F@kþ @F@b

@b@kþ @F@c

@c@k

� �ð55Þ

@F@k¼ bkb�1 Cs

ð1� Dþ AsÞ2� cð1� kÞc�1 Cn

ð1� Dþ AnÞ2ð56Þ

@F@D¼ kb Cs

ð1� Dþ AsÞ3þ ð1� kÞc Cn

ð1� Dþ AnÞ3ð57Þ

@F@b¼ Cs

ð1� Dþ AsÞ2

lnðkÞkb;

@F@c¼ Cn

ð1� Dþ AnÞ2lnð1� kÞð1� kÞc ð58Þ

The derivatives @b/@k and @c/@k are found numerically (Appen-dix A):

@b@k¼ ½bðkþ DkÞ � bðkÞ�=Dk;

@c@k¼ ½cðkþ DkÞ � cðkÞ�=Dk ð59Þ

In the above expressions, Ar and Cr are given, from (32) and (33),as:

Ar ¼ S2r = 2GrcTrcos2ar � S2

r

h i; Cr ¼ S2

r =ð2TrÞð1þ ArÞ2 ð60Þ

The derivatives of the yield function with respect to the trac-tions and the damage are given by:

@y@tr¼ @v@tr

1F� @F@tr

1F2 ;

@y@D¼ @v@D

1F� v @F

@D1F2 ð61Þ

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