1. IntroductionFiber reinforced thermoplastic composite laminatesare in high demand in advanced aerospace struc-tures owing to their light weight, high stiffness,high strength, good toughness, long durability,superior impact resistance and favorable damage-tolerance properties. Yet, despite many advantagescompared to their metal counterparts, these lami-nates are not exempt from deterioration and dam-age, especially in the presence of a stress concentra-tion. Of particular interest here is to study a com-mon type of thermoplastic composite known asAS4/PEEK laminate composed of poly-ether-ether-ketone (PEEK) reinforced by AS4 carbon fiber. Weseek to predict the strength of this type of laminatein the presence of stress concentration in the formof a hole or a notch in pinned- and bolt-joints. Thisinformation is useful at the design stage.

It is appropriate to begin with a brief survey of theopen literature since numerous investigations dedi-cated to the prediction of progressive damage andstrength of the laminates with stress concentrationsalready exist. For example, Chang et al. [1] studiedthe progressive damage for T300/976 graphite epoxycontaining a hole under tensile loading using themodified Hashin [2] failure criteria. Their approachwas to reduce the material properties to zero whenfailure criteria is satisfied by implementing the so-called ‘sudden degradation rule’. Lessard andShokrieh [3] modified Hashin failure criteria bytaking into account the non-linear characteristic ofmaterial as well as implementing the sudden mate-rial degradation rule. Dano et al. [4] investigated thebearing strength of glass/epoxy pinned-joint lami-nates using the commercial software ABAQUS™.Later, Icten and Karakuzu [5] investigated different

383

Stochastic analysis of inter- and intra-laminar damage innotched PEEK laminatesM. Naderi, M. M. Khonsari*

Department of Mechanical Engineering, Louisiana State University, LA 70803 Baton Rouge, USA

Received 21 October 2012; accepted in revised form 13 January 2013

Abstract. This paper presents a finite element model to predict the progressive damage mechanisms in open-hole PEEK(Poly-Ether-Ether-Ketone) laminates. The stochastic laminate’s properties with non-uniform stress distribution are consid-ered from element to element using the Gaussian distribution function. The failure modes considered are: fiber tension/compression, matrix tension/compression, fiber/matrix shear, and delamination damage. The onset of damage initiation andpropagation are predicted and compared with three different failure criteria: strain-based damage criterion, Hashin-baseddegradation approach and stress-based damage criterion which is proposed by the authors of the present work. The inter-laminar damage modes associated with fiber/matrix shearing and delamination are modeled using the cohesive elementstechnique in ABAQUS™ with fracture energy evolution law. Mesh sensitivity and the effect of various viscous regulariza-tion factors are investigated. Damage propagation and failure path are examined by re-running the program for severaltimes.

Keywords: polymer composites, modeling and simulation, damage mechanism, cohesive element, stochastic material prop-erties

eXPRESS Polymer Letters Vol.7, No.4 (2013) 383–395Available online at www.expresspolymlett.comDOI: 10.3144/expresspolymlett.2013.35

*Corresponding author, e-mail: khonsari@me.lsu.edu© BME-PT

failure modes and bearing strength in woven car-bon/epoxy laminates using the failure criteria pro-posed by Hoffman [6] and Hashin [2] both numeri-cally and experimentally. Maa and Cheng [7] devel-oped a failure model using the continuum damagemechanics (CDM)-based failure model using elas-tic-plastic constitutive equations implemented inABAQUS. Also reported was their experimentaldata for AS4/PEEK laminates containing a circularhole. In a similar way, Ding et al. [8] carried out athree-dimensional finite element analysis of anopen-hole thermoplastic AS4/PEEK laminate tointerpret the results of their experimental observa-tions.To consider delamination and inter-laminar damage,Lapczyk and Hurtado [9] developed an anisotropicdamage model for predicting failure and post-fail-ure behavior in fiber reinforced materials based onthe concept of fracture energy dissipation. Theyaddressed the convergence of the numerical modelin the softening regime by introducing a viscousregularization factor in the computations. Recently,Falzon and Apruzzese [10] carried out a three-dimensional CDM-based model in ABAQUS/Explicit to simulate the intra-laminar degradation offiber-reinforced laminates based on ply failuremechanisms. Their focus was on the non-linearresponse of the shear failure mode and the interac-tion with other failure modes.Recent literature in mechanic-based understandingand modeling of progressive damage analysis andprediction of strength of composite laminates spe-cially notched laminates in composite researchcommunity contains many noteworthy studies suchas the works reported by Camanho et al. [11], Abis-set et al. [12], van der Meer et al. [13] and Fang etal. [14], Daghia and Ladeveze [15]. Among them,Camanho et al. [11] examined a continuum damagemodel to predict the strength and size effects ofnotched carbon–epoxy laminates. They experimen-tally and analytically studied the effects of size andthe development of a fracture process zone beforefinal failure. Abisset et al. [12] investigated pro-gressive degradation in an open-hole IM7/8552 car-bon/epoxy laminate using a damage mesomodeldeveloped based on a micromechanical approach.They compared experimental and numerical resultsconsidering the ply’s thickness and the in-planescaling effect to capture the change in failure mode

and the effect of the specimen’s scale on tensilestrength. Van der Meer et al. [13] simulated pro-gressive failure, matrix cracking, interface elementsfor delamination, and a continuum damage modelfor fiber failure using phantom-node computationalmethod. They validated their computational frame-work against experimental observations for open-hole tests and compact tension tests. Fang et al. [14]presented a new augmented finite element method(A-FEM) which can account for arbitrary crackpath, and different intra-element discontinuities.They showed that within their new formulation oneis able to derive an explicit and fully condensed ele-mental equilibrium equations using augmented ele-ments without additional external nodes or degreeof freedom.A survey of the published works clearly reveals thatwhile substantial progress has been made especiallyin computational simulation of composites failure,predicting their progressive failure and arbitrarycrack path considering randomness in materialproperties distribution remain a challenging taskdue to the complexity of the interactions amongmultiple damage processes. While some computa-tional methods, i.e. phantom-node method [16],extended-finite element method (X-FEM) [17] andaugmented-finite element method (A-FEM) [18,19] to simulate progressive arbitrary failure in com-posite materials can yield more accurate resultsthan numerical methods considering reducing thematerial properties to zero at the time of failure andturning the element’s load carrying capacity off, theobjective of the present work is to investigate theprogressive degradation, delamination behavior, andstrength of an open-hole AS4/PEEK laminate withthe stochastic distribution of material propertiesusing a three-dimensional numerical simulation.The cohesive elements technique proposed byCamanho and Dávila [20] are considered betweentwo adjacent layers to model the failure due todelamination. The progressive intra-laminar dam-age model based on the sudden degradation rule iscompared to the other approaches with strain-basedcontinuum damage formulation proposed by Lindeet al. [21]. The procedure of the progressive inter-and intra-laminar damage evolution is implementedin commercial software ABAQUS™ [22] usingprogramming in UMAT (user-defined material) sub-routine.

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2. Progressive damage modelsIn order to establish a progressive damage analysis,a set of failure criteria and material property degra-dation rules are required. The failure criteria mustproperly take into account the damage mechanismssuch as fiber breakage, matrix cracking, fiber/matrixshearing and delamination. In the following sections,different failure methods used in the current workare explained in details. Two methods (Method I andMethod II) are according to the strain and stress-based continuum damage mechanics formulation.These methods consider the gradual material prop-erty degradation controlled by the individual frac-ture energies in both the fiber and the matrix. Theonset of damage initiation in Method II is based onthe Hashin failure criteria while the one in Method Iis based on an exponential damage initiation law (astrain-based evolution approach) developed byLinde et al. [21]. However, the damage evolution inMethod II is stress-based and incorporates a modi-fication of Method I proposed by the authors of thepresent work. The third method (Method III) is basedon Hashin failure criteria [2] and takes into accountthe effect of sudden material degradation rule.Inter-laminar damage or delamination in Methods Iand II are according to cohesive zone techniquewhile Method III uses the Hashin failure criteria.The details of these methods are described in thefollowing sections.

2.1. Method I2.1.1. Intra-laminar damage modelThe failure criteria of Method I is a strain-basedcontinuum damage formulation with different fail-ure criteria for matrix and fiber [21]. It takes intoaccount the gradual degradation of the materialcontrolled by the individual fracture energies ofmatrix and fiber. Matrix failure initiates if the fail-ure index, MF, defined below in Equation (1),exceeds the transverse tensile failure strain, !T

t. :

(1)where !T

c. and !s represent the transverse compres-sive failure strain and the shear failure strain,respectively. !22 and !12 are the strain componentsperpendicular to fiber direction and shear direction,respectively. The matrix failure strains can be

obtained from the following expressions shown asEquation (2):

;! ;! (2)

where E22 and E12 are the undamaged laminate’stransverse and shear stiffness, respectively. Yt and Ycrepresent transverse tensile and compressive failurestrength, respectively. The shear failure strength isS12.Damage evolution variable, dm, is a function of thestrain at failure, undamaged material stiffness, andthe current value of the failure initiation variable.Also considered in the damage variable calculationsare the fiber and matrix fracture energies (Gf and Gm)and the element characteristic length (Lc) to reducethe mesh sensitivity of the numerical model. Thematrix damage, dm, takes place in the direction per-pendicular to the fibers, and the fiber damage, df,occurs in the direction parallel to the fibers. Thematrix damage evolution parameter, dm, is calcu-lated by Equation (3) [21]:

(3)

where C22 is the stiffness matrix component in thedirection perpendicular to that of the fibers.The fiber failure, FF, initiates if the following crite-rion defined by Equation (4) [21]:

(4)

where !Lc. and !L

c. are longitudinal tensile and com-pressive failure strain, respectively and !11 is thelongitudinal strain component.The fiber failure strains can be obtained from Equa-tion (5):

,! (5)

where E11 is the initial laminate’s longitudinal stiff-ness. Xt and Xc represent longitudinal tensile andcompressive failure strength, respectively.Once Equation (4) is satisfied, the fiber damageparameter df evolves as shown in Equation (6) [21]:

(6)

where C11 is the component of stiffness matrix.

df 5 1 2eL

t

FFe3C11 e

tL1FF2e t

L2Lc >Gf4

eLc 5

Xc

E11

eLt 5

Xt

E11

FF25 ceLt

eLc 1e112 21aeL

t21e112 2eL

c b e11d71eLt 2 2

dm 5 1 2eT

t

MFe3C22 eT

t 1MF2eTt 2Lc>Gm4

es 5S12

E12

eTc 5

Yc

E22

eTt 5

Yt

E22

MF25 ceTteT

c 1e222 21aeTt21e222 2eT

c b e221aeTt

esb 21e122 2d71eT

t 2 2MF25 ceTteT

c 1e222 21aeTt21e222 2eT

c b e221aeTt

esb 21e122 2d71eT

t 2 2

eTt 5

Yt

E22

eTc 5

Yc

E22

es 5S12

E12

dm 5 1 2eT

t

MFe3C22 eT

t 1MF2eTt 2Lc>Gm4

FF25 ceLt

eLc 1e112 21aeL

t21e112 2eL

c b e11d71eLt 2 2

eLt 5

Xt

E11

eLc 5

Xc

E11

df 5 1 2eL

t

FFe3C11 e

tL1FF2e t

L2Lc >Gf4

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Once damage initiates in the laminate, the lami-nates’ stiffness degrades. Considering fiber andmatrix damage evolution and transverse isotropy,

the laminate’s stiffness matrix is written as shownby Equation (7) [21]:

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(7)Cd5 c 112df 2C11 112df 2 112dm 2C12 112df 2C13 0 0 0112dm 2C22 112df 2 112dm 2C23 0 0 0

C33 0 0 0112dm 2 112dm 2C44 0 0

C55 0symmetry C66

¥Cd5 c 112df 2C11 112df 2 112dm 2C12 112df 2C13 0 0 0112dm 2C22 112df 2 112dm 2C23 0 0 0

C33 0 0 0112dm 2 112dm 2C44 0 0

C55 0symmetry C66

¥

where

; ; C33 = C22;

; ;

; C44 = E23; C55 = E13;

C66 = E12;!

where "ij (i, j = 1, 2, 3) is the Poisson ratio.The damage growth reduces the stiffness and conse-quently the stress tends to redistribute in each ele-ment of the material.

2.1.2. Inter-laminar damage (delamination)model

Inter-laminar damage is simulated by placing cohe-sive elements between two adjacent layers. Theconstitutive response of the cohesive elements isdescribed by a linear traction-separation law as pre-sented in Figure 1. Figure 1 shows the schematic ofmaterial response in normal, shear and mixed modeloading. The triangles in the two vertical coordinateplanes ((#, $I) and (#, $Shear) planes) represent theresponse under pure normal and pure shear defor-mation, respectively. The intermediate vertical plane((#, $m) plane) represents the damage response undermixed mode condition. This figure shows that thematerial response is linear until it reaches a peakvalue of stress after which it softens linearly in thepost-peak region. The initiation under differentmodes of pure normal ($0

I), pure shear ($0Shear) and

mixed mode loading ($0m) occurs when the inter-

laminar stress reaches the inter-laminar pure normal

strength (#I), pure shear strength (#Shear), or mixedmode strength (#m) depending on the mode of theloading. The growth regime represents the softeningand separation behavior of the degraded material’selement. The material’s element is completely delam-inated once the area under traction-displacement(Figure 1) reaches the fracture energy, Gc, and thefailure displacement, #f

m, is reached. The cohesiveelements and their constitutive responses are pro-grammed in ABAQUS™ for the prediction of delam-ination initiation and propagation. A high initialstiffness (the so-called penalty stiffness; Kp) isdefined to avoid de-cohesion of two surfaces in thelinear elastic range.

Delamination initiation:The onset of delamination initiation is determinedbased on a quadratic nominal strain criterion [22].The strength of the adhesive in the normal and sheardirections are used as input data, as shown by Equa-tion (8):

D 51 2 n23

2 2 n13n31 2 2n21n32n13

E11E22E33

C23 5n23 1 n21n13

E22E11D

C13 5n13 1 n23n12

E22E11DC12 5

n12 1 n32n13

E22E11D

C22 51 2 n13n31

E22E11DC11 5

1 2 n232

E22E33D

C12 5n12 1 n32n13

E22E11DC13 5

n13 1 n23n12

E22E11D

C23 5n23 1 n21n13

E22E11D

D 51 2 n23

2 2 n13n31 2 2n21n32n13

E11E22E33

C22 51 2 n13n31

E22E11DC11 5

1 2 n232

E22E33D

Figure 1. Schematic of bilinear constitutive behavior ofcohesive zone for different modes of failure

with! ;

(8)

where !fn, !fs are inter-laminar tensile and shear strain.!s is tensile strain component, !s and !t are shearstrain component. Parameter N is the inter-laminartensile strength; S and T represent the inter-laminarshear strength. It is noted that for convenience, weconsidered a constitutive thickness of 1.0 mm suchthat there is no need to distinguish between thenominal strain and the separation displacement inthe above equation.

Delamination growth:The damage evolution or delamination propagationlaw postulates that the material stiffness is degradedonce the corresponding initiation criterion is met.The damage propagation prediction is usually stud-ied in terms of energy release rate and fracture tough-ness. Mixed-mode delamination growth is pre-dicted when mixed mode fracture energy (GT) isgreater than critical fracture energy (GC). Severallaws implemented in FEM codes have been consid-ered to compute the fracture toughness for mixed-mode failure, such as Benzeggagh-Kenane (B-K)criterion [23] the power law criterion [24], and theReeder criterion [25]. Demonstration of applica-tions of B-K criterion can be found in the work ofCamanho et al. [26] for PEEK and epoxy compos-ites. In this study, the ‘power law criterion’ proposedby Benzeggagh-Kenane – the so-called B-K crite-rion – [23] is used. This criterion is established interms of an interaction between the energy releaserates to predict the delamination evolution as shownby Equation (9):

, with

T = GI + GII + GIII (9)

where % is a material parameter and GIc and GIIc arethe fracture energies of Mode I and II at failure,respectively. Gc is critical fracture energy. GI, GIIand GIII are fracture energies of Mode I, II, and III.

2.2. Method IIMethod II is the modified of Method I and is a stress-based continuum damage formulation with the fiberand matrix failure proposed by the present authors.The main difference between Methods I and II is inthe definition of matrix and fiber failure indices(MF and FF) in which the former uses the strain-based approach and the latter II uses the stress-based approach. Damage initiation is based on theHashin failure criteria summarized in Table 1.As summarized in Table 1, seven sets of initiationcriteria are considered. For detecting fiber, matrixand delamination some indices are defined, i.e., MFfor matrix failure in either tension or compression,FF for fiber failure in either tension or compression,DELT delamination failure in tension and DELC fordelamination failure in compression. Once each ofthe indices reaches the unit value, failure is initiatedand softening process in the material begins.The matrix damage evolution parameter dm is themodified version of Equation (3) and based on stressapproach described by Equation (10) (Please seeSection 2.1. for more details):

(10)

The fiber damage evolution df is obtained usingEquation (11):

(11)

The procedure for evaluation of delamination orinter-laminar damage is the same as the delamina-tion method described in Section 2.1.2. (Please seeSection 2.1 for more details).

2.3. Method IIIMethod III uses the Hashin failure criteria as sum-marized in Table 1. This method is used for identify-ing matrix tensile/compressive cracking, fiber ten-sile/compressive breakage, fiber/matrix shear failure,and tension/compression delamination. If failurecriterion is satisfied in an element, the materialproperties of that failed element is changed by a setof material properties in accordance with the sud-den degradation rule. Corresponding to the type ofthe failure, the laminate response against the load ischanged in the case of the damage. The sudden stiff-

df 5 1 2Xt

E11FFecXt aFF2

Xt

E11

bLc>Gfd

dm 5 1 2Yt

E22MFecYt aMF2

Yte22

bLc>Gmd

GIc 1 1GIIc 2 GIc 2 aGII 1 GIII

GT

b h 5 Gc

esf 5

SKp

5TKp

enf 5

NKp

a en

enf b

2

1 a es

esf b

2

1 a et

esf b

2

7 1a en

enf b

2

1 a es

esf b

2

1 a et

esf b

2

7 1 enf 5

NKp

esf 5

SKp

5TKp

GIc 1 1GIIc 2 GIc 2 aGII 1 GIII

GT

b h 5 Gc

dm 5 1 2Yt

E22MFecYt aMF2

Yte22

bLc>Gmd

df 5 1 2Xt

E11FFecXt aFF2

Xt

E11

bLc>Gfd

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ness degradation of the failed element in the matrixand fibers is different. For example, once the matrixfails in tension, the transverse stiffness, E22, dropsto 0.2E22 while at the onset of the fiber failure thelongitudinal stiffness, E11, reduces to 0.07E11. There-fore, in the current work, the material degradationrules are based on the work of [27, 28] for matrix ten-sile/compressive failure and for fiber tensile/com-pressive failure (See Table 1). Note that fiber/matrixshear and delamination failure were not consideredin the work of [27, 28] and therefore material prop-erty is reduced to a small value for these damagemodes.

3. Model developmentTo perform stress and progressive damage analysisbased on Methods I, II, III, a three-dimensional finiteelement model is created in ABAQUS™ [22]. Oneend of the laminate is clamped in x, y, and z and theother end is subjected to the uniform displacementof v (Figure 2a). ABAQUS 6-node linear triangularprism (C3D6) is defined to mesh the composite lam-inate such that each ply has one element through thethickness direction. ABAQUS 6-node, three-dimen-sional cohesive element (COH3D6) is defined tomesh the cohesive layers with 0.001 mm thickness.Two types of laminates configuration are consid-ered in the model. The first one is [0/60/–45/+45]3sdescribed in details in Section 5. The second one is

[0/45/90/–45]2s for which experimental data andnumerical results are available in [7]. The latter ischosen to verify the results of the presented simula-tion; See Table 2. In each case, because of the sym-metry condition, only one quarter of the specimen ismodeled. That is, the model is in the shape of thespecimen is cut in half in XZ plane and then cut inhalf in YZ plane. Boundary conditions are shown inFigure 2a.

3.1. Stochastic material propertiesdistribution

The laminate’s stiffness and strength are generatedusing the Gaussian distribution function with around±1% of variation to consider spatial stochastic prop-erties in the model. A typical Gaussian distributionof the AS4/PEEK longitudinal stiffness is shown inFigure 2b. Gaussian distribution function is definedin user-subroutine SDVINI in ABAQUS whichenables the user to specify the initial solution-depen-dent state variables with stochastic material proper-ties.

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Table 1. Hashin failure criteria and material degradation rules [2, 3, 15]

Failure mode Failure index Failure approach Material degradation rule (for Method III)

Matrix tensile failure(#22 >"0) Method II and III E22 #0.2E22, E12 # 0.2E12,

E23#0.2E23

Matrix compression failure(#22 <"0) Method II and III E22 # 0.4E22, E12 # 0.4E12,

E23# 0.4E23

Fiber tensile failure(#11 >"0) Method II and III E11 # 0.07E11

Fiber compression failure(#11 <"0) Method II and III E11#0.14E11

Fiber/matrix shear failure Method III E11 # 0

Inter-laminar tensile failure(#33 >"0) Method III E33, E23, E13, # 0

Inter-laminar compression failure(#33 >"0) Method III E33, E23, E13, # 0

MF25 as22

Yt

b 2

1 as12

S12

b 2

1 as23

S23

b 2

7 1MF25 as22

Yt

b 2

1 as12

S12

b 2

1 as23

S23

b 2

7 1

MF25 ,as22

Yc

b 2

1 as12

S12

b 2

1 as23

S23

b 2

7 1MF25 ,as22

Yc

b 2

1 as12

S12

b 2

1 as23

S23

b 2

7 1

FF25 as11

Xt

b 2

1 as12

S12

b 2

1 as13

S13

b 2

7 1FF25 as11

Xt

b 2

1 as12

S12

b 2

1 as13

S13

b 2

7 1

FF2 5 as11

Xc

b 2

7 1FF2 5 as11

Xc

b 2

7 1

FF25 as11

Xc

b 2

1 as12

S12

b 2

1 as13

S13

b 2

7 1FF25 as11

Xc

b 2

1 as12

S12

b 2

1 as13

S13

b 2

7 1

DELT25 as33

Zt

b 2

1 as13

S13

b 2

1 as23

S23

b 2

7 1DELT25 as33

Zt

b 2

1 as13

S13

b 2

1 as23

S23

b 2

7 1

DELC25 as33

Zc

b 2

1 as13

S13

b 2

1 as23

S23

b 2

7 1DELC25 as33

Zc

b 2

1 as13

S13

b 2

1 as23

S23

b 2

7 1

Table 2. Mesh specifications of the model

Number ofelements

Average aspectratio

Mesh 1 8 937 2.45Mesh 2 24 854 1.7Mesh 3 54 374 1.4

3.2. Numerical procedureThe progressive damage model is implemented inABAQUS™ through the user-defined subroutineUMAT to describe the specific material characteris-tics. This subroutine is called to determine the mate-rial properties at each point. The stresses and solu-tion-dependent state variables are updated at the endof each iteration and the Jacobian matrix is recalcu-lated accordingly. The simulation starts with themodel preparation as presented in the flowchart ofthe procedure; See Figure 3. Then, a Gaussian distri-bution for the laminate properties is applied to all theelements. Stress analysis is then performed basedon the applied displacement followed by the failureanalysis. The values of the failure indexes are storedas the solution-dependent state variables. If the dam-age initiation criterion is satisfied, damage evolu-tion laws are applied for Methods I and II. InMethod III, if the damage initiation criterion is met,the material properties are treated according to thesudden material degradation rule. If there is no fail-ure, an incremental displacement, $v, is added to thelast displacement amplitude, vi. Next, a new Jaco-bian is computed, the stress is redistributed accord-ingly, and the calculations are repeated. The failureoccurs when the laminate cannot tolerate any moreload increment. Physically, this represents a large

deflection and excessive damage at which point theprogram is terminated. Program is stopped at thisstage and simulation cannot pass any new incre-ment. It is noted that the case in which there is nofailure does not have any inconsistency with equilib-rium solution and new increment successfully run.

4. Sensitivity analysis parametersSensitivity studies are performed to examine theinfluence of various parameters on the load-dis-placement curves of the progressive damage mod-els. The parameters of particular interest pertain tomesh density and viscous regularization (&).

Mesh dependency:The problem of strain localization induced by dam-age localization is treated by a method that mini-mizes the mesh dependency in the numerical results.This is done by using the fracture energy-baseddamage evolution and considering the characteristiclength of the element into the damage evolutionlaw. The calculation of the characteristic length with-out considering the crack direction depends on theelement geometry and formulations. Hence, the prob-lem of mesh dependency still exists [13]. In order tominimize the problem of mesh dependency, ele-ments’ aspect ratio (the ratio of shortest edge to the

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Figure 2. (a) 3D Finite element mesh and boundary conditions, (b) Gaussian distribution of longitudinal stiffness

longest edge of an element) should be close to one.The results of three different mesh refinements aresummarized in Table 2.

Viscous regularization factor (!):Material models that exhibits softening behaviorand stiffness degradation often tend to have severeconvergence difficulties. A common technique toalleviate the associated convergence difficulties isto implement the so-called viscous regularizationfactor which through the introduction of a viscosityterm in the damage evolution of Equations (10) and(11) forces the tangent stiffness matrix of the soft-ening material to be positive definite for adequatelysmall time increments [9, 22]. It is important tocheck that the energy associated with viscous regu-larization is small compared to the overall strain

energy. The time evolution equation of viscousdamage variable is as defined by Equations (12)and (13) [9, 22]:

(12)

(13)

where dvm and dv

f are the regularized matrix and fiberdamage variables used in the calculation of damagestiffness matrix and Jacobian matrix, respectively.

5. Material and experimentThe material of the present work is an AS4/PEEKquasi-isotropic laminate [0/90/–45/+45]3s with poly-ether-ether-ketone matrix and carbon fiber and fab-ricated using the autoclave method from the prepreg(APC-2). The volume fraction of the fibers is 60%according to the data provided by the manufacturer.The specimens are prepared according to ASTMD3039M with a centrally circular hole. The speci-men dimensions and mechanical properties ofAS4/PEEK laminate are summarized in Table 3 andTable 4, respectively. As presented in Table 3, L isthe length of specimen, W is the width and D is thehole diameter. Fracture properties are summarizedin Table 5 [29] and the values of fracture energies ofthe fiber and matrix in Table 5 are from [9]. Statictests were performed on a universal fatigue test

d?fv 51m1df 2 df

v 2

d?mv 51m1dm 2 dm

v 2d?mv 51m1dm 2 dm

v 2

d?fv 51m1df 2 df

v 2

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Figure 3. Damage flowchart implemented in ABAQUS™ Table 3. The laminate dimensions

Table 4. Material properties of AS4/PEEK

Laminateconfiguration

L[mm]

W[mm]

D[mm]

Thicknessof each layer

[mm][0/90/–45/+45]3s 254 25.4 6.35 0.139[0/+45/90/-45]2s 100 20 5 0.125

Material properties AS4/PEEKE11 [GPa]E22 = E33 [GPa]E12 = E13 [GPa]E23'12 = '13'23

138 10.2 5.7 3.7 0.3 0.45

Xt [MPa]Xc [MPa]Yt = Zt [MPa]Yc = Zc [MPa]S12 = S13 [MPa]S23 [MPa]

20701360

8623018686

machine (TESTRESOURCES Model 930LX50-T2000).

6. Results and discussionsNumerical results of the progressive damage analy-sis of two quasi-isotropic open-hole specimens withthe laminates’ configuration of [0/90/–45/+45]3sand [0/45/90/–45]2s are presented in this section.First, degradation and failure analysis of the lami-nate with the configuration of [0/45/90/–45]2s isperformed using the three different methods andcompared to the numerical and experimental resultsof [7] to validate the model. Also presented are theresults of the dependency of the present model toviscous regularization parameter and a mesh refine-ment study. Second, the predicted failure strengthand damage of the laminate with the configurationof [0/90/–45/+45]3s is compared to the experimen-tal results.

6.1. [0/45/90/–45]2s laminateFigure 4 shows the comparison of load-displace-ment curves of different progressive damageapproaches (Method I, II, and III) with the experi-mental and numerical data of [7] for [0/45/90/–45]2sAS4/PEEK laminate. It shows that once the damageaccumulation increases and the laminate is not ableto sustain more load, the strength reduces andnumerical simulation is stopped due to excessiveelement distortion. Three regions can be seen in

these curves: a linear response, softening trend rep-resenting the initiation and propagation of the dam-age such as fiber and matrix failure when damageoccurs in critical number of elements, followed by aregion in which the load capacity drops. Comparedto the experimental and numerical data, the predic-tion of the current numerical simulations are quiteclose.Mesh size study is performed to evaluate the effectof mesh density on the prediction of failure strength.Three different meshes ranges from a coarse(Mesh I), medium (Mesh II) to a fine (Mesh III) forthe area around the hole are shown in Figure 5. Theviscous regularization factor of 0.003 is used withthe degradation analysis of Method I. It can be seenthat for Mesh I the abrupt decrease of the ultimatestrength is slightly greater than the experimentalresult and that it overestimates the failure strengthby about 3 percent. By refining the mesh and reduc-ing the aspect ratio of the meshes to close to 1, theresults of Mesh II and III predictions show closeagreement with those obtained experimentally.Although the characteristic length is considered inthe damage analysis to minimize the mesh sizeeffect, a slight dependency to the mesh size stillexists in the numerical results.Figure 6 gives the load-displacement results of dif-ferent values of the viscosity parameter in Methods Iand II. It can be seen that the smaller the viscousregularization factor, the more abrupt the failureand the smaller failure strength become. Since two

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Table 5. Fracture properties of AS4/PEEK laminateGIc

[N/mm]GIIc = GIIIc

[N/mm]N

[MPa]S = T[MPa]

Gf[N/mm]

Gm[N/mm] ! Kp

[MPa/mm]1.7 2 80 100 12.5 1.0 2.284 106

Figure 4. Comparison of predicted load-displacementcurves of different progressive damage methodswith the experimental and numerical data of [7]for [0/45/90/–45]2s AS4/PEEK laminate

Figure 5. Comparison of load-displacement curves of dif-ferent mesh sizes using Method I with the experi-mental data of [7] for [0/45/90/–45]2s AS4/PEEKlaminate

approaches use different failure criteria, two differ-ent viscosity parameters are considered in the simu-lations. The intra-laminar damage initiation basedon stress criteria seems to have higher viscosity fac-tor than that of with strain criteria.

6.2. [0/90/–45/+45]3s laminateThe comparison of load-displacement curves of dif-ferent progressive damage approaches (Method I,II, and III) with the experiment for [0/90/–45/+45]3sAS4/PEEK laminate is presented in Figure 7. Thesame viscous regularization factor as in the case of[0/90/–45/+45]3s laminate is used for this simula-tion. The simulation results are consistent with theexperimental results.Figure 8 shows inter-laminar or delamination dam-age pattern at 0/90°, 90/–45°, and –45/45° interfaceof [0/90/–45/+45]3s AS4/PEEK laminate at the timewhen the laminate strength start to drop. As shownin this figure, the delamination at 90/–45° and –45/45° interfaces are more severe than the delami-

nation at 0/90° interface. Since the dominant failurestress in 0/90° interface is the normal stress, delam-ination pattern is in the zero degree direction. In thecase of 90/–45° and –45/45° interface, the delamina-tion patterns tend toward the 45° failure direction.A contour plot of fiber and matrix damage variablesdf and dm defined in Equations (11) and (12) as wellas the surfaces of the failed specimen for [0/90/–45/+45]3s AS4/PEEK laminate is shown in Fig-ure 9. Figure 9 also shows how implementation ofthe stochastic in material properties changes thefailure behavior and path of the laminate. As seen inFigures 9a–9c the intra-laminar damage of the 0°fibers which are the most catastrophic failure typeswithin the laminate initiates at the tip of the holeand propagates in the direction perpendicular to theload direction. In order to show the randomnessbehavior of failure due to probable distribution ofthe material properties, the scatter of the 0° fiberfailure path is obtained by re-running the model forthree times (paths 1, 2, 3). In each run, the failurepath differs in the direction perpendicular to thefiber directions. Comparing Figures 9a–9c with the

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Figure 6. Comparison of predicted load-displacement curvesof different values of viscosity parameters usingMethod I and II with the experimental data of [7]for [0/45/90/–45]2s AS4/PEEK laminate

Figure 7. Comparison of predicted load-displacementcurves of different progressive damage methodswith the current experimental data of [7] for[0/90/–45/+45]3s AS4/PEEK laminate

Figure 8. Delamination failure at the interface of 0/90° (a), 90/–45° (b), and –45/45° (c) for [0/90/–45/+45]3s AS4/PEEKlaminate using Method I

image of failed specimen (Figure 9d), the resultsindicate that the failure of 0° plies can be either inthe direction perpendicular to the fibers or in theinclined direction depending on sub-critical damagedevelopment in the adjacent plies. The numericalfailure paths in Figures 9a–9c are consistent withthe image of the failed laminate. Figures 9e–9f show

the damage patterns for matrix failure in 90 and 45°plies, respectively. The matrix strength, Yt, is smallcompared to the fiber strength, Xt, and matrix fail-ure is the sever type of the damage in 90, 45, and –45° plies. As seen in Figures 9e–9f, the matrix fail-ure path of 90 and 45° plies is in the 0 and 45 direc-tions, respectively (path 4, 5).

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Figure 9. Intra-laminar damage patterns of fiber and matrix for [0/90/–45/+45]3s AS4/PEEK laminate at the time of abruptstrength reduction. a, b, c) fiber damage image in 0° layer for different simulation runs. d) an open-hole failedspecimen under static test. e) matrix damage pattern in 90° layer. f) matrix damage pattern in 45° layer.

Although the presented simulation results of pro-gressive damage obtained from different methodsare in agreement with experiments, several pointsabout the comparison of different methods wouldbe in order. Methods I and II have the advantage ofmodeling delamination, fiber and matrix failureusing a damage parameter and embedded cohesiveelements in the model. However, their computationaltime is more expensive and numerical convergenceis more problematic than those of Method III.Method III uses the sudden material property degra-dation rule which may raises questions and difficul-ties in validation of numerical results for complexgeometry and laminate. However, from engineeringpoint of view, Method III is fast and may give arealistic approximation of the laminate’s strength.All of these methods depend on mesh type, meshnumber, viscous regularization factor, and materialproperty degradation rule. Future works consideringthe above-mentioned restrictions along with morerecent computational methods such as X-FEM andA-FEM with considerations of randomness in thedistribution of material properties are necessary totackle the design of open-hole or notched laminatesin a more realistic manner. Further, numerical resultsshow how the numerical results are sensitive tomesh type, viscous regularization factor and distri-bution of material properties. Questions such as howshould the variation of material properties be chosenand how sensitive are the mesh type and size to arandom function of material distribution can beanswered once a realistic material morphology isavailable. Future research is needed to study theeffect of material property distribution functionsalong with a realistic characterization of the lami-nate morphology that can be implemented in a robustcomputational method such as X-FEM or A-FEM.Capability to model arbitrary crack path and dis-crete damage in a composite laminate are thusneeded.

7. ConclusionsThe present paper presents a study of the progres-sive intra- and inter-laminar damage coupled withcohesive elements for the failure simulation of twoquasi-isotropic open-hole AS4/PEEK laminateswith [0/90/–45/+45]3s and [0/45/90/–45/]2s configu-rations. Three different approaches (Method I, IIand III) are implemented in ABAQUS™ user sub-

routine (UMAT) to simulate the degradation.Method I is a strain-based damage evolution whileMethod II is the modified of Method I and stress-based damage evolution proposed in this work.Method III obeys the Hashin-based degradationevolution criteria. Gaussian distribution is consid-ered to model the scatter of material properties dur-ing the simulation. The model is applied to predictthe strength of the open-hole laminate and the sim-ulation results correlate well with the experimentaldata of the present work and [7]. A parametric studyis performed for the effect of different mesh sizesand viscous regularization factors. The results showthat in Methods I and II the damage prediction issensitive to viscous regularization factor and meshsizes. However, in latter, the characteristic length ofthe elements is considered to minimize the effect ofthe mesh size. The results of different damage pat-terns of the 0° fibers indicates that depending on thestochastic material properties, the failure path dif-fers each time the model is run. The study on theeffects of Gaussian distribution of material proper-ties shows that depending on the scattering of mate-rial properties, the damage propagation and failurepath vary each time the numerical simulation is run.The failure paths obtained from re-running the sim-ulation for several times are in close agreementwith the image of fractured specimen.

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