r A

06-6

inaaquta. Tr E-qus

nd acgreatoring oplex ansile aatrix crigated,

free, and the remaining models available in the literature are for themost part not readily available to be used in conjunction with com-mercial FEA environments.

Therefore, the objective of this manuscript is to propose amethodology to determine values for the material parameters

required by the Abaqus PDA model. In this work, the values forthe parameters are found using available experimental data and

Damage initiation refers to the onset of degradation at a materialpoint. The criterion considers four different damage initiationmechanisms, which are assumed to be uncoupled, as follows:

Fiber tension (r11P 0)

Ftf r11F1t

2 a r12

F6

21

Fiber compression (r11 < 0)

Corresponding author. Tel.: +1 304 293 3176; fax: +1 304 293 6689.E-mail address: drb@cemr.wvu.edu (E.J. Barbero).

1 Abaqus and Simulia are trademarks or registered trademarks of Dassault Systmes

Composites: Part B 46 (2013) 211220

Contents lists available at Sci

Composites:

evor its subsidiaries in the United States and/or other countries.modes such as delamination and, due to load redistribution, mayeven lead to ber failure of adjacent laminas.

The earliest, simplest and least accurate modeling technique toaddress matrix damage is perhaps the ply discount method ([1],Section 7.3.1). Ply discount is used here as a baseline for contrast-ing predictions obtained with the progressive damage analysis(PDA) method implemented in Abaqus1 [2]. Although many othermodels exist [313], etc., and several plugins are available [14,15],this manuscript focuses on the PDA model available in Abaqus. Thisis because of the broad availability of Abaqus, while plugins are not

The Abaqus PDA model is a generalization of the approach pro-posed by Camanho and Davila [16]. It requires linearly elasticbehavior of the undamaged material and is used in combinationwith Hashins damage initiation criteria [17].

2.1. Damage initiation

In Abaqus [2] the onset of damage is detected by the Hashin andRotem damage initiation criteria [17] in terms of apparent(nominal, Cauchy) stresses r, which is calculated by the FEA code.1. Introduction

Prediction of damage initiation amatrix, laminated composites is ofproduction, certication, and monitvariety of structures. It is also a commatrix cracking due to transverse teis considered in this manuscript. Mrst mode of damage. If left unmit1359-8368/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compositesb.2012.09.069cumulation in polymerinterest for the design,f an increasingly largend difcult topic. Onlynd shear deformationsacking is normally theit often leads to other

a rational procedure. Once values are found, the PDA model is ap-plied for predicting other, independent results, and conclusions aredrawn about the applicability of the PDA model. This study focuseson E-glass epoxy laminates.

2. Progressive damage analysis (PDA)C. Finite element analysis (FEA)Parameter identicationDetermination of material parameters foof E-glass epoxy laminates

E.J. Barbero , F.A. Cosso, R. Roman, T.L. WeadonMechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 265

a r t i c l e i n f o

Article history:Received 2 August 2012Accepted 12 September 2012Available online 2 October 2012

Keywords:A. Polymermatrix composites (PMCs)B. Transverse crackingC. Damage mechanics

a b s t r a c t

A methodology for determmodel implemented in Abavailable experimental daset of experimental data foAlso, sensitivity of the Aba

journal homepage: www.elsll rights reserved.baqus progressive damage analysis

106, USA

tion of material parameters for the progressive damage analysis (PDA)s is presented. The methodology is based on tting PDA model results tohe applicability of the PDA model is studied by comparison to a broadglass epoxy laminates, as well as contrasting PDA with ply discount results.PDA model to h- and p-renement is studied.

2012 Elsevier Ltd. All rights reserved.

ier .com/locate /composi tesbVerse ScienceDirect

Part B

The same Eqs. (1)(4) are used as damage evolution criteria byintroducing effective stresses (10) in lieu of nominal stress. Therelation between the effective stress er and the apparent stress ris computed [18, (8.64)] as:

~r M1 : r 10Abaqus PDA uses the model proposed by Matzenmiller et al. [19] tocompute the degradation of the stiffness matrix coefcients. There-fore, the damage effect tensor in Voigt notation is given as

M1 1 df 1 0 0

0 1 dm1 01

26643775 11

tes: Part B 46 (2013) 211220Fcf r11F1c

22

Matrix tension and/or shear (r22P 0)

Ftm r22F2t

2 r12

F6

23

Matrix compression (r22 < 0)

Fcm r222F4

2 F2c

2F4

2 1

" #r22F2c

r12F6

24

where rij are the components of the stress tensor; F1t, F1c, are thetensile and compressive strengths in the ber direction; F2t, F2c,are the tensile and compressive strengths in the matrix direction;F6, F4, are the longitudinal and transverse shear strengths, and adetermines the contribution of the shear stress to the ber tensilecriterion. To obtain the model proposed by Hashin and Rotem [17]we set a = 0 and F4 = 1/2 F2C. Furthermore, Fft, Ffc, Fmt, Fmc, areindexes that indicate whether a damage initiation criterion in adamage mode has been satised or not. Damage initiation occurswhen any of the indexes exceeds 1.0.

The effect of damage is taken into account by reducing the val-ues of stiffness coefcients [2, Section 21.3.1] as follows:

r C : e 5where represents a tensor double contraction and the damagedstiffness is given by:

C1df E1=D 1df 1dmm21E1=D 0

1df 1dmm12E2=D 1dmE2=D 00 0 1dsG12

264375 6

D 1 1 df 1 dmm12m21ds 1 1 dtf

1 dcf

1 dtm

1 dcm 7

where r is the apparent stress, e is the strain, C is the damagedstiffness matrix, E1, E2, are the moduli in the ber direction andperpendicular to the bers, respectively, G12 is the inplane shearmodulus, m12, m21, are the inplane and Poissons ratios, anddtf ;d

cf ; d

tm;d

cm;ds, are damage variables for ber, matrix, and shear

damage modes, in tension and compression, respectively. Note thatthe shear damage variable is not independent but given by (7) interms of the remaining damage variables.

The damage variables dtf ; dcf ; d

tm; d

cm, that characterize ber and

matrix damage in tension and compression correspond to the fourdamage initiation modes (1)(4). At any time, each material pointis either in tension or compression. Therefore, the four damagevariables reduce to two variables [2, Section 21.3.2] as follows:

df dtf if r11 P 0dcf if r11 < 0

(8

and

dm dtm if r22 P 0

dcm if r22 < 0

(9

The six values of strength, F1t, F1c, F2t, F2c, F6, F4, are materialproperties that must be provided by the user. Due to differencesbetween testing and application conditions, as well as in situ ef-fects [1, Section 7.2.1], the strength values measured by standard

212 E.J. Barbero et al. / Composimethods may not predict damage initiation accurately. Therefore,in this work, the values of strength are found by correlating modelresults to laminate experimental data.0 0 1 dsPrior to damage initiation, the damage effect tensor M1 is equal tothe identity matrix, so er r. Once damage initiation and evolutionhas occurred for at least one mode, the damage effect tensor be-comes signicant in the criteria for damage initiation of othermodes. When df = 0 and dm = ds = 1, Eq. (11) represents the degrada-tion scheme of the ply discount method.

3. Damage evolution

Once any of the damage initiation criteria (1)(4) is satised,further loading will cause degradation of material stiffness coef-cients. The evolution of the damage variable employs the criticalenergy release rates Gci , which are material properties, with i = ft,fc, mt, mc, corresponding to the four damage modes: ber tension,ber compression, matrix tension, and matrix compression,respectively. Therefore, in addition to six values of strength, fourvalues of critical ERR must be provided, each one correspondingto the area of the triangle OAC shown in Fig. 1. This contrasts withother models where both onset and evolution of damage are pre-dicted in terms of critical ERR only [20].

Normally, the constitutivemodel is expressed in terms of stressstrain equations. When the material exhibits strain-softeningbehavior, as shown in Fig. 1, such formulation results in strongmeshdependency. In Abaqus, to alleviate the mesh dependency in thePDA constitutive model, a characteristic length is introduced intothe formulation. Formembranes and shells the characteristic lengthLc is computed as the square root of the area of the reference surfaceof the element. Using the characteristic length, the stressstrainconstitutive model is transformed into the stressdisplacementconstitutive model illustrated in Fig. 1.

The evolution of each damage variable is governed by anequivalent displacement deq. In this way, each damage mode is rep-resented as a 1D stressdisplacement problem (Fig. 1) even thoughthe stress and strain elds of the actual problem are 3D. Theequivalent displacement for each mode is expressed in terms ofFig. 1. Equivalent stress versus equivalent displacement for a linearly softeningmaterial.

the components corresponding to the effective stress componentsused in the initiation criterion for each damage mode. The equiva-lent displacement and stress for each of the four damage modes aredened as follows [2, Section 21.3.3]:

Fiber tension (r11P 0)

dfteq LC :he11i2 ae212

qrfteq

hr11ihe11i ar12e12dfteq=L

c

12

Fiber compression (r11 < 0)

dfceq LChe11i

rfceq hr11ihe11i

dfceq=Lc

13

Matrix tension and shear (r22P 0)

dmteq LChe11i2 e212

qrmteq

hr22ihe22i r12e12dmteq =L

c

14

Matrix compression (r22 < 0)

dmteq LChe22i2 e212

qrmteq

hr22ihe22i r12e12dmceq =L

c

15

where hi represents the Macaulay operator, which is dened ashgi 12 g jgj for every g 2 R.

Table 1Laminates considered in this study.

LSS Source

1 [02/904]s [22, Fig. 7]2 [15/904]s [22, Fig. 8]3 [30/904]s [22, Fig. 9]4 [40/904]s [22, Fig. 10]5 [0/908/01/2]s [21, Figs. 1,6, and 7]6 [0/70/01/2]s [21, Figs. 2,9, and 10]7 [0/55/01/2]s [21, Figs. 3,11, and 12]8 [0/40/01/2]s [21, Figs. 13 and 14]9 [0/25/01/2]s [21, Figs. 15 and 16]

Table 2Unidirectional ply properties for glass/epoxy berite HyE 9082Af. See Table 3 forin situ strengths.

Property Units Value Source

E1 (GPa) 44.7 [21]E2 (GPa) 12.7 [21]G12 (GPa) 5.8 [21]G23 (GPa) 4.5 [20]m12 (GPa) 0.297 [21]a1 (le/K) 8.42 [27]a2 (le/K) 18.42 [21]DT (K) 99 [21]F1t (MPa) 1020 [18]F2t (MPa) 40 [18]F1c (MPa) 620 [18]F2c (MPa) 140 [18]F6 (MPa) 60 [18]Ply thickness (mm) 0.144 [22]

E.J. Barbero et al. / Composites: Part B 46 (2013) 211220 213Fig. 2. Laminate #1 wFor each mode, the damage variable controls the reduction ofthe stiffness coefcients, and may assume values between zero(undamaged state) and one (fully damage state). The damage var-iable for a particular mode is derived using Fig. 1 and given by:

d dCeq deq d0eq

deq dCeq d0eq

16where dCeq is the maximum value of deq at point C in Fig. 1, for eachmode.

Values of interlaminar critical energy release rates Gc reportedin the literature are not directly applicable as input to AbaqusPDA because the damage addressed by PDA is intralaminar, notinterlaminar. Also, the PDA model computes equivalent 1D energyrelease rates as the area under the curve on Fig. 1, not 3D energyrelease rates, which are dened as:

G @U@A

17

where the strain energy U is a 3D function of e, and A is the crackarea. Furthermore, in PDA, there is no identiable crack but ratherith LSS [02/904]s.

an equivalent reduction of stiffness. Therefore, the crack area A isnot known and the ERR cannot be computed as in (17), but in termsof equivalent1D stress and displacement (14).

4. Ply discount

The ply discount method was used to compare to PDA results.The Hashin damage initiation criteria was implemented in a usersubroutine based on (1)(4), using nominal stress.

For damage evolution, the ply discount method simply reducesto zero the stiffness of the lamina that reaches the damage onset,

5. Experimental

laminates is subjected to ber modes (1) and (2) or matrixcompression (4). The 90 laminas in laminates #1 and #5 are sub-jected to pure traction and no shear, so damage initiation is con-

Table 3In situ strengths and apparent energy release rate.

Property Units Value

F2t (MPa) 80.8831F6 (MPa) 48.5725Gcmt (kJ/m

2) 11.5002

Fig. 4. Error vs. Gcmt .

Fig. 5. Error vs. F2t.

214 E.J. Barbero et al. / Composites: Part B 46 (2013) 211220Changes in the normalized longitudinal Youngs modulus andPoissons ratio with respect to the applied strain, measured exper-imentally and reported in the literature, were used to comparewith the progressive damage analysis and the Ply Discount meth-odology. The laminate stacking sequences reported by Varna et al.in [21,22], shown in Table 1, were considered. The dimensions ofthe specimens are 20 mm wide with a free length of 110 mm.The ply properties for the laminates are listed in Table 2.

All laminates were subjected to axial deformation ex, whichcoincides with e11 in the 0 laminas. None of the laminas in theseexcept in the ber direction which is left unchanged. In order toprevent convergence problems, the secant stiffness (6) is reducedusing df = 0, dm = ds = 0.999, abruptly as soon as the Hashin damageinitiation criteria (3) has been met.

Then, the stress is calculated with (5), where e is total strain. InAbaqus, ei = ei1 + Dei, where i is the current step. If damage has notinitiated, the damage initiation criteria is evaluated for the recentlycalculated values of stress to decide whether or not the damageinitiation criteria has been met. In ply discount, damage evolutionis sudden and abrupt, occurring simultaneously with damageinitiation.Fig. 3. Laminate #5 with LSS [0/908/01/2]s.

trolled by the parameter F2t. Therefore, the laminate experimentaldata for laminate #1 (Fig. 2a) was chosen to determine thein situ value of F2t, which is reported in Table 3. This value is100% higher than the UD strength reported in Table 2. When thein situ value is used to predict initiation for laminate #5, the pre-dicted rst ply failure (FPF) strain is 22.4% higher than the observedexperimental data (Fig. 3a). The difference is due to in situ effect,because the damage initiation criteria used in Abaqus does not ac-count for in situ effect.

6. Methodology

All the laminates considered for the study are symmetric andbalanced. Therefore a quarter of the specimen was used for theanalysis using symmetry b.c. and applying a uniform strain via im-posed displacements on one end of the specimen. A longitudinaldisplacement of 1.1 mmwas applied to reach a strain of 2%. AbaqusS8R elements were used for most of the study but the convergencestudy also considered S4R elements.

Fig. 6. Error vs. F6.

Fig. 8. Laminate #3 w

Fig. 7. Laminate #2 with LSS [15/904]s.

E.J. Barbero et al. / Composites: Part B 46 (2013) 211220 215ith LSS [30/904]s.

The value of apparent energy release rate Gcmt was obtainedsimultaneously with the value of F2t to provide a best t with thePDA model to the experimental modulus data for the laminate#1 (Fig. 2a). Laminate #1 was chosen because only F2t and G

cmt

are active. The experimental data and the tted PDA model resultsare shown in Fig. 2a.

Since numerical minimization algorithms only nd localminima, it is important to rst nd a good guess for the adjustableparameters F2t and G

cmt . For this, 804 simulations were performed

with random values of the parameters in the interval1:5 < Gcmt < 30:5 kJ=m

2 and 21 < F2t < 141 MPa. The results arevisualized in Figs. 4 and 5. The error was calculated using the usualformula:

Error 1N

XNi1

EeEAbaquseei

EeEexpeei

!vuut 2 18

where E and eE now stand for the laminate longitudinal elastic mod-ulus, damaged and intact respectively, and N is the number ofexperimental data points. Nineteen experimental data points are re-ported by Varna et al. [22].

Note that the shallow contour of error vs. Gcmt in Fig. 4 may ex-plain the great dispersion of values reported in the literature. InFig. 4, a broad range of Gcmt (1030 kJ/m

2 in this case), seem to adjustmore or less well the experimental degradation rate of modulus inFig. 2a. If the response measured (modulus) is weakly sensitive tothe parameter of interest Gcmt

, the experimental values of the

parameter will have large scatter. Nevertheless, by adjustingsimultaneously both parameters, F2t and G

cmt , the minimization

algorithm described next is able to converge to a global minimum.From Figs. 4 and 5, it can be seen that a good choice for initial

guess is Gcmt 12kJ=m2 and F2t = 80 MPa. Next, a MATLABscript

was executed to look for the minimum error (18), by repeatedlyexecuting Abaqus with parameters varying as per the Simplex

Fig. 9. Laminate #4 with LSS [40/904]s.

216 E.J. Barbero et al. / Composites: Part B 46 (2013) 211220Fig. 10. Laminate #6 with LSS [0/70/01/2]s.

tes:E.J. Barbero et al. / Composimethod [23]. The converged values of F2t and Gcmt are reported in

Table 3.According to (3), a combination of F2t and F6 determines the

damage onset for off-axis plies. Since the values of F2t and Gcmt

are already set, it only remains to adjust the in situ value ofF6, which is done by minimizing the error using the modulusdata for laminate #8 because this is the laminate where thecracking laminas experience the most shear. Since Abaqus PDAuses the Hashin damage initiation criterion, Eq. (3) is active,F2t is xed and only F6 can vary. A preliminary estimation ofthe error was obtained on 112 simulations by using randomnumbers in the range 27.5 < F6 < 93 MPa. Varna et al. reported13 experimental values [21, Fig. 13]. The error versus the longi-tudinal shear strength F6 is presented in Fig. 6. From this plot,the initial guess value for the search was chosen to beF6 = 49 MPa. Once again, the MATLAB script was used to ndthe minimum error, at which point the converged value of F6is found and reported in Table 3.

Fig. 11. Laminate #7 wi

Fig. 12. Laminate #8 witPart B 46 (2013) 211220 2177. Model assessment

7.1. Comparison between PDA and experimental data

In this section we compare modulus and Poissons ratio pre-dicted with PDA vs. experimental data for the nine laminates listedin Table 1. Note that the modulus vs. strain data of laminate #1was used previously to nd values for F2t and G

cmt , and that only

the damage initiation in laminate #8 was relevant in nding F6.Since the remaining data has not been used to adjust the PDAparameters, it can be used for assessing the quality of thepredictions.

The longitudinal modulus and Poissons ratio for the nine lam-inates are compared with experimental data in Figs. 2 and 7, 8, 9,10, 11, 12, 13. Fig. 2a merely states the fact that F2t and G

cmt were

adjusted using the data for this laminate. Fig. 2b is a new resultas it was not used to t any parameters. It shows that the laminatePoissons ratio is predicted well by the model.

th LSS [0/55/01/2]s.

h LSS [0/404/01/2]s.

7.3. Convergence

wi

Table 4Comparison of FPF laminate strain ex using Abaqus with in situ values and cadec-online.com with and without in situ correction of strength.

LSS Abaqus Cadec-online.com

With in situ With in situ Without in situ

1 [02/904]s 0.6420 0.6420 0.40532 [15/904]s 0.6622 0.6452 0.40733 [30/904]s 0.6622 0.6511 0.41114 [40/904]s 0.6420 0.6387 0.41215 [0/908/01/2]s 0.6420 0.6389 0.40346 [0/704/01/2]s 0.6219 0.6094 0.39727 [0/554/01/2]s 0.5817 0.5918 0.38378 [0/404/01/2]s 0.5616 0.5514 0.38799 [0/254/01/2]s 0.7427 0.7077 0.5069

tes:As long as the cracking lamina is a 90 lamina (Figs. 3 and 79),PDA predictions are quite good, because both the in situ F2t and G

cmt

are adjusted using a cracking 90 lamina. Even when the crackinglamina is at 70with respect to the load direction, the cracks prop-agate predominantly under mode I, so the predictions of modulusis good (Fig. 10a).

However, cracking lamina under signicant shear (Figs. 1113)are not modeled well by PDA. This may be the result of the sheardamage variable ds not being an independent variable in the mod-el. In fact, for the experimental data available, all the remainingmodes of damage are absent. Substituting dtf dcf dcm 0 in (7)yields ds dtm. That is, the reduction of shear modulus G12 repre-sented by ds is modeled as being equal to the reduction of trans-verse modulus E2. This contrast with other models in theliterature that use independent variables for ds and d

tm [24], etc.

In fact, analytical computation of the reduction of stiffness aftersolving the equations of elasticity for a cracked laminate in [20,(23)] for laminate #8 yields a constant ratio ds=d

tm 0:69 from

Fig. 13. Laminate #9218 E.J. Barbero et al. / Composicrack initiation to saturation crack density. In contrast, the PDAratio is ds=d

tm 1:0.

Note that the data for laminate #8 was used to adjust F6, an thusdamage initiation is predicted accurately in Fig. 12a, but Gcmt wasadjusted with laminate #1, and it cannot be readjusted with lam-inate #8, so the damage evolution is not predicted accurately.

7.2. In situ strength

Since the values of F2t and F6 reported in Table 3 were calculatedto match laminate data, the values obtained are in situ values. Tohighlight the error incurred when using unidirectional (UD) laminastrength values rather than in situ values, a comparison is presentedon Table 4 between the First Ply Failure (FPF) strain calculated withAbaqus and the FPF strain calculated using an online laminate anal-ysis software (cadec-online.com) with and without in situ correc-tion of the UD strength values. The UD strength values (withoutin situ correction) were back calculated using [1, (7.42)] in termsof the in situ values reported in Table 3 and assuming the transitionthickness of the E-glassepoxymaterial is 0.6 mm [25]. The Abaqusresults are found to be in good agreement with the FPF valuescalculated using in situ strength into the Hashin damage initiationcriterion implemented in the laminate analysis software [26]. How-ever, as shown in column 4 of Table 4, the FPF strain is grosslyunderestimated if the in situ correction is neglected.th LSS [0/25/01/2]s.

Part B 46 (2013) 211220In this section we assess the sensitivity of the Abaqus PDA mod-el to h- and p-renement, realized by mesh renement and bychanging the element type, respectively. Note that the specimenis subjected to a uniform state of membrane strain, so h-rene-ment should not be necessary to converge on a stress eld, unlessthe constitutive model is mesh dependent.

However, the mesh convergence study reveals mesh depen-dency, as shown in Fig. 14. Before rst ply failure (FPF), theforcedisplacement response is independent of the number ofelements used in the mesh. After FPF, the forcedisplacementresponse diverge, revealing mesh dependency.

With n = 140, asymptotic behavior at large strains is reachedwhen the damage in the 90 lamina is fully developed. At this stagethe laminate modulus reduces to one third of the undamaged va-lue, as shown in Fig. 15. Also in Fig. 15, it can be seen that for anapplied laminate strain ex = 2.2%, the 0 lamina fails due to bertension, which is consistent with the bers used in the laminates.

Modulus vs. strain for the [02/904]S laminate #1 are reported inFig. 15. It can be seen that the model predictions are affected by theelement type. Recall that the material parameters F2t ; F6;G

cmt , were

adjusted using quadratic elements (S8R). Reduced integration(S8R5) yields identical results, but when the element is changedto linear (S4, S4R), the model predictions are signicantly different,overestimating the modulus reduction.

tes:Displacement Ux [mm]

0.30 0.40 0.50 0.60 0.70 0.80

Forc

e Fx

[N]

2000

3000

4000

5000

6000

1 element2 elements22 elements140 elementsElastic

Fig. 14. Force vs. displacement for laminate #8 using different elements. The elasticline goes through the origin (not shown).

E.J. Barbero et al. / Composi8. Conclusions

A novel methodology is proposed to determine the materialparameters, namely in situ strength F2t, F6, and critical ERR G

cmt ,

using laminate experimental data. It is observed that PDA predic-tions are good for mode I matrix cracking but decient for mixedmode matrix cracking including shear (laminates #7, #8, and#9). Also, the PDA model is not very sensitive to changes in the va-lue of critical ERR, as shown by the shallow relationship of modelerror vs. critical ERR. This may explain the large dispersion of val-ues reported in the literature for the intra and interlaminar ERR.Compared to PDA, the ply discount method severely overestimatesthe stiffness changes and leads to large errors in the prediction ofenergy dissipation (toughness). However the asymptotic valuesof modulus reduction are reasonably accurate. Using PDA, all FPFpredictions are reasonably good as long as the ply thickness ofthe cracked lamina are similar (four plies in most of this study).However, a 20% in situ effect is shown when n = 8 if the values ofstrength are not readjusted for in situ effect taking into accountthe change in thickness from four to eight plies.

When the ber tension, ber compression, and matrix compres-sion are not active, which is normally the case due to the highstrength of the bers and sudden brittle failure in compression,the PDA shear damage is equal to the PDA transverse tensile

Fig. 15. Normalized modulus vs. applied strain for laminate #8 using differenttypes of elements eEx 68:22 GPa.damage. Such constraint is likely responsible for the poor predic-tion shown when the cracking ply is subjected to shear. The pres-ent study used E-glass laminates to identify the PDA parametersbecause the reduction of laminate modulus due to transverse ma-trix damage in carbon ber laminates is very small on account ofthe highmodulus of the bers. Therefore, it is unlikely that the pro-posed methodology would be applicable to carbon ber laminatessince the parameter identication relies on modulus vs. strain data.On the other hand, other models in the literature that are based onmeasurable state variables, such as crack density, can be identiedusing crack density, which is signicant regardless of the bermodulus.

Acknowledgements

This work was the result of a semester-long project in the 2012course MAE 646 Advanced Mechanics of Composite Materials atWest Virginia University, taught by the rst author. The authorswish to express their gratitude to their classmates for their com-ments and collaboration during the semester with special mentionand recognition to S. Koleti. Also, many thanks to Dr. A. Adum-itroaie for introducing us to Abaqus PDA, as well as to Rene Sprun-ger and Simulia for their support of our educational programs.

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220 E.J. Barbero et al. / Composites: Part B 46 (2013) 211220

Determination of material parameters for Abaqus progressive damage analysis of E-glass epoxy laminates1 Introduction2 Progressive damage analysis (PDA)2.1 Damage initiation

3 Damage evolution4 Ply discount5 Experimental6 Methodology7 Model assessment7.1 Comparison between PDA and experimental data7.2 In situ strength7.3 Convergence

8 ConclusionsAcknowledgementsReferences