Computational Materials Science 70 (2013) 37–50

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Modeling time-dependent and inelastic response of fiber reinforcedpolymer composites

0927-0256/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2012.12.022

⇑ Corresponding author.E-mail address: amuliana@neo.tamu.edu (A. Muliana).

Jaehyeuk Jeon, Jeongsik Kim, Anastasia Muliana ⇑Department of Mechanical Engineering, Texas A&M University, United States

a r t i c l e i n f o

Article history:Received 28 June 2012Received in revised form 4 December 2012Accepted 17 December 2012Available online 28 January 2013

Keywords:ViscoelasticViscoplasticTime-integration algorithmMicromechanicsPolymersFinite elementFiber reinforced composites

a b s t r a c t

This study presents two micromechanical modeling approaches for analyzing a time-dependent andinelastic response of fiber reinforced polymer (FRP) composites. The studied FRP composites consist ofunidirectional fibers, which are considered as linearly elastic and transversely isotropic with regards totheir mechanical response, and isotropic polymeric matrix, which shows viscoelastic–viscoplasticresponse. Due to the combined viscoelastic and viscoplastic behavior of the polymeric matrix, the overallFRP composites exhibit the time-dependent and inelastic behavior. The first micromechanical model isbased on a simplified unit-cell with four fiber and matrix subcells, which is formulated in terms of theaverage (uniform) stress and strain fields of each subcell. The unit-cell model is compatible with a dis-placement based finite element (FE), which is implemented at the Gaussian integration points withinthe continuum finite elements. The second micromechanics model considers detailed microstructuralmorphologies of the FRP microstructure, which incorporate the effects of fiber arrangements, stress con-centrations, and stress discontinuities at the fiber and matrix interphases on the overall response of theFRP composites. The representative microstructural models of the FRP with detailed microstructural mor-phologies are generated using continuum finite elements. The overall time-dependent and inelasticresponses of the FRP composites determined from the two micromechanical models are studied. The con-vergence behaviors in the above micromechanics models are also examined.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The response of fiber reinforced polymer (FRP) composites,when subjected to external mechanical loadings, depends on themicrostructural characteristics of the FRP, i.e., fiber arrangements,compositions- and properties of the constituents, and directions ofloading and the FRP composites are orthotropic, or at least trans-versely isotropic, with regards to their mechanical and physicalproperties. The FRP composites consisting of epoxy matrix showa pronounced time dependent behavior due to the viscoelastic(VE) nature of the polymers. When these composites are subjectedto high mechanical loadings and elevated temperatures, the poly-meric matrix could experience a viscoplastic (VP) or an inelasticdeformation, leading to the time-dependent and inelastic overallresponse of the FRP structures. Furthermore, the time-dependentand inelastic response becomes more significant when the FRP issubjected to off-axis mechanical loading due to the additionalshear effect presence in the composites.

There have been experimental studies on understanding theinelastic (viscoplastic) response of FRP composites. Tuttle et al.

[22] conducted cyclic thermo-mechanical tests on graphite–bisma-leimide (IM7/5260) composites at various stress levels and tem-peratures. The experiment on the multi-angle compositelaminate was done under a 50-h cyclic loading and isothermal con-ditions. The results showed nonlinear VE–VP responses under hightemperatures and stresses. The experimental data were comparedwith the response obtained using the Schapery [18] nonlinear VEintegral model and VP model proposed by Zapas and Crissman[23]. Guedes et al. [5] conducted experimental tests on T300/5208 and IM7/5260 composites under creep-recovery, ramp andmultiple relaxation loadings. They used the Schapery VE modeland Zapas and Crissman VP model to predict the experimentaldata. Megnis and Varna [11] characterized the time-dependentand inelastic response of glass–epoxy FRP composites from creeptests under isothermal condition. The difference between the creepstrains obtained from the experimental results of the off-axis spec-imens and predicted linear VE strains shows that the FRP compos-ite experienced a VP deformation. It was reported that the VPresponse is pronounced for the off-axis creep tests due to the highshear stresses.

Various micromechanical models have been developed to pre-dict the overall response of FRP composites with complex micro-structural geometry. The micromechanical modeling approachescan be classified into two groups based on the selection of the rep-

38 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

resentative volume element (RVE). The first micromechanicalmodeling approach deals with obtaining the overall response ofcomposites based on simplified microstructural characteristics, inwhich the RVE is considered as the smallest region that can givea reasonable prediction of the overall response. The size of theRVE depends on the responses being examined, i.e., elastic, plastic[9,20]. The second micromechanical modeling approach considersthe RVE of the composite as a domain containing all possiblemicrostructural configurations that exist in the composite. Thisconcept can be employed by discretizing the microstructural mod-el based on the experimentally obtained microstructural images(see Shan and Gokhale [19]), and the overall response can be ob-tained by solving the corresponding boundary value problems(BVPs) using for example FE. This approach leads to a relativelylarge RVE and high computational costs, especially for nonlinearinelastic problems.

Several micromechanics models based on the simplified micro-structural characteristics have been developed to predict elastic,viscoelastic and inelastic responses of FRP composites. Dvorakand Bahei-El-Din [3] used the self-consistent model to analyzeelastic–plastic response of fibrous composites. A method of cells(MOCs) by Aboudi [1] was used to evaluate the effective elasticand plastic responses of FRP composites comprising of periodicalrectangular fibers surrounded by matrix. Aboudi [2] developed amicromechanical model to predict the VE–VP responses of multi-phase materials. The VE–VP model developed by Frank and Brock-man [4] was implemented in the multiphase composites. Ohnoet al. [15] presented a homogenization method for analyzing anelastic–VP response of FRP composites. A periodic boundary condi-tion was imposed to the select unit-cells of the composite micro-structure. An incremental solution within FE was proposed fordetermining macro- and microscopic field variables. Matsudaet al. [10] presented a micromechanical model for analyzing elas-tic–VP response of unidirectional and woven laminated compos-ites. Representative unit-cell models were selected and FEmeshes were generated on the select unit-cells. Tsukamoto [21]formulated a mean-field micromechanical model for analyzingVE–plastic response of laminated composites comprising of unidi-rectional fiber and particle reinforcement embedded in a polymericmatrix undergoing an inelastic deformation. Recently, Jeon andMuliana [7] presented a simple micromechanical model for analyz-ing the VE–VP response of FRP composites. The micromechanicalmodel was used to predict the time-dependent and inelastic re-sponse of glass–epoxy FRP composites reported by Megnis andVarna [11].

This study presents two micromechanical models for analyzingtime-dependent and inelastic response of FRP composites. Theadvantages of using micromechanical modeling approaches are:they allow incorporating microstructural characteristics in predict-ing the overall response of composites, they are capable of deter-mining the nonlinear and inelastic response of the constituentsdue to the prescribed external stimuli, and they can provide a rig-orous and robust prediction of the complex nonlinear response ofthe FRP composites. In this study, the fiber is assumed linear elasticand transversely isotropic. The Schapery nonlinear VE model isadditively combined with the Perzyna VP model and used for thepolymeric matrix. The first micromechanical model considers asimple unit-cell consisting or four fiber and matrix subcells, whilethe second micromechanical model incorporates microstructuraldetails in terms of fiber arrangements of the FRP. The microme-chanical models are implemented in a general displacement basedFE. The time-dependent and inelastic responses of the FRP deter-mined from the two micromechanics models are compared. Thismanuscript is organized as follows. Section 2 presents a brief dis-cussion of the combined VE–VP constitutive model for the poly-meric constituent. Section 3 discusses the two-micromechanical

models followed by numerical results in Section 4. Finally Section 5is dedicated to a summary of the numerical results.

2. Viscoelastic–viscoplastic constitutive model for isotropicpolymeric materials

This chapter presents a constitutive model for a combined VEand VP response of an isotropic polymeric material undergoingsmall deformation gradients. The nonlinear Schapery integral mod-el is used for the VE part and the Perzyna model is considered forthe VP component; thus the nonlinearity is due to the stress-dependent material parameters. The total strains and incrementalstrains can be additively decomposed into the VE and VPcomponents:

etij ¼ eme;t

ij þ emp;tij

Detij ¼ Deme;t

ij þ Demp;tij 8t P 0

etij ¼ et�Dt

ij þ Detij ¼ eme;t�Dt

ij þ Deme;tij þ emp;t�Dt

ij þ Demp;tij

ð2:1Þ

where the superscripts ve, and vp denote the VE and VP compo-nents, respectively, and D indicates the incremental component.The superscript t denotes the current time. Thus, eme;t

ij and emp;tij are

the VE and VP strains at current time t, respectively, and Deme;tij

and Demp;tij are the incremental form of the VE and VP strains at cur-

rent time t, respectively. It is noted that this study neglects the dis-sipation of energy from the VE and VP deformations.

A nonlinear single-integral constitutive equation [18], which isused for the VE component, is modified for a multi-axial loading.The current total VE strain consists of the deviatoric and volumet-ric strain components, which for an isotropic case is written as

eme;tij ¼ eme;t

ij þ13

dijeme;tkk ð2:2Þ

eme;tij ¼

12

g0ð�rtÞJ0Stij þ

12

g1ð�rtÞZ t

0DJðw

t�wsÞ d½g2ð�rsÞSsij�

dsds ð2:3Þ

eme;tkk ¼

13

g0ð�rtÞB0rtkk þ

13

g1ð�rtÞZ t

0DBðw

t�wsÞ d½g2ð�rsÞrskk�

dsds ð2:4Þ

J0 and B0 are the instantaneous elastic shear and bulk compliances,respectively. DJ and DB are the transient shear and bulk complianc-es, respectively; dij is the delta Kronecker; Sij and rkk are the compo-nents of the deviatoric and volumetric stresses, respectively. Thenonlinear parameters g0, g1, and g2 depend on the current effectivestress �rt . In the linear VE responses, g0 = g1 = g2 = 1. The correspond-ing linear elastic Poisson’s ratio, m, is taken as constant and the shearand bulk compliances are

J0 ¼ 2ð1þ mÞD0; B0 ¼ 3ð1� 2mÞD0

DJwt¼ 2ð1þ mÞDDwt

; DBwt¼ 3ð1� 2mÞDDwt ð2:5Þ

For the transient part, a series of an exponential function is usedfor the time-dependent function, which for the uniaxial compli-ance is expressed as

DDwt¼XN

n¼1

Dnð1� exp½�knwt �Þ ð2:6Þ

where the reduced-time (effective time) is:

wt � wðtÞ ¼Z t

0

dnað�rnÞ ð2:7Þ

The parameter a (�rt) is the time shift factor measured with re-spect to the reference stress. The integral models in Eqs. (2.3) and(2.4) are solved numerically using a recursive-iterative approach.

J. Jeon et al. / Computational Materials Science 70 (2013) 37–50 39

Detailed recursive-iterative algorithm for the nonlinear VE re-sponses can be found in Haj-Ali and Muliana [6]. The final formof the incremental VE strain by using the recursive-iterativescheme is

Deve;tij ¼ Jtrt

ij þ13

dijfBt � Jtgrtkk

� �

� Jt�Dtrt�Dtij þ 1

3dijfBt�Dt � Jt�Dtgrt�Dt

kk

� �� At

ij

� 13

Btdij ð2:8Þ

Jt ¼ 12

gt0J0 þ gt

1gt2

XN

n¼1

Jn � gt1gt

2

XN

n¼1

Jn1� exp½�knDwt �

knDwt

" #ð2:9Þ

Bt ¼ 13

gt0B0 þ gt

1gt2

XN

n¼1

Bn � gt1gt

2

XN

n¼1

Bn1� exp½�knDwt �

knDwt

" #ð2:10Þ

Atij ¼

12

XN

n¼1

Jnðgt1 exp½�knDwt�Dt � � gt�Dt

1 Þqt�Dtij;n þ

12

gt�Dt2

XN

n¼1

Jn

� gt�Dt1

1� exp½�knDwt�Dt �knDwt�Dt

!� gt

11� exp½�knDwt�

knDwt

� �" #St�Dt

ij

ð2:11Þ

Bt ¼ 12

XN

n¼1

Bnðgt1 exp½�knDwt�Dt� � gt�Dt

1 Þqt�Dtkk;n þ

12

gt�Dt2

XN

n¼1

Bn

� gt�Dt1

1� exp½�knDwt�Dt �knDwt�Dt

!� gt

11� exp½�knDwt �

knDwt

� �" #rt�Dt

kk

ð2:12Þ

Jt and Bt are the shear and bulk compliances, respectively, that de-pend on the effective stress at the current time t. The history vari-ables are included in At

ij and Bt. The shear and volumetrichereditary variables, qt

ij;n and qtkk;n, respectively, are stored and up-

dated for the next time step, written as

qtij;n ¼ exp½�knDwt�qt�Dt

ij;n þ ðgt2St

ij � gt�Dt2 St�Dt

ij Þ

� 1� exp½�kij;nDwt �kij;nDwt ð2:13Þ

qtkk;n ¼ exp½�knDwt�qt�Dt

kk;n þ ðgt2r

tkk � gt�Dt

2 rt�Dtkk Þ

� 1� exp½�kij;nDwt�kij;nDwt ð2:14Þ

The Perzyna model is used for the VP component. The PerzynaVP strain [16,17] for an isotropic material is written as

_emp;tij ¼ _kt @Fð�rt ; ktÞ

@rtij

ð2:15Þ

where _emp;tij is the VP strain rate at current time t and _kt is the mag-

nitude of the VP strain rate. The stress dependent yield function

Fð�rt;jtÞ is expressed in terms of the effective stress �rt ¼ffiffiffiffiffiffiffiffiffiffiffiffi32 St

ijStij

q,

where Stij is the deviatoric stress components; the accumulated VP

strain jt ¼R t

0_jsds; _jt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23

_emp;tij

_emp;tij

q. The normal direction at the

stress point on the yield surface F is derived as vector @F=@rij whichis the direction of the VP strain rate. The VP yield function based onan overstress function for isotropic hardening materials at currenttime t is:

Fð�rt ; ktÞ ¼ �rt � roy � hkt ð2:16Þ

The parameter roy is the initial yield stress measured from the

uniaxial loading. The hardening material parameter, h, can also de-pend on the current effective stress �rt .

In this study, the following form for the plastic multiplier isused:

_kt ¼ 1gp

UðFÞh i ð2:17Þ

where gp is the viscosity constant during the viscoplastic deforma-tion, <> represents the Macauley bracket, and the function U(F) isgiven as:

UðFÞ ¼�rt � ro

y � hjt

roy

" #n

ð2:18Þ

The power n is the material constant that needs to be calibratedfrom experiments. The incremental form of the VP strain compo-nent is summarized as (see Kim and Muliana [8] for a detaileddiscussion):

UðFÞ ¼ �rt�DtþD�rt�roy�hjt�Dt�hDjt

roy

h in

Dkt ¼ Dtgp

�rt�DtþD�rt�roy�hjt�Dt�hDjt

roy

h in� �

Demp;tij ¼ Dt

gp

�rt�DtþD�rt�roy�hjt�Dt�hDjt

roy

h in� �

32�rt ðdikdjl � 1

3 dijdklÞStkl

ð2:19Þ

When the stress components are prescribed, the correspondingincremental VE and VP strains can be immediately calculated andthe total strain is then obtained from Eq. (2.1). In the case the straincomponents are prescribed like in a displacement based FE it isnecessary to calculate the current total stresses and VP strain com-ponents. The VP strains, however, depends on the current totalstresses. For this purpose, the trial incremental VP strains aredetermined from the stresses at the previous time:

Demp;tij ¼

Dtgp

�rt�Dt � roy � hjt�Dt

roy

" #n* +3

2�rtdikdjl �

13

dijdkl

� �St�Dt

kl

ð2:20Þ

The trial incremental stresses are now written asDrt;tr

ij ¼ Cijkl Detij � Demp;tðtrÞ

ij

and the total trial stresses are

rt;trij ¼ rt�Dt

ij þ Drt;trij .

Based on the trial stresses, the incremental VE and VP strainsare calculated using Eqs. (2.8) and (2.19), respectively. Finally,the residual incremental strains and plastic multiplier aredetermined:

Retij ¼ Deme;t

ij þ Demp;tij � Det

ij ð2:21Þ

Rkt ¼ Dkt � Dtgp

�rt�Dt þ D�rt � roy � hjt�Dt � hDjt

roy

" #n* +ð2:22Þ

The Perzyna model depends on the plastic multiplier Dkt , whichat current time remains as an unknown variable. To determine thetotal stress in the VE–VP model, we need to minimize each compo-nent of the residual tensors, Ret

ij and Rkt .This study uses the Newton–Raphson iterative method in order

to minimize the residual components. Once the convergence isachieved, the consistent tangent stiffness matrix is calculated:

Ctijkl ¼

@Retij

@Drtkl

" #�1

;@Ret

ij

@Drtkl

¼ Stijkl þ

23

Dkt

ð�rtÞ2I0ijmn �rtI0mnkl �

32�rt

StmnSt

pqI0pqkl

� �ð2:23Þ

where I0ijkl ¼ dikdjl � 13 dijdkl.

Fig. 3.1. Unit-cell micromechanical model for FRP composite.

Fig. 3.2. FE meshes of micromechanical models with detailed fiber arrangements.

40 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

3. Micromechanical models for FRP composites

This section presents two micromechanical models for unidi-rectional FRP composites undergoing VE–VP deformations. Thefirst micromechanical model is based on a unit-cell (UC) consisting

of one fiber and three matrix sub-cells (Fig. 3.1). The first sub-cellrepresents a fiber constituent, while sub-cells 2, 3, and 4 representa matrix constituent. The total volume of unit-cell is equal to one,and each volume of the subcell depends on the volume fraction offibers. The fibers are assumed to be transversely isotropic and lin-

Table 4.1Mechanical properties of carbon fibers.

Cabon fiber E11 (GPa) E22 = E33 (GPa) G12 = G13 (GPa) G23 (GPa) m12 = m13 m23

(T300) 230 10.4 27.3 3.08 0.256 0.3

Table 4.2Comparisons of the effective elastic moduli.

Vf (volume fraction)(%)

10 20 50

Fiber array Effective elastic modulus(GPa)

Difference%

Effective elastic modulus(GPa)

Difference%

Effective elastic modulus(GPa)

Difference%

E11 Unit cell 27.08 0 49.63 0 117.27 0Detailed FE [uniform] 26.49 2.19 48.46 2.35 112.23 4.49Detailed FE [random] 26.41 2.47 48.32 2.63 114.02 2.77

E22 Unit cell 5.22 0 5.64 0 7.13 0Detailed FE [uniform] 5.21 0.15 5.62 0.43 6.99 2.00Detailed FE [random] 5.20 0.33 5.59 0.96 6.98 2.10

Fig. 4.1. Effective elastic moduli along and transverse fiber directions.

J. Jeon et al. / Computational Materials Science 70 (2013) 37–50 41

ear elastic, while the matrix is assumed to be isotropic with VE–VPresponse. The outcome of the UC model is a homogenized VE–VPresponse of FRP composites and is compatible with a displacementbased FE. The second micromechanical model of the FRP, compris-ing of longitudinal fibers dispersed in a homogeneous VE–VP poly-meric matrix, is generated for the FRP composites at different fibervolume contents using FE, termed as the detailed FE microstruc-tural models (Fig. 3.2). The dimension of the fibers in the detailedFE microstructural model is kept constant. The VE–VP response

from the two micromechanical models are compared and numeri-cally examined.

3.1. Unit-cell formulation for FRP composites

The formulation of the UC model of a unidirectional FRP is ex-pressed in terms of the average stresses and strains in the subcells.The approximate (average) stresses and strains of the FRP areformed using a volume-averaging method:

Fig. 4.2. Stress contours of composites with 10% fiber volume content under uniaxial stress 10 MPa (a) Vf = 20%, uniform fibers (b) Vf = 20%, random fibers (c) Vf = 50%, uniformfibers and (d) Vf = 50%, random fibers.

42 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

�rtij ¼

1V

X4

a¼1

ZV ðaÞ

rðaÞ;tij ðxðaÞk ÞdV ðaÞ � 1

V

X4

a¼1

V ðaÞrðaÞ;tij i; j ¼ 1;2;3 ð3:1Þ

�etij ¼

1V

X4

a¼1

ZV ðaÞ

eðaÞ;tij ðxðaÞk ÞdV ðaÞ � 1

V

X4

a¼1

V ðaÞeðaÞ;tij i; j ¼ 1;2;3 ð3:2Þ

where rtijðxkÞ and et

ijðxkÞ are the components of stress and strainfields in the representative UC model. The stresses and deforma-tions are assumed spatially uniform in each sub-cell; thus thestress rðaÞ;tij and strain eðaÞ;tij are the average stress and strain ineach sub-cell at current time. �rt

ij and �etij indicate the effective

stresses and strains at current time. The unit-cell volume isV ¼

P4a¼1V ðaÞ.

In order to relate the effective stress and strain to the stress andstrain in each subcell, a concentration matrix is formulated. In thisstudy, the concentration matrices (B(a),t) are formulated to relatethe effective strains of the UC to the strains in the sub-cells:

eðaÞ;tij ¼ BðaÞ;tijkl�et

kl ð3:3Þ

The linearized constitutive equation in each sub-cell (Section 2)is used to determine the stresses in each subcell:

rðaÞ;tij ¼ CðaÞ;tijkl eðaÞ;tkl ¼ CðaÞ;tijkl BðaÞ;tklrs�et

rs ð3:4Þ

where C(a),t is the consistent tangent stiffness matrix of the sub-cell(a) at current time t. Substituting Eq. (3.4) into Eq. (3.1), the effec-tive stress is expressed as:

�rtij ¼

1V

XN

a¼1

V ðaÞCðaÞ;tijkl BðaÞ;tklrs�et

rs ð3:5Þ

The unit-cell effective tangent stiffness matrix Ct at time t is ex-press as follows:

Ctijrs ¼

1V

XN

a¼1

V ðaÞCðaÞ;tijkl BðaÞ;tklrs ð3:6Þ

A constraint for the concentration matrix is determined fromthe equilibrium and compatibility conditions. In this study, we alsoassume perfect bonds at the interfaces between the subcells andwe satisfy the traction and displacement continuity conditions. De-tailed discussion on the formulation of the concentration matrixand micromechanical relations can be found in Muliana and Haj-Ali [13] and Jeon and Muliana [7]. The linearized micromechanicalrelations, Eqs. (3.1)–(3.6), are only valid when the constituents in

Fig. 4.3. Stress contours of composites with 20% and 50% fiber volume contents loaded in the transverse fiber direction.

Table 4.3Comparisons of the effective elastic moduli for unit-cell and detailed micromechanical models with uniform fiber arrangements.

Loading direction Axial loading [E-Moduli] Transverse loading [E-Moduli]

Model Unit-cell (GPa) Detailed FE (GPa) % Difference Unit-cell (GPa) Detailed FE (GPa) % Difference

Vf 40 94.72 92.42 2.49% 6.60 6.49 1.58%Vf 54 126.63 123.55 2.49% 7.37 7.18 2.57%Vf 67 154.85 151.02 2.53% 8.11 7.92 2.44%Vf 75 173.63 169.33 2.54% 8.64 8.44 2.42%

J. Jeon et al. / Computational Materials Science 70 (2013) 37–50 43

the UC model exhibit a linear elastic response. Due to the VE–VPresponse of the polymeric matrix, the linearized micromechanicalrelations result in the following residual vector:

fRtg ¼ ½At �

Deð1Þ;t

Deð2Þ;t

Deð3Þ;t

Deð4Þ;t

8>>><>>>:

9>>>=>>>;�

D0

� �fD�etg ð3:7Þ

The above linearized relation is obtained by imposing the microme-chanical relations and the constitutive relation for each subcell. Thecomponents of At; D depend on the subcells’ volume contents V(a)

and consistent tangent stiffness matrix C(a),t, which are formed by

imposing the micromechanical relations. Comparing Eqs. (3.3) and(3.7), the strain concentration tensors are:

Bð1Þ;t

Bð2Þ;t

Bð3Þ;t

Bð4Þ;t

8>>><>>>:

9>>>=>>>;¼ ½At ��1 D

0

� �ð3:8Þ

The above linearized relation is used as a starting point withineach time increment and an iterative corrector (Newton–Raphson)scheme is formulated to minimize errors from the linearizationsuch that both micromechanical constraints and nonlinear consti-tutive equations are satisfied. In order to minimize the residual, an

Fig. 4.4. Effective elastic moduli of FRP composites with different fiber volume contents: (a) along the fiber direction and (b) transverse to the fiber directions.

Table 4.4CPU time for VE and VE + VP in homogeneous and heterogeneous models.

FE Models Vf (%) Total elements CPU time (s)

Fiber direction Transverse direction

VE Unit-cell model 10 1 2.1 75.420 1 2.0 75.650 1 1.6 168.7

Detailed FE model [uniform array] 10 73,728 41,170 (11 h) 534,776 (149 h)20 73,728 10,952 (3 h) 164,220 (46 h)50 137,490 25,724 (7 h) 20,632 (6 h)

VE + VP Unit-cell model 10 1 1.9 78.620 1 1.9 79.750 1 1.8 3836.6

Detailed FE model [uniform array] 10 73,728 92,753 (26 h) 110,59,224 (128 days)20 73,728 137,217 (38 h) 20,95,620 (24 days)50 137,490 24,582 (7 h) 35,39,790 (41 days)

44 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

iterative correction scheme is formulated at the micromechanical(global) and constitutive model (local) levels. The numericalscheme is briefly summarized as follows:

1. Input variables: D�etij; �rt�Dt

ij ; Histt�Dt

2. Determine trial stress: D�rt;trij ¼ Ct

ijklD�etij; �rt;tr

ij ¼ �rt�Dtij þ D�rt;tr

ij ;�rt;0

ij ¼ �rt;trij

3. Iterate for k = 0,1,2, . . . (k = iteration counter)3.1. Calculate DrðaÞ;t;ðkÞij ; DeðaÞ;t;ðkÞij ; DemeðaÞ;t;ðkÞ

ij ; DempðaÞ;t;ðkÞij ; DkðaÞ;t;ðkÞ;

D�rt;ðkÞij ; D�et;ðkÞ

ij

3.2. Define residual at the micromechanics levelRt;ðkÞ

ij ¼ f DrðaÞ;t;ðkÞij ; D�et;ðkÞij

and check for kRt;ðkÞk 6 Tol; yes

then go to 4 else

3.3. Correct the incremental stress DrðaÞ;t;ðkþ1Þij ¼ DrðaÞ;t;ðkÞij

þ@Rt;ðkÞ

ij

@DrðaÞ;t;ðkÞkl

� ��1

Rt;ðkÞkl and go to 3.1

4. Output variables: use Eq. (3.1) to determine D�rtij and

�rtij ¼ �rt�Dt

ij þ D�rtij; Histt

The history variables Histt consist of the viscoelastic and visco-plastic strains and consistent tangent stiffness for each subcell.

3.2. Detailed FE micromechanical models for FRP composites

In FRP composites, fibers are not necessarily of the same sizenor they are positioned regularly in the matrix medium. In order

(a)

(b)

Fig. 4.5. Creep-recovery for VE in fiber direction loading.

J. Jeon et al. / Computational Materials Science 70 (2013) 37–50 45

to examine the effects of fiber arrangements in the polymeric ma-trix, microstructural geometries of FRP composites are generatedusing FE. The detailed FE microstructural models describe the con-figuration of fibers and matrix. Fig. 3.2 shows the micromechanicalmodels with microstructural details for composites with 10%, 20%,50% fiber volume fractions. For each fiber volume fraction, twokinds of microstructural configurations, i.e., uniform and randomfiber distributions, are considered. The effects of different fiberarrangements on the overall time-dependent response andstress/strain field of the composites are studied. In addition, FEmicromechanical models are considered for composites with 40%,54%, 67%, 75% fiber volume contents with uniform fiber arrange-ments. The fibers in the micromechanical models with detailed fi-ber arrangements have the same diameter. The detailedmicromechanical models are generated using 3D continuum ele-ments. These micromechanical models have a total of 73,728–247,290 elements. A convergence study is conducted in order todetermine sufficient numbers of elements in each micromechani-cal model with detailed fiber arrangements. The numbers of fibersin the cubic RVE depend on the volume fraction of fiber. Like in theUC model, a linear elastic material response is used for the fibersand the VE–VP material response (Section 2) is considered forthe polymeric matrix. The interfaces between fibers and matrixare bonded perfectly. Incorporating detailed fiber arrangementsin a homogeneous matrix allows capturing the variations in thefield variables and localized stresses within the composite micro-

structures. The ultimate goal is to examine the effect of variationsin the field variables and localized stresses on the overall (average)time-dependent and inelastic response of FRP composites.

4. Comparison of the UC model and detailed FEmicromechanical models

The overall response obtained from the UC model, having foursubcells (Fig. 3.1), is compared with the one of the micromechan-ical models having detailed fiber arrangements generated usingFE (Fig. 3.2). Carbon fiber is used for the fiber constituent and thehigh-density polyethylene (HDPE) is used for the matrix. The linearelastic properties of the carbon fibers (T300) are given in Table 4.1,which are reported by Miyagawa et al. [12]. The properties of theHDPE are reported by Kim and Muliana [8].

First, the effective linear elastic moduli of the FRP compositeswith 10%, 20%, 50% fiber volume fractions generated from the UCand detailed FE micromechanical models are compared, whichare summarized in Table 4.2 and illustrated in Fig. 4.1. The re-ported moduli are along the fiber longitudinal direction and trans-verse to the fiber direction. It is seen that the effective elasticmoduli of the FRP composites obtained from the unit-cell and FEheterogeneous micromechanical models with uniform and randomfiber array are comparable. The differences (in percent) of theeffective elastic moduli of the two models, shown in Table 4.2,slightly increase with increasing the fiber volume contents from

(a)

(b)

Fig. 4.6. Creep-recovery for VE in transverse direction loading.

46 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

10% to 50%. The reason of the increasing differences is probablydue to interaction between fibers that could lead to stress–concen-tration effects. In the simplified unit-cell model, fiber is assumedfully surrounded by the matrix and interactions between fibersare not considered. In the composites with detailed microstruc-tural arrangements, the spacing between fibers decreases as the fi-ber volume contents increase, which can result in localized stressesas reported by Muliana and Sawant [14]. The uniform fiberarrangements for the composites with 10% and 20% volume frac-tions show less difference than the random fiber arrangements.The spacing between fibers in the composite models with randomfiber arrangements is irregular that can result in higher possibilityof the stress concentration due to narrow space between the fibers,which is more realistic microstructural representations. Fig. 4.2shows the effective stress (von Misses stress) fields of a compositewith 10% fiber volume content loaded along and transverse to thefiber axis. In both loadings, a uniaxial stress 10 MPa is applied. It isseen that when the composite is loaded in the transverse fiberdirection, more variations of the stress fields in the matrix are ob-served. However, for the composites with 10% fiber volume con-tent, the variations are relatively small. Fig. 4.3 illustrates theeffective stress contours for composites with 20% and 50% fibervolume contents loaded in the transverse fiber direction. As fibervolume content increases, the spacing between fibers decreases,resulting in higher stresses localized in these regions.

The axial and transverse elastic moduli of the detailed micro-mechanical models for 40%, 54%, 67%, 75% fiber volume contentsof FRP composites with uniform fiber configuration are shown inTable 4.3. The moduli of the FRP composites obtained from the de-tailed micromechanical models are comparable to the ones ob-tained from the unit cell model with relatively small differences.Fig. 4.4 shows the overall elastic moduli for all studied volume con-tents generated from the unit-cell model and detailed microme-chanical model with uniform fiber arrangements. It is seen thatthe effective elastic moduli from the two micromechanical modelsare in good agreement, indicating a minimum effect of the local-ized stresses on the overall elastic moduli of FRP composites.

Next, the creep-recovery response under a constant uniaxialstress of 10 MPa applied along the longitudinal fiber direction isconsidered. The creep is done for 1800 s followed by 1800-s recov-ery. The creep-recovery loading with stress of 3 MPa is consideredin the transverse fiber directions in order to avoid the high local-ized stresses in the matrix between the fibers. The VE–VP re-sponses obtained from the two micromechanical models arecompared. The UC model is integrated to one continuum 3D ele-ment and the homogenized (overall) response is sampled at eachmaterial (Gaussian) integration point. In the case of micromechan-ical models with detailed fiber arrangements, the VE–VP constitu-tive models discussed in Section 2 are implemented at the materialpoints within the matrix finite elements. In order to examine the

(a)

(b)

Fig. 4.7. Creep-recovery for VE in along and transverse to the fiber direction loading.

Table 4.5Percent difference of the strains from the unit-cell and FE micromechanical models.

Fiber array Time (s) Loading direction Strain Difference %

(10% Vf) (20% Vf) (50% Vf)

Unit-cell model /detailed FE model VE 1800 Fiber e11 2.43% 2.49% 4.45%e22 2.05% 1.82% 5.52%

Transverse e11 1.67% 1.79% 4.86%e22 1.99% 2.03% 5.37%

VE + VP 1800 Fiber e11 2.42% 2.49% 4.46%e22 2.05% 1.82% 5.54%

Transverse e11 0.02% 6.31% 5.02%e22 0.31% 3.90% 5.69%

J. Jeon et al. / Computational Materials Science 70 (2013) 37–50 47

efficiency of the two-micromechanical modeling approaches, com-puting (central processing unit, CPU) times during the creep-recov-ery analyses for the composites with 10%, 20%, and 50% fibervolume contents are monitored, as reported in Table 4.4. TheCPU times are reported for the creep-recovery analyzes with a uni-axial stress applied in the fiber and transverse directions. As ex-pected the computing time required in the UC model is muchless than the one of the FE micromechanical models with micro-structural details. It is noted that the micromechanical modelswith the combined VE–VP response results in much higher com-puting time than when only the VE response is considered, whichis due to more iterations needed in solving the VE + VP response.

The corresponding strains during the creep-recovery loadingsfor the composites with fiber volume contents 10–75%, when theVE is considered for the polymeric matrix, are shown in Figs. 4.5–4.7. The strains determined from the UC and micromechanicalmodels with detailed fiber arrangements are relatively close witha percentage difference at time 1800 s is given in Tables 4.5 and4.6. The detailed micromechanical models with uniform fiberarrangements are considered. When the composite is loaded alongthe axial fiber direction, the fiber carries the majority of the loadsand the time-dependent behavior of the matrix is insignificant. Onthe other hand, for loading in the transverse fiber direction the ma-trix carries relatively high mechanical loading, resulting in

Table 4.6Percent difference of the strains from the unit-cell and FE micromechanical models with uniform detailed fiber arrangements.

Loading direction Model Vf 40 Vf 54 Vf 67 Vf 75

Axial loading e11 (mm/mm) (1800 s)Unit-cell 1.08E-04 8.01E-05 6.51E-05 5.79E-05Detailed FE 1.11E-04 8.21E-05 6.68E-05 5.94E-05% Difference 2.53% 2.54% 2.53% 2.53%

Transverse loading e22 (mm/mm) (1800 s)Unit-cell 1.73E-03 1.26E-03 9.17E-04 7.27E-04Detailed FE 1.77E-03 1.34E-03 9.87E-04 7.58E-04% Difference 2.47% 6.05% 7.12% 4.10%

Number of elements 244,590 240,630 234,360 247,290

Fiber arrangement 12 � 12 13 � 15 15 � 16 15 � 18

Arrangement ratio 1.000 0.867 0.938 0.833

Fig. 4.8. Stress contours of composites with 54%, 67% and 75% fiber volume contents loaded in the transverse fiber direction for creep behavior at 1800 s.

48 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

pronounced time-dependent response. The effective stress con-tours for composites with 54%, 67%, and 75% fiber volume contentsloaded in the transverse direction under an overall stress level3 MPa are illustrated in Fig. 4.8. The contours are obtained at1800 s. It is seen that localized stresses are shown in the compos-ites. These localized stresses are higher than the nominal stressprescribed on the boundary of the micromechanical models. Thestress-dependent viscoelastic constitutive model for the matrix re-sults in higher strain response which significantly increases thelocalized stresses in the composites. The detailed micromechanicalmodels are capable in capturing the detailed variations of thestress and strain fields, while the UC model is limited in incorpo-

rating the different stress and strain fields in the matrix. It is notedthat in the UC model, the variations of the stress and strain fields inthe matrix are captured by the three subcells and each subcell hasuniform stress and strain fields. Table 4.6 presents the percent dif-ferences in the overall strains determined from the UC and detailedmicromechanical models for composites with 40%, 54%, 67%, and75% fiber volume contents. It is seen that the highest differenceis for the composites with 67% fiber volume contents loaded inthe transverse fiber direction which is probably due to the signifi-cant effects of the localized stresses on the matrix dominated re-sponse. However, for the composite with 75% fiber volumecontent under the transverse loading, percent differences between

(a)

(b)

Fig. 4.9. Creep-recovery for VE and VE–VP response in fiber direction loading.

J. Jeon et al. / Computational Materials Science 70 (2013) 37–50 49

the responses obtained from the UC and detailed micromechanicalmodels decrease. This is probably due to the relatively dense fibersin that the linear elastic fibers would carry significant amount ofthe mechanical load.

Fig. 4.9 shows the comparisons of the strains in the UC andmicromechanical models with detailed fiber arrangements consid-ering VE and VE–VP behaviors under 10 MPa applied along the ax-ial fiber direction. The responses are obtained for composites with10%, 20%, and 50% fiber volume contents. Again due to relativelysmall amounts of stresses experienced by the matrix, insignificantVP deformations are observed as seen by the nearly similar VE andVE–VP responses. However, when the load 3 MPa is applied in thetransverse fiber direction, as shown in Fig. 4.10, the effect of time-dependent and inelastic response on the overall performance ofFRP becomes significant. The higher the fiber volume fraction is,the less the time-dependent and inelastic effects are, which shouldbe expected. Table 4.5 also presents percent difference in the cor-responding strains from the UC and FE microstructural models.Consistent with the elastic moduli, the differences (in percent) ofthe effective time-dependent and inelastic strains obtained of thetwo models generally increase with increasing the fiber volumecontents from 10% to 50%. It is noted that for the 20% fiber volumefraction, the percent difference in the effective time-dependentand inelastic strains of two models are slightly larger than theone of the 50% fiber volume fraction which could be due toarrangement of fibers. Further investigation on understandingthe effect of microstructural arrangements on the overall response

of the FRP composites is necessary by varying the fiber distribu-tions and sizes, and conducting statistical analysis, which are notthe scope of this study. This study highlights that the localizedstresses within the microstructures of the composites can influ-ence the overall response of the composites.

5. Summary

The effective elastic moduli and VE–VP material responses cal-culated from the UC model have been compared with the onesfrom the detailed FE micromechanical models. The UC model is for-mulated based on an idealized and simplified microstructure ofFRP composites while the FE microstructural models incorporatemore realistic microstructural configurations. The overall (average)responses predicted by the two micromechanical models are rea-sonably close with some deviations are shown at high fiber volumecontents. In particular, for the lower volume fraction, the UC modelhas a good agreement with the detailed FE micromechanical mod-els due to the insignificant existence of the stress concentration(localized stress). The localized stresses within the microstructuresof the composites, which are more pronounced for compositeswith high fiber volume contents loaded along the transverse fiberdirection, can influence the overall response of the composites.The UC model is limited in capturing the localized stresses. TheUC model, however, is much more efficient in terms of the CPUtime used to complete the simulation than the FE microstructural

(a)

(b)

Fig. 4.10. Creep-recovery in transverse direction loading for VE–VP response.

50 J. Jeon et al. / Computational Materials Science 70 (2013) 37–50

models, which makes it suitable and appealing for performinglarge scale analyses of FRP composite structures.

Acknowledgment

This research is sponsored by the Air Force Office of ScientificResearch (AFOSR) under Grant FA 9550-10-1-0002.

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