Composites Science and Technology 68 (2008) 2792–2798

Contents lists available at ScienceDirect

Composites Science and Technology

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

Numerical generation of a random chopped fiber composite RVEand its elastic properties

Yi Pan, Lucian Iorga, Assimina A. Pelegri *

Mechanical and Aerospace Engineering, Rutgers, State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USA

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

Article history:Received 6 February 2008Received in revised form 12 May 2008Accepted 3 June 2008Available online 12 June 2008

Keywords:A. Short-fiber compositesB. Elastic propertiesC. Finite element analysis (FEA)C. ModelingC. Multiscale modeling

0266-3538/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compscitech.2008.06.007

* Corresponding author. Tel.: +1 732 445 0691; faxE-mail address: pelegri@jove.rutgers.edu (A.A. Pele

The elastic properties of random chopped fiber-reinforced composites (RAFCs) are of paramount impor-tance for their sound application in lightweight structures. Mass production of random chopped fiber-reinforced composite (RaFC) at a fraction of the cost of composite laminates establishes RaFCs as alternatecandidate materials for manufacturing lightweight components in the automotive industry. Nevertheless,understanding and modeling of their mechanical and fracture properties are still fields of active research,yet to be exhausted. In this paper, methods to generate an RVE for random fiber or particle reinforcedcomposites numerically are reviewed. A modified random sequential absorption algorithm is proposedto generate a representative volume element (RVE) of a random chopped fiber-reinforced composite(RaFC) material. It is assumed that the RVE represents the composite material within the framework ofelasticity. The RVE thus created is analyzed to obtain the mechanical properties of the composite materialby using finite element analysis (FEA). RVE generation uses both straight and curved fibers so as toachieve high fiber volume fractions (VFs) that are extremely difficult to obtain by using straight fibersalone. This work extends the capability of RVE generation of RaFCs to higher volume fractions, here35.1% is illustrated, which are in the range of values employed in industrial applications.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Significant efforts have been devoted toward the use of light-weight structures to increase energy efficiencies in various indus-trial and commercial sectors [1–4]. Fiber-reinforced compositeshave found numerous applications in aerospace industry for theirhigh specific strength and specific stiffness [5]. However, the costof traditional composite materials is also considerable. Randomchopped fiber-reinforced composites (RaFCs) have emerged aspromising alternative materials for lightweight structures due totheir low cost and mass production capabilities. Their potentialapplication in, for example, automotive industry has been docu-mented [1,2,4]. In order to expand their use, accurate materialcharacterization is required. The main difficulty in fully exploringthe capabilities of the RaFCs lies in the apparent impediment toeffectively model their geometry at the micro-level for high fibervolume ratios (35–40%). This difficulty becomes even more obviousat high aspect ratio (AR) fibers.

Among several notable methods to predict the constitutivebehavior of random-fiber composites the analytical homogeniza-tion method based on the Eshelby’s strain concentration tensor[6], where the fiber is treated as a second phase inclusion, enjoys

ll rights reserved.

: +1 732 445 3124.gri).

considerable support. Significant success has been enjoyed bytechniques based on the Mori-Tanaka’s mean field method [7].The aforementioned methods are employed to determine thematerial properties for unidirectional composites with fiber vol-ume fraction (VF) and aspect ratio similar to those of the randomcomposite analyzed. Orientation averaging, as developed byAdvani and Tucker [8], has to be performed in order to accountfor the random orientations. A different set of methodologies forthe characterization of RaFC response is based on the classical lam-inated plate theory (CLPT), as illustrated by ‘‘Laminated RandomStrand Method” proposed by Ionita and Weistman [9] for the anal-ysis of high volume fraction carbon fiber/urethane composites. Dueto their computational efficiency, such methods can readily be em-ployed in moving-window analyses to statistically assess the anal-ysis sample size effects on the simulated response of the material.However, this method does not provide any information on the mi-cro-mechanical stress–strain state. A third approach is enabled bythe availability of powerful computational hardware and advancedfinite element packages. It is possible to simulate the materialbehavior directly using direct 3D finite element analysis (FEA)[11–19] on a statistically representative geometric entity, referredto as representative volume element (RVE).

Of the aforementioned material characterization methods,namely, FEM, field based methods and quasi-analytical methods,the FEM based schemes are the most promising. The key issue of

Y. Pan et al. / Composites Science and Technology 68 (2008) 2792–2798 2793

the FEM approach is identification and generation of an RVE1

through which the effective homogeneous material properties arederived. The first definition of RVE is that an RVE is a statistical rep-resentation of material [20]. In this context, an RVE must be suffi-cient large such that it contains a large number of inclusions in aheterogeneous material, and the effective properties derived fromthe RVE represent the true material properties in macroscopic scale.As pointed out by Drugan and Willis [21], another pragmatic defini-tion of RVE is that ‘‘the smallest material volume element of thecomposite for which the usual spatially constant ‘overall modulus’macroscopic constitutive representation is a sufficiently accuratemodel to represent mean constitutive response”. Adopting the sec-ond definition of RVE, Drugan and Willis [21] showed that this def-inition yield small quantitative estimates of RVE size while incontrast to qualitatively large RVE size implied by the first definition.The determination of RVE size remains an open question.

In FEM analyses toward the determination of the effectivematerial properties, both of the above mentioned definitions comeinto effect. On one hand, the first definition guarantees the RVE islarge enough to represent the material macroscopically. On theother hand, a large RVE is computationally expensive and theRVE size should be minimized to increase the computational effi-ciency. In practical FEM simulation, which can be seen in worksby Iorga et al. [10], Gusev et al. [13], and Trias et al. [22] and Kariet al. [15], a series of various sizes of RVEs are generated and thematerial properties obtained from which are compared to deter-mine a relatively small size for the RVE within certain accuracy.

The remaining question is how the RVE micro-level geometryon which FEM analyses are performed is identified and generatednumerically. It depends on the microstructure of the material.While identification of an RVE (or UC) is immediate in woven fiber[23,24], and laminate [25] composites since they have a repeatablearchitecture. For composite materials with random inclusion phasearrangements the recognition and numerical construction of anRVE is not straightforward [13–16,18,26]. Knowledge of the shape,orientation distribution and/or location of the reinforcing fiber/particle is required a priori. Geometric shapes corresponding tothe reinforced phase particles are inserted in a volume box in sucha way that they follow the predefined location and/or orientationdistribution. In this paper, the composite is a two phase materialand the reinforced phase is referred to fiber.

There are three families of methods for numerically generatingRaFC (random-fiber composite) RVEs, namely, the random sequen-tial adsorption (RSA) schemes [15,16,26–29], the Monte Carlo pro-cedures [13,14,18] and image reconstruction technique [17]. AnRSA-type scheme sequentially adds fibers to a region by randomlygenerating its location and orientation angle. The new fiber is notallowed to intersect fibers previously accepted [15,16,26,30].Böhm et al. [16] employed a modified RSA approach to generateRVEs of metal matrix composites (MMC) with 15% volume fraction(VF) reinforced by random short fibers with aspect ratio (AR) equalto five (AR = 5) and by random spherical particles. Fibers or parti-cles were not allowed to overlap and the minimum distance be-tween neighboring fibers or particles was set at 0.0075 times theside length of the RVE. Geometrical periodicity was also applied,that is, parts of fiber that exceeded the surfaces of the cube arecut and shifted so that they enter the cube from the opposite sur-faces. Tu et al. [26] also used a similar RSA approach to generateRVEs for studying the thermo-conductivity and elastic modulusof composites reinforced by fibers (AR = 7) and spherical particles.Pan et al. [30] applied a modified RSA algorithm for a random-fiber

1 Note that, there is no consensus on the terminology used for the RaFCs in theliterature. This statistical depiction is referenced as a representative volume elemen(RVE), Kari et al. [15], Gusev [13,14], or unit cell (UC), Böhm et al. [16], andDuschlbauer et al. [18].

t

composite with composite with fiber AR = 10 and 13.5% fiber vol-ume fraction. We note that the volume fraction achievable throughRSA is much smaller than that predicted by existing analytical andphenomenological models. This phenomenon is called ‘‘Jamming”[29]. To overcome the jamming problem and to achieve higher vol-ume fraction, Kari et al. [15] used different sizes of fibers by depos-iting them inside the RVE in a descending manner. They firstdeposited the largest aspect ratio fibers and after reaching the jam-ming limit, then deposited the next largest possible aspect ratiofibers in the RVE. However, such an approach is not applicablefor composites with fibers of fixed aspect ratio.

Monte Carlo methods are two-step procedures that start from aconfiguration with arbitrary fiber locations and orientations withina large box and rearrange the fibers’ locations and orientation,without accepting intersections, until a predetermined desired ori-entation state is reached. Finally, it decreases the size of the box to-ward the designated fiber volume fraction without altering theorientation of the fibers. Intersection of fibers is not allowed ateither step during the process [13,14,18]. The MC procedure hasbeen extensively used by Gusev and his coworkers [13,14,31]and Duschlbauer et al. [18] for studying the material propertiesof short fiber and particle reinforced composites. A ‘‘jamming” lim-it similar to that occurring in RS algorithms was also observed inthis method. The fiber volume fraction of the RVE generated byMC was up to 21% for fiber aspect ratio 10 [18]. It is noted thatfor larger fiber aspect ratio, the maximum achievable volume frac-tion is even smaller.

A recent advancement in RVE generation is employing X-raytomography and 3D image analysis and reconstruction methodsas demonstrated for wood-based fiberboards [17]. In this method,microtomographic images of the material’s microstructure are ob-tained with a synchrotron radiation tomograph. 3D fiber networkare reconstructed and generated based on the X-ray absorptionradiographs of a sample using specialized software. However, theuse of such an approach for generating numerical representationsof carbon or glass random composites has yet to be demonstrated.

The main drawback of the RSA or Monte Carlo procedures is thesmall fiber volume fraction that can be achieved. Consequently,they cannot be applied directly for the study of composites withstraight fibers of fixed aspect ratio. This calls for a solution to over-come the jamming limit and in the present work we introduce amethod to account for fiber kinks that occur in high fiber volumefraction composites by making use of curved fiber geometries inthe RVE generation algorithm. This allows for a significant increasein the achievable fiber VF, close to values of 35–40%, representativeof typical production composites [32]. Such figures have so farproved elusive using the geometry generation approaches cur-rently employed in the literature. In the second part of the studywe employ a homogenization scheme to estimate the macro-scalematerial properties of the simulated material.

The RVE is generated by using an RSA-inspired scheme, whichaccepts new fibers in the volume according to the fiber geometry(aspect ratio 20) and orientation. The scheme is modified toaccount for curved fibers in order to achieve high fiber volumefractions (35.1% in the present study). For similar values of theaspect ratio, existing analytical [33,34], experimental [35] andnumerical [28] models predict a smaller maximum achievablevolume fraction (18.5–30%) for RVEs employing straight cylindricalfibers. The numerical results are compared and validated usingHalpin–Tsai’s analytical method [36].

2. Generation of 3D RVE using modified RSA

The generation of 3D representative volume element (RVE) of arandom chopped fiber composite requires the development of a

Fig. 2. Dodecagonal approximation of fiber bundle’s elliptical cross-section withmajor axis 2a and minor axis 2b.

2794 Y. Pan et al. / Composites Science and Technology 68 (2008) 2792–2798

comprehensive algorithm in order to capture the ‘‘random” and‘‘curved” characteristics of fibers existing in the real material. Theprotocol to generate the RVE is summarized in Section 2.1. Pene-tration of any two fibers is not allowed physically and that isreflected in the protocol. The mathematical and algorithmic imple-mentation of 2D and 3D fiber penetration tests is discussed inSections 2.2 and 2.3, respectively.

2.1. RVE generation protocol

In the study of particle packing and particle reinforced compos-ites the random sequential adsorption (RSA) technique has beenused for spheres [16,26,28,37], spherocylinders [16,28], ellipsoidsand rods [15,16,26,28]. For short fiber-reinforced composites withsmall aspect ratio (<10) and low volume fraction (<25%), the fibers,or fiber bundles, can be modeled as straight cylinders. The relationbetween the fiber aspect ratio and the maximum achievable fibervolume fraction has been studied by Evans and Gibson [33], Park-house and Kelly [35], Toll [34] and Williams and Phillipse [28],among others. Their results indicate that increase of the fiberaspect ratio results in decrease of the maximum achievable fibervolume fraction. For a fiber with aspect ratio 20, these models pre-dict maximum fiber volume fraction of 20% by Evans and Gibson[33], 30% by Parkhouse and Kelly [35], 18.5% by Toll [34], and27% by Williams and Phillipse [28]. However, the above predictedvolume fractions are relatively small compared to the values ob-served in industrial applications (35–40%) [32,38]. To increasethe fiber volume fraction, the bends of the fiber bundle has to beaccounted for. Based on our knowledge about the manufacturingprocess – during which E-glass fiber bundles (also referred to as fi-bers in this text) are chopped and sprayed into a mold – as well ason optical observations of material specimens, we assume that thefiber is uniformly distributed in the x–y plane and that the fibershave an elliptical cross-section. The in-plane angle of a fiber is inthe range of (0, 2p) while the location of its midpoint is randomlygenerated within a 3D box with uniform probability. The portion ofa fiber that is outside the box will be trimmed.

The in-plane (x–y) and through thickness (along z-direction)views of the E-glass/epoxy composite specimen (3 mm in thick-ness) are shown in Fig. 1a and b. The crossing fibers in the high-lighted region of Fig. 1a illustrate fibers that bend at the crossingregion with other bundles because they cannot physically pene-trate each other. After the crossing, the two fibers can continueto be lying within the same layer again (as shown in the high-lighted region.) Fig. 1a also indicates the irregular orientation ofthe fibers. The elliptical shape of the fibers crosssection is demon-strated in Fig. 1b. It is further demonstrated that the majority ofthe fibers are lying in a similar manner with their minor axis point-

Fig. 1. (a) In-plane and (b) through thickness views of the random chopped E-glass fibeintersecting (highlighted region). In (b) the elliptical cross-section of fibers lying with th

ing in the z-direction. Therefore, we assume the fibers can be eithercurved or straight, as well as, that a fiber can only bend into andout of the layer such that the project ion of the fiber on to x–y planeremains straight.

Based on the above observations and assumptions, the spine ofthe fiber is approximated by connected straight line segmentswhile its elliptical cross-section is approximated by a dodecagoninscribed to an ellipse with major axis 2a and minor axis 2b, seeFig. 2. This way, a fiber is generated by sweeping the cross-sec-tional profile along the spine. The protocol to generate the RVE isdescribed next:

(1) A fiber-rich sub-layer of 2b thickness (bundle’s minor axis)accommodates straight fibers and portions of the curvedfibers (other portions are with in a neighboring sub-layer).A thin matrix-rich sub-layer with the thickness of b/10 isintroduced to separate fiber-rich sub-layers.

(2) A fiber-mat layer has two fiber-rich sub-layers and threematrix-rich spacing sub-layers, as shown in Fig. 3. Thecurved fibers can bend away from one layer into the oppositesub-layer and bend back to the previous sub-layer whennecessary (for example, to avoid intersecting other existingfibers). In order to have a fiber population sufficiently large,three fiber-mat layers are stacked up to obtain the finalRVE, as shown in Fig. 3, ensuring that the two largest eigenvalues of the second order orientation tensor [8] for all fibersin the three layers are as close to 0.5 as possible. We note

r-reinforced composite. The highlighted region in (a) shows the fiber curve to avoideir minor axis pointing along the z-direction.

Fiber-rich layer Matrix-rich layer

b/10

2b

Fiber-mat-1

Fiber-mat-2

Fiber-mat-2

Fig. 3. Layer arrangement of the RVE. The RVE consists of three fiber-mat layerswhile each fiber-mat layers consists of two fiber-rich sub-layers and three matrix-rich sub-layers. The thickness of the fiber-rich and matrix-rich sub-layers are equalto 2b, and b/10, respectively, where b is the semi-minor axis of the ellipse.

IB

IAS1

S2

Top fiber-rich sub-layer

Fiber A Fiber B

r_a

(ξA=0) (ξA=1)

(ξB=0)

P1 P2

P3

P4

PA(ξA)

PB(ξB)

Δφ

nA

nB

(ξB=1)

IA

IB

y

x

x

z

a

b

Bottom fiber-rich sub-layer

Fig. 4. Schematic illustration of side (a) and top (b) views of intersecting fibers. (a)One fiber over-crossing another (side view). The generic lines of two fibers, A and B,originally intersect at point IB on the bottom layer. This point of fiber A was movedto the top sub-layer (noted as IA) and two side points S1 and S2 were also assignedto fiber A introduce a bend to avoid intersecting fiber B. (b) Relative position of twopotentially intersecting fibers A and B at points IA and IB on the projection x–y plane(top view).

Y. Pan et al. / Composites Science and Technology 68 (2008) 2792–2798 2795

that a second order fiber orientation tensor with eigenvalues(0.5, 0.5, and 0) corresponds to an ideally transversely iso-tropic RVE [14].

(3) The fiber is generated randomly in these two fiber-rich sub-layers for each fiber-mat layer. Initially, the spine of a fiber isrepresented by a line segment with center point coordinates(x, y, and z) and orientation (u, h), where u e (0, 2p) being thein-plane angle and h being the out-of-plane angle, which isequal to zero in this paper. The z coordinate can be either0 or t, with t being the sum of the thickness of a fiber-richsub-layer and a matrix-rich sub-layer.

(4) The fiber can be straight or curved. Straight fibers resideentirely in either the top or bottom fiber-rich sub-layers. Ifa newly generated fiber potentially intersects existing fibers,the new one has to bend away from the existing fiber to theopposite sub-layer to avoid intersection. Thus, the fiberbecomes a curved one. The curved fiber ‘‘interweaves” thestraight fibers within the fiber-mat layer.

(5) The generation of a curved fiber implies, first, the creation ofa straight line-segment (referred to as the generic line of thefiber) with random orientation and mid-point location. Allfibers previously included in the configuration are projectedon to the x–y plane and the minimum distance between thecurrent fiber and all the previous ones is computed. The pro-jection reduces the 3D geometric problem into a 2D one.Then, all the intersections on the generic lines are marked.Finally, the current generic line is ‘‘curved” by moving theintersecting points to the opposite sub-layer (i.e., by chang-ing the z coordinate of the intersecting point). Two addi-tional side points are added to each side of the shiftedpoint, one on the top and one on the bottom sub-layer, tosimulate the local bending of the fiber. This process gener-ates the spine of the fiber as illustrated in Fig. 4a.

(6) Each 3D fiber is obtained by sweeping the cross-sectionalprofile a long the spine. Each straight fiber is approximatedby a prism with two dodecagonal bases, while a curved fiberis approximated by a series of connected irregular polyhedra.In order to avoid interpenetration of fibers, a 3D intersectiontest is performed upon generation of a curved fiber. The dis-tance between each polyhedral segments of the newly gener-ated fiber and those of the existing ones is calculated and

checked so as not to exceed the minimum separation valueimposed. The mathematical implementation of the 2D and3D fiber penetration tests is described in Section 2.2.

The complete procedure for generating the RVE is outlined inthe flowchart in Fig. 5.

2.2. Mathematical implementation

In the following, the 2D fiber intersection will be discussed first,and second, the 3D intersection analysis algorithm will be pre-sented. For the 2D analysis, we will refer to Fig. 4b, which illus-trates the relative position of two fibers’ projection, A and B, onthe x–y plane. The length of the fibers has been normalized toone. Given one end point (~PE) and the unit direction (~n) of a fiber’sgeneric line segment, points on the line segment can be parameter-ized as

~PðxÞ ¼ ~PE þ nL~n; 0 6 n 6 1; ð1Þ

where L is the length of the fiber and the scalar n is the normalizeddistance measured along the segment from one end point. Thus, thedistance between two points on two different generic line segmentsis given by

dðn1; n2Þ ¼ k~P1ðn1Þ �~P2ðn2Þk; 0 6 n1 6 1 and 0 6 n2 6 1: ð2Þ

To determine the minimum distance between two fibers theconstrained on linear two-variable function d(n1, n2) described inEq. (2) needs to be minimized, i.e., solve

dmin 2D ¼minðdðn1; n2ÞÞ ð3Þ

subject to the constraints 0 6 n1 6 1 and 0 6 n2 6 1.

Error!Start

X, φ, Layer

Separation?(2D computation)

Save data and update variables

Vf < Vf_obj

Adjust the fiber to a Curved one

Separation?(3D computation)

Done!

Yes

No

Yes

Yes

No

Fig. 5. Flow chart of the RVE generation algorithm.

Fig. 6. RVE of the random chopped fiber-reinforced composite with curved fiberbundles in ABAQUS, fiber volume fraction of which is 35.1%: (a) RVE with curvedfibers, (b) fiber orientation distribution, and (c) mesh of the RVE with 4-nodetetrahedron elements.

2796 Y. Pan et al. / Composites Science and Technology 68 (2008) 2792–2798

For the 3D analysis, we note first that a straight fiber is repre-sented as a convex prism with two dodecagon end-faces, which acurve done consists of several convex irregular polyhedra. A con-vex polyhedron with s faces can be defined algebraically as theset of solutions to a system of linear in equalities:

A~x 6 b; ð4Þ

where A is a real s � 3 matrix, b is a real s � 1 vector and~x is a vec-tor corresponding to the point coordinates in 3D space [39]. TheLinear Matrix Inequality (LMI) in Eq. (4) can be computed fromthe vertices of the polyhedron. The distance between any twopoints from polyhedron A and polyhedron B is of the form:

dð~x1;~x2Þ ¼ k~x1 �~x2k; A1~x1 6 b1 and A2~x2 6 b2: ð5Þ

The minimum distance between two convex polyhedra can bedetermined by solving the convex optimization problem:

dmin 3D ¼minðdð~x1;~x2ÞÞ ð6Þ

subject to the constraints A1~x1 6 b1 and A2~x2 6 b2. The numericalsolution of the two convex optimization problems described inEqs. (3) and (6) can be obtained by applying the existing algorithmdescribed in next section.

2.3. Algorithm implementation

The RVE generation algorithm is implemented in Matlab [40].During the 2D geometry calculation, the constrained nonlinearoptimization function fmincon from the Optimization Toolbox iscalled to solve the problem described in Eq. (3). In the 3D case,the CVX modeling system developed by Grant et al. [41] for Matlabis employed to formulate and solve the convex optimization prob-lem described by Eq. (6). To choose the RVE size, we rely on resultsfrom previous studies for straight fiber RVEs with varying sizes[10] which suggest that L=‘ ¼ 2 yields satisfactory results, whereL and ‘ are the length of RVE and length of fibers, respectively. Sim-ilar results were also reported by Kari et al. [15] for transversely

randomly distributed short-fiber composites. In this work, we se-lect L=‘ ¼ 2 as the RVE size. The RVE has dimensions 80b �80b � 12.7b, where b is the semi-minor axis of the cross-sectionalellipse, and contains 174 straight and curved fibers with aspect ra-tio, ‘=2b ¼ 20. The cross section of a fiber is a dodecagon approxi-mation of an ellipse with a/b = 2, where a is the major semi-axis.Following the protocol, a RVE with fiber volume fraction of 35.1%is generated as shown in Fig. 6a. The in-plane fiber orientation dis-tribution is shown in Fig. 6b.

3. Finite element analysis results

Finite element analysis (FEA) is employed to determine thecomposite material’s linear elastic response. The FE model is built

Table 1Comparison of random-fiber composite in-plane elastic constants

FEM micro-mechanicalanalysis

Halpin–Tsai equations based empiricalpredictions [31]

E (GPa) 11.83

12.56G (GPa) 5.09

4.48

Fig. 7. Curved fiber formed by connected polyhedra.

Y. Pan et al. / Composites Science and Technology 68 (2008) 2792–2798 2797

and solved using the commercial FEA package ABAQUS [42], inwhich the Matlab geometry data is imported via the Python script-ing interface. It is assumed that the fiber and matrix are both linearelastic and isotropic and that they are perfectly bonded at theirinterface. Due to the in-plane random distribution of the fibers,the composite behaves, at the macroscopic scale, similar to ahomogeneous, transversely isotropic material. The condition forequivalence is formulated as the equality of the strain energy inthe two media subject to the same boundary conditions.

The determination of the equivalent homogeneous materialproperties implies a homogenization procedure (for detail see[38]) that requires the application of six independent loading con-ditions on the analyzed RVE. Each loading case consists of specify-ing displacement fields that render null all but one of the sixindependent components of the strain tensor. Resolving the stressfield in the heterogeneous material through a static equilibriumanalysis allows for the calculation of the average stress field com-ponents, �ri. Since the average strain �ej is imposed, the stiffness ten-sor components for the equivalent material, �C ij, can be directlycalculated from the generalized Hooke’s law:

�ri ¼ �C ij�ej; i; j ¼ 1;2; . . . ;6: ð7Þ

For the composite considered in this study, the fiber and matrixYoung modulus and Poisson ratios are Ef = 70 GPa and mf = 0.2,and Em = 3 GPa and mm = 0.35, respectively (note that the valuesprovided for E and m are generic values representative of the mate-rials used).

The meshing of the model is performed using ABAQUS’s built-infree meshing algorithm. The RVE geometry does not lend itself, dueto its complexity, to the use of hexahedra elements. Thus themeshing is performed with 4-node tetrahedron elements (C3D4in ABAQUS), with a total element count of 492,647, correspondingto 253,965 DOFs. The meshed RVE is shown in Fig. 6c.

Following the application of the homogenization procedure, theresulting material elastic constants of the homogeneous material�C ij are (in GPa)

�C ¼

14:47 5:06 3:84 0 0 05:06 14:22 3:87 0 0 03:84 3:87 9:49 0 0 0

0 0 0 5:09 0 00 0 0 0 3:06 00 0 0 0 0 3:10

2666666664

3777777775: ð8Þ

We note that the computed values of the off-diagonal entries in the4th–6th rows and columns of the matrix �C are two or three ordersof magnitude smaller than the other entries and, thus, can be con-sidered null for all practical purposes.

To investigate the in-plane isotropy, we employ the parameterdefined in Ref. [43]:

aXY ¼2�C44

�CXY11 � �C12

; ð9Þ

where

�CXY11 ¼

�C11 þ �C22

2ð10Þ

which takes the value aXY = 1 for a transversely isotropic materialwhose symmetry axis coincides with the z axis. Substituting the en-tries in �C from Eq. (8) into Eqs. (9) and (10), we have aXY ¼ 91:2%,confirming that the behavior of the generated RVE is close to that ofa transversely isotropic material.

Results validation is performed by comparing Young’s andShear moduli for the equivalent homogeneous material, computedusing the �C ij matrix obtained from previous section, against those

obtained using the traditional equations for short random-fibercomposites [36], based on the Halpin–Tsai estimations of the lon-gitudinal and transverse moduli of the equivalent aligned short-fi-ber composite. The results are listed in Table 1. Note that this resultis based on empirical relations and the model does not account forthe curved fibers, thus justifying the higher value for Young’s mod-ulus [9]. The higher value of the in-plane shear modulus for the FEAresult may be due to the interweaving regions formed by straightand curved fibers, which are stiffer when subjected to in-planeshear loading.

As mentioned before, the 3D volume of a fiber is generated bysweeping the cross-sectional profile along a fiber spine. In thecase of curved fibers, the spine is formed by a series of connectedline segments, with some of them oriented in out-of-plane direc-tions. Since the cross-sectional profile always remains normal tothe path during sweeping [42], the presence of the out-of-planespine segments leads to the creation of sharp edges in the regionof the two added side-points, as can be seen in Figs. 4(a) and 7.However, this is not true in the real sample, where a fibersmoothly bends away from another at the crossing region. Theartificial sharp turns of the fiber generated from the sweepingabout the connected line might introduce local stress concentra-tions which may influence the accuracy of the numerical solutionto a loading simulation. Thus, it may be desirable to avoid suchsharp ‘‘turns” in the fiber geometry. The influence of the localstress concentration on the overall RVE response is a subject ofon going research, as is the interrogation of the results for similar‘‘smooth” geometries.

4. Conclusions

By intermixing curved and straight fibers, the fiber volume frac-tion of the RVE can be increased drastically, comparable to valuesachieved in industrial-grade materials. This micro-geometry gen-eration technique allows for very detailed simulations of the re-sponse of high volume fraction RaFCs, which can capture andquantify effects at the micro-scale level.

The representativeness of the geometry generated using the de-scribed approach was investigated via a FE analysis based homog-enization approach that allows the characterization of the macro-

2798 Y. Pan et al. / Composites Science and Technology 68 (2008) 2792–2798

scale composite response. The finite element analysis results indi-cate that the generated RVE satisfactorily approximates the elastictransversely isotropic behavior of mostly in-plane randomchopped fiber-reinforced composites. These first results demon-strate that micro-scale geometries generated with the presentedalgorithm can be employed in in-depth studies such as the charac-terization of RaFC response beyond the elastic range or their dam-age initiation and propagation behavior.

Acknowledgements

This work was funded by NSF through the CMS-0409282 Grantand partially supported by the Department of Energy Cooperativeagreement No. DE-FC05-950R22363. Such support does not consti-tute an endorsement by the Department of Energy of the views ex-pressed herein. The authors would like to gratefully acknowledgethe support of NSF Program Manager Dr. Ken Chong, as well asthe insights and assistance of Drs. Libby Berger and Stephen Harrisof GM.

References

[1] Jacob GC, Starbuck JM, Fellers JF, Simunovic S, Boeman RG. Fracture toughnessin tandom-chopped fiber-reinforced composites and their strain ratedependence. J Appl Polym Sci 2006;100:695–701.

[2] Jacob GC, Starbuck JM, Fellers JF, Simunovic S, Beoman RG. Crashworthiness ofvarious rancom chopped carbon fiber reinforced epoxy composite materialsand their strain rate dependence. J Appl Polym Sci 2006;101:1477–86.

[3] Corum JM, Battiste RL, Ruggles MB, Ren W. Durability-based design criteria fora chopped-glass-fiber automotive structural composite. Compos Sci Technol2001;61:1083–95.

[4] Dahl JS, Smith GL, Houston DQ. The influence of fiber tow size on theperformance of chopped carbon fiber reinforced composites. In: materials andprocessing technologies for revolutionary applications, 37th ISTC conferenceproceedings. Seattle, Washington: 2005.

[5] Daniel IM, Ishai O. Engineering mechanics of composite materials. OxfordUniversity Press; 1994.

[6] Eshelby JD. The determining of the elastic field of an ellipsoidal inclusion andrelated problem. Proc Roy Soc Lond Ser A 1957;241:376–96.

[7] Mori T, Tanaka K. Average stress in matrix and average elastic energy ofmaterials with misfitting inclusions. Acta Metall Mater 1973;21:271–4.

[8] Advani SG, Tucker CL. The use of tensors to describe and predict fiberorientation in short fiber composites. J Rheol 1987;31:751–84.

[9] Ionita A, Weitsman WJ. On the mechanical response of randomly reinforcedchopped-fibers composites: data and model. Compos Sci Technol2006;66:2566–79.

[10] Iorga L, Pelegri AA, Pan Y. Numerical characterization of material elasticproperties for random fiber composites, in press.

[11] Garnich MR, Karami G. Finite element micromechanics for stiffness andstrength of wavy fiber composites. J Compos Mater 2004;38(4):273–92.

[12] Meraghni F, Desrumaux F, Benzeggagh MF. Implementation of a constitutivemicromechanical model for damage analysis in glass mat reinforcedcomposite structures. Compos Sci Technol 2002;62:2087–97.

[13] Gusev AA. Representative volume element size for elastic composites: anumerical study. J Mech Phys Sol 1997;45:1449–59.

[14] Gusev AA, Heggli M, Lusti HR, Hine PJ. Orientation averaging for stiffness andthermal expansion of short fiber composites. Adv Eng Mat 2002;4:931–3.

[15] Kari S, Berger H, Gabbert U. Numerical evaluation of effective materialproperties of randomly distributed short cylindrical fibre composites. ComputMat Sci 2007;39:198–204.

[16] Böhm HJ, Eckschlager A, Han W. Multi-inclusion unit cell models for metalmatrix composites with randomly oriented discontinuous reinforcements.Comput Mat Sci 2002;25:42–53.

[17] Faessel M, Delisée C, Bos F, Castéra P. 3D modelling of random cellulosicfibrous networks based on X-ray tomography and image analysis. Compos SciTechnol 2005;65:1931–40.

[18] Duschlbauer D, Böhm HJ, Pettermann HE. Computational simulation ofcomposites reinforced by planar random fibers: homogenization andlocalization by unit cell and mean field approaches. J Compos Mater2006;40:2217–34.

[19] Eason TG, Ochoa OO. Material behavior of structural reaction injection moldedcomposites under thermomechanical loading. J Compos Mater 2001;34:432.

[20] Hill R. Elastic properties of reinforce solid: some theoretical principles. J MechPhys Sol 1963;11:357–72.

[21] Drugan WJ, Willis JR. A mircomechanics-based nonlocal constitutive equationand estimates of representative volume element size for elastic composites. JMech Phys Sol 1996;44(4):497–524.

[22] Trias D, Costa J, Turon A, Hurtado JF. Determination of the critical size of astatistical representative volume element (SRVE) for carbon reinforcedpolymers. Acta Mater 2006;54:3471–84.

[23] Ivanov I, Tabiei A. Three-dimensional computational micro-mechanical modelfor woven fabric composite. Compos Struct 2001;54:489–96.

[24] Quek SC, Waas A, Shahwan KW, Agaram V. Compressive response and failureof braided textile composite: part 2 – computations. Int J Non-Linear Mech2004;39:649–63.

[25] Caizzo AA, Constanzo F. On the constitutive relations of materials withevolving microstructure due to microcracking. Int J Sol Struct2000;37:3375–98.

[26] Tu ST, Cai WZ, Yin Y, Ling X. Numerical simulation of saturation behavior ofphysical properties in composites with randomly distributed second-phase. JCompos Mater 2005;39(7):617–31.

[27] Torquato S. Random heterogeneous materials: microstructure andmacroscopic properties. In: Antman SS, Marsden JE, Sirovich L, Wiggins S,editors. Interdisciplinary applied mathematics, vol. 16. NewYork: Springer;2002.

[28] Williams SR, Philipse AP. Random packings of spheres and spherocylinderssimulated by mechanical contraction. Phys Rev E 2003;67. 051301-1-9.

[29] Widom B. Random sequential addition of hard sphere to a volume. J ChemPhys 1966;44(10):3888–94.

[30] Pan Y, Iorga L, Pelegri AA. Analysis of 3D random chopped fiber reinforcedcomposites using FEM and random sequential adsorption. Comput Mat Sci2008. doi:10.1016/j.commatsci.2007.12.016.

[31] Gusev AA, Hine PJ, Ward IM. Fiber packing and elastic properties of atransversely random unidirectional glass/epoxy composite. Compos SciTechnol 2000;60:535–41.

[32] Berger L. General motors. Personal communication; 2007.[33] Evans KE, Gibson AG. Prediction of the maximum packing fraction achievable

in randomly orientated short-fibre composites. Compos Sci Technol1986;25:149–62.

[34] Toll S. Packing mechanics of fiber reinforcements. Polym Eng Sci1998;38(8):1337–50.

[35] Parkhouse JG, Kelly A. The random packing of fibers in three dimensions. ProcMath Phys Sci 1995;451(1943):737–46.

[36] Agarwal BD, Broutman LJ. Analysis and performance of fiber composites. NewYork: John Wiley & Sons; 1990.

[37] Coelho D, Thovert JF, Adler PM. Geometrical and transport properties ofrandom packings of spheres and asperical particles. Phys Rev E1997;55(2):1959–78.

[38] Pan Y, Iorga L, Pelegri AA. Analysis of 3D random chopped fiber reinforcedcomposites using FEM and random sequential adsorption. Comput Mater Sci2007.

[39] Weisstein EW. Convex polyhedron. From mathworld – a Wolfram webresource; 2002, <http://mathworld.wolfram.com/ConvexPolyhedron.html>.

[40] The Mathworks, Matlab, The Language of Technical Computing, Version 7.5;2007.

[41] Grant M, Boyd S, Ye Y. CVX users’ guide, Version. 1.1; 2007.[42] ABAQUS/Standard, ABAQUS analysis user’s manual, Version 6.6. RI: ABAQUS

Inc.; 2006.[43] Kanit T, N’Guyen F, Forest S, Jeulin D, Reed M, Singleton S. Apparent and

effective physical properties of heterogeneous materials: representativity ofsamples of two materials from food industry. Comput Methods Appl Mech Eng2006;195:3960–82.