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Computational Materials Science 39 (2007) 198–204

Numerical evaluation of effective material properties ofrandomly distributed short cylindrical fibre composites

S. Kari *, H. Berger, U. Gabbert

Otto-von-Guericke-University of Magdeburg, Faculty of Mechanical Engineering, Institute of Mechanics,

Universitaetsplatz 2, D-39106 Magdeburg, Germany

Received 27 September 2005; received in revised form 10 January 2006; accepted 3 February 2006

Abstract

The aim of presenting this paper is to evaluate the effective material properties of randomly distributed short fibre (RDSF) and trans-versely randomly distributed short fibre (TRDSF) composites with change in volume fraction, and aspect ratio of fibres. A numericalhomogenization technique based on the finite element method (FEM) is used to evaluate the effective material properties with periodicboundary conditions. A modified random sequential adsorption algorithm (RSA) is applied to generate the three-dimensional unite cellmodels of randomly distributed short cylindrical fibre composites. The developed numerical homogenization technique is used to calcu-late effective material properties in order to systematically evaluate different material systems. The numerical results are also comparedand verified with different analytical methods.� 2006 Elsevier B.V. All rights reserved.

Keywords: Finite element method; Unite cell; Periodic volume element; Representative volume element; Homogenization; Short fibres; Random sequ-ential adsorption algorithm; Effective material properties

1. Introduction

Short fibre composites are have the advantage of easymanufacturing and good mechanical properties. Since shortfibres are easily mixed with the liquid matrix resin, and themixture can be injection or compression molded to producecomponents with complicated shapes, composites com-posed of spatially distributed short fibres have become pop-ular in a wide variety of applications. In addition, usingspatial short fibres as reinforcing elements in a controlledmanner can provide more balanced properties, which leadto an improved through-the-thickness stiffness/strengthand a better ability to form complex shapes. A classicalproblem in solid mechanics is the determination of effectiveelastic properties of a composite material made up of a sta-

0927-0256/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2006.02.024

* Corresponding author. Tel.: +49 391 67 12754; fax: +49 391 67 12439.E-mail addresses: kari@mb.uni-magdeburg.de, sreedhar.kari@mb.uni-

magdeburg.de (S. Kari), berger@mb.uni-magdeburg.de (H. Berger),gabbert@mb.uni-magdeburg.de (U. Gabbert).

tistically isotropic random distribution of isotropic andelastic short cylindrical fibres embedded in a continuous,isotropic and elastic matrix. Even if analytical and semianalytical models have been developed to homogenize fibrecomposites, they are often reduced to specific cases. Numer-ical models seem to be a well-suited approach to describethe behavior of these materials, because there is no restric-tion on the geometry, on the material properties, on thenumber of phases in the composite, and on the size. There-fore, finite element method has been used to determine theeffective properties of the short fibre composites. In orderto obtain realistic predictions of a new material macro-scopic behavior by computational means, three-dimen-sional numerical simulation of statistically representativemicro-heterogeneous material samples is most suitable.

A number of classical micro-mechanics theories havebeen developed and published. Using variational principles,Hashin and Shtrikman [9,10] established bounds on materi-als that could be considered as ‘‘Mechanical mixtures of anumber of different isotropic and homogeneous elastic

S. Kari et al. / Computational Materials Science 39 (2007) 198–204 199

phases’’ and, in bulk, regarded as statistically isotropic andhomogeneous. These two-point bounds were improved bythree-point bounds [20,16,17], which incorporate informa-tion about the phase arrangement through the statistical cor-relation parameters. The Mori–Tanaka method [18] wasdesigned to calculate the average internal stress in the matrixcontaining precipitates with eigenstrains. Benveniste [1]reformulated it so that it could be applied to composite mate-rials. He considered isotropic phases and ellipsoidal phases.Recently, Llorca et al. [14] and Bohm [5] have assessed theeffective coefficients of randomly distributed spherical parti-cles using random sequential adsorption method and com-pared them with Hashin–Shtrikman bounds and othermethods. Also Gusev et al. [7,8,11,15] made experiments ofrandomly and transversely randomly distributed short fibrecomposites and compared them with numerical results andgood agreement has been found. But, since the limitedamount of literature which deals with randomly distributedshort cylindrical fibres is available and is restricted to lowvolume fraction of fibres, we have been motivated to workin this direction. In our opinion micro–macro mechanicalapproaches offer new insights in the material behavior ofsuch fibre composites and may result in new procedures todevelop realistic material models for design and optimisationpurposes. In the paper a representative volume element(RVE) approach is used to calculate effective material prop-erties of randomly distributed short fibre composites. Aspointed out by Hill [21] a RVE is typical of the whole mixtureon average and contains a sufficient number of inclusions forthe apparent overall moduli to be effectively independent ofthe boundary conditions. The volume element we are using isstrictly speaking only a periodic volume element. But, we arepostulating the existence of a representative volume element,and, consequently, we are looking here at deterministic,homogeneous continuum theories, which do not clearlyaccount for random microstructures [21]. In generating theRVE first we have to ensure the statistical homogeneity toassure that the RVE is statistically representative and secondwe have to select a sufficiently large size of the RVE relativeto the size of the inclusion to ensure the independence of theboundary conditions (for details we refer to Ostoja-Starzew-ski [22]). These conditions we have carefully checked bynumerical investigations to be sure that the selected RVEfulfils the requirements. In the paper we are presenting sev-eral computational experiments to know the influence of dif-ferent parameters of short fibres in a composite like aspectratio a (length of fibre/diameter of fibre), volume fraction,fibre orientation angles on their effective material properties.

2. Numerical homogenization of randomly distributed

short cylindrical fibre composite

2.1. Generation of randomly distributed short cylindrical

fibre composite RVE model

The homogenized effective elastic constants of the com-posite are obtained by finite element analysis of a periodic

cubic RVE of volume L3 consisting of randomly distributedshort cylindrical non-overlapping fibres. The RVE can begenerated by using random sequential adsorption algorithm(RSA) modified to provide for a user specified minimumdistance between neighbouring inclusions, for uniformlydistributed fibre orientations, and for the periodicity ofthe volume elements. The distance between axis of thecylinder i and all the cylinders axes which are previouslyaccepted j = 1, . . . , i � 1 have to exceed a minimum value(2 * r + d), where r is the radius of the cylindrical short fibreand d is the minimum distance between any two cylindricalfibres, imposed by the practical limitations to create an ade-quate finite element mesh. If any surface of the cylinder i

intersects any of the cubic RVE surfaces, this conditionhas to be checked with the cylinder volumes on the oppositesurfaces because the microstructure of the composite is peri-odic. Also the cylinder surface should not be very close tothe cubic RVE surface as well as corners of the RVE inorder to avoid the presence of distorted finite elements dur-ing meshing. The RSA algorithm with the combination ofthe above conditions is used to generate the cylindrical vol-umes up to a desired volume fraction of fibres in a compos-ite with uniformly distributed random fibre orientations. Ingeneral with the identical aspect ratio of fibres (a 6 10), thedeveloped algorithm can generate up to 25% volume frac-tion RVE model. For higher volume fractions, differentsizes of fibres are used and these are deposited inside theRVE in descending manner, that is first depositing the larg-est aspect ratio fibres and after reaching the jamming limit(i.e., no more fibres with that aspect ratio can be deposited),again depositing the next largest possible aspect ratio fibresin the RVE. With this approach the volume fractionachieved is up to 40% with minimum distortion of the finiteelements and the adequate mesh.

2.2. Numerical homogenization in random media

The mechanical and physical properties of the constitu-ent materials are always regarded as a small-scale/micro-structure. One of the most powerful tools to speed up themodeling process, both the composite discretization andthe computer simulation of composites in real conditions,is the computational homogenization method. The mainidea of the method is to find a globally homogeneousmedium equivalent to the original composite, where thestrain energy stored in both systems is approximately thesame. The common approach to model the macroscopicproperties of fibre composites is to create a representativevolume element (RVE) or a unit-cell that captures themajor features of the underlying microstructure. Fig. 1shows the general procedure of homogenization method.

In this paper, two types of composites were considered toevaluate the influence on effective material properties byperforming the parametric study of fibres like variation ofvolume fraction and aspect ratio. First one is the randomlydistributed short fibre (RDSF) composite and second one isthe transversely randomly distributed short fibre (TRDSF)

Fig. 1. Procedure of homogenization method applicable to randomly distributed short fibre composites.

200 S. Kari et al. / Computational Materials Science 39 (2007) 198–204

composite. Figs. 2 and 3 show the RVE models and theircorresponding FE mesh of RDSF and TRDSF composites,respectively. All finite element calculations were performedwith the commercial FE package ANSYS. The matrix andthe fibres were meshed with 10 node tetrahedron elementswith full integration. To obtain the homogenized effectivematerial properties, periodic boundary conditions wereapplied to the RVE by coupling opposite nodes on theopposite boundary surfaces. For detailed description ofhomogenization techniques and boundary conditionsapplied to evaluate the effective material properties referto Berger et al. [2–4] and Kari et al. [19]. We used theANSYS Parametric Design Language (APDL) for the anal-

Fig. 2. Randomly distributed short fibre (RD

Fig. 3. Transversely randomly distributed short fibr

ysis, evaluation of needed average strains and stresses andevaluation of the effective material properties in the end.The developed APDL-Scripts in combination with theANSYS batch processing provide a powerful tool for thefast calculation of homogenized material properties of com-posites with a great variety of inclusion geometries.

3. Results and discussion

3.1. Influence of the size of the RVE

In the following several three-dimensional short fibrecomposite RVE models are presented, which are generated

SF) composites RVE and its FE mesh.

e (TRDSF) composites RVE and its FE mesh.

S. Kari et al. / Computational Materials Science 39 (2007) 198–204 201

by using the modified RSA algorithm. With identicalaspect ratio of fibres (length of fibre/diameter of fibre)using this algorithm, it is possible to generate up to 25%volume fractions RVE models. It is not possible to generatehigher volume fraction RVE models because of the jam-ming limit. In order to generate higher volume fractionRVE models, different aspect ratios of fibres were used.That means, first deposit all possible largest aspect ratioof fibres in the RVE, then deposit the next possible largestaspect ratio fibres inside the RVE by preserving the mini-mum distance between fibres and periodicity on the oppo-site boundary surfaces. This process would be continued upto achieving the desired volume fraction or maximumpossible volume fraction for the given aspect ratios of thefibres and minimum distance between the fibres. Since weused different aspect ratios of fibres for generating highervolume fraction RVE models, we studied the influence ofaspect ratios of fibres on effective material properties,which can be seen in the following sections. The materialproperties of constituents used for the analysis to evaluatethe effective material properties are, for the matrix material(Al2618-T4) Em = 70 GPa, mm = 0.3 and for the fibres (SiCreinforcements) Ef = 450 GPa, mf = 0.17 [5]. The results ofthe numerical methods are compared with different analyt-ical methods such as Hashin–Strikman two point bounds(HS) [10], Mori–Tanaka estimates (MTM) [18], self-consis-tent method (SCM) [13] and generalized self-consistentmethod (GSCM) [6]. Also, studies are presented to deter-mine the effect of the orientation of fibres and aspect ratioof fibres on the effective material properties of thesecomposites.

The RVE is generally considered as a volume V of het-erogeneous material that is sufficiently large to be statisti-cally representative of the composite, i.e., to effectivelyinclude a sampling of all microstructural heterogeneitiesthat occur in the composite [12]. The numerical studiesare conducted to investigate the influence of the RVE sizeon effective material properties of these composites. In thisstudy the aspect ratio of fibres is kept constant (a = 5) andby increasing the cubic RVE size, the effective material

Fig. 4. Variation of effective Young’s moduli with change in RV

properties are evaluated for both RDSF and TRDSF com-posites at 15% volume fraction. Fig. 4(a) and (b) shows thevariation of effective Young’s moduli with the change inRVE size for RDSF and TRDSF composites, respectively.In all graphs in this paper, the error means standard devi-ation of the ensemble averages (for each case three samplesare considered for the analysis) of effective material proper-ties and this is represented with vertical bars.

The variation of mean effective Young’s moduli alongthree co-ordinate directions is less than 1.5% for RDSFcomposites and also same for the transverse Young’s mod-uli of TRDSF composites. The error in the longitudinalYoung’s modulus for the RVE size between 0.9 and 1.3 isaround 5.5%. But by increasing the RVE size from 0.9 to1.3, the error (standard deviation of samples considered)is decreasing considerably. From 1.3 to 1.7 the variationsin the error are very small. From these results, we usedthe length of the cubic RVE as 1.5 for all remaining calcu-lations, which will give reasonably good effective materialproperties with less error.

3.2. Influence of the volume fraction

3.2.1. Randomly distributed short fibre composites (RDSF)

The elastic Young’s modulus E, Poison’s ratio m andshear modulus G are evaluated for different volume frac-tions from 10% to 40%. Five different RVE models withrandomly distributed short fibres are considered for eachvolume fraction, and subjected to uni-axial tensile as wellas shear deformation along the three axes of co-ordinates.The ensemble average of the effective material properties ateach volume fraction are considered as effective materialproperties of the total composite at that particular volumefraction with a certain error. Fig. 5(a)–(c) shows the varia-tion of effective Young’s modulus, Poison’s ratio and shearmodulus, respectively, with the change in volume fractionfor RDSF composites and comparison with different ana-lytical methods. The effective material properties, whichare obtained for the RDSF composites using the numericalhomogenization technique, are within the two point

E size: (a) RDSF composites and (b) TRDSF composites.

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Hashin–Shtrikman bounds. The results of analytical meth-ods of MTM and GSCM are same for the RDSF compos-ites and lower estimates the effective material properties.The results of RDSF composites are closer to the self-con-sistent method (SCM) for all volume fractions, the differ-ence between numerical method and SCM was about 3%.

The isotropy of the RVE models is achieved using themodified RSA algorithm and this is explained in terms ofthe effective Young’s moduli, which are obtained usingthree co-ordinate directions for different volume fractionsas shown in Fig. 5(d). The effective Young’s moduliobtained using the three co-ordinate directions, are thesame and variations are less than 1.5%.

3.2.2. Transversely randomly distributed short fibre

composites (TRDSF)

The effective material properties obtained for TRDSFcomposites are compared with RDSF composites.Fig. 6(a)–(c) shows the variation of the effective Young’smoduli for TRDSF composites with change in volumefraction in three co-ordinate directions and comparisonwith the results of RDSF composites. The transverseYoung’s moduli of TRDSF composites have slightly lowervalues when compared with RDSF composites. But alongthe longitudinal direction, the effective material propertieshave significantly higher values when compared withRDSF composites. This is because, in the case of the

Fig. 5. Variation of effective material properties of RDSF composites with chaYoung’s modulus, (b) Poisson’s ratio, (c) shear modulus and (d) isotropy of t

TRDSF composites, the fibres are aligned along the longi-tudinal direction (X3-axis). As shown in Fig. 6(d), the effec-tive Young’s moduli are the same across the transversedirection and satisfying transverse isotropy condition.

3.3. Influence of the aspect ratio of the fibres

Studies were made to investigate the influence of theaspect ratio of fibres on their effective material propertiesof RDSF and TRDSF composites. Here the size of theRVE is kept constant and by increasing the aspect ratioof fibres, the effective material properties are calculated at10% volume fraction of fibres. Table 1 represents the vari-ation of the effective material properties with the change inaspect ratio of the fibres for RDSF and TRDSF compos-ites. From Table 1, it can be observed that with the increaseof aspect ratio of fibres, there are no significant variationsin effective Young’s moduli along three co-ordinate direc-tions for RDSF composites. But, in case of TRDSF com-posites, there is a significant variation in E33 materialcoefficient with the increase of the fibre aspect ratio. Theeffective Young’s modulus E33 is increased significantlywith the increase of the fibre aspect ratio from 1 to 9.But from 9 to 15 the increase of E33 is very small and isabout less than 1%. The difference between the longitudinalYoung’s modulus E33 of composites with fibre aspect ratio15 and infinity (long fibres) is around 5%. Along the

nge in volume fraction and comparison with different analytical results: (a)he resultant material properties.

Fig. 6. (a)–(c) Variation of the effective Young’s moduli of TRDSF composites with change in volume fractions along the three co-ordinate directions andcomparison with RDSF composites. (d) Transverse isotropy of material properties for the TRDSF composite.

Table 1Variation of effective Young’s modulus with change in aspect ratio of fibre for RDSF and TRDSF composites at 10% volume fraction

Aspect ratio (L/D) E11 (GPa) (RDSF) E11 (GPa) (TRDSF) E22 (GPa) (RDSF) E22 (GPa) (TRDSF) E33 (GPa) (RDSF) E33 (GPa) (TRDSF)

1 83.73 83.53 83.79 83.21 83.85 83.953 83.81 82.68 83.77 82.34 84.80 92.966 84.57 82.54 84.27 81.98 82.94 98.859 85.17 82.16 83.53 82.19 84.44 103.98

12 83.74 81.96 83.85 82.01 83.92 104.3615 83.31 81.66 83.39 82.18 83.94 104.781 82.32 82.14 110.43

S. Kari et al. / Computational Materials Science 39 (2007) 198–204 203

transverse direction, the material properties E11 and E22 ofthe TRDSF composites are slightly less than the values ofRDSF composites, but variations in the transverseYoung’s moduli are not significant with the increase ofaspect ratio of fibres.

4. Conclusions

Numerical homogenization tools have been developedfor the evaluation of the effective material properties ofthe short fibres reinforced composite structures. The effec-tive material properties of randomly distributed short fibre(RDSF) and transversely randomly distributed short fibre(TRDSF) composites are obtained using these tools andcompared with the results of different analytical methods.Our numerical predictions are in between the Hashin–

Strikman bounds and close to the results of self-consistentapproximation. We also studied the influence of the aspectratio of fibres on the effective material properties. Thesestudies showed that there is not a significant influence oneffective material properties with increase of aspect ratiofor RDSF composites. But for the case of TRDSF compos-ites, only along the longitudinal direction of the fibres, thematerial properties are improved considerably for thelower aspect ratios and stabilized with further increase ofthe aspect ratios and the material properties are nearer tothe continuous fibre composites. From these studies, itcan be concluded that the effective material properties ofRDSF composites will depend mainly on the volume frac-tion of fibres and for the case of TRDSF composites theeffective material properties will also depend on the aspectratio of fibres particularly with lower values along with the

204 S. Kari et al. / Computational Materials Science 39 (2007) 198–204

volume fraction. Further studies have to make to know theinfluence of the orientation fibres on the effective materialproperties of these composites.

A generalized procedure has been developed to calculatedifferent effective coefficients for all desired volume frac-tions based on the ANSYS Parametric Design Language.This tool reduces the manual work and time and can beused as a template to evaluate the effective coefficients ofrandomly distributed RDSF and TRDSF composites upto 40% volume fraction. Finally, the tool, which we devel-oped, can be applied to any number of phases, i.e., thereare no restrictions regarding the number of materials,geometry and material symmetry and this can be used effec-tively to determine material coefficients of different types offibre and particle reinforced composites.

Acknowledgement

This work has been supported by DFG Germany, Grad-uiertenkolleg 828 ‘‘Micro–macro Interactions in StructuredMedia and Particle Systems’’. This support is greatlyacknowledged.

References

[1] Y. Benveniste, Mech. Mater. 6 (1987) 147–157.[2] H. Berger, S. Kari, U. Gabbert, R. Rodriguez-Ramos, R. Guinovart-

Diaz, J.A. Otero, J. Bravo-Castillero, Int. J. Solids Struct. 21–22(2005) 5692–5714.

[3] H. Berger, S. Kari, U. Gabbert, R. Rodriguez-Ramos, R. Guinovart-Diaz, J.A. Otero, J. Bravo- Castillero, J. Smart Mater. Struct. 15(2006) 451–458.

[4] H. Berger, U. Gabbert, H. Koppe, R. Rodriguez-Ramos, J. Bravo-Castillero, R. Guinovart-Diaz, J.A. Otero, G.A. Maugin, Comput.Mech. 33 (2003) 61–67.

[5] H.J. Bohm, A. Eckschlager, W. Han, Comput. Mater. Sci. 25 (2002)42–53.

[6] R.M. Christensen, K.H. Lo, Solutions for effective shear properties ofthree phase sphere and cylinder models, J. Mech. Phys. Solids. 27(1997) 315–330.

[7] A.A. Gusev, J. Mech. Phys. Sol. 45 (1997) 1449–1459.[8] A.A. Gusev, P.J. Hine, I.M. Ward, Compos. Sci. Technol. 60 (2000)

535–541.[9] Z. Hashin, J. Appl. Mech. 29 (1962) 143–150.

[10] Z. Hashin, S. Shtrikman, J. Mech. Phys. Solids 11 (1963) 127–140.[11] P.J. Hine, H.R. Lusti, A.A. Gusev, Compos. Sci. Technol. 62 (2002)

1445–1453.[12] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Int. J. Solids

Struct. 40 (2003) 3647–3679.[13] L.X. Li, T.J. Wan, Mater. Charact. 54 (2005) 49–62.[14] J. Segurado, J. Llorca, J. Mech. Phys. Solids 50 (2003) 2107–2121.[15] H.R. Lusti, P.J. Hine, A.A. Gusev, Compos. Sci. Technol. 62 (2002)

1927–1934.[16] G.W. Milton, J. Mech. Phys. Solids 30 (1982) 177–191.[17] G.W. Milton, N. Phan-Thien, Proc. Roy. Soc. Lond. A 380 (1982)

305–331.[18] T. Mori, K. Tanaka, Acta Metall. Mater. 21 (1973) 571–574.[19] S. Kari, H. Berger, U. Gabbert, R.R. Reinaldo, Compos. Struct., in

press.[20] M.J. Beran, J. Molyneux, Quart. Appl. Math. 24 (1966) 107–118.[21] R. Hill, J. Mech. Phys. Solids 11 (1993) 357–372.[22] M. Ostoja-Starzewski, J. Appl. Mech. 69 (2002) 25–35.