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

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1 Computational Materials Science 39 (2007) Numerical evaluation of effective material properties of randomly 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 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 transversely randomly distributed short fibre (TRDSF) composites with change in volume fraction, and aspect ratio of fibres. A numerical homogenization technique based on the finite element method (FEM) is used to evaluate the effective material properties with periodic boundary conditions. A modified random sequential adsorption algorithm (RSA) is applied to generate the three-dimensional unite cell models of randomly distributed short cylindrical fibre composites. The developed numerical homogenization technique is used to calculate effective material properties in order to systematically evaluate different material systems. The numerical results are also compared and verified with different analytical methods. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Finite element method; Unite cell; Periodic volume element; Representative volume element; Homogenization; Short fibres; Random sequential adsorption algorithm; Effective material properties 1. Introduction * Corresponding author. Tel.: ; fax: addresses: kari@mb.uni-magdeburg.de, sreedhar.kari@mb.unimagdeburg.de (S. Kari), berger@mb.uni-magdeburg.de (H. Berger), gabbert@mb.uni-magdeburg.de (U. Gabbert). Short fibre composites are have the advantage of easy manufacturing and good mechanical properties. Since short fibres are easily mixed with the liquid matrix resin, and the mixture can be injection or compression molded to produce components with complicated shapes, composites composed of spatially distributed short fibres have become popular in a wide variety of applications. In addition, using spatial short fibres as reinforcing elements in a controlled manner can provide more balanced properties, which lead to an improved through-the-thickness stiffness/strength and a better ability to form complex shapes. A classical problem in solid mechanics is the determination of effective elastic properties of a composite material made up of a statistically isotropic random distribution of isotropic and elastic short cylindrical fibres embedded in a continuous, isotropic and elastic matrix. Even if analytical and semi analytical models have been developed to homogenize fibre composites, they are often reduced to specific cases. Numerical models seem to be a well-suited approach to describe the behavior of these materials, because there is no restriction on the geometry, on the material properties, on the number of phases in the composite, and on the size. Therefore, finite element method has been used to determine the effective properties of the short fibre composites. In order to obtain realistic predictions of a new material macroscopic behavior by computational means, three-dimensional numerical simulation of statistically representative micro-heterogeneous material samples is most suitable. A number of classical micro-mechanics theories have been developed and published. Using variational principles, Hashin and Shtrikman [9,10] established bounds on materials that could be considered as Mechanical mixtures of a number of different isotropic and homogeneous elastic /$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi: /j.commatsci

2 S. Kari et al. / Computational Materials Science 39 (2007) phases and, in bulk, regarded as statistically isotropic and homogeneous. These two-point bounds were improved by three-point bounds [20,16,17], which incorporate information about the phase arrangement through the statistical correlation parameters. The Mori Tanaka method [18] was designed to calculate the average internal stress in the matrix containing precipitates with eigenstrains. Benveniste [1] reformulated it so that it could be applied to composite materials. He considered isotropic phases and ellipsoidal phases. Recently, Llorca et al. [14] and Böhm [5] have assessed the effective coefficients of randomly distributed spherical particles using random sequential adsorption method and compared them with Hashin Shtrikman bounds and other methods. Also Gusev et al. [7,8,11,15] made experiments of randomly and transversely randomly distributed short fibre composites and compared them with numerical results and good agreement has been found. But, since the limited amount of literature which deals with randomly distributed short cylindrical fibres is available and is restricted to low volume fraction of fibres, we have been motivated to work in this direction. In our opinion micro macro mechanical approaches offer new insights in the material behavior of such fibre composites and may result in new procedures to develop realistic material models for design and optimisation purposes. In the paper a representative volume element (RVE) approach is used to calculate effective material properties of randomly distributed short fibre composites. As pointed out by Hill [21] a RVE is typical of the whole mixture on average and contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the boundary conditions. The volume element we are using is strictly speaking only a periodic volume element. But, we are postulating the existence of a representative volume element, and, consequently, we are looking here at deterministic, homogeneous continuum theories, which do not clearly account for random microstructures [21]. In generating the RVE first we have to ensure the statistical homogeneity to assure that the RVE is statistically representative and second we have to select a sufficiently large size of the RVE relative to the size of the inclusion to ensure the independence of the boundary conditions (for details we refer to Ostoja-Starzewski [22]). These conditions we have carefully checked by numerical investigations to be sure that the selected RVE fulfils the requirements. In the paper we are presenting several computational experiments to know the influence of different parameters of short fibres in a composite like aspect ratio 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 composite are obtained by finite element analysis of a periodic cubic RVE of volume L 3 consisting of randomly distributed short cylindrical non-overlapping fibres. The RVE can be generated by using random sequential adsorption algorithm (RSA) modified to provide for a user specified minimum distance between neighbouring inclusions, for uniformly distributed fibre orientations, and for the periodicity of the volume elements. The distance between axis of the cylinder i and all the cylinders axes which are previously accepted j =1,...,i 1 have to exceed a minimum value (2 * r + d), where r is the radius of the cylindrical short fibre and d is the minimum distance between any two cylindrical fibres, imposed by the practical limitations to create an adequate finite element mesh. If any surface of the cylinder i intersects any of the cubic RVE surfaces, this condition has to be checked with the cylinder volumes on the opposite surfaces because the microstructure of the composite is periodic. Also the cylinder surface should not be very close to the cubic RVE surface as well as corners of the RVE in order to avoid the presence of distorted finite elements during meshing. The RSA algorithm with the combination of the above conditions is used to generate the cylindrical volumes up to a desired volume fraction of fibres in a composite with uniformly distributed random fibre orientations. In general with the identical aspect ratio of fibres (a 6 10), the developed algorithm can generate up to 25% volume fraction RVE model. For higher volume fractions, different sizes of fibres are used and these are deposited inside the RVE in descending manner, that is first depositing the largest 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 fibres in the RVE. With this approach the volume fraction achieved is up to 40% with minimum distortion of the finite elements and the adequate mesh Numerical homogenization in random media The mechanical and physical properties of the constituent materials are always regarded as a small-scale/microstructure. One of the most powerful tools to speed up the modeling process, both the composite discretization and the computer simulation of composites in real conditions, is the computational homogenization method. The main idea of the method is to find a globally homogeneous medium equivalent to the original composite, where the strain energy stored in both systems is approximately the same. The common approach to model the macroscopic properties of fibre composites is to create a representative volume element (RVE) or a unit-cell that captures the major features of the underlying microstructure. Fig. 1 shows the general procedure of homogenization method. In this paper, two types of composites were considered to evaluate the influence on effective material properties by performing the parametric study of fibres like variation of volume fraction and aspect ratio. First one is the randomly distributed short fibre (RDSF) composite and second one is the transversely randomly distributed short fibre (TRDSF)

3 200 S. Kari et al. / Computational Materials Science 39 (2007) Fig. 1. Procedure of homogenization method applicable to randomly distributed short fibre composites. composite. Figs. 2 and 3 show the RVE models and their corresponding FE mesh of RDSF and TRDSF composites, respectively. All finite element calculations were performed with the commercial FE package ANSYS. The matrix and the fibres were meshed with 10 node tetrahedron elements with full integration. To obtain the homogenized effective material properties, periodic boundary conditions were applied to the RVE by coupling opposite nodes on the opposite boundary surfaces. For detailed description of homogenization techniques and boundary conditions applied to evaluate the effective material properties refer to Berger et al. [2 4] and Kari et al. [19]. We used the ANSYS Parametric Design Language (APDL) for the analysis, evaluation of needed average strains and stresses and evaluation of the effective material properties in the end. The developed APDL-Scripts in combination with the ANSYS batch processing provide a powerful tool for the fast calculation of homogenized material properties of composites 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 fibre composite RVE models are presented, which are generated Fig. 2. Randomly distributed short fibre (RDSF) composites RVE and its FE mesh. Fig. 3. Transversely randomly distributed short fibre (TRDSF) composites RVE and its FE mesh.

4 S. Kari et al. / Computational Materials Science 39 (2007) by using the modified RSA algorithm. With identical aspect 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 generate higher volume fraction RVE models because of the jamming limit. In order to generate higher volume fraction RVE models, different aspect ratios of fibres were used. That means, first deposit all possible largest aspect ratio of fibres in the RVE, then deposit the next possible largest aspect ratio fibres inside the RVE by preserving the minimum distance between fibres and periodicity on the opposite boundary surfaces. This process would be continued up to achieving the desired volume fraction or maximum possible volume fraction for the given aspect ratios of the fibres and minimum distance between the fibres. Since we used different aspect ratios of fibres for generating higher volume fraction RVE models, we studied the influence of aspect ratios of fibres on effective material properties, which can be seen in the following sections. The material properties of constituents used for the analysis to evaluate the effective material properties are, for the matrix material (Al2618-T4) E m = 70 GPa, m m = 0.3 and for the fibres (SiC reinforcements) E f = 450 GPa, m f = 0.17 [5]. The results of the numerical methods are compared with different analytical methods such as Hashin Strikman two point bounds (HS) [10], Mori Tanaka estimates (MTM) [18], self-consistent method (SCM) [13] and generalized self-consistent method (GSCM) [6]. Also, studies are presented to determine the effect of the orientation of fibres and aspect ratio of fibres on the effective material properties of these composites. The RVE is generally considered as a volume V of heterogeneous material that is sufficiently large to be statistically representative of the composite, i.e., to effectively include a sampling of all microstructural heterogeneities that occur in the composite [12]. The numerical studies are conducted to investigate the influence of the RVE size on effective material properties of these composites. In this study the aspect ratio of fibres is kept constant (a = 5) and by increasing the cubic RVE size, the effective material properties are evaluated for both RDSF and TRDSF composites at 15% volume fraction. Fig. 4(a) and (b) shows the variation of effective Young s moduli with the change in RVE size for RDSF and TRDSF composites, respectively. In all graphs in this paper, the error means standard deviation of the ensemble averages (for each case three samples are considered for the analysis) of effective material properties and this is represented with vertical bars. The variation of mean effective Young s moduli along three co-ordinate directions is less than 1.5% for RDSF composites and also same for the transverse Young s moduli of TRDSF composites. The error in the longitudinal Young s modulus for the RVE size between 0.9 and 1.3 is around 5.5%. But by increasing the RVE size from 0.9 to 1.3, the error (standard deviation of samples considered) is decreasing considerably. From 1.3 to 1.7 the variations in the error are very small. From these results, we used the length of the cubic RVE as 1.5 for all remaining calculations, which will give reasonably good effective material properties with less error Influence of the volume fraction Randomly distributed short fibre composites (RDSF) The elastic Young s modulus E, Poison s ratio m and shear modulus G are evaluated for different volume fractions from 10% to 40%. Five different RVE models with randomly distributed short fibres are considered for each volume fraction, and subjected to uni-axial tensile as well as shear deformation along the three axes of co-ordinates. The ensemble average of the effective material properties at each volume fraction are considered as effective material properties of the total composite at that particular volume fraction with a certain error. Fig. 5(a) (c) shows the variation of effective Young s modulus, Poison s ratio and shear modulus, respectively, with the change in volume fraction for RDSF composites and comparison with different analytical methods. The effective material properties, which are obtained for the RDSF composites using the numerical homogenization technique, are within the two point Fig. 4. Variation of effective Young s moduli with change in RVE size: (a) RDSF composites and (b) TRDSF composites.

5 202 S. Kari et al. / Computational Materials Science 39 (2007) Hashin Shtrikman bounds. The results of analytical methods of MTM and GSCM are same for the RDSF composites and lower estimates the effective material properties. The results of RDSF composites are closer to the self-consistent method (SCM) for all volume fractions, the difference between numerical method and SCM was about 3%. The isotropy of the RVE models is achieved using the modified RSA algorithm and this is explained in terms of the effective Young s moduli, which are obtained using three co-ordinate directions for different volume fractions as shown in Fig. 5(d). The effective Young s moduli obtained using the three co-ordinate directions, are the same and variations are less than 1.5% Transversely randomly distributed short fibre composites (TRDSF) The effective material properties obtained for TRDSF composites are compared with RDSF composites. Fig. 6(a) (c) shows the variation of the effective Young s moduli for TRDSF composites with change in volume fraction in three co-ordinate directions and comparison with the results of RDSF composites. The transverse Young s moduli of TRDSF composites have slightly lower values when compared with RDSF composites. But along the longitudinal direction, the effective material properties have significantly higher values when compared with RDSF composites. This is because, in the case of the TRDSF composites, the fibres are aligned along the longitudinal direction (X 3 -axis). As shown in Fig. 6(d), the effective Young s moduli are the same across the transverse direction and satisfying transverse isotropy condition Influence of the aspect ratio of the fibres Studies were made to investigate the influence of the aspect ratio of fibres on their effective material properties of RDSF and TRDSF composites. Here the size of the RVE is kept constant and by increasing the aspect ratio of fibres, the effective material properties are calculated at 10% volume fraction of fibres. Table 1 represents the variation of the effective material properties with the change in aspect ratio of the fibres for RDSF and TRDSF composites. From Table 1, it can be observed that with the increase of aspect ratio of fibres, there are no significant variations in effective Young s moduli along three co-ordinate directions for RDSF composites. But, in case of TRDSF composites, there is a significant variation in E 33 material coefficient with the increase of the fibre aspect ratio. The effective Young s modulus E 33 is increased significantly with the increase of the fibre aspect ratio from 1 to 9. But from 9 to 15 the increase of E 33 is very small and is about less than 1%. The difference between the longitudinal Young s modulus E 33 of composites with fibre aspect ratio 15 and infinity (long fibres) is around 5%. Along the Fig. 5. Variation of effective material properties of RDSF composites with change in volume fraction and comparison with different analytical results: (a) Young s modulus, (b) Poisson s ratio, (c) shear modulus and (d) isotropy of the resultant material properties.

6 S. Kari et al. / Computational Materials Science 39 (2007) 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 and comparison with RDSF composites. (d) Transverse isotropy of material properties for the TRDSF composite. Table 1 Variation 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) E 11 (GPa) (RDSF) E 11 (GPa) (TRDSF) E 22 (GPa) (RDSF) E 22 (GPa) (TRDSF) E 33 (GPa) (RDSF) E 33 (GPa) (TRDSF) transverse direction, the material properties E 11 and E 22 of the TRDSF composites are slightly less than the values of RDSF composites, but variations in the transverse Young s moduli are not significant with the increase of aspect ratio of fibres. 4. Conclusions Numerical homogenization tools have been developed for the evaluation of the effective material properties of the short fibres reinforced composite structures. The effective material properties of randomly distributed short fibre (RDSF) and transversely randomly distributed short fibre (TRDSF) composites are obtained using these tools and compared with the results of different analytical methods. Our numerical predictions are in between the Hashin Strikman bounds and close to the results of self-consistent approximation. We also studied the influence of the aspect ratio of fibres on the effective material properties. These studies showed that there is not a significant influence on effective material properties with increase of aspect ratio for RDSF composites. But for the case of TRDSF composites, only along the longitudinal direction of the fibres, the material properties are improved considerably for the lower aspect ratios and stabilized with further increase of the aspect ratios and the material properties are nearer to the continuous fibre composites. From these studies, it can be concluded that the effective material properties of RDSF composites will depend mainly on the volume fraction of fibres and for the case of TRDSF composites the effective material properties will also depend on the aspect ratio of fibres particularly with lower values along with the

7 204 S. Kari et al. / Computational Materials Science 39 (2007) volume fraction. Further studies have to make to know the influence of the orientation fibres on the effective material properties of these composites. A generalized procedure has been developed to calculate different effective coefficients for all desired volume fractions based on the ANSYS Parametric Design Language. This tool reduces the manual work and time and can be used as a template to evaluate the effective coefficients of randomly distributed RDSF and TRDSF composites up to 40% volume fraction. Finally, the tool, which we developed, can be applied to any number of phases, i.e., there are no restrictions regarding the number of materials, geometry and material symmetry and this can be used effectively to determine material coefficients of different types of fibre and particle reinforced composites. Acknowledgement This work has been supported by DFG Germany, Graduiertenkolleg 828 Micro macro Interactions in Structured Media and Particle Systems. This support is greatly acknowledged. References [1] Y. Benveniste, Mech. Mater. 6 (1987) [2] H. Berger, S. Kari, U. Gabbert, R. Rodriguez-Ramos, R. Guinovart- Diaz, J.A. Otero, J. Bravo-Castillero, Int. J. Solids Struct (2005) [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) [4] H. Berger, U. Gabbert, H. Köppe, R. Rodriguez-Ramos, J. Bravo- Castillero, R. Guinovart-Diaz, J.A. Otero, G.A. Maugin, Comput. Mech. 33 (2003) [5] H.J. Böhm, A. Eckschlager, W. Han, Comput. Mater. Sci. 25 (2002) [6] R.M. Christensen, K.H. Lo, Solutions for effective shear properties of three phase sphere and cylinder models, J. Mech. Phys. Solids. 27 (1997) [7] A.A. Gusev, J. Mech. Phys. Sol. 45 (1997) [8] A.A. Gusev, P.J. Hine, I.M. Ward, Compos. Sci. Technol. 60 (2000) [9] Z. Hashin, J. Appl. Mech. 29 (1962) [10] Z. Hashin, S. Shtrikman, J. Mech. Phys. Solids 11 (1963) [11] P.J. Hine, H.R. Lusti, A.A. Gusev, Compos. Sci. Technol. 62 (2002) [12] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Int. J. Solids Struct. 40 (2003) [13] L.X. Li, T.J. Wan, Mater. Charact. 54 (2005) [14] J. Segurado, J. Llorca, J. Mech. Phys. Solids 50 (2003) [15] H.R. Lusti, P.J. Hine, A.A. Gusev, Compos. Sci. Technol. 62 (2002) [16] G.W. Milton, J. Mech. Phys. Solids 30 (1982) [17] G.W. Milton, N. Phan-Thien, Proc. Roy. Soc. Lond. A 380 (1982) [18] T. Mori, K. Tanaka, Acta Metall. Mater. 21 (1973) [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) [21] R. Hill, J. Mech. Phys. Solids 11 (1993) [22] M. Ostoja-Starzewski, J. Appl. Mech. 69 (2002)