Homogenization methods for multi-phase elastic composites with non-elliptical reinforcements: Comparisons and benchmarks B. Klusemann ∗ ,a , H.J. B ¨ ohm b , B. Svendsen a a Material Mechanics, RWTH Aachen University, 52062 Aachen, Germany b Institute of Lightweight Design and Structural Biomechanics, Vienna University of Technology, 1040 Vienna, Austria Abstract The purpose of this work is comparing three strategies for dealing with inhomogeneities of non-elliptical shape in the context of homogenization methods. First, classical mean field methods and two relatively new approaches, IDD and ESCS, are used in combination with analytical expressions for the Eshelby tensor based on its irreducible decomposition. The second strategy to be investigated is the Mori-Tanaka method in combination with the replacement tensor approach, which uses numerical models of dilute inhomogeneities embedded in large matrix regions. The third approach consists of the direct Finite Element discretization of microstructures. The elasticity tensors and directional Young’s moduli are first studied for arrangements of aligned inhomogeneities of three different shapes and of combinations of these shapes. Subsequently the three modeling strategies are applied to a real microstructure. Comparisons are not only carried out with respect to phase volume fractions, but also with respect to the contrast in the elastic phase properties. All calculations are restricted to plane strain conditions and to isotropic material behavior. 1. Introduction The prediction of the macroscopic stress-strain response of composite materials is related to the description of their com- plex microstructural behavior exemplified by the interaction between the constituents. In this context, the microstructure of the material under consideration is basically taken into ac- count by representative volume elements (RVE). In previous decades and especially in the absence of computers, analytical and semi-analytical approximations based on RVEs and mean- field homogenization schemes were developed. Mean-field ho- mogenization methods were first developed in the framework of linear elasticity for ellipsoidal inhomogeneities and are now well-established. These schemes provide efficient and straight forward algorithms for the prediction of, among other proper- ties, the elastic constants (e.g., the Mori-Tanaka method (Mori and Tanaka, 1973), Double Inclusion method (Hori and Nemat- Nasser, 1993), Interpolative Double Inclusion model (Pierard et al., 2004) and Self Consistent schemes, e.g., CSCS (Hill, 1965) and GSCS (Christensen and Lo, 1979)). Moreover, the results obtained can be shown to correspond to or to lie be- tween upper or lower bounds to the true solution of the un- derlying problem in most cases, e.g., the Voigt-Reuss and the Hashin-Shtrikman bounds (see, e.g., Gross and Seelig, 2006; Nemat-Nasser and Hori, 1999). A further important group of analytical models for the macroscopic elastic response of inho- mogeneous materials are estimated based on n-point statistics. For example, Torquato (1998) developed three-point estimates ∗ Corresponding author. Fax: +49 241 80 92007. Email address: benjamin.klusemann@rwth-aachen.de (B. Klusemann) for the thermoelastic properties of two-phase materials that re- quire statistical information on the phase arrangement in the form of n-point correlations. In this context, improved bounds which are significantly tighter than Hashin-Shtrikman bounds were formulated by Torquato (1991). For reviews of higher- order bounds for elastic properties of inhomogeneous materials see, e.g., Torquato (2002). These improved bounds can provide highly useful information for low and moderate phase contrasts (B¨ ohm, 2011). Sevostianov and Kachanov (2007) studied the effects of inhomogeneities of equal shape but different elastic contrast on the macroscopic behavior of inhomogeneus materi- als, bringing out the importance of the elastic contrast. Zheng and Du (2001) listed the main requirements on ho- mogenization methods for predicting the effective properties of inhomogeneous materials as a) a simple structure which can be solved explicitly, such that a physical interpretation for the behavior of all the compo- nents involved is possible; b) a valid structure for multiphase composites with various inhomogeneity geometries, isotropy and anisotropies; c) an accurate model for the influence of various inhomo- geneity distributions and interactions between inhomo- geneities and their immediate surrounding matrix. However, none of the aforementioned methods is fully able to fulfill these requirements completely. The major limitations of these methods are exemplified by the fact that descriptors of the microstructure such as orientation, size, shape or alignment dis- tributions, are unaccounted for and that the influence of the ma- trix material on the inhomogeneity does not enter these methods directly. Preprint submitted to European Journal of Mechanics A/Solids published in European Journal of Mechanics A/Solids 34 (2012) 21-37 http://dx.doi.org/10.1016/j.euromechsol.2011.12.002