Homogenization methods for multi-phase elastic composites: Comparisons and benchmarks

Usually homogenization methods are investigated regardin g the volume fraction of the inclusion. In this paper classical homogenization methods are recalled and compare d on the basis of the contrast in the elastic properties of the constituents. This has a significant influence on the accuracy of the homogenization method. In addition two relatively new approaches, the ESCS and IDD method, are i ntroduced and compared to more standard homogenization approaches. The analysis of these methods sho ws that the IDD method is an improvement due to its simple but universally applicable structure. A number of co mparisons of these and other analytical approaches are carried out with the corresponding finite element result s. The prediction of the macroscopic stress-strain response of composite materials is related to the description of their complex microstructural behavior exemplified by the i nteraction between the constituents. In this context, the microstructure of the material under consideration is b asically taken into account 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 homogenization methods were first developed in the framework of lin ear elasticity and are now well-established. These schemes provide efficient and straight forward algorithms f or the prediction of, among other properties, the elastic constants (e.g. Mori-Tanaka method, Lielens method (Lielens, 1999), self consistent scheme). Moreover, the results obtained can be shown to be upper or lower bounds to the true solution of the underlying problem in most cases (e.g. Voigt-Reuss, Hashin-Shtrikman bounds). (see e.g., Gross and Seelig, 2001; Nemat-Nasser and Hori, 1999; Pierard et al., 2004). All these methods are based on two steps to predict the macroscopic response. In a first step, a local problem for a single inclusion is solved in order to obtain approximations for the local field behavior as outlined by Eshelby for elastic fields of an ellipsoidal inclusion (Eshelby, 1957). The second step consists of averaging the local fields to obtain the global ones (e.g. Mercier and Molinari, 2009). In this context, the main requirements on homogenization methods for predicting the effective properties, according to Zheng and Du (2001) are