Micromehcanics of Heterogenous Materials pp 315-346 | Cite as

# Generalization of the MEFM in Random Structure Matrix Composites

The outline of this chapter (where the results in [147] are presented) is as follows. Based on the results of Chapter 4, we present the numerical solution by the volume integral equation (VIE) method of the problem for two inclusions in an infinite medium, subjected to the homogeneous effective field. The iteration the VIE method used is a standard iteration method of the solution of the Fredholm integral equation of the second order involved which led to an impenetrable barrier of computer costs if the number of inclusions with near-to-dense packing is large enough. These solutions fulfill the role of some building blocks used in the subsequent calculations. Just with some additional assumptions (such as an effective field hypothesis) they can be expressed through both the Green function, Eshelby tensor, and external Eshelby tensor. By the use of these building blocks, a *generalization of the MEFM* is proposed for the estimation of effective properties (such as compliance, thermal expansion, stored energy) and the first statistical moments of stresses varying along a cross-section of inclusions. No restriction on the homogeneity of the effective fields (similar to the hypothesis H1) acting on the individual inclusions is used. However, we will use the hypothesis of homogeneity of the effective field acting on each pair of inclusions. The last hypothesis is modified taking into account the dependence of these fields on the mutual location of each pair of inclusions estimated. After that the particular cases of the general proposed approach such as the composites with identical aligned inclusions as well as the simplifications produced by both the effective field hypothesis H1 and quasi-crystalline approximation are analyzed. Finally, we employ the proposed explicit relations and some related ones for numerical estimations of effective elastic moduli in an isotropic composite made of the isotropic matrix and aligned fibers.

## Keywords

Eshelby Tensor Constant Tensor Radial Distribution Function Strain Polarization Volume Integral Equation## Preview

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