Estimations of the elastic moduli of three-phase composites with two different ellipse inclusions randomly distributed in the matrix

Abstract

A new method called the polarization approximation (PA) is constructed from the minimum energy principle; in this work, we construct a new approximation formula based on PA to determine the elastic moduli for three-phase composite materials with two different ellipse inclusions randomly distributed in the matrix. Some solutions with different approaches including PA, Mori–Tanaka approximation (MTA), differential approximations (DA), and fast Fourier transformation (FFT) method are constructed to estimate the elastic bulk and shear modulus of the three-phase composites in 2D. The numerical solutions using FFT analysis are compared with PA, DA, MTA, and Hashin–Shtrikman’s bounds. The comparison results show the effectiveness of the approximation methods, which makes MTA, PA, and DA useful with simplicity and ease of application.