Effective Elastic Properties of 3-Phase Particle Reinforced Composites with Randomly Dispersed Elastic Spherical Particles of Different Sizes

Abstract

Higher-order multiscale structures are proposed to predict the effective elastic properties of 3-phase particle reinforced composites by considering the probabilistic spherical particles spatial distribution, the particle interactions, and utilizing homogenization with ensemble volume average approach. The matrix material, spherical particles with radius a₁, and spherical particles with radius a₂, are denoted as the 0^th phase, the 1^st phase, and the 2^nd phase, respectively. Particularly, the two inhomogeneity phases are different particle sizes and the same elastic material properties. Improved higher-order (in ratio of spherical particle sizes to the distance between the centers of spherical particles) bounds on effective elastic properties of 3-phase particle reinforced proposed Formulation II and Formulation I derive composites. As a special case, i.e., particle size of the 1^st phase is the same as that of the 2^nd phase, the proposed formulations reduce to 2-phase formulas. Our theoretical predictions demonstrate excellent agreement with selected experimental data. In addition, several numerical examples are presented to demonstrate the competence of the proposed frameworks.