Effective Elastic Properties of 3-Phase Particle Reinforced Composites with Randomly Dispersed Elastic Spherical Particles of Different Sizes Dedicated to Professor Karl Stark Pister for his 95th birthday

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 0th phase,the 1st phase,and the 2nd 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 1st phase is the same as that of the 2nd 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.