Variational estimates for the effective properties and field statistics of composites with variable particle interaction strengths

Abstract

This work provides analytical estimates for the macroscopic elastic, viscous and viscoplastic response of composite materials with particulate microstructures consisting of various families of ellipsoidal inclusions with given properties, volume fractions, shapes and orientations that are randomly distributed with ‘ellipsoidal symmetry’ in a matrix of a different material. The estimates follow from a simultaneous generalization of the variational estimates of Ponte Castañeda and Willis (1995) (PCW) and the self-consistent (SC) estimates of Budiansky (1965) and Hill (1965b) by means of an interpolating parameter λ in the selection of the so-called ‘uniform reference medium.’ When λ = 0 , the PCW estimates are recovered, while for λ = 1 a generalization of the SC estimates is obtained that can consistently handle multiple families of inclusions. For more general values of λ , the new estimates lie in between the PCW and SC estimates and can account approximately for strong particle interactions when the particles do not satisfy the ‘well-separated’ hypothesis. In addition, the method provides estimates for the averages and covariances of the field fluctuations in the matrix and inclusion phases. In particular, it is found that the particle field fluctuations tend to increase with increasing value of λ , thus providing a significant improvement over the PCW estimates, which predict zero fluctuations in the inclusions for all volume fractions — and are therefore not able to account for microstructural features leading to strong particle interactions (e.g., cluster formation). When the estimates are specialized to the simplest case of isotropically distributed spherical inclusions that are either rigid or void, the resulting estimates exhibit ‘rigidity’ or ‘decohesion’ thresholds at critical particle volume fractions, depending on the value of λ , which can be interpreted physically as macroscopic manifestations of spanning clusters of (1-D) particle chains or (2-D) void surfaces, respectively.