Average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity: Effective elastic properties of composites

Abstract

The effective mechanical properties of composites are strongly influenced by the morphology, volume fraction, and properties of the inclusion phase. Extensive theoretical models for the evaluation of effective properties are mainly based on the continuum micromechanics derived from the well-known Eshelby’s solution of an ellipsoidal inclusion problem, which are subjected to the limitations on the Eshelby tensor for a non-ellipsoidal inclusion. This work devises a three-step method to derive the average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity in an elastic field. The proposed three-step method is user-friendly and avoids the singularity that arises from solving the complicated integrals for an arbitrarily shaped inclusion. The calculated average Eshelby tensors of inclusions with different morphologies, including Platonic solids, superballs, superellipsoids, doughnuts, helices, and non-convex particles, are thereafter compared against the existing data reported in the literature, in order to verify the reliability of the present method. Moreover, we also incorporate the proposed method into a micromechanical model to predict the effective elastic moduli of two-phase composites, in which congruent non-ellipsoidal inclusions of a specific volume fraction are randomly distributed in a homogeneous matrix. Comparisons with the direct finite element method (FEM) simulations indicate that the developed micromechanical model is a reliable means to evaluate the elastic properties of composites. The proposed average Eshelby tensor of an arbitrarily shaped inclusion can also be generalized to the theoretical predictions of the effective transport properties of composites with inclusions of diverse types and shapes, such as particles, rocks, granules, pores, fibers, and cracks.