Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

TL;DR

This paper derives higher-order non-local elastic properties for dilute two-phase composites, showing how shape and symmetry influence the effective behavior of the equivalent Mindlin second-gradient elastic solid.

Contribution

It provides explicit formulas for non-local parameters in various inclusion geometries, extending first-order homogenization to second-gradient elasticity.

Findings

Second-gradient model is positive definite only with negative definite discrepancy tensor.

Nonlocal material symmetries match those of the discrepancy tensor.

Shape of the RVE influences nonlocal behavior but not the first-order response.

Abstract

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i.) is positive definite only when the discrepancy tensor is negative defined; (ii.) the non-local material symmetries are the same of the discrepancy tensor, and (iii.) the nonlocal effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthortropic matrix.