Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions

Abstract

We consider gradient models of elasticity which permit taking into account the characteristic scale parameters of the material. We prove the Papkovich–Neuber theorems, which determine the general form of the gradient solution and the structure of scale effects. We derive the Eshelby integral formula for the gradient moduli of elasticity, which plays the role of the closing equation in the self-consistent three-phase method. In the gradient theory of deformations, we consider the fundamental Eshelby–Christensen problem of determining the effective elastic properties of dispersed composites with spherical inclusions; the exact solution of this problem for classical models was obtained in 1976.

This paper is the first to present the exact analytical solution of the Eshelby–Christensen problem for the gradient theory, which permits estimating the influence of scale effects on the stress state and the effective properties of the dispersed composites under study.We also analyze the influence of scale factors.