Anti-plane elastic field of a cylindrical inhomogeneity within first strain-gradient theory: exact solution vs. extended equivalent inclusion method

Abstract

Determination of the elastic field developed in a heterogeneous material provides useful information for calculating its overall elastic properties. When dimensions of inhomogeneities in such a material are comparable to the intrinsic length scale of its constituents, classical elasticity ceases to produce reliable solutions. Mindlin’s first strain-gradient elasticity, as an enhanced continuum mechanics theory, has proved its success in dealing with such a problem. Hence, this theory is utilized in the present paper to obtain an exact solution of the elastic displacement field induced in an infinite isotropic medium that contains a circular cylindrical inhomogeneity and is subjected to an anti-plane loading. The obtained solution demonstrates the size effect on the elastic field of the medium. On the other hand, an extended version of the equivalent inclusion method which is adapted to Mindlin’s first strain-gradient theory is developed to attack the same problem, leading to an approximate solution. Subsequently, by solving some numerical examples, a comparison between these exact and approximate solutions is provided in this paper.