Plane deformation of circular inhomogeneous inclusion problems with non-uniform symmetrical dilatational eigenstrain

Abstract

This paper presents the solution of a class of plane elasticity problems for circular inhomogeneous inclusions with non-uniform radially varying axis-symmetrical dilatational eigenstrain. First, the fundamental solution for a point-wise eigenstrain in an infinite plane solid is presented. Next the circular homogeneous inclusion problem is formulated using Green's function method. By using the principle of equivalent eigenstrain recently proposed (Ma and Korsunsky, 2014, International Journal of Solids and Structures, 51, 4477–4484), the main difficulty in solving inhomogeneous inclusion problems is overcome through the use of the equivalent uniform eigenstrain formulation, allowing the general explicit analytical solution to be derived. Based on these results, two illustrative examples of practical significance are solved: (i) the thermo-elastic problem of a point heat source at the centre of a circular inclusion, and (ii) the problem of a circular inclusion with interface eigenstrain that applies in the case of nano-scale inclusions. The fundamental formulation introduced here will find application in other aspects in the mechanics of fibre composites, thermoelasticity, and nano-mechanics of defects in solids.