Analytical solution for circular inhomogeneous inclusion problems with non-uniform axisymmetric eigenstrain distribution

Abstract

This paper presents the analytical solution of a class of plane elasticity problems for circular inhomogeneous inclusions with non-uniform axisymmetric eigenstrain distribution, which includes both radial and hoop eigenstrains. The complex variable potential solution for a point-wise eigenstrain in an infinite plane solid is presented first, in which two principal strains and their directions are taken into account. Directly employing it as influence function, the complex variable potential for the circular homogeneous inclusion problem is formulated with Green's function method. The novelty of this approach is that it is able to take intrinsic advantage of complex variable approach and effectively tackle the mathematical difficulties encountered during formulation. Next, by using the principle of equivalent eigenstrain, the main challenge in solving inhomogeneous inclusion problems is overcome, allowing the general explicit analytical solution to be derived. Based on these results, three illustrative examples of practical significance are given: (i) dissimilar cylinder interference-fits within an infinite body, (ii) a pure dilatational eigenstrain problem within a circular inclusion, and (iii) a circular inclusion problem with a wedge disclination eigenstrain distribution. The fundamental formulation introduced here will find application in other aspects in the mechanics of fiber composites, thermoelasticity, and nano-mechanics of defects in solids.