Solution for Eshelby’s elliptic inclusion with polynomials distribution of the eigenstrains in plane elasticity

Abstract

Keywords:Eshelby's elliptic inclusion Plane elasticity Polynomials distribution of eigenstrains Complex variable method Closed form solution a b s t r a c t This paper provides a closed form solution for the Eshelby's elliptic inclusion in plane elasticity with the polynomials distribution of the eigenstrains. The complex variable method and the conformal mapping technique are used. The continuity conditions for the traction and displacement along the interface in the physical plane are reduced to a similar condition along the unit circle of the mapping plane. From those continuity conditions, we can obtain two sets of the complex potentials for the region outside of the unit circle and for the ring region, respectively. Further, we can obtain the complex potentials in the physical plane, or z-plane (z = x + iy). The mapping function maps the ring region in the mapping plane into a finite elliptic region with a crack on the real axis in the physical plane.

Analysis of a finite matrix with an inhomogeneous circular inclusion subjected to a non-uniform eigenstrain

“…Thus, in this section, we will apply the solutions to consider (24) It can be found from Eqs. (23) and (24) that the value of radial stress inside the inclusion is equal to the hoop stress at the same position, while they are not equal outside.…”

“…These studies, which can be mainly categorized into homogenous and inhomogeneous inclusion problems, have largely released the assumptions existed in the pioneering works of Eshelby [1][2]. Besides elliptical and ellipsoid inclusions and uniform eigenstrains distributed in inclusions, the studies have considered line [3][4][5], polygonal (or polyhedral) [6][7][8][9][10][11] and even arbitrary-shaped [12][13][14][15][16][17][18] inclusions and also extended the eigenstrains to non-uniformity [19][20][21][22][23]. A comprehensive review work can be found in literature [24].…”