A General Solution for Anti-plane Elastic Field in Anisotropic Media Containing an Eshelby’s Elliptic Inhomogeneity

A general complex variable approach is presented for anti-plane problem for a system of one anisotropic elliptic inhomogeneity embedded in an infinite anisotropic medium (matrix). The system is subject to far-field shear stresses in the matrix at infinity and eigenstrains inside the inhomogeneity. The solution is derived based on conformal mapping for the complex variable and Laurent series expansion for the stress functions in which the coefficients in the series are determined by using continuous conditions at the interface. The shear stress distribution at the interface and displacements for the entire system are shown for the case of polynomial eigenstrains with the forms of uniform, linear and quadratic distributions, respectively. The translational deformation of the inhomogeneity and the effect of the orientation of principal axes of the anisotropic materials on the displacements of both the matrix and inhomogeneity are illustrated.