Solution for elliptic inclusion in an infinite plate with remote loading and Eshelby’s eigenstrain of polynomial type

Abstract

In this paper, a particular inhomogeneous inclusion problem is studied. In the problem, Eshelby’s eigenstrain takes the type $x^m{y}^n$ , where m+n = 2, and the remote loadings ${\sigma }_x^{\infty }$ , ${\sigma }_y^{\infty }$ are applied. In the solution, the complex variable method is used. The continuity conditions along the interface of the matrix and the inclusion are formulated exactly. Because the stress field is no longer uniform in inclusion in this case, the studied problem has an inherent difficulty. After some manipulation, the final result for stress components ${\sigma }_x^{\mathrm{in}}$ , ${\sigma }_y^{\mathrm{in}}$ and ${\sigma }_{\mathrm{xy}}^{\mathrm{in}}$ in inclusion are obtainable. In the present study, ${\sigma }_x^{\mathrm{in}}$ , ${\sigma }_y^{\mathrm{in}}$ and ${\sigma }_{\mathrm{xy}}^{\mathrm{in}}$ are no longer uniform.