Inclusions inside a bounded elastic body undergoing anti-plane shear

Abstract

As a problem of theoretical and practical interest, anti-plane shear is mathematically simple and prone to analytical solutions. However, almost no analytical solutions have been reached for the problem of inclusions embedded in a bounded domain and undergoing anti-plane shear eigenstrain. In this work, using some techniques of complex analysis and integral formulae derived previously, Eshelby’s anti-plane shear problem of inclusions in a finite isotropic elastic body is solved analytically. The cross-section of the inclusion with a prescribed uniform anti-plane eigenstrain can be of polygonal shape or any smooth one characterized by a Laurent polynomial, while the cross-section of the prismatic bar submitted to displacement or traction boundary conditions is required to be smooth. In particular, in the case where the cross-section of the prismatic bar and the one of the inclusion are both circular, explicit analytical expressions are derived for the strain and stress fields in different ways. Some examples are provided to illustrate how to implement the solutions based on the conformal maps.