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On the Elliptic Inclusion in Anti-Plane Shear

In this paper, we consider the anti-plane shear problem of an elliptic inclusion embedded in an infinite, isotropic, elastic medium, subjected at infinity to a uniform stress field. Using complex variable methods and the theory of analytic functions, we prove that the state of deformation in the inclusion is a simple shear if and only if the curve enclosing the inclusion is an ellipse.