Evaluation of the Eshelby tensor for polygonal inclusions

Abstract

Abstract The Eshelby tensor field is evaluated analytically for polygonal inclusions of arbitrary shape. The formula contributed in the paper is directly expressed as function of the coordinates defining the vertices of the polygon, thus avoiding the use of complex variables and anomalies exploited in previous contributions on the subject. It has been obtained by evaluating analytically the displacement field induced by a uniform eigenstrain within the inclusion and differentiating its expression in order to derive the infinitesimal strain tensor and, hence, the Eshelby tensor. This allows us to deal only with the first derivative of the Green tensor, appearing in the expression of the displacement field, rather than addressing its second derivative as commonly done in the literature. The proposed formulation has been implemented in a Matlab code and we report the numerical results concerning inclusions whose boundary is defined by Laurent polynomials or by quasi-circular shapes.