On the evaluation of the Eshelby tensor for polyhedral inclusions of arbitrary shape

We derive the analytical expression of the Eshelby tensor field for inclusions of arbitrary polyhedral shape. The formula contributed in the paper is directly expressed as function of the coordinates defining the vertices of the polyhedron thus avoiding the use of complex variables and anomalies exploited in previous contributions on the subject. It has been obtained by evaluating analytically the integrals appearing in the very definition of the Eshelby tensor by means of two consecutive applications of the Gauss theorem. The first one allows one to express the original volume integrals as a sum of 2D integrals extended to the faces of the polyhedron, while the second application transforms each 2D integral into the line integrals extended so the edges of each face. The effectiveness of the proposed formulation is numerically assessed by comparing the results provided by its implementation in a Matlab code with results available in the literature.