Analytical expression of the Eshelby tensor for arbitrary polygonal inclusions in two-dimensional elasticity

We derive the analytical expression of the Eshelby tensor field for inclusions of arbitrary polygonal shape. Differently from previous contributions in the literature, the formula proved in the paper has a particularly compact expression since it is based on the analytical evaluation of integrals depending by lower exponents of the scalar product of the vector connecting the source and observation points. Moreover, the formula is directly expressed as function of the coordinates defining the vertices of the polygon, thus avoiding the use of complex variables and anomalies. The effectiveness of the proposed formulation is numerically assessed by comparing the results provided by its implementation in a Matlab code with available results specifically contributed by different authors for weakly non-circular inclusions.