New explicit expressions of the Hill polarization tensor for general anisotropic elastic solids

Except for particular cases, the classical expressions of the Eshelby or Hill polarization tensors, depend, respectively, on a simple or double integral for a fully anisotropic two-dimensional or three-dimensional elastic body. When the body is two-dimensional, we take advantage of Cauchy's theory of residues to derive a new explicit expression which depends on the two pairs of complex conjugate roots of a quartic equation. If the body exhibits orthotropic symmetry, these roots are explicitly given as a function of the independent components of the elasticity tensor. Similarly, the double integral is reduced to a simple one when the body is three-dimensional. The corresponding integrand depends on the three pairs of complex conjugate roots of a sextic equation which reduces to a cubic one for orthotropic symmetry. This new expression improves significantly the computation times when the degree of anisotropy is high. For both two and three-dimensional bodies, degenerate cases are also studied to yield valid expressions in any events.