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

Abstract

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.