Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

In this paper, the derivation of irreducible bases for a class of isotropic 2nth-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an nth-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2nth-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.