Evaluation of Conjugate Stresses to Seth’s Strain Tensors

Abstract

An explicit expression providing the symmetric stress tensor T^(m) conjugate to the Seth’s strain measure E^(m) for each integer m unequal 0 is presented. The result is obtained by exploiting an original approach for the solution of a tensor equation in the unknown T^(m) expressed as function of the powers of the right stretch tensor U. The proposed approach is based upon the spectral decomposition of U and exploits some peculiar features of the set of fourth-order tensors obtained as linear combination of dyadic and square tensor products of the eigenprojectors of U. On the basis of such properties it is shown that the unknown T^(m) can be expressed in the given reference frame as linear combination of six fourth-order tensors scaled through coefficients which are rational functions of the eigenvalues of U.