Integrity basis for a second-order and a fourth-order tensor

Josef Betten

doi:10.1155/s0161171282000088

Abstract

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

Highlights

In this paper a scs!ar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material
Many mathematicians have studied the theory of algebraic invariants in detail
An integrity basis is a set of polynomials, each invariant under the group of transformations, such that any polynomial function invariant under the group is expressible as a polynomial in elements of the integrity basis

Summary

This function is said to be isotropic if the condit ion

From the theory of isotropic tensor functions [5, 7, 8], it is evident that in an isotropic medium the plastic potential F can be expressed as a single-valued function of the irreducible basic invariants. The invariants (2.2) or, alternatively, (2.3a,b,c) form o the integrity basis for the stress tensor under the proper orthogonal group, i.e. aij. The function F is merely required to be invariant under the group of transformations (SikSjk ij) associated with the symmetry properties of the material [8], where s a. For a particular crystal class [ii] the potential F may be represented as a s polynomial in the stresses which is invariant under the subgroup of transformtions associated with the symmetry properties of the crystal class considered. The central problem is: to construct an irreducible integrity basis for the tensors o ij

Together with the invariants of the single argument

Aijkl Ajikl Aijlk lij or alternatively exgressed by

OipOpj Aij klOkl Aij klOikOj i

Epqopq o ij AijpqOpq’