Irreducible invariants of fourth-order tensors

Abstract

In solid mechanics of isotropic and anisotropic materials representing scalar-valued tensor functions or symmetric second-order tensor-valued tensor functions is of major concern, For instance, the plastic potential is scalar-valued, whereas constitutive equations are tensor-valued. In this paper scalar-valued functions have been represented. Anisotropic effects have been characterized by material tensors of rank two and four. For instance, the effect of damage or the behaviour of oriented solids have been characterized by second-order tensors, and with respect to more general cases inborn anisotropy has been described by using a rank four. In representing scalar-valued tensor functions, a set of irreducible invariant involving the above mentioned tensor variables has been constructed. The central problem is: to find an integrity basis for the argument tensors. Together with the invariants of the single argument tensors the system of simultaneous or joint invariants is considered. Such invariants are given by traces of outer products formed from different argument tensors considered. In finding irreducible invariants of a fourth-order tensor the characteristic polynomial for a fourth-order tensor is derived by the definition of the eigenvektor . Furthermore, Hamilton-Cayley's theorem is applied to a fourth-order tensor. Thus irreducible invariants of a fourth-order tensor can be expressed through sums of principal minors.