A Methodology to Find Relations between Invariants of n Symmetric Second-Order Tensors

Abstract

A methodology is proposed to find either implicit or explicit relations between invariants in a minimal integrity basis for n symmetric secondorder tensors defined on a three-dimensional euclidean space. The implicit relations are called syzygies. In particular, the methodology i) allows the construction of a set of 6 n − 3 independent invariants, ii) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis, and iii) allows the construction of syzygies. The results of this investigation are important in the modeling of biological structures, which, in general, are non-homogeneous and made of anisotropic viscoelastic materials that are subjected to large deformations.