Construction of Invariant Relations of \(n\) Symmetric Second-Order Tensors

Abstract

A methodology is presented to find either implicit or explicit relations, called syzygies, between invariants in a minimal integrity basis for \(n\) symmetric second-order tensors defined on a three-dimensional euclidean space. The methodology i) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis and ii) allows the construction of the implicit relations. The results of this investigation are important in modeling biological structures, which, in general, are non-homogeneous and made of anisotropic viscoelastic materials that are subjected to large deformations and are modeled through constitutive relations that depend on symmetric tensors.