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.

提出了一种方法，用于在三维欧几里得空间上定义的\(n\)对称二阶张量的最小完整性基础上找到不变量之间的隐式或显式关系（称为 syzygies） 。该方法 i) 根据完整性基础中的剩余不变量产生某些不变量的显式非多项式表达式，并且 ii) 允许构造隐式关系。这项研究的结果对于生物结构的建模非常重要，生物结构通常是非均质的，由各向异性粘弹性材料制成，会发生大变形，并通过依赖于对称张量的本构关系进行建模。