Constitutive modelling of nonlinear isotropic elastic materials requires a general formulation of the strain–energy function in terms of invariants, or equivalently in terms of the principal stretches {λ1,λ2,λ3}. Yet, when choosing a particular form of a model, the representation in terms of either the principal invariants or stretches becomes important, since a judicious choice between one or the other can lead to a better encapsulation and interpretation of much of the behaviour of a given material. Here, we introduce a family of generalised isotropic invariants, including a member Jα=λ1α+λ2α+λ3α, which collapses to the classical first and second invariant of incompressible elasticity when α is 2 or -2, respectively. Then, we consider incompressible materials for which the strain–energy can be approximated by a function W that solely depends on this invariant Jα. A natural question is to find α that best captures the finite deformation of a given material. We first show that there exist pseudo-universal relationships that are independent of the choice of W, and which only depend on α. Then, on using these pseudo-universal relationships, we show that one can obtain the exponent α that best fits a given dataset before seeking a functional form for the strain–energy function W. This two-step process delivers the best model that is a function of a single invariant. We show, on using specific examples, that this procedure leads to an excellent and easy to use approximation of constitutive models.

中文翻译：

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超弹性材料的广义不变量和伪通用关系：一种本构建模的新方法


非线性各向同性弹性材料的本构建模需要根据不变量或等效的主拉伸 {λ1，λ2，λ3} 来概括应变-能量函数。然而，当选择模型的特定形式时，主不变量或拉伸的表示变得很重要，因为在其中一种或另一种之间做出明智的选择可以更好地封装和解释给定材料的大部分行为。在这里，我们引入了一个广义各向同性不变量族，包括一个成员 Jα=λ1α+λ2α+λ3α，当 α 为 2 或 -2 时，它分别坍缩为不可压缩弹性的经典第一和第二不变量。然后，我们考虑不可压缩材料，其应变-能量可以用一个函数 W 来近似，该函数仅取决于这个不变的 Jα。一个自然而然的问题是找到最能捕捉给定材料有限变形的α。我们首先表明存在独立于 W 选择且仅取决于 α 的伪普遍关系。然后，在这些伪通用关系上，我们表明，在寻找应变-能量函数 W 的函数形式之前，可以获得最适合给定数据集的指数α。这个两步过程提供了作为单个不变量函数的最佳模型。我们通过使用具体示例来表明，此过程导致了出色且易于使用的本构模型近似。