Generalised invariants and pseudo-universal relationships for hyperelastic materials: A new approach to constitutive modelling

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.