Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

Abstract

In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.

Introduction

Most solid materials show in the finite strain range nearly incompressible, i.e. weakly compressible material behaviour. The constitutive models concerned usually utilize hyperelasticity relations which describe one part of the model. This is done in the case of rate-dependent and rate-independent constitutive theories such as viscoelasticity, elastoplasticity or viscoplasticity. Naturally, the investigations below are also of interest in the case of purely elastic material behaviour. In analytical derivations the weakly compressible behaviour, which can be observed in most experiments, is replaced by the assumption of incompressibility in order to obtain particular solutions. On the other hand, it is much more convenient in the numerical treatment of these constitutive models, for example, utilizing the finite element method, to employ the nearly incompressible extension. In this case there are three usually employed constitutive models in use, namely the generalized polynomial-type elasticity due to Rivlin and Saunders, the Arruda and Boyce as well as Ogden’s model (Rivlin and Saunders, 1951; Arruda and Boyce, 1993; Ogden, 1972a). The strain-energy functions concerned are originally formulated in the case of incompressibility as functions of the first and second invariant or in eigenvalues of the left (or right) Cauchy–Green tensor. One way of extending these models to the nearly incompressible case is to exchange the invariants or eigenvalues of the original Cauchy–Green tensors, i.e. in the modified model one uses the unimodular part of the Cauchy–Green tensors which are based on the multiplicative decomposition of the deformation gradient into a volume-changing and a volume-preserving part. This decomposition goes back to Flory (1961). Furthermore, additional parts of the strain-energy function depend merely on the determinant of the deformation gradient, i.e. the strain energy decomposes additively into two parts: one part depends on the volume-changing part via the determinant of the deformation gradient and the other part on the invariants or eigenvalues of the right Cauchy–Green tensor built up by the volume-preserving part of the deformation gradient. This article discusses this modification of the models with respect to physical and mathematical aspects.

A much debated question in the area of finite elasticity is the correct formulation of constitutive inequalities ensuring reasonable solutions to physical problems. Since we are focusing on hyperelastic material behaviour these constitutive inequalities translate into conditions on the free energy. The so-called Baker–Ericksen inequalities, the Coleman–Noll condition or Hill’s inequality may serve as examples for the earlier attempts to establish these correct formulations. For a discussion on these inequalities see, for example, Baker and Ericksen (1954), Marsden and Hughes (1983), Truesdell and Noll (1965), Wang and Truesdell (1973), Hill (1970) and Ogden (1984). However, these inequalities could not be proved to guarantee the well-posedness of the problem. Moreover, some criteria cannot be shown to be satisfied a priori.

The mathematical treatment of the corresponding boundary-value problem (the structural mechanics problem) is mainly based on the direct methods of variation, i.e. to find a minimizing deformation of the elastic free energy subject to the specific boundary conditions. This minimizing deformation is found by constructing infimizing sequences of deformations and then showing that the sequence converges in some sense to the sought minimizer. The main ingredient for carrying out this program is a quasi-convexity hypothesis on the free energy, Morrey (1952): roughly, it ensures that the functional to be minimized is weakly lower semi-continuous. However, this condition is difficult to handle since it is a non-local integral condition. A much more tractable condition has been introduced by Ball in his seminal article (Ball, 1977a), it is the so-called polyconvexity condition. There exists a vast literature on polyconvexity (see e.g., Ball, 1977a, Ball, 1977b; Marsden and Hughes, 1983; Ciarlet, 1988; Charrier et al., 1988 and the literature cited there) and fortunately some energy expressions already introduced are covered by this concept (Ogden’s, Mooney-Rivlin and Neo-Hookean model). For isochoric-volumetric decouplings some forms of polyconvex energies have been proposed by Charrier et al. (1988) or Dacorogna (1989, p. 134 and pp. 256ff.). There are some simple stored energies of St. Venant-Kirchhoff type (Ciarlet, 1988; Raoult, 1986) or energies involving the so-called (logarithmic) Hencky tensor which, however, do not satisfy the polyconvexity condition (Neff, 2000). Moreover, it can be shown that neither St.Venant-Kirchhoff nor strain-energy functions based on Hencky strains lead to elliptic equilibrium conditions.

It can be shown that polyconvexity of the stored energy implies that the corresponding acoustic tensor is elliptic for all deformations whatsoever, moreover strict ellipticity is sufficient for the Baker–Ericksen inequalities to hold. The precise difference between the local property of ellipticity and quasi-convexity is still an active field of research, since counterexamples exist which are elliptic throughout but not quasi-convex. However, these examples are neither frame indifferent nor isotropic; from a purely mechanical point of view this difference might be negligible. Polyconvexity as such does not conflict with the possible non-uniqueness of equilibrium solutions, since it only guarantees the existence of at least one minimizing deformation. It is possible that several metastable states (local equilibria) and several absolute minimizers exist, even so, one might conjecture that apart from trivial symmetries the absolute minimizer is unique, at least for the pure Dirichlet boundary-value problem. In general, under polyconvexity assumptions, no claim can be made as to the stability or smoothness of the solution, apart from the natural statement that the minimizer lies in the Sobolev space considered. Moreover, it is not known that the minimizing deformation is a weak solution of the local balance equation, due to possible singularities in the deformation gradient. We remark, following (Ball, 1977a, p. 398) that polyconvexity implies the existence for all boundary conditions and body forces which might be somewhat unrealistic. The conclusion that one particular form of the stored energy is not polyconvex does not mean that this energy should be ruled out from the outset. Indeed, the corresponding failure of ellipticity at large deformation gradients only (see the examples below) might be physically correct, indicating, for example, the onset of fracture or some other local instability like the formation of microstructure. On the other hand, the proof that some energy is elliptic for a reasonable range of deformation is presently not enough to establish an existence theorem. Since we will be concerned with nearly incompressible isotropic hyperelasticity only where we expect neither fracture nor microstructure, the polyconvexity assumption seems to be a convenient mathematical tool to ascertain the existence of a minimizer of an elastic free energy.

Our main contribution consists of enlarging the class of known polyconvex energies including expressions which bear resemblance to the generalized polynomial-type elasticity relations due to Rivlin and Saunders (1951), defined by modified invariants mentioned above. To our knowledge general polynomial energy expressions like those of Rivlin and Saunders (1951) or Arruda and Boyce (1993) have not been investigated with respect to polyconvexity. The polyconvexity conditions will then translate into a requirement on the structure of the polynomial terms and the restriction to positive material parameters.