Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky's logarithmic strain tensor

Hencky's strain-energy function for finite isotropic elasticity is obtained by the replacement of the infinitesimal strain measure occurring in the classical strain-energy function of infinitesimal isotropic elasticity with the Hencky or logarithmic strain measure. It has been shown recently by Anand that this simple strain-energy function, with two classical Lame elastic constants, is in good agreement with a wide class of materials for moderately large deformations. Very recently, it has been shown by these authors that the hyperelastic relation with the foregoing Hencky strain-energy function may enter as a basic constituent into the Eulerian rate formulation of finite elastoplasticity for metals, etc. Now it is commonly used in finite-element method (FEM) computations and in commercial packets of FEM codes, etc. For this useful strain-energy function, there is a need to study the restrictions and consequences resulting from certain well-founded constitutive inequality conditions. Here, we consider the well-known Legendre-Hadamard, or the ellipticity, condition. We first derive simple, explicit necessary and sufficient conditions for ellipticity in terms of the principal stretches. Then we determine the largest common region for ellipticity in the principal stretch space, which applies to all Hencky strain-energy functions with non-negative Lame constants. In particular, we find out the largest cube contained in the common region just mentioned. We prove that the Hencky strain-energy function fulfils the Legendre-Hadamard condition whenever every principal stretch falls within the range [alpha, (3)roote], where the lower bound alpha = 0.21162... is the unique root of a certain transcendental equation involving the natural logarithmic function, and e = 2.718 28..., in the upper bound, is the base of the natural logarithm. The range mentioned above, i.e. [0.21162, 1.395 61], covers the range [0.7,1.3] set by Anand for moderately large deformations. Moreover, it is shown that Hencky's strain-energy function obeys the well-known Baker-Ericksen inequality and Hill's inequality over the whole range of deformations.