On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain

This paper discusses various constitutive restrictions on the strain energy function for an isotropic hyperelastic material derived from the condition of strong ellipticity. The strain energy function is assumed to be a function of a novel set of invariants of the Hencky (logarithmic or natural) strain tensor introduced by Criscione et al. (J. Mech. Phys. Solids 48 (2000) 2445). A key step in the analysis is the derivation of an expression for the Fréchet derivative of the Hencky strain with respect to the deformation gradient that is convenient for analyzing the quadratic form over the space of second order tensors central to establishing strong ellipticity. The theory is illustrated by applying the restrictions to a model for rubber proposed by Criscione et al. (J. Mech. Phys. Solids 48 (2000) 2445) It is shown that while that model can be made to violate strong ellipticity, it does so only for very large strains.