A natural generalization of hypoelasticity and Eulerian rate type formulation of hyperelasticity

According to the classical hypoelasticity theory, the hypoelasticity tensor , i.e. the fourth order Eulerian constitutive tensor, characterizing the li near relationship between the stretching and an objective stress rate, is d ependent on the current stress and must be isotropic. Although the classica l hypoelasticity in this sense includes as a particular case the isotropic elasticity, it fails to incorporate any given type of anisotropic elasticit y. This implies that one can formulate the isotropic elasticity as an integ rable-exactly classical hypoelastic relation, whereas one can in no way do the same for any given type of anisotropic elasticity. A generalization of classical theory is available, which assumes that the material time derivat ive of the rotated stress is dependent on the rotated Cauchy stress, the ro tated stretching and a Lagrangean spin, linear and of the first degree in t he latter two. As compared with the original idea of classical hypoelastici ty, perhaps the just-mentioned generalization might be somewhat drastic. In this article, we show that, merely replacing the isotropy property of the aforementioned stress-dependent hypoelasticity tensor with the invariance p roperty of the latter under an R-rotating material symmetry group R star G( 0), one may establish a natural generalization of classical theory, which i ncludes all of elasticity. Here R is the rotation tensor in the polar decom position of the deformation gradient and G(0) any given initial material sy mmetry group. In particular, the classical case is recovered whenever the m aterial symmetry is assumed to be isotropic. With the new generalization it is demonstrated that any two non-integrable hypoelastic relations based on any two objective stress rates predict quite different path-dependent resp onses in nature and hence can in no sense be equivalent. Thus, the non-inte grable hypoelastic relations based on any given objective stress rate const itute an independent constitutive class in its own right which is disjoint with and hence distinguishes itself from any class based on another objecti ve stress rate. Only for elasticity, equivalent hypoelastic formulations ba sed on different stress rates may be established. Moreover, universal integ rability conditions are derived for all kinds of objective corotational str ess rates and for all types of material symmetry. Explicit, simple, integra ble-exactly hypoelastic relations based on the newly discovered logarithmic stress rate are presented to characterize hyperelasticity with any given t ype of material symmetry. It is shown that, to achieve the latter goal, the logarithmic stress rate is the only choice among all infinitely many objec tive corotational stress rates.