Hypo-Elasticity Model Based upon the Logarithmic Stress Rate

Recently these authors have proved 46, 47] that a smooth spin tensor Ω^log can be found such that the stretching tensor D can be exactly written as an objective corotational rate of the Eulerian logarithmic strain measure ln V defined by this spin tensor, and furthermore that in all strain tensor measures only ln V enjoys this favourable property. This spin tensor is called the logarithmic spin and the objective corotational rate of an Eulerian tensor defined by it is called the logarithmic tensor-rate. In this paper, we propose and investigate a hypo-elasticity model based upon the objective corotational rate of the Kirchhoff stress defined by the spin Ω^log, i.e. the logarithmic stress rate. By virtue of the proposed model, we show that the simplest relationship between hypo-elasticity and elasticity can be established, and accordingly that Bernstein's integrability theorem relating hypo-elasticity to elasticity can be substantially simplified. In particular, we show that the simplest form of the proposed model, i.e. the hypo-elasticity model of grade zero, turns out to be integrable to deliver a linear isotropic relation between the Kirchhoff stress and the Eulerian logarithmic strain ln V, and moreover that this simplest model predicts the phenomenon of the known hypo-elastic yield at simple shear deformation. This revised version was published online in August 2006 with corrections to the Cover Date.