The choice of objective rates in finite elastoplasticity: general results on the uniqueness of the logarithmic rate

A Eulerian rate formulation of finite elastoplasticity is a composite one composed of a rate equation for elastic behaviour and a flow rule for plastic behaviour as well as an evolution equation for hardening behaviour, in which objective Eulerian tensor rates are used. Among a large variety of objective rates (actually infinitely many), how to choose suitable ones has been one of the crucial points in finite elastoplasticity. It is realized that the foregoing composite formulation of elastoplasticity should fulfil certain consistency criteria in order to avoid inconsistencies or contradictions. These criteria narrow the choice of objective rates. These authors (Bruhns and co–workers and Xiao and co–workers) have recently introduced the self–consistency criterion: in a composite formulation of elastoplasticity, the rate equation intended for characterization of elastic behaviour should be exactly integrable to deliver an elastic relation. It has been demonstrated in the work of the aforementioned authors that if a composite formulation of elastoplasticity with objective rates is required to fulfil the just–stated self–consistency criterion, as it should be, then the newly discovered logarithmic rate is the only possible choice among all objective corotational rates, including the Zaremba–Jaumann rate and the Green–Naghdi rate, etc. In the afore–mentioned result, non–corotational rates, including Oldroyd rates, Cotter–Rivlin rate and Truesdell rate, etc., are not taken into consideration in general. It is the main goal of this paper to further establish the uniqueness of the logarithmic rate among all corotational and non–corotational objective rates. Essential to the attainment of this goal is the use of the yielding–stationarity criterion: the vanishing of the stress rate implies that the yield function is stationary. It is shown that the just–stated criterion is necessary for a composite Eulerian rate formulation of finite elastoplasticity to be consistent and means that the stress rate must be a corotational rate. The main goal of this article is, thus, attained by combining the just–stated result and the established result stated before.