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 c omposed of a rate equation for elastic behaviour and a flow rule for plasti c behaviour as well as an evolution equation for hardening behaviour, in wh ich objective Eulerian tensor rates are used. Among a large variety of obje ctive rates (actually infinitely many), how to choose suitable ones has bee n one of the crucial points in finite elastoplasticity. It is realized that the foregoing composite formulation of elastoplasticity should fulfil cert ain consistency criteria in order to avoid inconsistencies or contradiction s. These criteria narrow the choice of objective rates. These authors (Bruh ns and co-workers and Xiao and co-workers) have recently introduced the sel f-consistency criterion: in a composite formulation of elastoplasticity, th e rate equation intended for characterization of elastic behaviour should b e exactly integrable to deliver an elastic relation. It has been demonstrat ed in the work of the aforementioned authors that if a composite formulatio n of elastoplasticity with objective rates is required to fulfil the just-s tated self-consistency criterion, as it should be, then the newly discovere d logarithmic rate is the only possible choice among all objective corotati onal rates, including the Zaremba-Jaumann rate and the Green-Naghdi rate, e tc. In the aforementioned result, non-corotational rates, including Oldroyd rates, Cotter-Rivlin rate and Truesdell rate, etc., are not taken into con sideration in general. It is the main goal of this paper to further establi sh the uniqueness of the logarithmic rate among all corotational and non-co rotational objective rates. Essential to the attainment of this goal is the use of the yielding-stationarity criterion: the vanishing of the stress ra te 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 mu st be a corotational rate. The main goal of this article is, thus, attained by combining the just-stated result and the established result stated befo re.