Choice of objective rate in single parameter hypoelastic deformation cycles

Hill [Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; 1950] demonstrated that “infinitesimal-displacement theory, used in classical elastoplasticity, may no longer be valid in elastic–plastic analysis because the convected terms in the rate of change of the stress acting on material particle may then not be negligible” [Lee EH. Some anomalies in the structure of elastic–plastic theory at finite strain. In: Carroll MM, Hayes M. editors. Nonlinear effects in fluids and solids, New York: Plenum Press; 1996. p. 227–49]. From this we may deduce that
• the elastic deformation part may have considerable influence on the total deformation, even when it is relatively small, i.e. we are confronted with the vital requirement of properly computing elastic deformations.
• Further, finite deformation kinematics should be applied, i.e. they should take account of possibly large rotations, e.g. through a formulation in an Eulerian frame.Xiao et al. [Xiao H, Bruhns OT, Meyers ATM. Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate. Int J Plast 1999;15:479–520] gave the mathematical proof, that Bernstein’s consistency criterion [Bernstein B. Relation between hypo-elasticity and elasticity. Trans Soc Rheol 1960;4:23–8; Bernstein B. Hypoelasticity and elasticity. Arch Ration Mech Anal 1960;6:90–104] is fulfilled in a hypoelastic law of grade zero if, and only if, the objective logarithmic stress rate [Xiao H, Bruhns OT, Meyers A. Hypo-elasticity model based upon the logarithmic stress rate. J Elasticity 1997;47:51–68] has been applied. This proof is of complicated mathematical nature. Here, we compare several objective Eulerian stress rates of corotational and non-corotational type for the hypoelastic law cited above in closed single parameter deformation cycles. It is found that the logarithmic stress rate returns the element to its stress-free original state after the closed cycle, thus confirming the findings in Xiao et al. (1999). We show that for some other objective rates the errors are accumulating to considerable amounts after several cycles, even when the deformation in investigation is relatively small. Interestingly, for Jaumann stress rate, the error may vanish for specified deformation measures.