Keywords: elasto-plasticity, strain cycle, objectivity, large deformation.

Hill [1] 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 [2]). 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 & al. [5] gave the mathematical proof, that Bernstein's consistency criterion [3,4] is fulfilled in a hypoelastic law of grade zero if, and only if, the objective logarithmic stress rate [6] 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 strain 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 [5]. 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 some stress rates, the error may vanish for specified deformation measures.

1

R. Hill. "The Mathematical theory of Plasticity" Clarendon Press, Oxford, England, 1950.

2

E. H. Lee. "Some anomalies in the structure of elastic-plastic theory at finite strain". In M. M. Carroll and M. Hayes, editors, Nonlinear Effects in Fluids and Solids, pages 227-249. Plenum Press, New York, 1996.

3

B. Bernstein. "Relation between hypo-elasticity and elasticity". Trans. Soc. Rheol., 4:23-28, 1960. doi:10.1122/1.548874

4

B. Bernstein. "Hypoelasticity and elasticity", Arch. Rat. Mech. Anal., 6:90-104, 1960.

5

Heng Xiao, Otto Timme Bruhns, and Albert Thomas Marie Meyers. "Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate", International Journal of Plasticity, 15:479-520, 1999. doi:10.1016/S0749-6419(99)00003-0

6

Heng Xiao, Otto Timme Bruhns, and Albert Thomas Marie Meyers. "Logarithmic strain, logarithmic spin and logarithmic rate". Acta Mechanica, 124:89-105, 1997. doi:10.1007/BF01213020

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