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This paper is devoted to the question, whether there is an order barrier$p\leq2$ for time integration in computational elasto-plasticity. In theanalysis we use an implicit Runge-Kutta (RK) method of order $p=3$ forintegrating the evolution equations of plastic flow within a nonlinear finiteelement framework. We show that two novel algorithmic conditions are necessaryto overcome the order barrier, (i) total strains must have the same order intime as the time integrator itself, (ii) accurate initial data must becalculated via detecting the elastic-plastic switching point (SP) in thepredictor step. Condition (i) is for a \emph{consistent} coupling of the globalboundary value problem (BVP) with the local initial value problems (IVP) viadisplacements/strains. Condition (ii) generates consistent initial data of theIVPs. The third condition, which is not algorithmic but physical in nature, isthat (iii) the total strain path in time must be smooth such that condition (i)can be fulfilled at all. This requirement is met by materials showing asufficiently smooth elastic-plastic transition in the stress-strain curve. Wepropose effective means to fulfil conditions (i) and (ii). We show in finiteelement simulations that, if condition (iii) is additionally met, the presentmethod yields the full, theoretical convergence order 3 thus overcoming thebarrier $p\leq 2$ for the first time. The observed speed-up for a 3rd order RKmethod is considerable compared with Backward Euler.

Is there an order-barrier $p\leq2$ for time integration in computational elasto-plasticity?