  This paper is devoted to the question, whether there is an order barrier
$p\leq2$ for time integration in computational elasto-plasticity. In the
analysis we use an implicit Runge-Kutta (RK) method of order $p=3$ for
integrating the evolution equations of plastic flow within a nonlinear finite
element framework. We show that two novel algorithmic conditions are necessary
to overcome the order barrier, (i) total strains must have the same order in
time as the time integrator itself, (ii) accurate initial data must be
calculated via detecting the elastic-plastic switching point (SP) in the
predictor step. Condition (i) is for a \emph{consistent} coupling of the global
boundary value problem (BVP) with the local initial value problems (IVP) via
displacements/strains. Condition (ii) generates consistent initial data of the
IVPs. The third condition, which is not algorithmic but physical in nature, is
that (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 a
sufficiently smooth elastic-plastic transition in the stress-strain curve. We
propose effective means to fulfil conditions (i) and (ii). We show in finite
element simulations that, if condition (iii) is additionally met, the present
method yields the full, theoretical convergence order 3 thus overcoming the
barrier $p\leq 2$ for the first time. The observed speed-up for a 3rd order RK
method is considerable compared with Backward Euler.
