  We analyse a numerical scheme for a system arising from a novel description
of the standard elastic--perfectly plastic response. The elastic--perfectly
plastic response is described via rate-type equations that do not make use of
the standard elastic-plastic decomposition, and the model does not require the
use of variational inequalities. Furthermore, the model naturally includes the
evolution equation for temperature. We present a low order discretisation based
on the finite element method. Under certain restrictions on the mesh we
subsequently prove the existence of discrete solutions, and we discuss the
stability properties of the numerical scheme. The analysis is supplemented with
computational examples.
