  Implicit rate-type constitutive relations utilizing discontinuous functions
provide a novel approach to the purely phenomenological description of the
inelastic response of solids undergoing finite deformation. However, this type
of constitutive relations has been so far considered only in the purely
mechanical setting, and the complete thermodynamic basis is largely missing. We
address this issue, and we develop a thermodynamic basis for such constitutive
relations. In particular, we focus on the thermodynamic basis for the classical
elastic--perfectly plastic response, but the framework is flexible enough to
describe another types of inelastic response as well.
