  We study a model for rate-dependent gradient plasticity at finite strain
based on the multiplicative decomposition of the strain tensor, and investigate
the existence of global-in-time solutions to the related PDE system. We reveal
its underlying structure as a generalized gradient system, where the driving
energy functional is highly nonconvex and features the geometric nonlinearities
related to finite-strain elasticity as well as the multiplicative decomposition
of finite-strain plasticity. Moreover, the dissipation potential depends on the
left-invariant plastic rate and thus, depends on the plastic state variable.
  The existence theory is developed for a class of abstract, nonsmooth, and
nonconvex gradient systems, for which we introduce suitable notions of
solutions, namely energy-dissipation-balance (EDB) and
energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox
of the direct method of the calculus of variations to check that the specific
energy and dissipation functionals for our viscoplastic models comply with the
conditions of the general theory.
