  We study a model for a fluid showing viscoelastic and viscoplastic behavior,
which describes the flow in terms of the fluid velocity and an internal stress.
This stress tensor is transported via the Zaremba--Jaumann rate, and it is
subject to two dissipation processes: one induced by a nonsmooth convex
potential and one by stress diffusion. We show short-time existence of strong
solutions as well as their uniqueness in a class of Leray--Hopf type weak
solutions satisfying the tensorial component in the sense of an evolutionary
variational inequality. The global-in-time existence of such generalized
solutions has been established in a previous work. We further study the limit
when stress diffusion vanishes. In this case, the above notion of generalized
solutions is no longer suitable, and we introduce the concept of
energy-variational solutions, which is based on an inequality for the relative
energy. We derive general properties of energy-variational solutions and show
their existence by passing to the non-diffusive limit in the relative energy
inequality satisfied by generalized solutions for non-zero stress diffusion.
