Basic difference in viscoplasticity and elastoplasticity I am trying to create a Finite element based code to solve dynamic Plasticity problem. I recently started reading about Plasticity and I have come to understand that Rate-independent plasticity is also called elasto-plasticity and Rate-dependent platicity is called visco-plasticity. 
I am sticking to a rate-independent case. Does that mean that my FEM weak (variational) form is not supposed to have a viscosity, ie., damping term? Can one model a dynamic elastoplastic problem without damping? I feel that both the classifications can have a damping term, but then, why is only the rate - dependent case called viscoplastic? Am I missing something trivial here?
 A: Physically, elasto-plasticity does act like damping, in the sense that creating the plastic deformation requires energy which is then "locked up" in the material as residual stresses and strains. 
Depending on your problem, you probably want to include some other source of damping in your dynamics model, otherwise you are likely to get undamped elastic vibrations of the structure which continue for ever, after there is no further change in the plastic deformation.
If there is no physical source of damping that you want to model explicitly, you could use Rayleigh damping (damping matrix = $\alpha M + \beta K$ for some constants $\alpha$ and $\beta$), or use a time-stepping algorithm which includes some numerical damping.
Including "non-plastic" damping that is equivalent to a Q factor of say 50 for the lowest linear-elastic vibration mode of the structure would be a reasonable starting point. Many real-world structures have damping of that order of magnitude caused by things like friction between the structure and its restraints (which are nominally modeled as perfectly rigid), aerodynamic damping caused by the motion of the surface of the structure transferring energy into the air, etc, etc.
Note that "hysteretic damping" caused by the changes in the elastic strain energy of the material is typically much smaller (by one or two orders of magnitude) than the effects mentioned in the previous paragraph, at least for typical metals used in engineering.
A: I have stumbled upon two books covering the topic in detail performing a quick search:


*

*Dvorkin Goldschmit: Nonlinear Continua; Springer Berlin Heidelberg 2006

*Francois et al., Mechanical Behaviour of Materials Vol 1; Springer Dordrecht 2012


From them I have taken that:
Material properties of an Elastoplastic material are ([1] p. 135): 


*

*below a certain limit load, the material is (hyper-)elastic [the stress tensor is the derivative of the elastic energy function with respect to the strain tensor]

*at the limit load, permanent/plastic deformations start

*the limit load is influenced by the plastic deformations, described by a hardening law

*the material behaviour is rate independent

*the material initially has a prefered configuration


Viscoplasticity differs from this in apparently mainly two points:


*

*the limit load is rate-dependent ([1] (p. 176))

*the flow rate above the limit load is also rate dependent ([2] (p. 364))


As I said, metal continuum mechanics is not my field, so for more information I will advise you to read e.g. those books. But from what I have understood - the two behaviours are named differently because they really are different. But both behaviours seem to allow for dissipation in the flow regime.
