  An elastoplastic theory is not volume conserved if it improperly sets an
arbitrary plastic strain rate tensor to be deviatoric. This paper discusses how
to rigorously realize volume conservation in finite strain regime, especially
when the unloading stress free configuration is not adopted or unique in the
elastoplastic theories. An accurate condition of volume conservation is
clarified and used in this paper that the density of a volume element after the
applied loads are completely removed should be identical to that of the initial
stress free states. For the elastoplastic theories that adopt the unloading
stress free configuration (i.e. the intermediate configuration), the accurate
condition of volume conservation is satisfied only if specific definitions of
the plastic strain rate are used among many other different definitions. For
the elastoplastic theories that do not adopt the unloading stress free
configuration, it is even more difficult to realize volume conservation as the
information of the stress free state lacks. To find a universal approach of
realizing volume conservation for elastoplastic theories whether or not adopt
the unloading stress free configuration, we propose a single assumption that
the density of material only depends on the trace of the Cauchy stress, and
interestingly find that the zero trace of the plastic stress rate is equivalent
to the accurate condition of volume conservation. Two strategies are further
proposed to satisfy the accurate condition of volume conservation: directly and
slightly revising the tangential stiffness tensor or using a properly chosen
stress/strain measure and elastic compliance tensor. They are implemented into
existing elastoplastic theories, and the volume conservation is demonstrated by
both theoretical proof and numerical examples.
