Volume conservation during finite plastic deformation
  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. 
He-Ling Wang, Dong-Jie Jiang, Li-Yuan Zhang, Bin Liu
cond-mat.soft