  Finite plasticity theories are still a subject of controversy and lively
discussions. Among the approaches to finite elastoplasticity two became
especially popular. The first, implemented in the commercial finite element
codes, is based on the introduction of a hypoelastic constitutive law and the
additive elastic-plastic decomposition of the deformation rate tensor.
Unfortunately, the use of hypoelasticity may lead to a nonphysical creation or
dissipation of energy in a closed deformation cycle. In order to replace
hypoelasticity with hyperelasticity the second popular approach based on the
multiplicative elastic-plastic decomposition of the deformation gradient tensor
was developed. Unluckily, the latter theory is not perfect as well because it
introduces intermediate plastic configurations, which are geometrically
incompatible, non-unique, and, consequently, fictitious physically. In the
present work, an attempt is made to combine strengths of the described
approaches avoiding their drawbacks. Particularly, a tensor of the plastic
deformation rate is introduced in the additive elastic-plastic decomposition of
the velocity gradient. This tensor is used in the flow rule defined by the
generalized isotropic Reiner-Rivlin fluid. The tensor of the plastic
deformation rate is also used in an evolution equation that allows calculating
an elastic strain tensor which, in its turn, is used in the hyperelastic
constitutive law. Thus, the present approach employs hyperelasticity and the
additive decomposition of the velocity gradient avoiding nonphysical
hypoelasticity and the multiplicative decomposition of the deformation gradient
associated with incompatible plastic configurations. The developed finite
elastoplasticity framework for isotropic materials is specified to extend the
classical -theory of metal plasticity to large deformations and the simple
shear deformation is analyzed.
