  A system of partial differential equations (PDEs) is derived to compute the
full-field stress from an observed kinematic field when the flow rule governing
the plastic deformation is unknown. These equations generalize previously
proposed equations that assume pure plastic behavior without elasticity. A
method to numerically solve these equations is also presented. In addition to
force balance, the equations are derived from the elastic-plastic decomposition
of the deformation gradient, the assumption of isotropy, and the assumption
that the function mapping the elastic strain to stress is known. The system of
equations can be directly applied to complex geometries, finite deformation,
non-linear elasticity and plasticity, compressible materials, rate dependent
materials, and a variety of hardening laws. This system of PDEs is non-linear
and time dependent. Furthermore, it overcomes an important prior limitation: it
can be directly applied to cases where some regions of a body are elastically
deforming while others are elasto-plastically deforming. A two-dimensional case
study of necking in a uniaxial tensile specimen is investigated to illustrate
and validate the method. The governing equations are numerically solved using
strain fields output from a finite element simulation and validated against
this same simulation showing accurate results.
