whereand stand for the shear and bulk moduli, and
is the deviatoric part of the fourth-rank identity tensor, . The current UVP model includes both non-linear kinematic hardening back-stresses, , related to the motion of center of yield surface, and isotropic hardening drag stress, , related to the yield surface expansion, after cyclic plastic deformation. Thus, the inelastic visco-plastic strain-rate flow rule,, based on the von-Mises yield function, , can be assumed to be independent of hydrostatic pressure and defined such that: