A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=FeFpF=FeFp [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and Fp=FenpFdisp [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion FeFe contributes to a standard elastic free-energy ψ(e)ψ(e), while Fenp, the energetic part of FpFp, contributes to a defect energy ψ(p)ψ(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since Fe=FFp-1Fe=FFp-1 and Fenp=FpFdisp-1, the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for FpFp and Fdisp – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.