Variational Formulation, Discrete Conservation Laws, and Path-Domain Independent Integrals for Elasto-Viscoplasticity

A discrete variational formulation of plasticity and viscoplasticity is developed based on the principle of maximum plastic dissipation. It is shown that the Euler-Lagrange equations (spatial conservation laws) emanating from the proposed discrete Lagrangian yield the equilibrium equation, the strain-displacement relations, the stress-strain relations, the discrete flow rule and hardening law in the form of closest-point-projection algorithm, and the loading/unloading conditions in Kuhn-Tucker form. Lack of invariance of the discrete Lagrangian relative to the group of material translations precludes the classical Eshelby law from being a conservation law. However, a discrete inhomogeneous form of Eshelby’s conservation law is derived which leads to a path-domain independent integral that generalizes the classical J-integral to elasto-viscoplasticity. It is shown that this path-domain independent integral admits a physical interpretation analogous to Budiansky and Rice interpretation of the classical J-integral.