A dynamic flow rule for viscoplasticity in polycrystalline solids under high strain rates

For visco-plasticity in polycrystalline solids under high strain rates, we introduce a dynamic flow rule (also called the micro-force balance) that has a second order time derivative term in the form of micro-inertia. It is revealed that this term, whose physical origin is traced to dynamically evolving dislocations, has a profound effect on the macro-continuum plastic response. Based on energy equivalence between the micro-part of the kinetic energy and that associated with the fictive dislocation mass in the continuous dislocation distribution (CDD) theory, an explicit expression for the micro-inertial length scale is derived. The micro-force balance together with the classical momentum balance equations thus describes the viscoplastic response of the isotropic polycrystalline material. Using rational thermodynamics, we arrive at constitutive equations relating the thermodynamic forces (stresses) and fluxes. A consistent derivation of temperature evolution is also provided, thus replacing the empirical route. The micro-force balance, supplemented with the constitutive relations for the stresses, yields a locally hyperbolic flow rule owing to the micro-inertia term. The implication of micro-inertia on the continuum response is explicitly demonstrated by reproducing experimentally observed stress-strain responses under high strain-rate loadings and varying temperatures. An interesting finding is the identification of micro-inertia as the source of oscillations in the stress-strain response under high strain rates.