Nonequilibrium statistical thermodynamics of thermally activated dislocation ensembles: part 2—ensemble evolution toward correlation of enthalpy barriers

In part 1, we formally defined important physical concepts in statistical thermodynamics that undergird internal state variable (ISV) theory such as configurational subsystems (e.g., individual grains or phases), constrained local equilibrium, and thermally activated dislocation reactions in the context of crystal plasticity. The primal importance of the Gibbs free energy barrier to dislocation reactions within each subsystem was emphasized since the enthalpy barriers are affected by local constraint and resulting long-range and short-range athermal internal stresses acting within subsystems. In this work, we formally apply Gibbsian thermodynamics up to the saddle point of pending dislocation barrier reactions. The partition function based on reaction probability serves as the statistical mechanics basis for the change of Gibbs free energy that drives pending ensemble reactions; it is employed to define both the changes of the thermal entropy and configurational entropy of pending reactions; the latter enumerates reactions not only within subsystems, but also over permutations of subsystems, consistent with a multiscale hierarchy. Intrinsic entropy produced via post-reaction dynamic rearrangement and glide is of purely thermally dissipative character, being coupled to the thermal bath via dispersive phonon dynamics, and is distinguished from the entropy change of pending reactions. Building on the important concept of degree of correlation of thermally activated dislocation processes both within each subsystem and across the ensemble of subsystems introduced in part 1, we argue that nonequilibrium trajectories progressively trend toward increased correlation of enthalpy barriers across subsystems of the ensemble by virtue of internal stress redistribution among hard (unfavorable for extended dislocation reactions) and soft (more favorable) subsystems. The degree of correlation is a many-body concept involving populations of dislocations within and among various configurational subsystems. The present statistical thermodynamics framework invokes the constrained local equilibrium approach that fully respects heterogeneity of subsystems and distributions of barrier strengths, in common with ISV theory. However, literature forms of ISV theories are almost universally of reduced-order character to reduce complexity. Accordingly, reduced-order representations of ISV theory are presented for fully correlated conditions, as in standard representations of the hardening laws and flow kinetics of crystal plasticity, along with potential variations relevant to decorrelated behavior, including transients induced by unloading or changes in direction of the external applied stress. The origin of ensemble kinematic hardening with multiple back stress components due to interactions of hard and soft subsystems emerges naturally.