Microstructure evolution as a game

Abstract

We develop a whole-trajectory relaxation framework for dissipative evolutions with non-convex, state- and history-dependent kinetics. The framework arises naturally from a game-theoretical formulation of energy--dissipation evolution. State--rate consistency is eliminated by causal Volterra reconstruction, reducing the problem to a diagonal equilibrium condition on complete rate trajectories. We establish a direct method based on vanishing-defect approximability, compactness, and complementary semicontinuity. The relaxation mechanism is computed exactly for a reduced simple-shear model with a non-convex kinetic potential and a local memory variable. The enriched state is a Young measure on complete local histories. We identify explicitly the sequential lower-semicontinuous envelope of the pure diagonal defect, characterize its zero set, construct pure recovery sequences for every relaxed zero, and prove existence through a frozen-best-response time discretization. The resulting coexistence of distinct rate populations with distinct memories provides a reduced mechanism for kinetic shear banding and connects naturally with negative strain-rate sensitivity and dynamic strain aging such as underlie the Portevin--Le Chatelier effect.