Plasticity as Directional Stationarity: Yielding, Flow, and Hardening from One Functional

Abstract

Traditional plasticity theory is commonly organized through an elastic law, a yield condition, a flow rule, hardening relations, and loading--unloading conditions. This paper formulates these relations through directional stationarity of one scalar functional evaluated over one-sided admissible plastic paths. The first variation determines stress, internal-variable forces, and activity resistance. Restriction to an admissible plastic tangent defines a reduced directional force for each mechanism; its sign and one-sided stationarity give the elastic inequality, complementarity, loading--unloading conditions, and active consistency. Associated response corresponds to alignment between the admissible tangent and the normal to the resulting yield boundary, whereas a nonparallel tangent represents non-associated flow. A self-similarity analysis identifies positively homogeneous stress gauges as a broad associated family and separates the roles of yield-surface shape, isotropic expansion, and kinematic translation. The storage and resistance terms cover isotropic, kinematic, coupled, and gradient hardening, while multiple activity variables describe independently activated mechanisms. A viscous potential extends the construction to rate-dependent evolution. Four closed-form solutions illustrate multi-activity regions, kinematic-hardening fields, pressure-sensitive limit states, and the distinct displacement fields associated with normal and non-normal flow. Time-discrete constitutive integration and dimensional checks are collected in the appendices.