An arXiv preprint proposes a way to build several parts of a plasticity model, the mathematics used to describe permanent deformation, from one scalar functional. The key move is to examine the functional’s first variation, or change under a small permitted step, only along one-sided admissible plastic paths. In the proposed formulation, that directional calculation supplies stress, forces associated with internal variables, resistance to plastic activity and complementarity conditions that link activity to resistance.
The formulation is intended to keep these outputs within one constitutive construction. Its central result is a rule for generating the quantities that determine how a plastic mechanism is directed and whether its activity satisfies the model’s complementarity conditions.
The direction of plastic flow matters
Flow is handled as a geometric choice. Associated response is defined when the admissible plastic tangent aligns with the normal, or perpendicular direction, to the resulting yield boundary. If the tangent is not parallel to that normal, the response is non-associated. The distinction allows the framework to describe normal and non-normal flow within the same one-functional setup.
Separating shape, size and movement
A separate self-similarity analysis focuses on positively homogeneous stress gauges, meaning stress measures with a consistent scaling rule. It identifies them as an associated family and separates three ingredients of the yield surface: its shape, its isotropic expansion and its kinematic translation. The construction therefore treats the surface’s form, size and movement as distinct parts of the model.
Hardening is not limited to one mechanism. The framework represents isotropic, kinematic, coupled and gradient hardening, and it uses separate activity variables for independently activated mechanisms. Coupling and gradient terms are described as changing hardening or resistance while preserving separate activation decisions.
Four analytical examples
To demonstrate the construction, the preprint gives four closed-form analytical solutions: multi-threshold torsion, a non-uniform bar with kinematic hardening, a Hill-type annulus with a non-associated tangent, and elastic-plastic cavity expansion.
Across these examples, the analysis presents multi-activity regions, an explicit kinematic-hardening field, pressure-sensitive limit states and distinct displacement fields for normal and non-normal flow. The solutions are framed as complementary checks of active regions, hardening fields, pressure-sensitive strength and the separation of strength from dilatancy, the tendency associated with volume change during plastic deformation.
In the tensorial example, a von Mises gauge, a stress measure used in the construction, selects the plastic support direction.
The model can also include time dependence
The framework is extended to rate-dependent evolution through a viscous potential. The paper states that when the viscosity scale tends to zero while active rates remain bounded, active directional forces approach zero and the viscous graph converges to the rate-independent KKT graph, the complementarity graph associated with the core formulation.
Time-discrete constitutive integration, a stepwise numerical formulation, and dimensional checks are documented in Appendices A and B.
What the examples can and cannot show
The result should be read as a modeling proposal supported by mathematical illustrations. Because the demonstrations are closed-form analytical solutions presented as complementary checks, they show what kinds of active regions, hardening fields, pressure sensitivity and strength-dilatancy separation the framework can represent; they do not by themselves establish predictive accuracy for real materials.
The document is arXiv version 1, dated 26 August 2026.
Paper data and sources
Original title: Plasticity as Directional Stationarity: Yielding, Flow, and Hardening from One Functional
Authors: Huilong Ren
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text