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wiki/concepts/Plastic Flow Rules and Hardening.md

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+---
+type: concept
+title: "Plastic Flow Rules and Hardening"
+complexity: advanced
+domain: computational-mechanics
+created: 2026-06-02
+updated: 2026-06-02
+address: c-000135
+aliases:
+  - associated plasticity
+  - non-associated plasticity
+  - isotropic hardening
+  - kinematic hardening
+tags:
+  - concept
+  - finite-element-method
+  - plasticity
+  - constitutive-modeling
+status: current
+related:
+  - "[[Finite Element Plasticity]]"
+  - "[[Plasticity Yield Criteria]]"
+  - "[[Abaqus Constitutive Integration]]"
+  - "[[Abaqus Metal Plasticity Models]]"
+  - "[[Abaqus Geomaterial and Concrete Plasticity]]"
+sources:
+  - "[[Finite-Elements-in-Plasticity-Theory-and-Practice|Finite Elements in Plasticity: Theory and Practice]]"
+---
+
+# Plastic Flow Rules and Hardening
+
+## Definition
+
+Plastic flow rules and hardening laws define what happens after a stress state reaches a yield surface. The flow rule gives the direction of plastic strain increment, and the hardening law evolves the yield condition as plastic deformation accumulates.
+
+## Flow Rules
+
+In associated plasticity, the plastic potential is the same as the yield function, so the plastic strain increment is normal to the yield surface. In non-associated plasticity, the plastic potential differs from the yield function, which is often important for pressure-dependent frictional materials where dilatancy must be controlled separately from yield.
+
+## Hardening
+
+The source distinguishes hardening ideas that are central to implementation:
+
+- Isotropic hardening expands or contracts the yield surface.
+- Kinematic hardening translates the yield surface and is important for reversed or cyclic loading.
+- Work hardening links yield evolution to accumulated plastic work or equivalent plastic strain.
+
+## Solver Consequences
+
+Flow and hardening choices determine which internal variables must be stored at integration points. They also determine the material tangent used by implicit global iteration and the pseudo-load corrections used by simpler incremental schemes.
+
+## Connections
+
+[[Plasticity Yield Criteria]] gives the elastic/plastic boundary. [[Abaqus Constitutive Integration]] is the production stress-update layer that turns flow and hardening rules into state updates and tangent terms. [[Abaqus Metal Plasticity Models]] and [[Abaqus Geomaterial and Concrete Plasticity]] are user-facing model families built from these ideas.
+
+## Sources
+
+- [[Finite-Elements-in-Plasticity-Theory-and-Practice|Finite Elements in Plasticity: Theory and Practice]]
+