---
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]]"
source_refs:
  - source: "[[Finite-Elements-in-Plasticity-Theory-and-Practice|Finite Elements in Plasticity: Theory and Practice]]"
    raw_path: ".raw/FiniteElementsinPlasticityTheoryandPractice/"
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      - "FiniteElementsinPlasticityTheoryandPractice_050.md"
      - "FiniteElementsinPlasticityTheoryandPractice_001.md"
    md_indices:
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    confidence: high
---

# 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]]