Kinematic Hardening Model
Kinematic Hardening: Theoretical Foundations
What is Kinematic Hardening?
Professor, how is kinematic hardening different from isotropic hardening?
Isotropic hardening causes the yield surface to expand. Kinematic hardening causes the yield surface to translate (parallel shift) in stress space. The size of the yield surface does not change.
Representation of the Bauschinger Effect
After plastic deformation in tension, the yield stress in the compression direction decreases (Bauschinger effect).
- Isotropic Hardening — Tensile yield 400 MPa → Compressive yield also 400 MPa (no Bauschinger effect)
- Kinematic Hardening — Tensile yield 400 MPa → Compressive yield decreases to 250 MPa or less (Bauschinger effect present)
The Bauschinger effect is important for repeated loading (fatigue), right?
Exactly. In low-cycle fatigue (LCF), tension-compression repeats for hundreds to thousands of cycles. Ignoring the Bauschinger effect makes the stress-strain hysteresis loop inaccurate.
Back Stress
In kinematic hardening, the back stress $\alpha_{ij}$ describes the translation of the yield surface center:
The yield surface translates in the direction of $\alpha_{ij}$.
Prager/Ziegler Linear Kinematic Hardening
The simplest kinematic hardening:
$C$ is the kinematic hardening coefficient. It is linear, so it becomes inaccurate at large strains. In practice, nonlinear kinematic hardening (Chaboche model) is used.
Summary
Key points:
- The yield surface translates in stress space — its size does not change
- Represents the Bauschinger effect — essential for repeated loading (fatigue)
- Back stress $\alpha_{ij}$ describes the translation
- Linear kinematic hardening (Prager) — simple but inaccurate at large strains
- Nonlinear kinematic hardening (Chaboche) → recommended for practical use
Discovery of the Bauschinger Effect
Johann Bauschinger experimentally confirmed in 1886 at the Technical University of Munich that after tensile deformation, the yield stress in compression decreases. To reproduce this "Bauschinger effect," kinematic hardening rules were developed, formulated as a model where the center of the yield surface (back stress α) translates with plastic flow.
Computational Methods for Kinematic Hardening
Abaqus (Linear Kinematic Hardening)
```
*PLASTIC, HARDENING=KINEMATIC
250., 0.0
350., 0.05
```
Abaqus (Chaboche Nonlinear Kinematic Hardening)
```
*PLASTIC, HARDENING=COMBINED
250., 0.0
*CYCLIC HARDENING
250., 0.0
300., 0.1
```
Nastran
```
MATS1, 1, , PLASTIC, , , 3 $ TYPE=3 Kinematic Hardening
```
Summary
Armstrong-Frederick Evolution Rule
The nonlinear kinematic hardening rule proposed by Armstrong and Frederick in 1966 uses α̇=C(σ-α)ε̇ₚ - γα|ε̇ₚ| for the evolution of back stress. The γ term (recovery term) allows back stress to saturate, partially representing ratcheting. Chaboche improved the accuracy of cyclic fatigue analysis by superimposing N of these rules.
Kinematic Hardening in Practice
Kinematic Hardening in Practice
Used for stress-strain hysteresis in low-cycle fatigue (LCF), shakedown analysis, and thermal fatigue.
Practical Checklist
Fatigue Analysis of Nuclear Power Plant Piping
In thermal fatigue analysis of high-temperature piping (SUS304) in nuclear power plants, linear kinematic hardening rules (Prager rule) were found to overestimate ratcheting, leading to the adoption of the Chaboche multi-kinematic hardening rule since the 1990s. The RCC-M standard (French) and ASME Section III recommend using nonlinear kinematic hardening models for fatigue evaluation.