Isotropic Hardening Model
Isotropic Hardening: Theoretical Foundations
What is Isotropic Hardening?
Professor, what is isotropic hardening?
Isotropic hardening (isotropic hardening) is a model where the yield surface expands uniformly during plastic deformation. The yield stress increases with the accumulation of plastic strain.
$H$ is the hardening coefficient. Arbitrary hardening curves can be defined using a table format (correspondence table of stress vs. plastic strain).
Characteristics
What is the Bauschinger effect?
The phenomenon where, after plastic deformation in tension, the yield stress in the compression direction decreases. Since isotropic hardening expands the yield stress to the same value for both tension and compression, it cannot represent this effect. For cyclic loading (fatigue), kinematic hardening (kinematic hardening) is necessary.
Setting in FEM
Summary
Key points:
- Yield surface expands uniformly — Same yield stress in all directions
- Optimal for monotonic loading — Use kinematic hardening for cyclic loading
- Cannot represent Bauschinger effect — Same yield stress for tension → compression
- Default for all solvers — The simplest
Yield Surface Expansion in Isotropic Hardening
In the isotropic hardening rule, the yield surface expands isotropically with plastic deformation, while its center position remains unchanged. The yield stress σy(εₚ) is updated with the accumulated plastic strain εₚ. Since Prandtl formulated the plastic flow rule in 1934, it has been used for over 90 years as the simplest choice for monotonic loading problems.
Computational Methods for Isotropic Hardening
Numerical Processing of Isotropic Hardening
Processed with the Return Mapping algorithm. Elastic predictor → yield check → radial return. Same as von Mises plasticity.
Table Input
```
*PLASTIC
250., 0.0
300., 0.02
400., 0.1
450., 0.2
```
Correspondence table of true stress vs. true plastic strain. Intermediate values are linearly interpolated.
Summary
Hardening Curve Input Format
Many solvers accept isotropic hardening in a table format of "true stress vs. accumulated plastic strain". In Abaqus, up to 200 points can be input on the *PLASTIC card. In practice, the basic procedure is to convert engineering stress-strain from tensile tests using σ_true=σ_eng(1+ε_eng), ε_true=ln(1+ε_eng), and then subtract elastic strain to obtain εₚ.
Isotropic Hardening in Practice
Isotropic Hardening in Practice
Most widely used for monotonic loading of metals (tensile tests, forming, pressure tests).
Practical Checklist
Staple for Press Forming Analysis
In stamping analysis for automotive body panels, combining the isotropic hardening rule with the Swift equation (σ=Cεₙ) has been a standard method since the 1980s. Representative values for high-strength steel DP980 are C≈1600MPa, n≈0.12. Prediction accuracy for thickness reduction rate generally deviates from experimental values by 3-5%, making it widely used for initial die design studies.
Isotropic Hardening: Software & Solver Comparison
Tools for Isotropic Hardening
Standard in all solvers. Only the setup method differs.
| Solver | Setting |
|---|---|
| Abaqus | *PLASTIC table |
| Nastran | MATS1 + TABLES1 |
| Ansys | TB, BISO (bilinear) or TB, MISO (multilinear) |
| LS-DYNA | *MAT_24 |
Solver-Agnostic Implementation
While the physical model is identical across solvers, the input card format differs significantly. Abaqus uses a straightforward table format, Nastran requires linking separate TABLES1 cards, and Ansys distinguishes between bilinear (BISO) and multilinear (MISO) options. LS-DYNA's *MAT_24 is optimized for explicit dynamics and handles strain rate effects differently. Understanding these differences is critical for CAE workflows involving multiple software platforms.