von Mises Plasticity Theory
von Mises Plasticity Theory: Theoretical Foundations
What is von Mises Plasticity?
Professor, the von Mises plasticity theory is fundamental for material nonlinearity in FEM, right?
The von Mises yield criterion is the most basic theory describing the plastic deformation of metals. It states that "plastic deformation begins when the equivalent stress (von Mises stress) reaches the yield stress."
von Mises Equivalent Stress
Or in component form:
Yield Condition
If $f < 0$, it's in the elastic region. $f = 0$ means on the yield surface (plastic deformation). $f > 0$ is not allowed (cannot go outside the yield surface).
So the "sphere" in stress space is the yield surface, right?
When viewed in deviatoric stress space, the von Mises yield surface is a cylinder. Its characteristic is that it does not depend on hydrostatic pressure (volumetric stress). This is physically reasonable because plastic deformation of metals does not involve volume change (incompressible plastic flow).
Hardening Rule
The stress-strain relationship after yielding (hardening rule):
| Hardening Type | Change in Yield Surface | Application |
|---|---|---|
| Perfectly Elastic-Plastic (Perfect Plasticity) | Yield surface is fixed | Evaluation of collapse load |
| Isotropic Hardening | Yield surface expands | Monotonic loading |
| Kinematic Hardening | Yield surface translates | Cyclic loading (Fatigue) |
| Mixed Hardening | Expansion + Translation | Most general |
Abaqus
```
*MATERIAL, NAME=steel
*ELASTIC
200000., 0.3
*PLASTIC
250., 0.0 $ Yield stress 250 MPa, plastic strain 0
400., 0.1 $ 400 MPa, plastic strain 10%
500., 0.3 $ 500 MPa, plastic strain 30%
```
Nastran
```
MAT1, 1, 200000., , 0.3
MATS1, 1, , PLASTIC, , , 1, 1
TABLES1, 1, , ,
, 0.0, 250., 0.1, 400., 0.3, 500., ENDT
```
So the hardening curve is defined by a stress-plastic strain table, right?
The nominal stress-nominal strain curve from a tensile test must be converted to true stress-true strain before input into FEM. True stress-true strain is essential for large deformation analysis.
Summary
Key points:
- Yielding occurs when $\sigma_{vm} = \sigma_Y$ — The basis of metal plasticity
- Independent of hydrostatic pressure — No volume change (characteristic of metals)
- Hardening rules — Perfect plasticity/Isotropic hardening/Kinematic hardening/Mixed hardening
- Input true stress-true strain into FEM — Conversion from nominal values is necessary
- Standard in all FEM solvers — The most basic material nonlinear model
von Mises' 1913 Paper
In 1913, Richard von Mises proposed a criterion in the journal of the Göttingen Scientific Society, expressing the yield condition as J₂=k² (the second deviatoric stress invariant). The physical interpretation that yielding occurs when the shear strain energy reaches a critical value was added by Hencky (1924). In principal stress space, it appears as a cylindrical surface and is now the most widely used yield criterion.
Computational Methods for von Mises Plasticity Theory
Return Mapping Algorithm
Professor, how is plasticity handled numerically?
The Return Mapping (Stress Return) Algorithm is standard:
1. Elastic Predictor — Calculate a trial stress assuming the entire strain increment is elastic.
2. Yield Judgment — Does the trial stress lie outside the yield surface?
3. Plastic Corrector — If outside, "return" the stress onto the yield surface.
So it's a two-step process: "assume elastic for trial calculation → return to yield surface"?
For von Mises plasticity, this return can be calculated exactly using radial return. It's very efficient and stable. Implemented in all commercial solvers.
Tangent Stiffness Matrix (CTO)
The tangent stiffness in the plastic state (Consistent Tangent Operator, CTO):
$H$ is the hardening coefficient. $\{n\}$ is the normal to the yield surface. The CTO guarantees quadratic convergence for the Newton-Raphson method.
Summary
- Return Mapping (Stress Return) — Elastic prediction → Plastic correction
- Radial Return — Exact for von Mises
- CTO (Consistent Tangent Operator) — Guarantees quadratic convergence
- Standard implementation in all solvers — Users only need to input material data
Radial Return Mapping
"Radial Return Mapping" is used for FEM implementation of von Mises + isotropic hardening. It consists of three steps: elastic prediction → check for yield surface overshoot → return in the tangential direction. Simo & Taylor (1985) proved its linear convergence. Since an exact solution can be obtained in one iteration (if the tangent modulus is used), computational cost is low, and it is adopted in almost all general-purpose solvers.
von Mises Plasticity Theory in Practice
Plastic Analysis in Practice
In what situations is von Mises plasticity used?
All nonlinear strength evaluations for metal structures:
| Application | Purpose |
|---|---|
| Elastoplastic analysis for pressure tests | ASME Div.2 Part 5 for pressure vessels |
| Evaluation of plastic collapse load | Limit load method (does it converge under 2x design load?) |
| Sheet metal forming (pressing) | Post-deformation shape and springback |
| Welding residual stress | Thermo-elastoplastic analysis of welding → cooling |
| Elastoplastic time history for earthquakes | Formation of plastic hinges |
True Stress-True Strain Conversion
When performing elastoplastic analysis, always convert nominal stress-strain data from material tests to true stress-true strain for input:
This conversion is essential to capture large deformation behavior correctly.
Summary
- von Mises plasticity is the standard for metal strength evaluation
- Applicable to pressure vessel design, collapse analysis, forming analysis, and seismic design
- True stress-true strain conversion is mandatory for accurate large deformation simulation
- All FEM solvers provide built-in implementation with return mapping algorithms
- Practical workflow: Material test data → True conversion → Input to solver → Result interpretation