von Mises塑性理論
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic inhomogeneities.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behaviors like plasticity and creep require constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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):
$$ [D_{ep}] = [D_e] - \frac{[D_e]\{n\}\{n\}^T[D_e]}{\{n\}^T[D_e]\{n\} + H} $$
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
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.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix every iteration. Provides quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, apply it in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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
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