Kinematic Hardening Model
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when braking suddenly? 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, assuming "forces are applied slowly enough that acceleration can be ignored." It 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. 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 force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it 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. That's because 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 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 equilibrium between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/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
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.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. 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. Recommendation: 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. Quadratic convergence within the convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Lower cost per iteration, but convergence 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, it is applied in small increments. The arc-length method (Riks method) can track beyond extremum points in the load-displacement relationship.
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"—initially a rough answer, but accuracy improves 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 at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
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.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work." If the mesh quality is poor, the results will be a mess no matter how good the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the results are correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct."
Thinking About Boundary Conditions
Setting boundary conditions is the same as "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraints is often the most critical step in the entire analysis.
Software Comparison
Tools for Kinematic Hardening
Related Topics
なった
詳しく
報告