Isotropic Hardening Model
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "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 is "left behind". In static analysis, this term is set to zero, assuming "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 stretch a spring, you feel a "force trying to return it", right? That is 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". Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body force $f_b$ (gravity, etc.) 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 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 are rotated 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 vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — deliberately absorbing vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue 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 heterogeneity.
- 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: 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 loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic modulus $E$ | Pa | Steel: approx. 210 GPa, Aluminum: approx. 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
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.
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. Recommended: when stress evaluation is important.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass mode (zero-energy mode). 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 each 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 $\epsilon = 10^{-3}$〜$10^{-6}$). 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" — 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 at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy the 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 mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Easily Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some kind of answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across 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 exam question". What if the question 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 constraint conditions is actually the most important step in the entire analysis.
Software 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 |
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