Ogden Hyperelastic Model
Theory and Physics
What is the Ogden Model?
Professor, what kind of hyperelastic model is the Ogden model?
The Ogden Model (1972) directly describes the strain energy using principal stretch ratios ($\lambda_1, \lambda_2, \lambda_3$):
$N = 1 \sim 3$ term sum. $\mu_i, \alpha_i$ are material constants.
More parameters than Mooney-Rivlin?
$N = 3$ gives 6 parameters. More accurate at large strains (200%+) than Mooney-Rivlin's 2 parameters. Especially capable of simultaneous fitting for both tension and compression.
Summary
The Birth of the Ogden Model
This model, published by Ray Ogden in 1972, expresses strain energy with a polynomial of arbitrary order. It was said to reproduce synthetic rubber within 5% error for 2nd order and natural rubber within 1% for 3rd order. After its publication in the Journal of Mechanics that same year, its adoption spread rapidly in the tire design field.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? 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 "acceleration is negligible because forces are applied slowly". 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", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. 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" — 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), the force of a tire pressing 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 typical pitfall here: getting the load direction wrong. Intending "tension" but actually applying "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. That's because vibration energy turns into heat due to air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibration energy for a smoother ride. What if damping were zero? Buildings would keep swaying 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, 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/damping forces, considers only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behaviors like plasticity/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/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note 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
FEM Settings for Ogden
```
*HYPERELASTIC, OGDEN, N=3
mu1, alpha1, D1
mu2, alpha2, D2
mu3, alpha3, D3
```
Or:
```
*HYPERELASTIC, OGDEN, TEST DATA INPUT
*UNIAXIAL TEST DATA
stress, strain
*BIAXIAL TEST DATA
stress, strain
```
Automatic fitting is recommended.
Summary
Parameter Identification Sequence
The standard procedure for Ogden model parameter identification is uniaxial → equibiaxial → pure shear in that order. Treloar's 1944 experimental data for natural rubber is still used as a benchmark today. The standard procedure is to simultaneously identify all 6 parameters (μ₁~μ₃ and α₁~α₃) using least squares for a 3rd-order Ogden model.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with 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 tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Lower cost per iteration but linear convergence speed.
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
Applies total load not all at once but in small increments. The arc-length method (Riks method) can track beyond limit 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: opening to an estimated page and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
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
Ogden in Practice
Used for tires, O-rings, large-deformation rubber components. For large strain problems where Mooney-Rivlin is insufficient.
Practical Checklist
Input Order in Abaqus
When defining the Ogden model in Abaqus/CAE, declare the order N first, then input μᵢ, αᵢ, Dᵢ in sequence. For N=3, there will be 3 parameter lines. In practice, setting Dᵢ≈0 (incompressibility assumption) can reproduce behavior equivalent to Poisson's ratio 0.4995 while reducing computational cost by about 15%.
Analogy: Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate (visualize with 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, results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often 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 answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Verify that results stabilize across at least 3 mesh densities — neglecting this leads to the dangerous assumption that "the computer's answer must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "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 actually the most important step in the entire analysis.
Software Comparison
Tools for Ogden
Selection Guide
Supported Order by Solver
The supported order of the Ogden model varies by solver. Abaqus・LS-DYNA support up to N=6, MSC Marc supports theoretically unlimited (practical N=9), Nastran supports up to N=3 in SOL 400. NX Nastran provides cards OGDEN1~OGDEN3, using separate cards for each order.
The 3 Most Important Questions for Selection
- "What are you solving?": Does the required physical model/element type for the Ogden hyperelastic model have support? For example, in fluids, LES support availability; in structures, contact/large deformation capability makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission in cars.
- "How far will it expand?": Selection considering future analysis scale expansion (HPC support), expansion to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Ogden
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