Mooney-Rivlin Hyperelastic Model
Theory and Physics
What is Hyperelasticity?
Professor, is "hyperelasticity" a model for materials like rubber?
Yes. Hyperelasticity (hyperelasticity) describes materials that return to their original shape even after large deformations. Examples include rubber, elastomers, and biological tissues. Stress is determined by the derivative of the strain energy function $W$.
Mooney-Rivlin Model
Mooney-Rivlin is the most widely used hyperelastic model. Strain energy:
$I_1, I_2$ are invariants of the deformation tensor. $C_{10}, C_{01}$ are material constants.
Can the large deformation of rubber be represented with just two constants?
It provides good accuracy up to about 100% strain. For strains above 200%, models like Ogden or Arruda-Boyce are more accurate. Mooney-Rivlin is the "simplest and most widely used" model.
Abaqus
```
*HYPERELASTIC, MOONEY-RIVLIN
C10, C01, D1
```
$D1$ is the bulk modulus (for incompressible materials, $D1 \to 0$).
Nastran
```
MATHE, 1, MOONEY
, C10, C01, , , , , D1
```
Material Testing
Mooney-Rivlin constants are determined from the following tests:
- Uniaxial Tension Test — Most basic
- Biaxial Tension Test — More accurate
- Planar Tension (Pure Shear) — Complementary
Mooney-Rivlin constants are determined from the following tests:
In Abaqus, use *HYPERELASTIC, TEST DATA to input test data for automatic fitting.
Summary
Key Points:
- $W = C_{10}(I_1-3) + C_{01}(I_2-3)$ — 2-constant strain energy function
- Good accuracy up to 100% strain — Use Ogden, etc., for higher strains
- Note incompressibility — Hybrid elements (e.g., C3D8RH) are essential
- Fit from test data — *HYPERELASTIC, TEST DATA
Mooney's 1940 Paper and Rivlin's 1948 Extension
When Melvin Mooney published his 1940 paper first expressing isotropic hyperelasticity of rubber with two parameters (C₁, C₂), the mathematical framework for finite deformation was still under development. In 1948, Ronald Rivlin provided a general expansion theory for the strain energy function using invariants I₁, I₂, I₃, proving that the Mooney function is its lowest-order approximation. This is the origin of the name "Mooney-Rivlin" bearing both names.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "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 is "left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration is negligible". It 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. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? 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"—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 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 typical 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. 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 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, and the stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): 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 plasticity, creep, and other nonlinear material behaviors, 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: ~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 Implementation of Hyperelasticity
Points to note for hyperelastic materials in FEM:
1. NLGEOM=YES is mandatory — Rubber undergoes large deformation.
2. Hybrid elements — Rubber is incompressible ($\nu \approx 0.5$). Countermeasure for volumetric locking.
3. Fitting of $C_{10}, C_{01}$ — From test data.
Which elements should be used?
Summary
Practical C₁・C₂ Fitting
Mooney-Rivlin parameters C₁ and C₂ are typically determined by simultaneous least-squares fitting to three types of test data: uniaxial tension, plane strain, and equibiaxial tension. Typical values for natural rubber (NR) are C₁≈0.16 MPa, C₂≈0.04 MPa, providing a good fit up to an extension ratio λ≈3. Abaqus's "Evaluate" feature allows checking predicted curves for each test mode on a single screen and automatically checks whether the parameters are stable (Drucker stability).
Linear Elements (1st-order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Capable of representing curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is critical.
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 each iteration. Provides quadratic convergence within the convergence radius but has 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 linear convergence speed.
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 trace 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 can take 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 open it at an estimated location 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
Hyperelasticity in Practice
O-rings, tires, rubber bushings, vibration isolators, medical devices, etc.
Determining Material Constants
Stability Check
Mooney-Rivlin must have positive stiffness in all deformation modes (tension, compression, shear). Fitting results can sometimes be unstable (negative stiffness). Check with Abaqus STABILITY CHECK.
Practical Checklist
The Main Model for Tire Sidewall Analysis
For large deformation analysis of automotive tire sidewalls (natural rubber-based compounds), the Mooney-Rivlin 2-parameter model remains the industry-standard first choice. The FEM analysis departments of Bridgestone and Michelin have been analyzing contact pressure distribution and deformation shape using Abaqus and Mooney-Rivlin since the 1990s, establishing fitting procedures that keep errors within 5% compared to actual tire measurements. Switching to the Ogden model is recommended for large extension regions with λ≧4.
Analogy for the Analysis Flow
The analysis flow is actually very similar to cooking. First, you 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 the mesh quality is poor, the results will be a mess no matter how excellent the solver is.
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