Mooney-Rivlin Hyperelastic Model
Mooney-Rivlin Hyperelastic: Theoretical Foundations
What is Hyperelasticity?
Professor, is "hyperelasticity" a model for materials like rubber?
Yes. 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.
Computational Methods for Mooney-Rivlin Hyperelastic
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).
Mooney-Rivlin Hyperelastic in Practice
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.