Mullins Effect (Softening of Rubber)
Mullins Effect (Softening of Rubber): Theoretical Foundations
What is the Mullins Effect?
Professor, what is the Mullins effect?
A phenomenon where rubber, when stretched significantly for the first time, exhibits reduced stress (softens) in subsequent loadings. Also called "stress softening".
Physical mechanism: The bond between filler (carbon black, etc.) and the rubber matrix is partially destroyed during the initial loading.
Ogden-Roxburgh Model
Abaqus's Mullins effect model (Ogden-Roxburgh, 1999):
$\eta$ is the damage variable ($0 < \eta \leq 1$). $\eta < 1$ as long as the peak stress of the initial loading is not exceeded.
Summary
- Softening during the first loading of rubber — Stress decreases in subsequent loadings
- Ogden-Roxburgh Model — Abaqus's *MULLINS EFFECT
- Tires, anti-vibration rubber, seals — Important for repeated use of rubber components
Discovery History of the Mullins Effect
Leonard Mullins quantified the softening phenomenon of carbon black-filled rubber in 1947 at the British Rubber Producers' Research Association. He described it as "stress softening" where the stress-strain curves for the first loading and subsequent loadings do not match. This phenomenon is now called the "Mullins effect". The primary physical cause is considered to be the dissociation of filler-polymer chains.
Computational Methods for Mullins Effect (Softening of Rubber)
FEM for Mullins Effect
```
*HYPERELASTIC, OGDEN, N=3
...
*MULLINS EFFECT
r, m, beta
```
Simply add *MULLINS EFFECT to the hyperelastic model (Ogden, etc.).
Summary
- Hyperelastic + *MULLINS EFFECT — Abaqus standard setting
- Determine $r, m, \beta$ from repeated loading tests
Ogden-Roxburgh Damage Variable η
The Ogden-Roxburgh (1999) model expresses the Mullins effect using a scalar damage variable η(r). r depends on the maximum strain energy value Wmax. Upon complete unloading, η → η_min (0~1), and η recovers upon reloading. Parameter identification requires at least 4 cycles of uniaxial testing, and the standard procedure is to sequentially identify the three parameters r, μ, and β.
Mullins Effect (Softening of Rubber) in Practice
Practical Checklist
- [ ] Is the hyperelastic model correctly defined (Ogden, etc.)?
- [ ] Are the Mullins effect parameters determined from repeated tests?
- [ ] Verify that stress softens correctly during initial and repeated loading.
Tire Bead Seal Durability Analysis
In the assembly durability analysis of automotive tire bead seal rubber (SBR compound), ignoring the Mullins effect can lead to overestimating compressive stress by 20-30% in some cases. A case study reported in a Continental paper shows that hyperelastic analysis combined with the MULLINS_EFFECT option in Abaqus 6.7 and later predicted permanent deformation after 100,000 cycles within ±8% of actual measurements.
Mullins Effect (Softening of Rubber): Software & Solver Comparison
Tools
Solver Implementation Status 2024
Implementation of the Mullins effect varies among major solvers. Abaqus has had it as standard since 2003 (v6.3), LS-DYNA has supported it since around 2010 with MAT_181 (SIMPLIFIED_RUBBER_WITH_DAMAGE), MSC Marc implemented it in Marc2014 from 2014. On the other hand, as of 2024, Nastran SOL 400 still lacks direct implementation of the Mullins effect, requiring custom implementation via UMAT.
Advanced Mullins Effect (Softening of Rubber): Modern Research & Trends
Advanced
- Anisotropic Mullins — Direction-dependent softening (fiber-reinforced rubber)
- Mullins + Viscoelasticity — Rate-dependent softening
- Multiscale — Predict Mullins effect from filler-matrix interface damage
Separate Modeling from Permanent Deformation
The Mullins effect and permanent deformation (Permanent Set) have physically different mechanisms. Bergström and Boyce (1999) proposed a model separating both based on molecular chain network theory. In Abaqus, combining *MULLINS EFFECT and *PERMANENT SET as independent keywords can improve accuracy in high-cycle fatigue analysis.
Mullins Effect (Softening of Rubber): Common Issues & Debugging
Troubleshooting
- No softening occurs → Is *MULLINS EFFECT defined? The Mullins effect does not manifest under monotonic loading.