Mullins Effect (Softening of Rubber)
Theory and Physics
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):
$$ \sigma = \eta \cdot \sigma_{primary} $$
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
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.
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 pulled" 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 "gets left behind". In static analysis, this term is set to zero, assuming "acceleration can be ignored because the force is applied slowly". It absolutely cannot be omitted in impact loading 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——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"; 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), the force of the tire 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, right? Because the 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 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 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 for Mullins Effect
```
*HYPERELASTIC, OGDEN, N=3
...
*MULLINS EFFECT
r, m, beta
```
Simply add *MULLINS EFFECT to the hyperelastic model (Ogden, etc.).
Summary
OgdenRoxburgh 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.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower 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. 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. Achieves 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 $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Incrementation Method
Instead of applying the full load at once, it is applied in small increments. The arc-length method (Riks method) can trace 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 accuracy improves 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) than to search 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 judgment should be based on total cost-effectiveness.
Practical Guide
Practical Checklist
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.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare 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——in cooking, which step 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 good the solver is.
Common Pitfalls for Beginners
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 with 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 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 often the most critical step in the entire analysis.
Software 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.
The 3 Most Important Questions for Selection
- "What are you solving?": Does it support the necessary physical models/element types for the Mullins effect (rubber softening)? For example, for fluids, the presence of LES support; for structures, the capability for contact/large deformation 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 cars.
- "How far will it be extended?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced
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.
Troubleshooting
Troubles
Related Topics
なった
詳しく
報告