Neo-Hookean Hyperelastic Model
Theory and Physics
Neo-Hookean Hyperelasticity
Professor, what kind of model is Neo-Hookean?
The simplest hyperelastic model. Strain energy:
Defined by a single parameter $C_{10}$ ($= G/2$, where $G$ is the shear modulus). A special case of Mooney-Rivlin with $C_{01} = 0$.
Just one parameter! It can be used when there's limited test data.
Neo-Hookean is reasonable up to moderate strains (around 50%). For large strains (over 100%), Mooney-Rivlin or Ogden models are needed.
Summary
The Naming of Neo-Hookean
The name Neo-Hookean (New Hookean) was given with the meaning of "modernizing (neo)" Hookean, i.e., Hooke's linear elastic law, for the nonlinear finite deformation regime. Ronald Rivlin established this term in his 1948 paper. The strain energy function W = C₁(I₁−3) is the simplest hyperelastic model with only one parameter, and because the stress-strain relationship can be solved analytically, it is still frequently used as a benchmark for verifying numerical algorithms.
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, which assumes "forces are applied slowly enough that acceleration can be ignored". 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 it", right? That's Hooke's law $F=kx$, and it's the essence of the stiffness term. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (gravity, etc.) 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 interior" (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 common mistake here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it 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 away, right? That's because the vibration energy is converted into 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 and large rotation problems, geometric nonlinearity is required. For nonlinear material behaviors like plasticity and creep, 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 inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the 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 Neo-Hookean
```
*HYPERELASTIC, NEO HOOKE
C10, D1
```
Nastran: MATHE, 1, MOONEY + $C_{01}=0$.
Summary
Small Strain Limit and Shear Modulus Relationship
The parameter C₁ of the Neo-Hookean model matches C₁ = μ/2 (μ: shear stiffness) in the small strain limit. That is, C₁ can be calculated from the elastic modulus E and Poisson's ratio ν without experiments (μ = E/2(1+ν)). This property justifies the practical approximation of directly using linear elastic material data to run Neo-Hookean analysis in the preliminary design stage of rubber components where detailed large deformation test data is unavailable.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: 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 the tangent stiffness matrix every 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 Increment Method
Instead of applying the full load at once, it is applied 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 can take too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—the initial answer is rough, but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it 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
Neo-Hookean in Practice
Used for conceptual design of rubber components, modeling of biological tissues (brain, muscle). First choice when test data is limited.
Practical Checklist
Medical Implants and Neo-Hookean
FEM analysis of silicone rubber breast implants and heart valve prostheses most frequently uses Neo-Hookean and Mooney-Rivlin models. The US FDA, in its guidance on structural analysis of medical devices (2019), recommends using hyperelastic models with confirmed Drucker stability for soft tissue/rubber-like material computational models. Neo-Hookean is designated as a First-choice model due to its proven property of always being stable if C₁>0.
Analogy of 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.
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 can be far from reality. Confirm that results stabilize across 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".
How to Think About Boundary Conditions
Setting boundary conditions is the same as "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 truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling the real-world constraints is often the most critical step in the entire analysis.
Software Comparison
Tools
Supported by all solvers. The simplest hyperelastic model.
Comparison of Hyperelastic Model Implementations Across Vendors
ABAQUS, ANSYS, and Nastran all support Neo-Hooke, Mooney-Rivlin, and Ogden models, but their material constant fitting UIs differ. ABAQUS can simultaneously fit Mooney-Rivlin C10/C01 to tensile, compressive, and shear test data, while ANSYS uses uniaxial fitting as standard. A case study published in the Journal of Constitutive Models of Rubber reported that differences in fitting methods led to up to a 2x variation in predicted durability life for tire sidewall analysis.
The Three Most Important Questions for Selection
- "What are you solving?": Does the necessary physical model/element type for the Neo-Hookean hyperelastic model have support? For example, in fluids, the presence of LES support, and in structures, the capability to handle contact/large deformation can differ.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between an automatic transmission car (GUI) and a manual transmission car (script).
- "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
Origin of the Neo-Hooke Model: Statistical Mechanics of Rubber Elasticity
The Neo-Hookean hyperelastic model was derived in 1943 by Flory and Rehn from the molecular chain network theory of vulcanized natural rubber. It is a practical model with errors within 10% even at an elongation ratio of λ=3 for natural rubber with shear modulus G≈0.4 MPa. Today, it is widely used in ABAQUS and ANSYS for applications ranging from medical silicone catheters (G≈0.1 MPa) to tire compounds (G≈1.2 MPa).
Troubleshooting
Troubles
Accuracy Degradation at Excessive Strains
It is well known from many experiments that Neo-Hookean underestimates stress when the stretch ratio λ≧2.5. For uniaxial tension of natural rubber, the stress difference between Mooney-Rivlin and Neo-Hookean at λ=3 can reach 20-40%. This discrepancy is due to the absence of the C₂ term (I₂ dependence). For parts where large deformation is expected (tire bead, O-ring compression), switching to a multi-parameter model like Mooney-Rivlin or higher should always be considered.
If You Think "The Analysis Doesn't Match"
- First, take a deep breath—Panicking and randomly changing settings can make the problem more complex.
- Create a minimal reproducible case—Reproduce the Neo-Hookean hyperelastic model problem in its simplest form. "Subtractive debugging" is the most efficient.
- Change one thing and re-run—Making multiple changes simultaneously makes it unclear what worked. The principle of "controlled experiment", same as in scientific experiments.
- Return to the physics—If the calculation result is non-physical, like "objects floating against gravity", suspect a fundamental mistake in the input data.
Related Topics
なった
詳しく
報告