Coulomb friction model
Theory and Physics
What is Coulomb Friction?
Professor, please teach me the basics of friction models in FEM.
Coulomb friction is the most basic friction model. The tangential friction force is proportional to the normal force:
- $\tau$ — Friction force (tangential stress)
- $p_n$ — Contact pressure (normal stress)
- $\mu$ — Coefficient of friction
Sticking and Sliding
Two states:
- Sticking — $|\tau| < \mu p_n$. No relative sliding.
- Sliding — $|\tau| = \mu p_n$. Friction force reaches its limit and sliding occurs.
So the transition from sticking to sliding is nonlinear, right?
Typical Values of Friction Coefficients
| Material Pair | $\mu$ (dry) |
|---|---|
| Steel-Steel | 0.15–0.3 |
| Steel-Aluminum | 0.2–0.4 |
| Steel-Rubber | 0.5–0.8 |
| Steel-Teflon | 0.04–0.1 |
| Concrete-Steel | 0.3–0.5 |
Friction Settings in FEM
Summary
Key points:
- $|\tau| \leq \mu p_n$ — Upper limit of friction force
- Binary state of stick/slip — Adds additional nonlinearity to contact
- Friction coefficient depends on material pair — Use measured values
- Friction handled by penalty method — Penalty spring also in sliding direction
Coulomb's 1781 Experiment
French military engineer Charles-Augustin de Coulomb systematically conducted hundreds of friction experiments in 1781 using combinations of wood, metal, and stone, and derived "Coulomb's law of friction," which states that friction force is proportional to the normal load and independent of contact area. Looking further back, it was discovered in 1967 that Leonardo da Vinci had illustrated the same relationship in his secret notebook from 1495, causing a surprise in the history of science.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when braking suddenly? 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 "gets left behind." In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored." It cannot be omitted in impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. So, 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 force $f_b$ (e.g., gravity) 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), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A typical pitfall 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. That's 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 keep 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 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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behaviors like plasticity and creep require constitutive law extensions.
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 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
Penalty Method for Friction
Friction is also handled by the penalty method. Tangential "stick penalty":
$k_t$ is the tangential penalty stiffness, $\delta_t$ is the tangential elastic slip. Transition to sliding occurs when $|\tau| = \mu p_n$.
Does "elastic slip" mean there is a tiny slip before actual sliding?
It's an artificial "elastic slip" in the penalty method. Physically, there is zero slip in the stick state, but numerically, a finite tangential penalty stiffness causes a tiny slip. A larger $k_t$ results in smaller elastic slip.
Static and Dynamic Friction
In LS-DYNA, static friction coefficient $\mu_s$ and dynamic friction coefficient $\mu_d$ can be set separately. When sliding starts, it transitions from $\mu_s → \mu_d$ ($\mu_d < \mu_s$). The transition is smoothed using an exponential function.
Summary
Stick-Slip Determination
In FEM implementation of Coulomb friction, it's necessary to branch each contact point into "stick" or "slip." The return mapping method introduced by Zienkiewicz et al. in the 1970s adopted a two-step operation: calculating a trial stress and then projecting it onto the friction cone, significantly reducing computational cost per iteration. This method remains the standard algorithm in ABAQUS and ANSYS today.
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 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 important.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass mode (zero-energy mode). Choose appropriately.
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 tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence rate 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 full load at once, apply 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"—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 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 judge based on total cost-effectiveness.
Practical Guide
Friction in Practice
Problems where friction is important:
- Clamping force in bolted joints — Load transfer via friction
- Press forming — Friction between die and blank affects deformation
- Brakes — Friction force = braking force
- Pipe supports — Friction in sliding supports
Uncertainty in Friction Coefficient
Friction coefficient has very high variability (±30% or more). Strongly depends on surface condition (roughness, lubrication, oxide film).
Countermeasures:
- Sensitivity analysis of friction coefficient (two cases: $\mu_{low}$ and $\mu_{high}$)
- Use measured values (test results)
- Literature values are for reference only
Practical Checklist
Brake Squeal Analysis
Automotive brake squeal occurs when slight velocity dependence of the Coulomb friction coefficient generates negative damping. In a complex eigenvalue analysis using Nastran conducted jointly by Ford and TRW in the 2000s, simply changing the friction coefficient from 0.35 to 0.40 tripled the number of unstable modes, providing insights directly linked to pad shape optimization.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). 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 is 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."
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 truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most critical step in the entire analysis.
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