Coulomb friction model
Coulomb friction: Theoretical Foundations
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.
Computational Methods for Coulomb friction
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.
Coulomb friction in Practice
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.