Rolling Resistance

The following rolling resistance methods are applicable to DEM particles:

Force Proportional Method
Constant Torque Method
Displacement Damping Method

If you are using the Hertz-Mindlin, Walton-Braun, or Bonded Particle contact models together with the Rolling Resistance model, the Rolling Resistance model applies a rolling friction coefficient to the contact models.

Force Proportional Method

This method only supplies a coefficient of rolling resistance, $μ_{r}$ , also called the coefficient of rolling friction. It can take on any positive real value. The default value is 0.001.

The moment due to rolling resistance, which is acting on the particle, is given by:

Figure 1. EQUATION_DISPLAY

M_{r} = - (μ_{r} | F_{n} | | r_{c} |) \frac{ω_{p}}{| ω_{p} |}

(3275)

where:

$F_{n}$ is the contact force.
$r_{c}$ is the position vector from the particle centroid to the point of contact.
$ω_{p}$ is the component of the particle angular velocity parallel to the contact plane.

Constant Torque Method

The Constant Torque method applies a torque to resist rolling, calculated as:

Figure 2. EQUATION_DISPLAY

M_{r} = \frac{- ω_{r e l}}{| ω_{r e l} |} μ_{r} R_{e q} F_{n}

(3276)

where:

$ω_{r e l}$ is the relative rotation, given as $ω_{r e l} = ω_{i} - ω_{j}$ , where:
- $ω_{i}$ is the rotation of the particle for which the resistance moment $M_{r}$ is being calculated.
- $ω_{j}$ is the rotation of the other particle in the contact.
$μ_{r}$ is the coefficient of rolling friction, also called coefficient of rolling resistance. The default value is 0.1.
$R_{e q}$ is the particle equivalent radius, that is, the harmonic mean of the radii of the two particles.
$F_{n}$ is the magnitude of the contact force.

This method also uses a “Moment Limiter.” This function limits the size of the angular momentum of the particle, measured around the axis of moment. This maximum is 0.08 of the moment that negates the lesser of the two rotations $ω_{i}$ and $ω_{j}$ in the next time-step.

This limit can control specious rotations in particle compacts. These rotations can arise from a large number of contacts or from high compact pressure.

Displacement Damping Method

The Displacement Damping method combines a viscous damping resistance with a spring-like resisting torque generated by elastic energy stored in the particle as it tries to turn around a finite contact area.

The spring resistance torque at time $t$ is:

Figure 3. EQUATION_DISPLAY

{M^{k}}_{r, t + δ t} = {M^{k}}_{r, t} - k_{r} ω_{r e l} Δ t

(3277)

where $| {M^{k}}_{r, t} | \leq k_{r} {θ_{r}}^{m a x}$ and $k_{r} = (μ_{r} R_{e q} F_{n}) / {θ_{r}}^{m a x}$ , and where:

${θ_{r}}^{m a x}$ is the maximum angular displacement before rolling.
$ω_{r e l}$ is the relative rotation, given as $ω_{r e l} = ω_{i} - ω_{j}$ , where:
- $ω_{i}$ is the rotation of the particle for which the resistance moment $M_{r}$ is being calculated.
- $ω_{j}$ is the rotation of the other particle in the contact.
$μ_{r}$ is the coefficient of rolling friction, also called coefficient of rolling resistance. The default value is 0.1.
$R_{e q}$ is the particle equivalent radius, that is, the harmonic mean of the radii of the two particles.
$F_{n}$ is the magnitude of the contact force.

You can activate the viscous damping torque or leave it off. When active, it is:

Figure 4. EQUATION_DISPLAY

{M_{r}}^{d} = {\begin{matrix} - ν C^{crit} ω_{rel} & if (| {M^{k}}_{r, t} | \leq k_{r} {θ_{r}}^{m a x}) \\ - f ν C^{crit} ω_{rel} & if (| {M^{k}}_{r, t} | > k_{r} {θ_{r}}^{m a x}) \end{matrix}

(3278)

where:

$ν$ is the damping factor parameter.
$f$ is a damping factor multiplier
$C^{crit}$ is the critical viscous damping constant, computed as:
${C_{r}}^{crit} = 2 \sqrt{I_{r} k_{r}}$

where $I_{r}$ is the equivalent moment of inertia for the relative rotational vibration mode about the contact point.