Models for rolling friction in granular interactions

If the rolling keyword is omitted, or if rolling none is specified, no rolling friction model is used.

sds rolling model

Parameters: \(k_{roll}\), \(\gamma_{roll}\), \(\mu_{roll}\)

Example:

pair_style granular
pair_coeff * * hertz 1000.0 50.0 tangential mindlin 1000.0 1.0 0.4 rolling sds 500.0 200.0 0.5

For rolling sds, rolling friction is computed via a spring-dashpot-slider, using a ‘pseudo-force’ formulation, as described by Luding. Unlike the formulation in Marshall, this allows for the required adjustment of rolling displacement due to changes in the frame of reference of the contacting pair. The rolling pseudo-force is computed analogously to the tangential force:

\[\mathbf{F}_{roll,0} = k_{roll} \mathbf{\xi}_{roll} - \gamma_{roll} \mathbf{v}_{roll}\]

Here, \(\mathbf{v}_{roll} = -R(\boldsymbol{\Omega}_i - \boldsymbol{\Omega}_j) \times \mathbf{n}\) is the relative rolling velocity, as given in Wang et al and Luding. This differs from the expressions given by Kuhn and Bagi and used in Marshall; see Wang et al for details. The rolling displacement is given by:

\[\mathbf{\xi}_{roll} = \int_{t_0}^t \mathbf{v}_{roll} (\tau) \mathrm{d} \tau\]

A Coulomb friction criterion truncates the rolling pseudo-force if it exceeds a critical value:

\[\mathbf{F}_{roll} = \min(\mu_{roll} F_{n,0}, \|\mathbf{F}_{roll,0}\|)\mathbf{k}\]

Here, \(\mathbf{k} = \mathbf{v}_{roll}/\|\mathbf{v}_{roll}\|\) is the direction of the pseudo-force. As with tangential displacement, the rolling displacement is rescaled when the critical force is exceeded, so that the spring length corresponds to the critical force. Additionally, the displacement is adjusted to account for rotations of the frame of reference of the two contacting particles in a manner analogous to the tangential displacement.

The rolling pseudo-force does not contribute to the total force on either particle (hence ‘pseudo’), but acts only to induce an equal and opposite torque on each particle, according to:

\[\tau_{roll,i} = R \mathbf{n} \times \mathbf{F}_{roll}\]

\[\tau_{roll,j} = -\tau_{roll,i}\]

References

(Kuhn and Bagi, 2005) Kuhn, M. R., & Bagi, K. (2004). Contact rolling and deformation in granular media. International journal of solids and structures, 41(21), 5793-5820.

(Wang et al, 2015) Wang, Y., Alonso-Marroquin, F., & Guo, W. W. (2015). Rolling and sliding in 3-D discrete element models. Particuology, 23, 49-55.