Models for twisting friction in granular interactions

A history-dependent spring-dashpot-slider is used to compute the twisting torque. Because twisting displacement is a scalar, there is no need to adjust for changes in the frame of reference due to rotations of the particle pair. The formulation in Marshall therefore provides the most straightforward treatment:

\[\tau_{twist,0} = -k_{twist}\xi_{twist} - \gamma_{twist}\Omega_{twist}\]

Here \(\xi_{twist} = \int_{t_0}^t \Omega_{twist} (\tau) \mathrm{d}\tau\) is the twisting angular displacement, and \(\Omega_{twist} = (\mathbf{\Omega}_i - \mathbf{\Omega}_j) \cdot \mathbf{n}\) is the relative twisting angular velocity. The torque is then truncated according to:

\[\tau_{twist} = \min\left(\mu_{twist} F_{n,0}, |\tau_{twist,0}|\right)\operatorname{sgn}(\tau_{twist,0})\]

Similar to the sliding and rolling displacement, the angular displacement is rescaled so that it corresponds to the critical value if the twisting torque exceeds this critical value:

\[\xi_{twist} = \frac{1}{k_{twist}} (\mu_{twist} F_{n,0}\operatorname{sgn}(\Omega_{twist}) - \gamma_{twist}\Omega_{twist})\]

The twisting torque on each particle is then given by:

\[\mathbf{\tau}_{twist,i} = \tau_{twist}\mathbf{n}\]

\[\mathbf{\tau}_{twist,j} = -\mathbf{\tau}_{twist,i}\]

sds twisting friction model

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

Example:

pair_style granular
pair_coeff * * hertz 1000.0 50.0 tangential mindlin 1000.0 1.0 0.4 twisting sds 100.0 50.0 0.2

For twisting sds, the coefficients \(k_{twist}, \gamma_{twist}\) and \(\mu_{twist}\) are simply the user input parameters that follow the twisting sds keywords in the pair_coeff command.

marshall twisting history model

Parameters: none beyond the tangential model parameters.

Example:

pair_style granular
pair_coeff * * hertz 1000.0 50.0 tangential mindlin 1000.0 1.0 0.4 twisting marshall

For twisting marshall, the coefficients \(k_{twist}, \gamma_{twist}\) and \(\mu_{twist}\) are expressed in terms of the tangential friction coefficients \(k_t, \gamma_t, \mu_t\), as discussed in Marshall (see equations 32 and 33 of that work, and the doc page for Models for tangential friction in granular interactions):

\[k_{twist} = 0.5k_ta^2\]

\[\gamma_{twist} = 0.5\gamma_ta^2\]

\[\mu_{twist} = \frac{2}{3}a\mu_t\]

Here, \(k_t\), \(mu_t\) and \(\gamma_t\) are the tangential stiffness, tangential friction coefficient, and tangential damping coefficient, respectively.

References

(Marshall, 2009) Marshall, J. S. (2009). Discrete-element modeling of particulate aerosol flows. Journal of Computational Physics, 228(5), 1541-1561.