rolling_friction model epsd

Purpose

Elasto-plastic spring-dashpot model in its standard form with two parameters for general materials.

Syntax

rolling_friction epsd [other model_type/model_name pairs as described here ] settings keyword values

zero or one keyword/value pair may be appended after the keyword settings (after all models are specified)

torsionTorque values = 'on' or 'off'
  on = also the normal, relative rotation (torsion) contributes to the resulting torque
  off = only the tangential, relative rotation contributes to the resulting torque
rollFrictionFactor values = f
  f = factor for the roll friction damping when in the full mobilization regime [default: 0]

Associated material properties

Material interaction properties

coefficientRollingFriction ( $\mu_r$ ): Coefficient of rolling friction [–]
coefficientRollingViscousDamping ( $\eta_r$ ): viscous damping coefficient for the rolling friction model [–]

Description

This model can be used in the rolling_friction argument of both particle_contact_model and wall_contact_model.

The elasto-plastic spring-dashpot (EPSD) model (Ai et al. 2011) adds the rolling friction torque contribution $M_r$ [Nm] to the angular momentum equation. In this model the rolling friction torque is decomposed in two components, the elastic torque $M^k_r$ [Nm] and the viscous damping torque $M^d_r$ [Nm], i.e.

$M_r = M^k_r + M^d_r$

The calculation of the elastic component $M_r^k$ [Nm] is done incrementally, where its value at a time instant $t + \Delta t$ [s] is obtained from the previous value according to the following expressions

$M^k_{r,t+\Delta t} = M^k_{r,t} -k_r\Delta\theta_r, \ \text{if} \ |M^k_{r,t+\Delta t}| \leq |M^m_r| M^k_{r,t+\Delta t} = M^m_r, \ \text{if} \ |M^k_{r,t+\Delta t}| > |M^m_r|$

where $\Delta\theta_r$ [radians] is the incremental relative angle of rotation between the two particles, $M^m_r$ [Nm] is the full mobilization torque (see below), and the rotation rolling stiffness $k_r$ [Nm/radians] is obtained from the following equation:

$k_r = 2.25 k_n \mu_r^2 {r^*}^2,$

where $k_n$ [N/m] is the spring stiffness of the normal contact model, $r^*$ [m] is the effective radius (see e.g. here) and $\mu_r$ is the (non-dimensional) coefficient of rolling friction.

The full mobilization torque is given by

$M^m_r = \mu_r r^* F_n,$

where $F_n$ [N] is the normal contact force. If the normal model Luding, is used, $F_n$ [N] is replaced by $F_{n, modified}$ : see here.

The viscous damping torque $M_r^d$ is implemented as

$M^d_{r,t+\Delta t} = -C_r\dot{\theta}_r\;\mbox{ if }|M^k_{r,t+\Delta t}| \leq |M_r^m| M^d_{r,t+\Delta t} = -f C_r \dot{\theta}_r\;\mbox{ if }|M^k_{r,t+\Delta t}| > |M_r^m|$

where $\dot{\theta}_r$ [radians/s] is the rate of change of $\theta$ [radians], and $C_r$ [Nms/radians] is the damping scaling factor. By default damping is disabled in case of full mobilisation ( $f = 0$ ), which can be adjusted by the rollFrictionFactor setting.

The scaling factor $C_r$ [Nms/rad] is obtained from

$C_r = \eta_r C_r^{crit}$

where $\eta_r$ [–] is the non-dimensional viscous damping coefficient (set by the user), and $C_r^{crit}$ [Nms/radians] is given by

$C_r^{crit} = 2 \sqrt{I_r k_r}$

and the effective moment of inertia $I_r$ [Kg m ²] is

$I_r = \left(\frac{1}{I_i + m_i r_i^2} + \frac{1}{I_j + m_j r_j^2}\right)^{-1}$

where $I_{i/j}$ [Kg m ²] is the moment of inertia and $m_{i/j}$ [Kg] is the mass of the particles $i$ and $j$ , respectively.

Torque information:

By default the relative normal rotation (torsion), also called twisting friction sometimes, is subtracted and does not contribute to the resulting rolling friction torque. This allows the particles to rotate in the contact-normal (twisting) direction without any resistance. In rare cases, this can lead to high angular velocities that make the simulation unstable, where particles appear to jump (pop) unrealistically. By setting the torsionTorque keyword to ‘on’, the full relative rotation contributes to the rolling friction torque.

Coarse-graining information:

Using coarsegraining in combination with this command might lead to different dynamics or system state and thus to inconsistencies.

Default

torsionTorque = ‘off’

References

Jun Ai, Jian-Fei Chen, J. Michael Rotter, Jin Y. Ooi, Powder Technology, 206 (3), p 269-282 (2011).