Popov JKR - msolids/musen GitHub Wiki

Velocities

$\bar{v}_{rel} = \bar{v}_2 - \bar{v}_1 + \bar{\omega}_1 \times \bar{r}_c - \bar{\omega}_2 \times \bar{r}_{c_2}$

$\bar{v}_{rel,n}=\bar{r}_n\cdot(\bar{r}_n\cdot\bar{v}_{rel})$

$\bar{v}_{rel,t}=\bar{v}_{rel}-\bar{v}_{rel,n}$

Additional parameters

$\gamma_{12} = \sqrt{\gamma_1\gamma_2}$

$R^* = \frac{r_1 \cdot r_2}{r_1 + r_2}$

$M^* = \frac{m_1 \cdot m_2}{m_1 + m_2}$

Normal force

$s_{crit}^{adh}=\left[\frac{3(\pi\, \cdot\, \gamma_{12} )^2 R^*}{64(E^* )^2 }\right]^{\frac{1}{3}}$

$F_{crit}^{adh}=\frac{3}{2}\,\gamma_{12}\,\pi\,R^*$

$F_n^{JKR-Popov} = F_{crit}^{adh} \cdot \left[ -1 + 0.12 \left( \frac{\xi_n}{s_{crit}^{adh}} + 1 \right)^\frac{5}{3} \right] - \bar r_n \cdot sgn(\bar v_{rel,n} \cdot r_n ) \cdot 1.8257 \alpha \cdot |\bar v_{rel,n} \cdot |\sqrt{k_n \cdot M^* }$

Tangential force

$\Delta\bar{\xi}_t = \bar{v}_{rel,t}\cdot \Delta t$

$k_t = 8 \cdot G^* \cdot \sqrt{R^* \xi_n }$

$\Delta \bar{F}_t = [ k_t \cdot \Delta \bar{\xi}_t - 1.8257 \cdot \alpha \cdot \bar{v}_{rel,t} \cdot \sqrt{k_t\cdot M^* }]$

$\bar{F}_{t,pr}^{cor} = \bar{F}_{t,pr}-\bar{r}_n \cdot (\bar{r}_n \cdot \bar{F}_{t,pr} )$

$\bar{F}_{t,pr}^{cor} = \bar{F}_{t,pr}^{cor} \cdot |\bar F_{t,pr}|\, / \,|\bar F_{t,pr}^{cor} |$

$\bar F_t = \bar F_{t,pr}^{cor} + \Delta \bar F_t$

if $|\bar F_t| > \mu_{sl}\cdot |\bar F_n |$

then $\bar F_t = \mu_{sl} \cdot |\bar F_n | \cdot \frac{\bar F_t}{|\bar F_t|}$

Rolling friction

$\bar M_{ro,1} = -\mu_{ro} \cdot |\bar F_n| \cdot r_1 \cdot \frac{\bar\omega_1}{|\bar\omega_1|}$

$\bar M_{ro,2} = -\mu_{ro} \cdot |\bar F_n| \cdot r_2 \cdot \frac{\bar\omega_2}{|\bar\omega_2|}$

Summarized forces and moments acting on particle

$\bar F_{tot} = \bar F_n + \bar F_t$

$\bar F_{1} = \bar F_n + \bar F_t$

$\bar F_{2} = -\bar F_n + \bar F_t$

$\bar M_{tot,1} = \bar r_n \times \bar F_t \cdot r_1 + \bar M_{ro,1}$

$\bar M_{tot,2} = -\bar r_n \times \bar F_t \cdot r_2 + \bar M_{ro,2}$

Literature

Popov V. (2010). Kontaktmechanik und Reibung, Springer-Verlag, Berlin.

Symbol	Description
CP	Contact point [m]
$\bar F_n,\, \bar F_t$	Force in normal and tangential directions [N]
$\bar F_{t,pr}$	Tangential force on previous iteration [N]
$G^*$	Equivalent shear modulus [Pa]
$M^*$	Equivalent mass [kg]
$\bar M_{ro}$	Moment due to the rolling friction [N]
$m_1,\,m_2$	Particle masses [kg]
$O_1,\,O_2$	Centers of contact partners [m]
$R^*$	Equivalent radius [m]
$r_1,\,r_2$	Particle radii [m]
$\bar r_c$	Contact vector [m]
$\bar r_n$	Normalized contact vector [-]
$\bar v_{rel}$	Relative velocity [m/s]
$\bar v_1,\, \bar v_2$	Velocities of contact partners [m/s]
$\Delta\bar\xi_t$	Tangential displacement in the current step [m]
$\gamma$	Surface tension [-]
$\mu_{ro},\,\mu_{sl}$	Coefficient of rolling friction and sliding friction [-]
$\xi_n$	Normal overlap [m]
$\omega_1,\,\omega_2$	Rotation velocities of particles [rad/s]