Hertz Mindlin - msolids/musen GitHub Wiki

This is soft-sphere model which is used to calculate particle-particle or particle wall contact. In both cases calculations are similar, however, when particle-wall contact is calculated, then particle radius and mass are considered as equivalent radius $R^*$ and mass $M^*$ .

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

$\alpha = \frac{\ln(e)}{\sqrt{\pi^2 + \ln^2(e)}}$

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

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

$E^*=\left(\frac{1-\nu^2_1}{E_1}+\frac{1-\nu^2_2}{E_2}\right)^{-1}$

Normal force

$\xi_n = r_1 + r_2 - |O_2 - O_1|$

$k_n=2E^*\sqrt{\xi_{n}\cdot R^*}$

$\bar F_n = - \bar r_n \cdot \frac{2}{3} \cdot \xi_n \cdot k_n -\bar{r}_n \cdot sgn(\bar{v}_{rel,n} \cdot \bar{r}_n) \cdot 1.8257 \cdot \alpha \cdot |\bar{v}_{rel,n}|\cdot \sqrt{k_n \cdot M^*}$

Tangential (shear) force

$\Delta\bar\xi_t = \bar v_{rel,t} \cdot \Delta t$

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

$\Delta \bar{F}_t = k_t \cdot \Delta \bar{\xi}_t$

$\bar{F}_{t,damp} = - 1.8257 \cdot \alpha \cdot |\bar{v}_{rel,t}|\cdot \sqrt{k_t \cdot M^*}$

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

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

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

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

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

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

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

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

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

Literature

Hertz H. (1882). Über die Berührung fester elastischer Körper. Journal die reine und angewandte Mathematik, 92, 156-171.
Tsuji Y., Tanaka T., Ishida T. (1992). Lagrangian numerical simulation of plug flow of cohesionless particles in horizontal pipe. Powder Technology, 71 239-250.

Symbol	Description
$E^*$	Equivalent Young’s modulus [Pa]
$E_1,\,E_2$	Young’s moduli of contact partners [Pa]
e	Restitution coefficient [-]
$\bar F_n,\, \bar F_t$	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]
$\bar r_c$	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]
$r_1,\,r_2$	Particle radii [m]
$\Delta\bar\xi_t$	Increment of tangential displacement in the current step [m]
$\mu_{ro},\,\mu_{sl}$	Coefficient of rolling friction and sliding friction [-]
$\xi_n$	Normal overlap [m]
$\omega_1,\,\omega_2$	Rotation velocities of particles [rad/s]