Hertz-Mindlin

Силы взаимодействия

[math]{F_n} = -\frac{4}{3}E^*\sqrt{R^*}\delta^{\frac{3}{2}}[/math]

[math]{F^d_n} = -2 \sqrt{\frac{5}{6}}\beta \sqrt{S_n m^*}{v^{rel}_n}[/math]

where [math]{F_n}[/math] - normal force, [math]E^*[/math] - the equivalent Young’s Modulus, [math]R^*[/math] - the equivalent radius, [math]{\delta}_n[/math] - normal overlap; [math]F^d_n[/math] - normal damping force, [math]m^*[/math] - the equivalent mass, [math]\vec{v^{rel}_n}[/math] - the normal component of the relative velocity and [math]\beta[/math] and [math]S_n[/math](the normal stiffness) are given by

[math]\beta=\frac{\ln e}{\sqrt{\ln^2 e + \pi^2}}[/math]

[math]S_n=2Y^*\sqrt{R^*{\delta}_n}[/math]

with [math]e[/math] the coefficient of restitution.

The tangential force, [math]{F_t}[/math], depends on the tangential overlap [math]{\delta_t}[/math] and the tangential stiffness [math]{S_t}[/math].

[math]\vec{F_t}=-S_t\vec{\delta_t}[/math]

with

[math]S_t=8G^*\sqrt{R^*\delta_n}[/math]

Additionally there is a tangential damping force,[math]\vec{F^d_t}[/math]

[math]\vec{F^d_t} = -2 \sqrt{\frac{5}{6}}\beta \sqrt{S_t m^*}\vec{v^{rel}_t}[/math]

where, [math]\vec{v^{rel}_t}[/math], is the relative tangential velocity. The tangential force is limited by Coulomb friction, [math]\mu_s F_n[/math] , where [math]\mu_s[/math] is the coefficient of static friction. For simulations in which rolling friction is important, this is accounted for by applying a torque to the contacting surfaces.

[math] \vec{\tau_i}=-\mu_r F_n R_i \vec{\hat{\omega_i}} [/math]

with [math]\mu_r[/math] the coefficient of rolling friction, [math]R_i[/math] the distance of the contact point from the centre of mass for object [math]i[/math] and [math]\vec{\hat{\omega_i}}[/math] the unit angular velocity vector of object [math]i[/math] at the contact point.

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