Hertz-Mindlin — различия между версиями
Материал из Department of Theoretical and Applied Mechanics
Строка 1: | Строка 1: | ||
− | Силы взаимодействия | + | ==Силы взаимодействия== |
:<math>{F_n} = -\frac{4}{3}E^*\sqrt{R^*}\delta^{\frac{3}{2}}</math> | :<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. | ||
+ | |||
+ | ==Ссылки== | ||
+ | Первоистоник<ref name="HM"/> | ||
<references> | <references> | ||
− | + | ||
− | |||
<ref name="HM"> Alberto Di Renzo, Francesco Paolo Di Maio, ''Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes''. Chemical Engineering Science, 59,(2004) pp. 525–541, | <ref name="HM"> Alberto Di Renzo, Francesco Paolo Di Maio, ''Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes''. Chemical Engineering Science, 59,(2004) pp. 525–541, | ||
</ref> | </ref> | ||
− | </references | + | </references> |
[[Category:Взаимодействия]] | [[Category:Взаимодействия]] | ||
[[Category:Непотенциальные взаимодействия]] | [[Category:Непотенциальные взаимодействия]] |
Версия 13:15, 1 сентября 2011
Силы взаимодействия
where
- normal force, - the equivalent Young’s Modulus, - the equivalent radius, - normal overlap; - normal damping force, - the equivalent mass, - the normal component of the relative velocity and and (the normal stiffness) are given bywith
the coefficient of restitution.The tangential force,
, depends on the tangential overlap and the tangential stiffness .with
Additionally there is a tangential damping force,
where,
, is the relative tangential velocity. The tangential force is limited by Coulomb friction, , where 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.with
the coefficient of rolling friction, the distance of the contact point from the centre of mass for object and the unit angular velocity vector of object at the contact point.Ссылки
Первоистоник[1]
- ↑ Alberto Di Renzo, Francesco Paolo Di Maio, Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chemical Engineering Science, 59,(2004) pp. 525–541,