ABSTRACT
The elementary units of granular materials are mesoscopic grains which deform under the stress developing at their contacts. Since the realistic modeling of the internal deformations of the particles is much too complicated, we relate the normal interaction force to the overlap δ of two spherical particles. If all forces f i, acting on particle i, either from other particles, from boundaries or from external forces, are known, the problem is reduced to the integration of Newton’s equations of motion for the translational and rotational degrees of freedom
with the mass mi of particle i, its position ri the total force f i = ∑c f ci acting on it due to contacts with other particles or with the walls, its moment of inertia Ii, its angular velocity ωi = dϕi/dt and the total torque t i. 2.1.1 Linear normal contact law The force acting on particle i from particle j can be decomposed into a normal and a tangential part, where the simplest normal force is a linear spring and a linear dashpot f ni = kδ+ γ0δ˙, with spring constant k and some damping coefficient γ0. The consequences of more realistic normal force laws, involving non-linearities, plastic deformation, and attractive cohesion forces will be discussed elsewhere. The halfperiod of a (damped) vibration around the equilibrium position can be computed, and one obtains a typi-
cal response time tc =π/ω, with ω= √
(k/mij) − η20,
the eigenfrequency of the contact, the reduced mass mij = mimj/(mi + mj), and the rescaled damping coefficient η0 = γ0/(2mij). The energy dissipation during a collision, as caused by the dashpot, leads to a (constant) restitution coefficient r =−v′n/ vn = exp (−η0tc), where the prime denotes the normal velocity after a collision.
