Equivalent Normal Stiffness of the Ball in Granular Dynamics

摘要

The outcome of any discrete granular system simulation and its validity depend, among other factors, on the model of the ball used. When the balls form clusters, the transfer of energy and momentum within this cluster as a result of collision can only be found by treating the cluster as an interconnected system. This poses a computational problem due to the incompatibility of time scales for dynamic processes in clusters and those for the detached balls. Any attempt to circumvent the problem of time scales by, for example, considering collisions as pairwise [1], may achieve computational efficiency at the expense of accuracy. The existence of different time scales reflects the physical phenomenon in discrete dynamical systems, while the error accumulation is a numerical phenomenon. The mathematical model of the ball affects both of these phenomena and thus the outcome and the validity of simulations. The interconnectivity between the balls in a cluster is, in addition to the ball model, another factor affecting the validity of simulations. The most popular approach to speed up the computations while taking into account the interactions in multi-ball systems is to treat each ball as disconnected from other balls in the system during a time step in numerical simulations. This approach is known as the distinct element method (DEM) [2] and the governing equations in this case are reduced to a decoupled system of equations. The decoupling is a system simplification. This factor was investigated in [3] using a string of balls as a sample system. Since the effect of system decoupling is independent from the ball model, for the purpose of this paper a system of balls will be treated as coupled. To investigate the effect of the normal stiffness model we consider t he s implest c luster of b alls, a c ollinear c hain o f balls touching each other and struck by a cue ball. We use it because it allows simple verification of energy and momentum conservation principles, and thus verification of the system and ball models and of the accuracy of numerical procedures. The properties of the ball that are of interest in this paper is the effect of ball's stiffness on accuracy. We investigate two ball models: one with nonlinear and another one with linearized contact stiffness properties. We introduce a concept of an equivalent linear stiffness and show that it allows reduction of accumulated error while preserving the energy and momentum conservation laws, and thus, as a result, an increase of the time step without loss of accuracy.