Analytical solution for the problem of frictional-elastic collisions of spherical particles using the linear model

In the present paper, the problem of a frictional-elastic impact of two spheres is addressed with a novel approach. The set of equations arising from the linear model of the contact mechanics is analytically integrated considering the combined effects of the elastic and frictional mechanisms. The linear model is very commonly used in simulations based on the soft-sphere distinct element method (DEM) (Geotechnique 29 (1979) 47), where numerical methods are used for the integration of the equations of motion of the particles. The analytical approach presented in this work allows the examination of many important aspects related to the use of the linear model in dynamic simulation of multi-particle systems. The impact characteristics, in terms of the mechanisms governing the evolution of the force-displacement relation, can be classified in terms of the initial conditions, showing the same subdivision as that obtained with the more complex model of Maw et al. (Wear 3 8 (1976) 101). It is demonstrated how the values of many interesting variables at the end of the impact can be directly related to the impact initial conditions through a one-step calculation procedure. The model results of the tangential coefficient of restitution, rebound angles of the contact point and center of mass are validated with the experimental data on frictional-elastic collisions of Kharaz et al. (Powder Technol. 120 (2001) 281), showing, despite the simplicity of the considered model, an excellent agreement. However, it is demonstrated that the force evolution and time duration of the collisions strongly depend on the model parameters and can be improperly evaluated with incorrect material constants. Further analyses are carried out, for various impact angles, on the amount of energy loss due to the frictional mechanism. Also, an analysis of the direct influence of each model parameter on the properties of the particles at the end of the collision is carried out, with special emphasis on the not-Mal elastic spring constant K(n). This helps clarifying why, in the literature, realistic macroscopic results were obtained even with very small values of K(n). (C) 2004 Elsevier Ltd. All rights reserved.