A semi-analytical model for oblique impacts of elastoplastic spheres

Abstract

Results of finite-element analysis (FEA) of oblique impacts of elastic and elastic, perfectly plastic spheres with an elastic flat substrate are presented. The FEA results are in excellent agreement with published data available in the literature. A simple model is proposed to predict rebound kinematics of the spheres during oblique impacts. In this model, the oblique impacts are classified into two regimes: (i) persistent sliding impact, in which sliding occurs throughout the impact, the effect of tangential (elastic or plastic) deformation is insignificant and the model reproduces the well-established theoretical solutions based on rigid body dynamics for predicting the rebound kinematics and (ii) non-persistent sliding impact, in which sliding does not occur throughout the impact duration and the rebound kinematics depends upon both Poisson's ratio and the normal coefficient of restitution (i.e. the yield stress of the materials). For non-persistent sliding impacts, the variation of impulse ratio with impact angle is approximated using an empirical equation with four parameters. These parameters are sensitive to the values of Poisson's ratio and the normal coefficient of restitution, but can be obtained by fitting numerical data. Consequently, a complete set of solutions is obtained for the rebound kinematics, including the tangential coefficient of restitution, the rebound velocity at the contact patch and the rebound rotational speed of the sphere during oblique impacts. The accuracy and robustness of this model is demonstrated by comparisons with FEA results and data published in the literature. The model is capable of predicting complete rebound behaviour of spheres for both elastic and elastoplastic oblique impacts.

Eulerian–Eulerian simulation of dense solid–gas cylindrical fluidized beds: Impact of wall boundary condition and drag model on fluidization

“…Several analytical and semi-empirical models have been developed to evaluate µ and e w in terms of flow parameters such as the impact velocity, impact angle, particle spin and the particle and wall properties [25][26][27][28][29]. While it has been shown that the frictional coefficient significantly affects the collision regime (sliding/non-sliding) [25], fluidization predictions are not sensitive to e w in the range 0.7-1.0 [1,30].…”

“…In an attempt to determine φ in situ, Li and Benyahia [1] used the classic rigid-body theory in conjunction with the semi-analytical model for oblique collisions by Wu et al [29] to obtain φ in terms of µ and e w through numerical integration. The derived polynomial expression was further used in a series of 2D simulations for bubbling and circulating fluidized beds and found to be consistent with trends in literature [34].…”

Modeling of combined slip and finite sliding at spherical grain contacts

“…3a) should be regarded as the reference path, which, in turn, represents the actual path for the spheres of equal elastic moduli. Using formulae (13,14), any path can be reconstructed from the specified overlap and used for calculation of frictional dissipation referred to the contact path.…”

“…However, for a non-proportional loading, Elata and Berryman [13] found that the slip regime has to be taken into account, even for identical spheres, and solution becomes path-dependent. The semi-analytical model for oblique impacts of elasto-plastic spheres was proposed and verified by FEM and experiments in [14]. In [15,16], analyzing the relative sphere motion along a specified linear trajectory without the initially activated contact zone, it was demonstrated that both normal and tangential forces vary proportionally, generating the continuing evolution of the sliding regime without the initial slip mode.…”

A finite sliding model of two identical spheres under displacement and force control. Part II: dynamic analysis

“…In Maw's model, an elastic oblique impact was considered describing the normal forces by Hertz contact theory without any energy loss in this direction, resulting in the coefficient of restitution equal to one. By examination of the FEM-based results from the rebound kinematics of elastic and elastoplastic spheres, a simple semi-analytical model accounting for impact velocity, impact angle and degree of plastic deformation, was proposed by Wu et al [14].…”