Dynamic Analysis And Modeling To Elastic-Plastic Collision Between Two Spheres

Impact is a forcc interaction between collision bodies; it is a widespread nature phenomenon influencing humanity activities. According to the type of the force interaction, it can be divided into non-contact (e.g. interaction between electrons) and contact impact (e.g. collision between two sands, billiards and bird-impact to aircraft) totally. Contact impact is an important and basic subject in disaster, engineering and military science etc. Generally speaking, when the two bodies impact together, the contact process is very complex. It is different from normal dynamic problems for two reasons:(1) its loading force and final response (e.g. relative velocity after impact and the residual deformation) are unknown; (2) the loading force and final response are coupled with material properties, geometric parameters and local deformation. All of these constitute a complex nonlinear dynamic system.Collision between two elastic-plastic spheres is the most basic cell and model in granular dynamics analysis using Discrete Element Methods. It requires not only the representation of collision model be simple and easy to calculate, but also requires model can accurately predict the coefficient of restitution and contact force. Based on "Hertz Contact Law" and "Thornton elastic-perfectly plastic model", a new elastic-plastic model for normal collisions between two soft spheres is proposed in this paper to predict the coefficient of restitution and other information during the collision process. The model considered the sensitivity of strain rate and the effects of strain hardening synthetically. The main works are concluded as follows:Firstly, we thought that the contact force consistent with the "Hertz contact Law" in the elastic loading stage during compression process. When the plastic yield occurred inside the collision bodies, the type of Hertz contact pressure will be changed and soften with the dilatation of plastic zone. After analyzing the character of contact pressure distribution from FEM simulation, a simple function with an unknown parameter is proposed in this paper to characterize the contact pressure over the contact area. An equivalent "Ramberg-Osgood" strain-stress relation is employed to characterize the effects of stress strengthening and ensure the unknown parameter in pressure distribution function. Now, we give out the contact force during the loading process. Secondly, we thought that the unloading process is accord with Hertz Elastic Contact Law with a changed curvature radius Rp caused by the local permanent residual deformation. We assume that the final local shape of the contact area is cambered and its curvature radius is Rp. After considering the continuity condition of the radius of contact area during compression and recovery stage, we give out the equation of the changed curvature radius Rp. Now, we give out the contact force during the unloading process.Finally, we give out a comparison between model predictions in this paper and experimental measurements from the relevant literatures for five spheres of different materials. It is shows that the coefficient of restitution and other information (e.g. collision duration, residual deformation and final radius of contact area) of the two collision spheres are in good agreement quantitatively. We also give out comparisons of "Force-Displacement Curve" and "Force-Time Curve" between relevant model predictions and indirect measurements qualitatively. The result shows that the model predictions can accurately reflect the characters and information of "Force-Displacement Curve" and "Force-Time Curve".