Generalized dynamic theory of a solid particle grinding by pulse-force compression with two non-deformable balls

Machines such as ball mill, vibrating mill, planetary mill, and stirred mill using spherical grinding media as a destruction energy transmitter are widely used among the manufacturing equipment for materials grinding. Providing a necessary product particle size at an optimal energy efficiency and productivity during grinding makes it necessity to investigate new ways of improving the mentioned types of mills, supplementing them with a system for regulating the grinding regime, taking into account the physical and mechanical properties of the feed particle. As the grinding media motion is quite complicated, it is necessary to develop a new theory of a particle grinding for an accurate and adequate evaluation of a particle stress-strain state with the grinding media using simple physical and mathematical relations. So, we obtained a new general solution of the fundamental-applied multifunctional problem of materials resistance during a direct inelastic collision of two identical isotropic weightless balls of a small diameter and high rigidity with a given form moving towards each other. The solution is based on a complex application of classical Hertz-Staerman analytic dependences on the force contact of the spherical grinding media and the energy conservation law (the Reilly-Young method). In the corresponding quasi-static physical and mathematical model we considered local deformations within the framework of Hooke's law, own weight of spherical media, the impact time and the radial dimensions of the interacting elements of the constructively nonlinear mechanical system. To calculate the bearing capacity of the grinded particle and determine the crushing force, the well-known Galilei-Leibniz strength criterion is used. The criterion is interpreted by the largest tensile stress, which adequately characterizes the limiting state when a wide class of brittle homogeneous solids is destroyed. The results of the developed theory are presented by the formulae aimed to regulate and optimize the grinding process of stone materials.