The applied model of grinding a spherical solid particle with a direct impact on a non-deformable flat surface

A grinding process using a free impact breakage mechanism is used in industries. In order to make calculations, predict grinding results, and evaluate mills functioning, it is necessary to assess the parameters of the grinding process and interrelations between the process parameters, mills parameters and materials properties, i.e. it is necessary to use an adequate mechanical-mathematical model of the process. However it is difficult to model due to some phenomena occurring in this process. Nowadays, various researchers have established the basis for the structure of the grinding process, but the application of the existing hypotheses and methods to evaluate the grinding process is quite difficult. This paper solves the problem of a spherical shape particle impacting an absolutely rigid half-space. It proposes a refined mechanical and mathematical model describing the process of destruction of the particle using the free direct impact breakage mechanism on an absolutely rigid, stationary, and flat surface. By using the Hertz-Staerman's classical analytical dependencies on the force contact interaction of the spherical bodies and the technical theory of the longitudinal waves’ propagation in the elastic continuous medium, we obtained a new refined solution of the applied dynamic problem related to a direct impact of a ball simulating a particle of a feeding material (an absolutely rigid surface simulating the working body of the mill) taking into account local physically linear deformations, the time parameter and radial particle size. The improved theoretical model of the spherical particle destruction was brought to applicable analytical calculations, tested and illustrated by a numerical example. It made it possible to describe the fracture of the material particles, predict the result and calculate the grinding process depending on its parameters providing the required quality of grinding by regulating and selecting characteristics, designing and selecting the grinding equipment, and modeling the grinding process using the free impact breakage mechanism.