Forecasting nonlinear vibrations of patches of granular materials by elastic interactions between spheres

A particle model devoted to the study of three-dimensional grain packages in a dynamic setting is presented and discussed. The proposed approach considers each grain as a rigid sphere immersed in an elastically deformable medium. The motion of the grains is described by using as Lagrangian parameters the grain centroid displacement and the grain rotation. In this framework, strain and kinetic energies are described in a fully nonlinear way. The motion evolution is reconstructed by using a stepwise time integration scheme based on a quadratic evolution of the displacement in the chosen time step and on a discrete form of the momentum–impulse relationship. A numerical simulation of a package containing few grains gives an idea of the proposed model performances.