Two-dimensional model of a granular medium

A two-dimensional model of a granular medium is represented as a square lattice composed of elastically interacting round particles with translational and rotational degrees of freedom. In the long-wave approximation, we derive linear partial differential equations describing the propagation and interaction of waves of various types in such a medium. If microrotations of particles in the lattice and the related moment interactions are taken into account, then a microrotation wave (a spin wave) appears in the medium. We establish the one-to-one correspondence between the parameters of the microstructure and the elastic constants of second order. We analyze the dependence of the medium elasticity constants on the grain dimensions. In the continuum approximation, we compare the model proposed here with the model of two-dimensional Cosserat continuum.