Modeling of granular media by the 2D discrete lattice

Abstract

A two-dimensional model of a granular medium is considered that represents a square lattice consisting of uniform round particles. The particles have both translational and rotational degrees of freedom. Each particle (granule) is supposed to interact directly with eight nearest neighbours in the lattice [1]. The nonlinear governing equations describing propagation and interaction of waves of various types in such a medium have been derived for a cubic potential of elastic interactions between the particles in the discrete and continuum approximations. The discrete equations are valuable, particularly, for numerical simulation of nonlinear wave processes in granular media. The continuum approximations of the model at issue are convenient for its comparison with known theories of solids. The continuum equations do not coincide with the classical theory of elasticity due to additional equation for the rotational wave. Such a wave exists when the frequency is larger than a threshold value. Its dispersion properties are similar to dispersion properties of the spin wave in a magnetelastic medium. From the numerical estimations of the rotational wave velocity in some cubic crystals follows that, as a rule, it is less than the translational wave velocities. When microturns of the particles are absent, the linear parts of the governing equations of the first continuum approximation degenerate into Lame equations for anisotropic medium with the cubic symmetry. The governing equations are structurally similar to equations of the anisotropic Cosserat continuum with centrally symmetric particles. However, the longitudinal wave velocity does not depend on the medium structure in the Cosserat continuum, while such dependence presents in the considered model. The last fact enables to explain, particularly, experimentally observing variations of this wave velocity when size of granules grows. The governing equations of the second (quasicontinuum) approximation contain summands with higher-order derivatives and give explanation to appearance of the longitudinal wave dispersion. The Cosserat theory is unable to explain this effect.