On elastic waves in granular assemblies: From a continuumnization viewpoint

The characteristics of elastic waves in granular assemblies of spherical particles, which translate and spin, are analyzed by a continuumnization technique. The continuumnized governing equations for equal spheres in three-dimensional lattices are first obtained, accompanied by auxiliary tensors corresponding to the macroscopic material properties. The property tensors, i.e., the stiffness, coupling, and spinning tensors, given in neat analytical form, are solely determined by the micro-mechanics (the normal and tangential spring constants) and the microstructures of the lattices (the unit vectors connecting the centers of neighboring particles). The general formulas are then applied for two types of closest packed lattices. Theoretical wave velocities for p- and s-waves in such lattices are derived, together with the frequency for the pure spin mode and the group velocity for the spin motion coupled with s-waves. Two interesting findings of the physics of lattices of spheres are revealed: the s-wave coupled with spin motion in the lattices can be understood as acoustic and optical branches of waves in the diatomic model in solid-state physics; the anisotropic closest packed lattices would degenerate into a special isotropic state with zero Poisson's ratio and a constant velocity ratio of 2 between p- and s-waves. How the wave velocities and the anisotropy are influenced by the spring constant ratios are also analyzed. The theoretical results obtained can be used for quantitative verification and for parameter calibration of discrete-element-method simulations using spherical particles.