Frequency-dependent attenuation and elasticity in unconsolidated earth   materials: effect of damping
  We use the Discrete Element Method (DEM) to understand the underlying attenuation mechanism in granular media, with special applicability to the measurements of the so-called effective mass developed earlier. We consider that the particles interact via Hertz-Mindlin elastic contact forces and that the damping is describable as a force proportional to the velocity difference of contacting grains. We determine the behavior of the complex-valued normal mode frequencies using 1) DEM, 2) direct diagonalization of the relevant matrix, and 3) a numerical search for the zeros of the relevant determinant. All three methods are in strong agreement with each other. The real and the imaginary parts of each normal mode frequency characterize the elastic and the dissipative properties, respectively, of the granular medium. We demonstrate that, as the interparticle damping, $\xi$, increases, the normal modes exhibit nearly circular trajectories in the complex frequency plane and that for a given value of $\xi$ they all lie on or near a circle of radius $R$ centered on the point $-iR$ in the complex plane, where $R\propto 1/\xi$. We show that each normal mode becomes critically damped at a value of the damping parameter $\xi \approx 1/\omega_n^0$, where $\omega_n^0$ is the (real-valued) frequency when there is no damping. The strong indication is that these conclusions carry over to the properties of real granular media whose dissipation is dominated by the relative motion of contacting grains. For example, compressional or shear waves in unconsolidated dry sediments can be expected to become overdamped beyond a critical frequency, depending upon the strength of the intergranular damping constant. 
Yanqing Hu, Hernan A. Makse, John J. Valenza, David L. Johnson
physics.geo-ph cond-mat.soft