Frequency-dependent attenuation and elasticity in unconsolidated earth materials: Effect of damping

We used 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 considered that the particles interacted via Hertz-Mindlin elastic contact forces and that the damping was describable as a force proportional to the velocity difference of contacting grains. We determined 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 were in strong agreement with each other. The real and the imaginary parts of each normal-mode frequency characterized the elastic and the dissipative properties, respectively, of the granular medium. We found that as the interparticle damping. increased, the normal modes exhibited nearly circular trajectories in the complex frequency plane, and that for a given value of., they all lay on or near a circle of radius R centered on the point -iR in the complex plane, where R proportional to 1/xi. We found that each normal mode became critically damped at a value of the damping parameter xi approximate to 1/omega(0)(n), where omega(0)(n) was the (real-valued) frequency when there was no damping. The strong indication was that these conclusions carried over to the properties of real granular media whose dissipation was dominated by the relative motion of contacting grains. For example, P- or S-waves in unconsolidated dry sediments can be expected to become overdamped beyond a critical frequency, depending upon the strength of the intergranular damping constant.