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Our study builds on the work of Buckingham JASA (2000) which employed a grain-shearing (GS) model to describe the propagation of elastic waves in saturated, unconsolidated granular materials. He ensemble averages the random stick-slip process which follows the velocity gradient set up by the wave. The stick-slip process is due to the presence of micro-asperities between the contact surfaces of the grains. This is a strain hardening process which is represented by a time-dependent coefficient in the Maxwell element, besides the coefficient also gives the order of the loss term in the wave equations. We find that the material impulse response derived from the GS model is similar to the power-law memory kernel of fractional calculus. The GS model then gives two equations; a fractional Kelvin-Voigt wave equation for the compressional wave and a fractional diffusion equation for the shear wave. These equations have already been analyzed extensively in the framework of fractional calculus. Since the Kelvin-Voigt model is used in biomechanics of living tissue, we believe the GS theory could offer insights into ultrasound and elastography as well. The overall goal is to understand the role of different material parameters which affect wave propagation.
Pandey et al. (Tue,) studied this question.