Grain shearing models and relevance beyond sediment acoustics

Abstract

The grain shearing (GS) and viscous grain shearing (VGS) models for saturated, unconsolidated marine sediments fit well to real data. But the models have much further relevance. The inherent convolution is in fact the definition of a non-integer derivative. Therefore, the GS theory is exactly described by the fractional diffusion-wave equation for shear waves and the fractional Kelvin-Voigt wave equation for the compressional mode. Both have been studied extensively in fractional calculus [Pandey and Holm, “Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations,” JASA (2016)]. The GS theory demonstrates that fractional derivatives arise naturally from physical processes and thus are more than just clever mathematics. These derivatives appear in the relaxation modulus, the stress response to a strain step, of the GS process. The variation in time of the viscosity makes it a time variant non-Newtonian process. But when strain and stress change roles, a surprising result appears. The creep response is the Lomnitz law, an empirically defined logarithmic relationship for such diverse materials as igneous rocks and wood [Pandey and Holm, “Linking the fractional derivative and the Lomnitz…,” Phys. Rev. E (2016)]. In the VGS model, the power-law response is exponentially tempered. The relaxation modulus is then that of the Cole-Davidson model, known from dielectric materials. This relationship has yet to be explored [Holm, Waves with Power-Law Attenuation (Springer, 2019), Chap. 8].