Causal Damping Ratio Spectra and Dispersion Functions in Geomaterials from the Exact Solution of Kramers-Kronig Equations of Viscoelasticity

Abstract

The constitutive parameters controlling the response of geomaterials to low-strain dynamic loading are important in a variety of situations in Earthquake Geotechnical Engineering (e.g. ground response analysis) and Soil Dynamics (e.g. propagation of ground-borne vibrations). Linear viscoelasticity is the simplest constitutive theory able to satisfactory capture the mechanical response of soils and rocks undergoing small-amplitude oscillations. An important result predicted by this theory is the functional dependence of the speed of propagation \({{\mathrm{V}}_{P}}\) and \({{\mathrm{V}}_{S}}\) of mechanical P and S waves from the corresponding material damping ratios \({{\mathrm{D}}_{P}}\) and \(\mathrm{D}_{S}\). Yet, in the current practice of experimental Soil Mechanics, these parameters are measured independently using different and inconsistent procedures. Furthermore, the frequency-dependence of \({\mathrm{V}_{P}, \mathrm{V}_{S}}\) and \({\mathrm{D}_{P}}\), \({\mathrm{D}_{S}}\) is disregarded in most practical applications. This study thoroughly investigates the causal relationship existing in soils between damping ratio spectra and dispersion functions by exploiting a recently obtained, exact solution of the Kramers-Kronig equations. A number of cases associated to realistic damping ratio spectra for geomaterials have been analyzed, from which the corresponding dispersion functions have been rigorously calculated.