On the implications of the principle of physical causality in the propagation of seismic waves in geomaterials

摘要

The objective of this paper is to illustrate some important implications of the principle of physical causality in the propagation of seismic waves in geomaterials at low-strain levels. Under these conditions the simplest constitutive model able to capture the capacity exhibited by geomaterials, subjected to small amplitude dynamic excitations, to absorb and dissipate strain energy is linear viscoelasticity. An important result implied by this theory of material behaviour is that the phase velocities of P and S waves and material damping ratio are not independent quantities but they are related by the Kramers-Kronig dispersion equations, which are nothing but a statement of the necessary and sufficient conditions required of a viscoelastic continuum so that a pulse propagating through it satisfies the principle of physical causality Approximate and recently obtained exact solutions of the Kramers-Kronig equations, which from a mathematical point of view are a pair of linear, singular integral equations with Cauchy kernel, are illustrated. The exact solutions were derived with no simplifying assumptions beyond the so-called fading memory hypothesis, a rather weak conjecture well fulfilled by geomaterials which states that the current stress tensor depends more strongly on the recent rather than on the distant strain history. These rigorous solutions of the Kramers-Kronig relations are attractive as they allow, at least in principle, the calculation of frequency-dependent damping ratio from phase velocity dispersion and, inversely, frequency-dependent phase velocity of P and S waves from the spectrum of damping ratio. Thus these solutions conjure up a new approach in determining the small-strain dynamic properties of geomaterials with measurements only of one material function. The theoretical results were validated using experimental data from non-resonant column tests carried out on fine-grained soils,. Furthermore, they were shown to be consistent with previously derived approximate solutions of the Kramers-Kronig relations and in particular with the well-known dispersion relation widely used in seismology based on the assumption that material damping ratio is rate-independent over the seismic frequency band.