Continuum model of wave propagation in fragmented media: Linear damping approximation

Abstract

Energy dissipation during wave propagation in fragmented geomaterials can be caused by independent movement of fragments leading to energy loss on their impact. By considering a pair of impacting fragments at times much greater than the period of their oscillations, we show that at a large timescale, the dynamics of the pair can be described by a linear viscous model with damping coefficients expressed through the restitution coefficient representing energy loss on impact. Wave propagation in fragmented geomaterials is also considered at the large timescale assuming that the wavelengths are much larger than the fragment sizes such that the attenuation associated with wave scattering on the fragment interfaces can be neglected. These assumptions lead to the Kelvin-Voigt model of damping during wave propagation, which allows the determination of a dispersion relationship. As the attenuation and dispersion are not related to the rate dependence of rock deformation, but rather to the interaction of fragments, the increased energy dispersion at low frequencies can be seen as an indication of the fragmented nature of the geomaterial and the capacity of the fragments for independent movement.