When a wave propagates the wave is greatly attenuated due to the presence of fractures. This paper presents a study on S-wave attenuation across single fracture with the Coulomb slip behaviour. In the theoretical formulation, the method of characteristics combined with the Coulomb slip model is used to develop a set of recurrence equations with respect to particle velocities and shear stress. These equations are then numerically solved. To compare with the theoretical study, numerical modelling using Universal Distinct Element Code (UDEC) is conducted. A general agreement between UDEC modelling and theoretical analyses is achieved. The magnitude of transmission coefficient is calculated as a function of shear stress ratio and normalized shear stiffness. The study shows that the shear stress ratio is the important factor influencing wave transmission. The Influence of normalized shear stiffness becomes obvious, when shear stress ratio is small.
Multiple parallel fractures exist commonly in fractured rock masses and can be grouped as joint sets. Fractures have displacements (opening, closure and slip) under normal and shear stresses. Therefore, rock fractures cannot be treated as welded interfaces, but should be referred as displacement discontinuities in the rock masses. Thus, the stresses across the interface are continuous, but displacements across the interface are discontinuous. The discontinuity in displacement is equal to the average applied stress divided by the specific stiffness of the interface (k, and k, are fracture specific stiffness in the normal and shear directions, respectively).
The effects of single fracture on wave propagation have been extensively studied with full consideration of different fracture deformational behaviours (e.g. Miller 1977, 1978, Schoenberg 1980, Pyrak-Nolte et al, 1990a, Zhao & Cai 2001). By comparison, the effects of multiple parallel fractures on wave propagation become complicated due to multiple reflections and transmissions occurring between fractures. A simplified method was proposed by ignoring the multiple reflections as a short-wavelength approximation (e.g. Pyrak-Nolte et al. 1990b). Cai & Zhao (2000) used the method of characteristics to study wave attenuation across linearly deformational fractures, where the multiple reflections have been taken into account. They found that the magnitude of transmission coefficient across parallel fractures depends not only on fracture specific stiffness and wave frequency, but also on fracture spacing and number of fractures.
When incident wave amplitude is small, the magnitude of stress is too small to mobilize nonlinear deformation of the fractures, so linearly deformational behaviour is adopted in these studies. However, it has been found that the complete deformational (either normal or shear) behaviours of rock fractures are generally nonlinear (e.g. Bandis et al. 1983). In practice, there is a requirement for considering the nonlinearly deformational behaviour of the fractures in solving the problems of large amplitude wave propagation, i.e. blasting wave propagation from the sources of explosion. The static BB model was used by Zhao & Cai (2001) to analyze large amplitude P-wave attenuation across single fracture. In addition, studies on S-wave attenuation across single fracture with slip rate-dependent frictional behaviour have been conducted by many researchers.