When stress wave propagates through fractured rock masses, the wave will be slowed and attenuated due to the presence of these fractures. If the amplitude of stress wave is large enough to mobilize nonlinear deformation of these fractures, they will present different effects on one-dimensional P-wave propagation compared with those of linear deformational fractures. The static BB model is adopted to describe the nonlinear deformational behavior. Method of characteristics combined with displacement discontinuity theory is used to build analytical solutions of wave particle velocities and wave stresses. These equations are numerically solved with sufficient accuracy.


Fractured rock masses consist of rock material and fractures. The presence of fractures makes the rock masses a discontinuous and inhomogeneous medium. When stress waves propagate through fractured rock masses, the waves will be slowed and attenuated. Usually, the effects of fractures on wave propagation have been modeled by equivalent medium theories (Schoenberg & Muir 1989). Effective elastic moduli for the fractured rock masses are calculated by some developed expressions (based on different mechanisms) and are directly related to velocity and attenuation through the elastic-dynamic equation. These theories treat problems of wave propagation from the viewpoint of entirety. This assumption inherently results in two limitations: One is that the method may lose the discreteness of wave amplitude attenuation at individual fracture; the other is the loss of fracture's intrinsic frequency-dependent property. Recently, displacement discontinuity theories are frequently used to simulate fracture deformational properties. In these theories, fractures are physically treated as displacement discontinuities, but stress continuous boundaries. They can be treated as an extension of traditional boundary and can be combined with elastic-dynamic equation directly. Displacement discontinuity theories are usually utilized in the study of fracture whose thickness is much smaller than the wavelength. Effects of single fracture on small-amplitude wave propagation have been widely studied. The complicated multiple reflections on wave propagation through multiple parallel fractures have been experimentally observed and addressed by Pyrak-Nolte et al. (1990b) and Myer et al (1995). It is often difficult to explicitly determine the complex process of the superposition of these multiple reflected and transmitted waves. A simplified method was adopted by ignoring the multiple reflections as a short-wavelength approximation. Thus, the wave transmitted coefficient through multiple parallel fractures can be calculated by the product of transmission coefficients of individual fractures and this method is valid only when fractures have a large spacing relative to the wavelength. Cai and Zhao (2000) used the method of characteristics to study the effects of multiple parallel fractures with linear deformational behavior on one-dimensional wave propagation by considering the explicit multiple reflections between fractures. Two important indices of ξ (threshold value ξ thr and critical value ξ cri) are found, where ξ is the ratio of fracture spacing to wavelength and it is named as non-dimensional fracture spacing. These two important ξ divide the area of nondimensional fracture spacing into three parts: individual fracture area, transition area and small-spacing area.

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