The purpose of this paper is to incorporate the dynamic BB model of rock fracture normal behaviour into Universal Distinct Element Code (UDEC), and to conduct UDEC modelling to predict stress wave propagation across rock fractures. With the incorporation of new dynamic model of fractures, the UDEC is improved in the capability of modelling wave propagation across fractures. In this study, the case of normally incident P-wave transmission across a fracture is modelled, and the effect of dynamic normal behaviour of fracture is concentrated on. Wave transmission coefficient for wave transmission across the fracture with the dynamic BB model is obtained for various combinations of fracture dynamic parameters, as well as different wave amplitudes and frequencies.


Stress waves propagating in fractured rock masses are attenuated (and slowed) due to the reflection and transmission at rock fractures. To account for the effects of fractures on wave propagation, displacement discontinuity theories (or termed as non-welded or slip interface theories by some researchers) have been developed by treating mathematically the fracture constitutive relation as displacement discontinuity boundary conditions in the wave equation. In these theories, fractures are assumed to be large in extent and small in thickness relative to wavelength. Based on the displacement discontinuity theories, extensive studies on wave propagation across fractures have been conducted from various considerations of the fracture deformational models. By assuming the dynamic linear elastic model of dry fractures or linear rheologic model of wet fractures, analytical solutions of reflection and transmission coefficients for arbitrarily incident wave propagation across fractures have been derived and experimentally verified by Schoenberg, 1980 and 1983; Kitsunezaki, 1983; Myer et al., 1985 and 1997; King et al., 1986; Pyrak- Nolte et al., 1990a and 1990b; Rossmanith et al. 1993; Gu et al., 1996; Daehnke and Rossmanith, 1997; Nakagawa et al., 2000; Cai and Zhao, 2000). The assumption of linearity of fracture deformational model is reasonable if the magnitude of the stress waves is insufficient to mobilize nonlinear normal deformation and fictional slip of the fractures. Such a situation is typical in engineering practices such as ultrasonic wave non-destructive evaluation of engineering materials and in most seismic investigations in fractured rock masses.

If the amplitude of the stress wave is relatively large, such as blasting waves initiated by blasting, the nonlinear deformational behaviour of fractures may be excited within a certain distance from the blasting source. By considering the nonlinear (hyperbolic) model of fracture normal behaviour, Zhao and Cai (2001) derived a theoretical solution of transmission coefficient for normally incident P-waves to transmit across a fracture. In their study, the nonlinear constitutive model of fractures was assumed to be the quasi-static BB model, namely, the loading rate independent model. However, the fracture deformation tests conducted by Cai (2001) revealed that this assumption is valid only for the cases of:

  • the loading rate induced by incident waves is low and hence the effect of loading rate can be ignored.

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