Transmission of Normal P-Waves across a Single Joint Based ong‐λModel

This paper investigates the wave transmission and reflection of an elastic P-wave at a single joint for normal incidence. First, considering a coupled joint (correction parameterλ,0<λ<1), a normal deformation constitutive model of the joint (g‐λmodel) under static or quasi-static loading is introduced and then extended to dynamic loading. The nonlinearity of the joint stress-deformation curve increases with increasingλ. Second, the interaction between the P-wave and the joint is investigated by using the method of characteristics and the displacement discontinuity method to deduce the differential expression of the transmitted wave’s particle velocity. The approximate analytical expressions of the transmission and reflection coefficients are obtained according to the Lemaitre equivalent strain assumption. Third, parametric studies are conducted to evaluate the effects ofλon transmission characteristics for a normally incident P-wave at a single joint. The results show that the particle velocity of the transmitted wave depends onλ. Whenλtakes the limit values 0 and 1, the transmitted wave’s particle velocities are then consistent with the conclusions of the classical exponential model and the Barton–Bandis model. In addition, the transmission and reflection coefficients are discussed with respect toλand also to the ratio of the joint closure to the maximum allowable joint closure.

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abstract: Wave attenuation across fractured rock masses is a great concern of rock engineers to assess the safety of underground structures in and on rocks under dynamic loads. Due to the discreteness of rock masses, the universal distinct element code (UDEC) has been adopted for the study of rock mass problems. In this paper, the calibration work of UDEC modelling on P-wave propagation across single linearly and nonlinearly deformable fractures is conducted. Subsequently, numerical studies of P-wave propagation across multiple nonlinearly deformable fractures are carried out. The magnitude of transmission coefficient is calculated as a function of nondimensional fracture spacing for different numbers of fractures. The results reveals that under some circumstances, the magnitude of transmission coefficient not only increases with increasing number of fractures, but also is larger than 1.

title: Mechanics of Solid Materials
abstract: 1. Elements of the physical mechanisms of deformation and fracture 2. Elements of continuum mechanics and thermodynamics 3. Identification and theological classification of real solids 4. Linear elasticity, thermoelasticity and viscoelasticity 5. Plasticity 6. Viscoplasticity 7. Damage mechanics 8. Crack mechanics.

title: Seismic response of a single and a set of filled joints of viscoelastic deformational behaviour
abstract: Rock joints are often filled with weak medium, for example, saturated clay or sand, of viscoelastic nature. Their effects on wave propagation can be modelled as displacement and stress discontinuity conditions. The viscoelastic behaviour of the filled joint can be described by either the Kelvin or the Maxwell models. The analytical solutions for wave propagation across a single joint are derived in this paper by accounting for the incident angle, the non-dimensional joint stiffness, the non-dimensional joint viscosity and the acoustic impedance ratio of the filled joint. It is shown that the viscoelastic behaviour results in dissipation of wave energy and frequency dependence of the reflection and transmission coefficients. Based on curve fitting of the experimental data of P-wave propagation across a single joint filled with saturated sand, both the Kelvin and Maxwell models are found to reproduce the behaviour of the filled joint, in terms of the amplitude and frequency contents. Then, wave transmission across a filled joint set is studied with the virtual wave source method and the scattering matrix method, where multiple wave reflections among joints are taken into account. It is shown that the non-dimensional joint spacing and the number of joints have significant effects on the transmission coefficients.

title: Obliquely incident wave propagation across rock joints with virtual wave source method
abstract: Due to the presence of joints, waves are greatly attenuated when propagating across rock masses. Zhu et al. (2011) (Normally incident wave propagation across a joint set with virtual wave source method.  Geophys.73, 283–288.) studied normally incident wave propagation across a joint set with the virtual wave source method (VWSM). The introduced VWSM has merits in some aspects, especially the capability of separating differently arriving transmitted waves. However, normal wave incidence is only the special case for wave incidence with arbitrary incident angles. Obliquely incident wave propagation across a joint set is more complicated than normally incident wave propagation due to wave transformation at the joints. As a continuation of the previous paper, this work is extended to analytically study obliquely incident wave propagation across joints with VWSM. Complete theoretical reflection and transmission coefficients across single joint described by displacement discontinuity model are derived through plane wave analysis. The superposition of P wave and S wave is for the first time mathematically expressed and studied. The VWSM is verified through comparison with the propagation matrix method. Through extensive parametric studies on wave transmission across single and multiple parallel joints, it is shown that transmitted wave energy is mainly constrained in the transmitted wave of the same type as the incident wave. And with increasing joint stiffness, the transmission coefficients across single joint increases except those whose wave type is different from the incident wave. The amplitude of superposed transmitted wave for P wave incidence increases with incident angle, which is coincident with field observations. Both joint spacing and number of joints have significant effects on transmission coefficients. We find that when joint spacing is sufficiently large, the transmission coefficient is no longer a constant as the normally incident wave propagation case (Zhu et al., 2011). And when joints are very closely spaced, wave attenuation depends little on the number of joints, which is different from the conclusions from equivalent medium method.

title: Analytical Time-Domain Solution of Plane Wave Propagation Across a Viscoelastic Rock Joint
abstract: The effects of viscoelastic filled rock joints on wave propagation are of great significance in rock engineering. The solutions in time domain for plane longitudinal (P-) and transverse (S-) waves propagation across a viscoelastic rock joint are derived based on Maxwell and Kelvin models which are, respectively, applied to describe the viscoelastic deformational behaviour of the rock joint and incorporated into the displacement discontinuity model (DDM). The proposed solutions are verified by comparing with the previous studies on harmonic waves, which are simulated by sinusoidal incident P- and S-waves. Comparison between the predicted transmitted waves and the experimental data for P-wave propagation across a joint filled with clay is conducted. The Maxwell is found to be more appropriate to describe the filled joint. The parametric studies show that wave propagation is affected by many factors, such as the stiffness and the viscosity of joints, the incident angle and the duration of incident waves. Furthermore, the dependences of the transmission and reflection coefficients on the specific joint stiffness and viscosity are different for the joints with Maxwell and Kelvin behaviours. The alternation of the reflected and transmitted waveforms is discussed, and the application scope of this study is demonstrated by an illustration of the effects of the joint thickness. The solutions are also extended for multiple parallel joints with the virtual wave source method and the time-domain recursive method. For an incident wave with arbitrary waveform, it is convenient to adopt the present approach to directly calculate wave propagation across a viscoelastic rock joint without additional mathematical methods such as the Fourier and inverse Fourier transforms.

title: An equivalent viscoelastic model for rock mass with parallel joints
abstract: An equivalent viscoelastic medium model is proposed for rock mass with parallel joints. A concept of "virtual wave source (VWS)'' is proposed to take into account the wave reflections between the joints. The equivalent model can be effectively applied to analyze longitudinal wave propagation through discontinuous media with parallel joints. Parameters in the equivalent viscoelastic model are derived analytically based on longitudinal wave propagation across a single rock joint. The proposed model is then verified by applying identical incident waves to the discontinuous and equivalent viscoelastic media at one end to compare the output waves at the other end. When the wavelength of the incident wave is sufficiently long compared to the joint spacing, the effect of the VWS on wave propagation in rock mass is prominent. The results from the equivalent viscoelastic medium model are very similar to those determined from the displacement discontinuity method. Frequency dependence and joint spacing effect on the equivalent viscoelastic model and the VWS method are discussed.