Recent experimental studies focused on the evaluation of the seismic and static stiffnesses of rock joints have demonstrated that the seismic stiffness measured using seismic wave propagation is greater than the static stiffness determined by static stress-strain measurements. Although this difference has been quantified in laboratory experiments, the underlying physical mechanisms responsible for the difference are still poorly understood. In this study, the increase in the rock joint stiffness from static to seismic conditions is attributed to the frequency dependence of the joint stiffness. Three velocity discontinuity models of H/N (Kelvin), H-N (Maxwell) and H-H/N were used for interpretation of experimentally collected shear wave transmitted through a rock joint to quantify the stiffness-frequency dependence of the rock joint. It was found that the frequency-dependent stiffness is an inherent property of the rock joint with a velocity discontinuity. Velocity discontinuity models demonstrated a better agreement with the experimental observations for shear wave transmission across a joint than the displacement discontinuity model. This study shows that the H-H/N model has the advantage of quantifying the frequency dependence of joint stiffness with the potential to determine both seismic and static stiffnesses of rough joints by simple shear wave transmission experiments. The low-frequency stiffness can provide an approximate value of the joint static stiffness while the high-frequency stiffness provides the seismic stiffness.
A detailed classification of joint stiffness has been made by Zhao and Cai (2001), who suggested that static stiffness and dynamic stiffness, determined by static and dynamic stress-strain measurements, are ‘large-deformation’ stiffnesses (ε≥0.1%), and seismic stiffness, determined by ultrasonic wave propagation, is ‘small-strain' stiffness (ε≤0.0001%). Many experimental studies have found that the dynamic and seismic stiffnesses of a rock joint is typically larger than the static one, due to strain rate effects (Barton, 1988; Cui et al., 2017; Hencher, 1981; Kana et al., 1996). Cai (2001) demonstrated that the seismic shear stiffness of rock joints was approximately 2 to 3 times as high as the dynamic one, and the dynamic stiffness was 1.2 to 2 times larger than the static/quasi-static stiffness. A similar trend was observed by Pyrak-Nolte et al. (1990), who found that the value of joint seismic stiffness measured using seismic/ultrasonic shear wave propagation was 4 to 8 times larger than the static values determined by static stress-strain measurements. The difference between the static and seismic joint stiffnesses can be attributed to dynamical effects concerning alterable physical properties of joints at different loading rates (Stowe and Ainsworth, 1968). An alternate simple explanation for the difference in the stiffnesses can be the frequency-dependent joint stiffness (Pyrak-Nolte and Nolte, 1992), i.e., joint stiffness is smaller at low frequencies than at high frequencies. Pyrak-Nolte and Nolte (1992) demonstrated that the joint stiffness is frequency dependent as different frequencies sample different subsets of the joint geometry. Thus, the frequency dependence may be a simple consequence of joint geometry and is an inherent property of the rough rock joints.