Research on dispersion relation and dynamic properties of pendulum‑type waves in 1D blocky rock masses with complex hierarchical structures

Rock masses are characterized by the complex hierarchical structures involving various scale levels. The deformation of rock masses is primarily controlled in weak structural layers between rocks, whereas the rock block can be regarded as a non-deformable block and can move as a whole. In consequence, a new dynamic phenomenon, namely the pendulum-type wave, has emerged, which is a kind of nonlinear displacement wave caused by the overall movement of relatively intact large-scale rock blocks. Aiming at the complex hierarchical structures of rock masses and low-frequency characteristics of pendulum-type waves, the blocky rock masses composed of granite blocks and rubber interlayers are simplified into the block-spring model and wave motion model. Based on Bloch theorem and d’Alembert’s principle, the dispersion relation and equations of motion of 1D blocky rock masses are determined. Research shows that with the increase of the rock size and geomechanical invariant, the initial frequency of the first attenuation zone gradually decreases, and only the low-frequency waves lower than that frequency can propagate in blocky rock masses, which reveals the mechanism of low-frequency characteristics of pendulum-type waves theoretically. The equivalent substitution for the two models and their errors are given, and the results show that the equivalent substitution of the two models is not universal and unconditional. Finally, the influence of hierarchical structures on the dispersion relation and dynamic response is further studied. The larger the stiffness ratio, or the higher the order of hierarchical structures, the smaller is the effect of ignoring the high-order hierarchical structures.