Research on pendulum-type and rotational waves in 2D discrete blocky rock masses with complex hierarchical structures

Rock masses consist of nesting rock blocks with various scales separated by weak structural layers, and their complex hierarchical structures play a significant role in dynamic deformation and stress wave propagation. Based on the Cosserat theory, a dynamic model of pendulum-type and rotational waves in blocks rock mass with complex hierarchical structures is established to determine the influence of hierarchical structures on dynamic deformation. Then, aiming to low-frequency and low-velocity characteristics of pendulum-type waves, dispersion equations of waves are determined and solved in different hierarchical structures based on the Bloch theorem, and furthermore, the dispersion relation and velocity characteristics of waves are investigated. Finally, mechanism of low-frequency characteristics of pendulum-type waves is revealed on the basis of solid energy band theory, and the possibility of pendulum-type and rotational waves inducing rock bursts is discussed based on the research results. It is indicated that ignoring higher-order hierarchical structures of rock masses may underestimate displacement and overall deformation of rock masses, resulting in unsafe numerical results. Under the action of long wave disturbance, for the first mode pendulum-type waves (the acoustic branches), the dispersion is not significant and propagation velocity decreases, and higher-order hierarchical structures inside rock masses hinder the wave propagation. However, the dispersion of other waves (the optical branches) is significant so that they hardly exist and propagate independently. The low-frequency pendulum-type waves are dominant, which have slower attenuation and longer propagation distance than the high-order mode waves and traditional P and S-waves.