Pendulum waves on the surface of block rock mass under dynamic impact

Under numerical investigation is propagation of surface pendulum waves in 3D block medium. The medium is modeled by 3D lattice of masses connected with elastic springs and viscous dampers. The surface vertical pulsed concentrated loading is considered. The displacements and velocities of the surface masses are calculated. The numerical results obtained for the block medium are compared with the similar data on elastic medium and in situ experiments carried out by other researchers.

💡 Research Summary

The paper investigates the propagation of surface pendulum waves in a three‑dimensional (3‑D) block‑structured rock mass. The authors model the medium as a regular 3‑D lattice of point masses connected by elastic springs and viscous dampers, representing the rigid blocks and the compliant interlayers, respectively. The interlayer behavior follows a Kelvin‑Voigt viscoelastic law, with identical stiffness (k₁ = k₂) and damping (λ₁ = λ₂) in axial and diagonal directions, which simplifies the governing equations while retaining essential anisotropic effects.

The dynamic problem considered is the classic Lamb’s problem: a vertical, time‑limited pulsed load applied at the surface (a half‑sinusoid of duration t₀). The equations of motion for interior masses (Eq. 1) and surface masses (Eq. 2) are derived using Newton’s second law and the force contributions from neighboring springs and dampers. Zero initial conditions are assumed. The authors discretize the time derivatives with a second‑order central difference scheme and integrate the system explicitly. Stability requires the time step τ to satisfy τ ≤ √(M/k)·l, where M is the mass of a lattice node and l the lattice spacing.

Dispersion analysis performed in a prior work (reference