Transient response of a cylindrical cavity in three-dimensional elastic space under impact loading: an analytical solution

The transient dynamic response of a cylindrical cavity in an elastic medium under impact loading is a fundamental problem in geotechnical engineering, with direct applications to tunnel and pipeline safety. While most existing analytical studies adopt two-dimensional plane-strain assumptions or steady-state harmonic incidence, the present work establishes a genuinely three-dimensional analytical model for a cylindrical cavity subjected to a triangular impulse load. Using the wave function expansion method and Laplace transform, we derive general frequency-domain solutions for stresses and displacements in cylindrical coordinates. Time-domain solutions are obtained via Durbin’s numerical inverse Laplace transform and validated against pendulum impact test data. The results show good agreement in first-wave trends and peak values, with peak errors within 20%. Parametric analysis reveals that doubling the cavity diameter reduces the peak longitudinal vibration velocity by 13.4% and the peak longitudinal stress by 3.2%, indicating that velocity is more sensitive to aperture changes. Halving the shear modulus increases the peak longitudinal velocity by 25.5% and decreases the peak longitudinal stress by 23.1%. The distributions of vibration velocity and stress around the cavity edge at different heights (z-axis positions) are presented: radial velocity peaks at 0° and 180° and increases with height; hoop stress exhibits four peak values at 30°, 150°, 210° and 330°, first increasing then decreasing with height. This three-dimensional transient solution provides a theoretical foundation for the dynamic design of underground cavities and highlights limitations of 2D approximations.