Evolution of the radiated seismic wave energy of underground explosion in visco-elastic solids

The energy coupling between the underground explosion and the medium and the wave propagation mechanism in the medium are important bases for understanding the physics of the underground explosion source. In order to study the law of the propagation and attenuation for the seismic wave energy of underground explosion, the composition of the radiated energy of underground explosion in viscoelastic medium was analyzed, and the formulas for calculating inflow energy, outflow energy, radiated kinetic energy, potential energy and dissipation energy in a limited observation region were given. Based on the theory of viscoelastic spherical wave in infinite medium, the theoretical solutions of velocity, displacement, stress and strain in the Laplace domain were given by using the exponential attenuation pressure model and the generalized Maxwell viscoelastic model. The numerical solutions of velocity, displacement, stress and strain were given by using the Laplace numerical inverse method, and the inflow energy, outflow energy, radiated kinetic energy, potential energy and dissipation energy were calculated by these numerical results. The numerical results of different components of seismic wave energy are consistent with the theoretical results, and the correctness of this method is proved. Using the dry loess as typical viscoelastic material, the radial stress and particle velocity at different radii were calculated, and the relationship between the inflow energy at different radii and the wave propagation distance was obtained. The spatial distributions of the radiated kinetic energy of seismic wave were calculated by using the spatial distributions of the radial particle velocity at different times, and the propagation law of the radiated kinetic energy was obtained. The changes of the inflow energy and the radiated kinetic energy with the propagation distance in the limited observation area were analyzed, and the results show as follows: (1) In a viscoelastic medium, the energy flowing into a sphere surface decreases gradually with the increase of radius. In an ideal elastic medium, the energy flowing into a sphere surface at the elastic radius of about several times can be stabilized to a constant value. (2) The potential energy and the dissipative energy tend to constant values when the observation time is long enough in a fixed limited observation region, and the radiated kinetic energy tends to zero. (3) When a limited observation area can hold the seismic waves with complete wave length, the steady-state value of the radiated kinetic energy of seismic waves decreases with the increase of the wave propagation distance. In general, exponential function and power function can be used for piecewise fitting of the attenuation law for the steady-state value of the radiated kinetic energy of the seismic wave.