MATHEMATICAL MODELING OF NON-STATIONARY PROCESSES IN SEISMIC ACTIVITY ZONE

The processes of rock transition from undisturbed to dynamic fracture before earthquakes, under conditions of limited source zone volume and compression at great depths, are slow. Such a slowdown can be used for predictive purposes. According to natural waveguides, which are fault zones, these processes in the depths of the lithosphere cause variations in geophysical fields on the Earth's surface, in the atmosphere and ionosphere. In turn, these variations can already be seen by modern ground and satellite observations.
This article presents the results of mathematical and computer modeling of dynamic and quasi-static processes in earthquake focal zones. The motion of an elastic medium in the event of a sudden rupture along a finite strip under longitudinal shear conditions, taking into account contact viscous friction, is investigated. The use of an exact solution to this problem, constructed by the superposition method, is convenient for the first arrivals of reflected waves and is difficult with multiple reflections. In this regard, a different approach has been applied in this paper, which consists in reducing the boundary value problem to the Fredholm integral equation of the 2nd kind in images, the solution of which, under certain conditions, allows us to obtain the parameters of the motion of the medium at an arbitrary time. These conditions are sufficiently large values of the effective viscosity at break when a quasi-static non-stationary process is realized. Applying the method of solving integral equations in images, which uses the Hilbert-Schmidt formula for a resolvent kernel and decomposition of a normalized symmetric kernel into a bilinear series by eigenfunctions, a solution of a quasi-static integral equation in images is obtained. Further, by inverting the solution in the images, the original solution is obtained.
From the conditions of proximity of the approximate (quasi-static) solution and the general (dynamic) solution of the corresponding integral equation, an estimate of the lower bound of the effective viscosity at break is obtained for the transition from solving a dynamic non-stationary problem to solving a quasi-static non-stationary problem. The correspondence of the obtained condition of quasi-static processes to the available experimental laboratory and geophysical data is investigated.
With the help of analytical methods, together with numerical calculation, graphs of quasi-static nonstationary displacements of the rupture shores and the stress intensity coefficient at its ends, depending on background stresses, effective viscosity at the rupture, coordinates and time are obtained. A general solution of the quasi-static problem for displacements and displacement velocities of the rupture shores in the form of a static surface in a normalized coordinate system is obtained. A general solution of the quasi-static problem for the intensity coefficient of tangential stresses at the ends of the rupture in normalized coordinates is obtained.