Mathematical models for numerical solution of nonstationary problems of geomechanics

Abstract

The study of the problems of wave propagation in a layered half-space with a cavity lying on an elastic foundation is necessary to create models, calculation methods and substantiate hazard assessments in such various areas as the propagation of seismic waves in ground and underground media during earthquakes, during underground explosions, and during vibration. Recently, in seismic and geomechanics, approaches have been widely used to describe the deformation of a layered half-space under the influence of external forces. Thus, the study of wave propagation in a layered half-space with a cavity lying on an elastic foundation is of great importance for seismology, seismic exploration, and other applications. In particular, a model of the earth’s crust is widely applicable in seismology, representing it as a pack of layers, limited, on the one hand, by a free surface and lying on an elastic half-space, the model of wave propagation in a layered half-space with a cavity lying on an elastic foundation is a special case of the model of the earth bark. On the basis of a theoretical solution to the problems of wave propagation in a layered half-space with a cavity lying on an elastic foundation, the interpretation of field seismograms obtained during earthquakes and explosions is carried out. The result of interpretation is data on the structure of the object under study and the processes occurring in it. The aim of our work is to develop models that describe the dynamic behavior of inhomogeneous media and structures, methods for solving non-stationary problems in rigid body mechanics and analysis of the results obtained to predict the behavior of disturbances under non-stationary action for specific systems and their application in applied problems of mechanics. The following research methods are used in the work: mathematical modeling of mechanical processes using an explicit finite-difference method for solving partial differential equations and a numerical solution method consisting in the use of integral transformations. In this paper, we consider the problem of wave propagation in a layered half-space with a cavity lying on an elastic foundation under the action of a dynamic load from the day surface above the cavity. To solve the set problem, we use the method of “discontinuity decay” by S.K. Godunov, which makes it possible to analyze the parameters of the process under study in the stress-strain state of the medium under initial and boundary conditions by known values, which ultimately makes it possible to determine the optimal parameters of the process under study [1], [2]. The obtained results of the solutions can be used in the design and assessment of aboveground and underground structures, as well as in the development of mineral deposits.