Mathematical Modeling of Non-Stationary Elastic Stress Waves (Transient Process) at Influence (Vertical Concentrated as a Triangular Pulse) on the Surface of a Half-Plane (Lamb’s Problem)

Abstract

Relevance. The problem of numerical modeling of longitudinal, transverse and surface waves on the free surface of an elastic half-plane under the influence of a vertical concentrated load in the form of a triangular pulse is considered. The obtained results of the problem under study are relevant, since they allow us to identify the types of waves on the surface of an elastic half-plane that are used in seismology and seismic stability of geo objects. The aim. To assess the load-bearing capacity of geo objects, you need information about the stress state. Therefore, obtaining information about the amplitudes and shapes of longitudinal, transverse, and Rayleigh waves in the lamb problem in the form of contour stresses is an urgent fundamental scientific task. Method. To solve a non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements is used. Using the finite element method in displacements, the linear problem with initial and boundary conditions was led to the linear Cauchy problem. A quasi-regular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the domain under study is proposed. The method is based on the following diagrams: point, line, and plane. The study area is divided by spatial variables into triangular and rectangular finite elements of the first order. According to the time variable, the study area is divided into linear finite elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is provided on numerical modeling of elastic stress waves in an elastic half-plane under concentrated wave action in the form of a triangular pulse (Delta-function). The estimated area under study has 12008001 nodal points. A system of equations consisting of 48032004 unknowns is solved. The change in the elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of surface Rayleigh waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves when a concentrated vertical action is performed in the form of a triangular pulse on the surface of an elastic half-plane. After surface Rayleigh waves, a dynamic process is observed in the form of standing waves.