Mathematical modeling of stress waves under concentrated vertical action in the form of a triangular pulse: Lamb’s problem

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Abstract


The aim of the work. The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. Methods. To solve the non-stationary dynamic problem of elasticity theory with initial and boundary conditions, the finite element method in displacements was used. Using the finite element method in displacements, a linear problem with initial and boundary conditions was led to a linear Cauchy problem. A quasiregular approach to solving a system of second-order linear ordinary differential equations in displacements with initial conditions and to approximating the area under study is proposed. The method is based on the schemes: 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 end elements with two nodal points. The Fortran-90 algorithmic language was used in the development of the software package. Results. Some information is given about numerical modeling of elastic stress waves in an elastic half-plane with a concentrated wave action in the form of a Delta function. The estimated area under study has 12 008 001 nodal points. A system of equations consisting of 48 032 004 unknowns is solved. The change of elastic contour stress on the free surface of the half-plane at different points is shown. The amplitude of Rayleigh surface waves is significantly greater than the amplitudes of longitudinal, transverse, and other waves with a concentrated vertical action 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.


About the authors

Vyacheslav K. Musayev

Russian University of Transport; Moscow State University of Civil Engineering (National Research University); Mingachevir State University

Author for correspondence.
Email: musayev-vk@yandex.ru
ORCID iD: 0000-0003-4336-6785
9 Obraztsova St, bldg 9, Moscow, 127994, Russian Federation; 26 Yaroslavskoye Shosse, Moscow, 129337, Russian Federation; 21 Dilyara Alieva St, Mingachevir, AZ4500, Republic of Azerbaijan

Professor of the Department of Technosphere Safety of the RUT (MIIT), Professor of the Department of Integrated Safety in Construction of the NRU MGSU, Professor of the Department of Higher Mathematics of MSU (Azerbaijan), Doctor of Technical Sciences

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