Kazakh Mathematical Journal. 2019; : 6-19
Тензор грина дозвуковой транспортной краевой задачи для упругого полупространства
Аннотация
Рассматривается первая краевая задача теории упругости для упругого полупространства при движении по его поверхности дозвуковых транспортных нагрузок. Предполагается, что скорость их движения меньше или больше скорости распространения упругих поверхностных волны Рэлея. На основе обобщенного преобразования Фурье построено фундаментальное решение задачи – тензор Грина, который описывает динамику массива при движении сосредоточенной силы вдоль его поверхности. Построено аналитическое решение краевой задачи для произвольных транспортных нагрузок, распределенных по поверхности полупространства, при дорелеевских и сверхрелеевских скоростях. Показано, что при превышении скорости волны Рэлея транспортные нагрузки генерируют поверхностные волны Рэлея. Задача является модельной для исследования напряженно-деформированного состояния породного массива в непосредственной близости от дорожных сооружений под действием транспортных нагрузок, движущихся с высокими скоростями.
Список литературы
1. Erzhanov Zh.S., Aytaliev Sh.M., Alexeyeva L.A. Dynamics of tunnels and underground pipelines, Almaty: Nauka, 1989.
2. Alekseyeva L.A. Fundamental solutions in an elastic space in the travelling load case applied mathematics and mechanics, Journal of Applied Mathematics and Mechanics, 55:5 (1991), 840-848.
3. Alexeyeva L.A., Kayshibaeva G.K. Transport solutions of Lame equations, Shock waves Computational mathematics and mathematical physics, 56:7 (2016), 1343-1354.
4. Alekseyeva L.A. Somigliana’s formulae for solving the elastodynamics equations for travelling loads, Applied mathematics and mechanics, 58:1 (1994), 109-116.
5. Alekseyeva L.A. Boundary value problems of elastodynamics under stationary moving forces using boundary integral equation method, Engineering Analysis with Boundary Element, 22:11 (1998), 327-331.
6. Alexeyeva L.A. Singular border integral equations of the BVP of elastodynamics in the case of subsonic running loads, Differential equations, 46:4 (2010), 512-519.
7. Alexeyeva L.A. Singular Boundary Integral Equations of Boundary Value Problems of the Elasticity Theory under Supersonic Transport Loads, Differential equations, 53:3 (2017), 317-332.
8. Alexeyeva L.A. Dynamics of elastic half-space by the action of running load, Applied mathematics and mechanics, 71:4 (2007), 561-569.
9. Cole J., Huth J. Stresses produced in a half plane by moving loads, Applied Mechanics, 25 (1958), 433-436.
10. Brehovskih L.M. Waves in multilayered media, Moscow: Nauka, 1973.
11. Nowazky V. Theory of elasticy, Moscow: Mir, 1975.
12. Kech V., Teodoresku P. Introduction to the theory of distributions with application in engeneering, Moscow: Mir, 1978.
Kazakh Mathematical Journal. 2019; : 6-19
Green’s tensor of subsonic transport boundary value problem for elastic half-space
Abstract
The first boundary value problem of the theory of elasticity for an elastic half-space at the movement on its surface of subsonic trans loads is considered. The speed of motion is less or more than the speed of distribution of elastic Rayleigh waves. On the basis of the generalized Fourier’s transformation the fundamental solution of the task is constructed which describes the dynamics of the massif at the movement of the concentrated force on and along its surface. Based on this, the analytical solution is constructed for arbitrary transport loads distributed over the surface, moving with the pre-Rayleigh and super-Rayleigh velocities. It is shown that when the Rayleigh wave velocity is exceeded, the transport loads generate surface Rayleigh waves. The task is a model for research of the stress-strain state of the massif in the vicinity of road constructions under the action of trans loads moving with high velocities.
References
1. Erzhanov Zh.S., Aytaliev Sh.M., Alexeyeva L.A. Dynamics of tunnels and underground pipelines, Almaty: Nauka, 1989.
2. Alekseyeva L.A. Fundamental solutions in an elastic space in the travelling load case applied mathematics and mechanics, Journal of Applied Mathematics and Mechanics, 55:5 (1991), 840-848.
3. Alexeyeva L.A., Kayshibaeva G.K. Transport solutions of Lame equations, Shock waves Computational mathematics and mathematical physics, 56:7 (2016), 1343-1354.
4. Alekseyeva L.A. Somigliana’s formulae for solving the elastodynamics equations for travelling loads, Applied mathematics and mechanics, 58:1 (1994), 109-116.
5. Alekseyeva L.A. Boundary value problems of elastodynamics under stationary moving forces using boundary integral equation method, Engineering Analysis with Boundary Element, 22:11 (1998), 327-331.
6. Alexeyeva L.A. Singular border integral equations of the BVP of elastodynamics in the case of subsonic running loads, Differential equations, 46:4 (2010), 512-519.
7. Alexeyeva L.A. Singular Boundary Integral Equations of Boundary Value Problems of the Elasticity Theory under Supersonic Transport Loads, Differential equations, 53:3 (2017), 317-332.
8. Alexeyeva L.A. Dynamics of elastic half-space by the action of running load, Applied mathematics and mechanics, 71:4 (2007), 561-569.
9. Cole J., Huth J. Stresses produced in a half plane by moving loads, Applied Mechanics, 25 (1958), 433-436.
10. Brehovskih L.M. Waves in multilayered media, Moscow: Nauka, 1973.
11. Nowazky V. Theory of elasticy, Moscow: Mir, 1975.
12. Kech V., Teodoresku P. Introduction to the theory of distributions with application in engeneering, Moscow: Mir, 1978.
