Аннотації
25.12.2017
Метод граничних інтегральних рівнянь застосовується для розв’язання задачі про тривимірні гармонічні коливання поропружного масивного тіла. Наведені основні розрахункові співвідношення та проаналізовано склад матриці фундаментальних розв’язків. Аналітично та чисельно розв’язана тестова задача про змушені коливання поропружного шару.
Метод граничных интегральных уравнений применяется для решения задачи про трехмерные гармонические колебания пороупругого массивного тела. Приведены основные расчетные соотношения и проанализировано состав матрицы фундаментальных решений. Аналитически та численно решена тестовая задача про вынужденные колебания пороупругого слоя.
In this paper vibration of fluid-saturated porous solids under the equal distribution load is studied using two different approaches. One of them is analytical way. Biot’s equations in terms of displacement, pore pressure, porosity and effective densities are used for one-dimensional column. Using boundary conditions analytical expressions for parameters of stress-strain state: the solid and fluid displacements and stresses are obtained. Another way is Boundary Integral Equation Method. Equilibrium equations for 3-D linear dynamic poroelasticity are presented. Also required components of fundamental solution tensors as weighting displacement fields are obtained and analyzed with the help of the analogy between poroelastisity and thermoelastisity. The solution of the porous solid vibration problem for two types of boundary conditions is presented in the figures. Graphs present the comparison of the normalized solid displacement u3 at the top and normalized pressure σ33 in elastic region and in porous solid of poroelastic region depending on frequency ω that are computed using Boundary Integral Equation and analytical methods. Figures show that graphs of the displacements and pressure in poroelastic and elastic region have the same character but different values. The numerical solution of this problem was calculated using material properties which are corresponding to the Barea Sandstone. It shows that massive porous bodies cannot be modeling as homogeneous elastic media but it is necessary to use two phase model and equations of poroelasticity. Since the agreement between the BEM results and the analytical solution is good so such an approach can be used for development and testing of numerical techniques for analyzing of 3-D porous solids vibration.
1. Kovtun Al.A. Ob uravneniah modeli Bio i ih modifikatsiyah (On the equations of Biot’s theory and their modifications / Al.A.Rovtun // Voprosy geofiziki (Problems of geophysics). – 2011. – V.44. – С. 3-24.2. Igumnov L.A. Chislenno-analiticheskoe modelirovanie dinamiki tryehmernyh sostavnyh porouprugih tel (Number-analytical modeling dynamic three-dimensional composite poroelastic bodies). [electronic study-methodical manual] / L.A.Igumnov, C.Y.Litvinchuk, A.V.Amenickiy., A.A.Belov. – Nighniy Novgorod: University of Nighniy Novgorod, 2012. – 52 p.3. Li P. Boundary element method for wave propagation in partially saturated poroelastic continua / P.Li. - Verlag der Technischen Universität Graz, 2012. – 143 p.4. Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-Frequency Range / M.A.Biot // J. Acoust. Soc. Amer. – 1956. – V. 28, №2. – P.168-178.5. Dominguez J. Boundary elements in dynamics / J.Dominguez. – Computational Mechanics Publications. Southampton Boston, 1993. – 689p.6. Detornay E. Fundamentals of poroelasticity. Chapter 5 in Comprehensive Rock Engineering: Principles, Practice and Projects / E Detornay, A.H.-D.Cheng // Analysis and Design Method, ed. C. Fairhurst, Pergamon Press. – 1993. – V.II. – P. 113-171.7. Tryehmernye zadachi matematicheskoy teorii uprugosti I termouprugosti (Three-dimensional problems of mathematical theory of elasticity and thermoelasticity) [monograph] / [V.D.Kupradze, T.G.Hehelia, M.O.Basheleyshvili, G.V.Burchuladze] / Edited by V.D.Kupradze.; M: Mir, 1976. – 664p.8. Nowacki W. Dinamicheskiye zadachi termouprugosti (Dynamic problems of thermoelasticity) / Edited by G.S. Shapiro – M: Mir, 1970. – 256 p.