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The Solution of Boundary Value Problems of Various Types with Consideration of Volume Forces for Anisotropic Bodies of Revolution

Авторы: Ivanychev D.A., Levina E.Yu. Опубликовано: 26.08.2021
Опубликовано в выпуске: #4(97)/2021  
DOI: 10.18698/1812-3368-2021-4-57-70

 
Раздел: Математика | Рубрика: Математическая физика  
Ключевые слова: the method of boundary states, boundary value problems of mechanics, anisotropy, bulk forces, state space, elastic equilibrium

In this work, we studied the axisymmetric elastic equilibrium of transversely isotropic bodies of revolution, which are simultaneously under the influence of surface and volume forces. The construction of the stress-strain state is carried out by means of the boundary state method. The method is based on the concepts of internal and boundary states conjugated by an isomorphism. The bases of state spaces are formed, orthonormalized, and the desired state is expanded in a series of elements of the orthonormal basis. The Fourier coefficients, which are quadratures, are calculated. In this work, we propose a method for forming bases of spaces of internal and boundary states, assigning a scalar product and forming a system of equations that allows one to determine the elastic state of anisotropic bodies. The peculiarity of the solution is that the obtained stresses simultaneously satisfy the conditions both on the boundary of the body and inside the region (volume forces), and they are not a simple superposition of elastic fields. Methods are presented for solving the first and second main problems of mechanics, the contact problem without friction and the main mixed problem of the elasticity theory for transversely isotropic finite solids of revolution that are simultaneously under the influence of volume forces. The given forces are distributed axisymmetrically with respect to the geometric axis of rotation. The solution of the first main problem for a non-canonical body of revolution is given, an analysis of accuracy is carried out and a graphic illustration of the result is given

The study was carried out with the financial support of the Russian Foundation for Basic Research and the Lipetsk Region in the framework of the research (KAIK project no. 19-41-480003 "p_a")

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