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")


[1] Struzhanov V.V., Sagdullaeva D.A. On the solution of boundary value problems of the theory of elasticity by orthogonal projections method. Matematicheskoe modelirovanie, 2004, no. 12, pp. 89--100 (in Russ.).

[2] Fukalov A.A. [Problems on the elastic equilibrium of composite thick-walled transversally isotropic spheres under mass forces and internal pressure effect and their applications]. XI Vseros. s”ezd po fundamental’nym problemam teoreticheskoy i prikladnoy mekhaniki [XI Russ. Cong. on Fundamental Problems of Theoretical and Applied Mechanics]. Kazan, KFU Publ., 2015, pp. 3951--3953 (in Russ.).

[3] Agakhanov E.K., Magomedeminov N.S. Equivalence impact conditions for displacements. Vestnik DGTU. Tekhnicheskie nauki, 2006, no. 12, pp. 27--28 (in Russ.).

[4] Vestyak V.A., Tarlakovskii D.V. Unsteady axisymmetric deformation of an elastic thick-walled sphere under the action of volume forces. J. Appl. Mech. Tech. Phy., 2015, vol. 56, no. 6, pp. 984--994. DOI: https://doi.org/10.1134/S0021894415060085

[5] Pozharskiy D.A. Contact problem for transversely isotropic half-space with unknown contact zone. Doklady AN, 2014, vol. 455, no. 2, pp. 158--161 (in Russ.).

[6] Privalikhin R.S. Tension in contact zone of two cylindrical final length bodies. Izvestiya Samarskogo nauchnogo tsentra Rossiyskoy akademii nauk [Izvestia RAS SamSC], 2011, vol. 13, no. 1-3, pp. 599--603 (in Russ.).

[7] Algazin O.D., Kopaev A.V. Solution to the mixed boundary-value problem for Laplace equation in multidimensional infinite layer. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2015, no. 1 (58), pp. 3--13 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2015-1-3-13

[8] Sobol’ B.V. On the asymptotic solutions of static problems of three-dimensional elasticity with mixed boundary conditions. Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo [Vestnik of Lobachevsky University of Nizhny Novgorod], 2011, no. 4-4, pp. 1778--1780 (in Russ.).

[9] Stankevich I.V. Numerical solution of mixed problems of the theory of elasticity with one-sided constraints. Matematika i matematicheskoe modelirovanie, 2017, no. 6, pp. 40--53 (in Russ.). DOI: https://doi.org/10.24108/mathm.0517.0000078

[10] Bozhkova L.V., Ryabov V.G., Noritsina G.I. The Hertzian problem for the ring layer of the optional thickness subject to friction forces in the contact area. Izvestiya MGTU MAMI, 2011, no. 1, pp. 217--221 (in Russ.).

[11] Ivanychev D.A., Levin M.Yu., Levina E.Yu. The boundary state method in solving the anisotropic elasticity theory problems for a multi-connected flat region. TEST Engineering & Management, 2019, vol. 81, no. 11-12, pp. 4421--4426.

[12] Ivanychev D.A., Levina E.Yu., Abdullakh L.S., et al. The method of boundary states in problems of torsion of anisotropic cylinders of finite length. ITJEMAST, 2019, vol. 10, no. 2, pp. 183--191. DOI: https://doi.org/10.14456/ITJEMAST.2019.18

[13] Ivanychev D.A., Levina E.Y. Solution of thermo elasticity problems for solids of revolution with transversal isotropic feature and a body force. J. Phys.: Conf. Ser., 2019, vol. 1348, art. 012058. DOI: https://doi.org/10.1088/1742-6596/1348/1/012058

[14] Pen’kov V.B., Pen’kov V.V. Boundary conditions method for solving linear mechanics problems. Dal’nevostochnyy matematicheskiy zhurnal [Far Eastern Mathematical Journal], 2001, vol. 2, no. 2, pp. 115--137 (in Russ.).

[15] Lekhnitskiy S.G. Teoriya uprugosti anizotropnogo tela [Elasticity theory of anisotropic body]. Moscow, Nauka Publ., 1977.

[16] Satalkina L.V. [Basis expansion of state space with severe limitations on computation energy consumption]. Sb. tez. dokl. nauch. konf. studentov i aspirantov LGTU [Proc. Sc. Conf. of Student and Post-Graduates of LGTU]. Lipetsk, LGTU Publ., 2007, pp. 130--131 (in Russ.).

[17] Aleksandrov A.Ya., Solov’yev Yu.I. Prostranstvennye zadachi teorii uprugosti [Spatial problems of elasticity theory]. Moscow, Nauka Publ., 1978.

[18] Ivanychev D.A. The boundary states for axisymmetric problems for anisotropic bodies. Vesti vysshikh uchebnykh zavedeniy Chernozem’ya [News of Higher Educational Institutions of the Chernozem Region], 2014, no. 1, pp. 19--26 (in Russ.).

[19] Levina L.V., Kuz’menko N.V. [Inverse method of elastic body state effective analysis from mass forces from the class of continuous]. XI Vseros. s”ezd po fundamental’nym problemam teoreticheskoy i prikladnoy mekhaniki [XI Russ. Cong. on Fundamental Problems of Theoretical and Applied Mechanics]. Kazan, KFU Publ., 2015, pp. 2278--2280 (in Russ.).

[20] Levina L.V., Novikova O.S., Pen’kov V.B. Full-parameter solution of theory of elasticity problem for simply connected bounded body. Vestnik LGTU [Vestnik LSTU], 2016, no. 2, pp. 16--24 (in Russ.).

[21] Ivanychev D.A. The method of boundary states in the solution of the second fundamental problem of the theory of anisotropic elasticity with mass forces. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State University Journal of Mathematics and Mechanics], 2019, no. 61, pp. 45--60 (in Russ.). DOI: https://doi.org/10.17223/19988621/61/5

[22] Ivanychev D.A. Solving the mixed type axisymmetric problems for anisotropic bodies with mass forces. Trudy MAI, 2019, no. 105 (in Russ.). Available at: http://trudymai.ru/published.php?ID=104014

[23] Ivanychev D.A. The contact problem solution of the elasticity theory for anisotropic rotation bodies with mass forces. Vestnik PNIPU. Mekhanika [PNRPU Mechanics Bulletin], 2019, no. 2, pp. 49--62 (in Russ.). DOI: https://doi.org/10.15593/perm.mech/2019.2.05