Axisymmetric thermoelastic deformation of transversely isotropic rotation bodies
UDC
539.3DOI:
https://doi.org/10.31429/vestnik-19-2-17-28Abstract
The paper presents a technique for constructing elastic fields for anisotropic plates by means of the energy method of boundary states. Forces are set on the side surface of the plates, leading to the problems of bending and torsion. The developed theory for constructing bases for the spaces of internal and boundary states is based on a general approximate solution of the problem of plate bending. Relations are formed that determine the desired elastic state. The bases of the state spaces that form the basis of the method are formed according to the fundamental system of Weierstrass polynomials. An isomorphism of the spaces of internal and boundary states is proved, which makes it possible to establish a one-to-one correspondence between the elements of these spaces. The isomorphism of spaces makes it possible to reduce the search for an internal state to the study of a boundary state isomorphic to it. Mechanical characteristics are presented in the form of Fourier series.
The solution of the test first main problem of bending with torsion for a rectangular fiberglass plate with the corresponding conclusions, the problem of torsion for a plate of a non-trivial shape, and the problem with mixed boundary conditions for a rectangular plate, where both twisting and bending forces are set on one face, and the opposite face pinched. Explicit and indirect signs of convergence of problem solving and graphical visualization of the results are presented.
Keywords:
anisotropy, anisotropic plates, boundary state method, bending, torsion, equilibriumReferences
- Недорезов П.Ф. Численное исследование напряженно-деформированного состояния в задачах изгиба тонкой анизотропной прямоугольной пластинки. Изв. Сарат. ун-та. Сер. Математика. Механика. Информатика, 2009, т. 9, вып. 4, ч. 2, с. 142–148. [Nedorezov P.F. Numerical study of the stress-strain state in the problems of bending of a thin anisotropic rectangular plate. Izvestiya Saratovskogo universiteta. Seriya Matematika. Mekhanika. Informatika = Bulletin of the Saratov University. Series Mathematics. Mechanics. Informatics, 2009, vol. 9, iss. 4, pt. 2, pp. 142–148. (in Russian)]
- Ромакина О.М., Шевцова Ю.В. Метод сплайн-коллокации и его модификация в задачах статического изгиба тонкой ортотропной прямоугольной пластинки. Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика, 2010, т. 10, вып. 1, с. 78–82. [Romakina O.M., Shevtsova Yu.V. Spline-collocation method and its modification in problems of static bending of a thin orthotropic rectangular plate. Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika. Mekhanika. Informatika = Bulletin of the Saratov University. New series. Series Mathematics. Mechanics. Informatics, 2010, vol. 10, iss. 1, pp. 78–82. (in Russian)]
- Максименко В.Н., Подружин Е.Г. Фундаментальные решения в задачах изгиба анизотропных пластин. Прикладная механика и техническая физика, 2003, т. 44, № 4, с. 135–143. [Maksimenko V.N., Podruzhin E.G. Fundamental solutions in problems of bending of anisotropic plates. Prikladnaya mekhanika i tekhnicheskaya fizika = Applied Mechanics and Technical Physics, 2003, vol. 44, no. 4, pp. 135–143. (in Russian)]
- Рябчиков П.Е. Напряженно-деформированное состояние анизотропных пластин сложной формы при изгибе: дисс. ...канд. физ.-мат. наук. Новосибирск, 2007. [Ryabchikov P.E. Napryazhenno-deformirovannoe sostoyanie anizotropnykh plastin slozhnoy formy pri izgibe = Stress-strain state of complex-shaped anisotropic plates under bending. Diss. ... Cand. Phys.-Math. Sciences. Novosibirsk, 2007. (in Russian)]
- Космодамианский А.С. Напряженное состояние анизотропных сред с отверстиями и полостями. Издательское объединение "Вища школа", Киев–Донецк, 1976. [Kosmodamiansky A.S. Napryazhennoe sostoyanie anizotropnykh sred s otverstiyami i polostyami = Stress state of anisotropic media with holes and cavities. Publishing association "Vishcha school", Kyiv, Donetsk, 1976. (in Russian)]
- Подружин Е.Г. Приложение метода сингулярных интегральных уравнений к задачам изгиба анизотропных пластин с многосвязным контуром. Дисс. д-ра техн. наук. Новосибирск, 2007. [Podruzhin E.G. Prilozhenie metoda singulyarnykh integral'nykh uravneniy k zadacham izgiba anizotropnykh plastin s mnogosvyaznym konturom = Application of the method of singular integral equations to problems of bending of anisotropic plates with a multiply connected contour. Diss. Dr. Tech. Sciences. Novosibirsk, 2007. (in Russian)]
- Максименко В.Н., Подружин Е.Г. Изгиб конечных анизотропных пластин, содержащих гладкие отверстия и сквозные криволинейные разрезы. Сиб. журн. индустр. матем., 2006, т. 9, № 4, с. 125–135. [Maksimenko V.N., Podruzhin E.G. Bending of end anisotropic plates containing smooth holes and through curvilinear cuts. Sibirskiy zhurnal industrial'noy matematiki = Siberian Journal of Industrial Mathematics, 2006, vol. 9, no. 4, pp. 125–135. (in Russian)]
- Амбарцумян С.А. Теория анизотропных пластин. Наука, Москва, 1967. [Ambartsumyan S.A. Teoriya anizotropnykh plastin = Theory of anisotropic plates. Nauka, Moscow, 1967. (in Russian)]
- Лехницкий С.Г. Анизотропные пластинки. ГИТТЛ, Москва, 1957. [Lekhnitsky S.G. Anizotropnye plastinki = Anisotropic plates. GITTL, Moscow, 1957. (in Russian)]
- Лехницкий С.Г. О некоторых вопросах, связанных с теорией изгиба тонких плит. Прикладная математика и механика, 1938, т. II, вып. 2, с. 181–210. [Lekhnitsky S.G. On some questions related to the theory of bending of thin slabs. Prikladnaya matematika i mekhanika = Applied Mathematics and Mechanics, 1938, vol. II, iss. 2, pp. 181–210. (in Russian)]
- Пеньков В.Б., Пеньков В.В. Метод граничных состояний для решения задач линейной механики. Дальневосточный математический журнал, 2001, т. 2, № 2, с. 115–137. [Penkov V.B., Penkov V.V. Boundary State Method for Solving Problems of Linear Mechanics. Dal'nevostochnyy matematicheskiy zhurnal = Far Eastern Russian Mathematical Journal, 2001, vol. 2, no. 2, pp. 115–137. (in Russian)]
- Пеньков В.Б., Пеньков В.В. Метод граничных состояний для основной смешанной задачи линейного континуума. Всероссийская конференция. Тезисы докладов. ТулГУ, Тула, 2000. С. 108–110. [Penkov V.B., Penkov V.V. Boundary State Method for the Basic Mixed Linear Continuum Problem. Vserossiyskaya konferentsiya. Tezisy dokladov = All-Russian Conference. Abstracts of reports. TulGU, Tula, 2000, pp. 108–110. (in Russian)]
- Саталкина Л.В. Наращивание базиса пространства состояний при жестких ограничениях к энергоемкости вычислений. Сборник тезисов докладов научной конференции студентов и аспирантов Липецкого государственного технического университета. ЛГТУ, Липецк, 2007, с. 130–131. [Satalkina L.V. Increasing the basis of the state space under severe restrictions on the energy consumption of calculations. Sbornik tezisov dokladov nauchnoy konferentsii studentov i aspirantov Lipetskogo gosudarstvennogo tekhnicheskogo universiteta = Collection of abstracts of the scientific conference of students and post-graduate students of the Lipetsk State Technical University. LGTU, Lipetsk, 2007, pp. 130–131. (in Russian)]
- Лехницкий С.Г. Теория упругости анизотропного тела. Наука, Москва, 1977. [Lekhnitsky S.G. Teoriya uprugosti anizotropnogo tela = Theory of elasticity of an anisotropic body. Nauka, Moscow, 1977. (in Russian)]
Downloads
Submitted
Published
How to Cite
Copyright (c) 2022 Ivanychev D.A.
This work is licensed under a Creative Commons Attribution 4.0 International License.
