Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University1097010.18287/2541-7525-2022-28-1-2-46-54Research ArticleGeneral theory of orthotropic shells. Part IVelikanovPeter G.<p>Candidate of Physical and Mathematical Sciences, assistant professor of the Department of Theoretical Mechanics; assistant professor of the Department of Jet Engines and Power Plants</p>pvelikanov@mail.ruhttps://orcid.org/0000-0003-0845-2880ArtyukhinYuri P.<p>Doctor of Physical and Mathematical Sciences, professor of the Department of Theoretical Mechanics</p>ArtukhinYP@mail.ruhttps://orcid.org/0000-0002-6243-9145Kazan (Volga Region) Federal UniversityKazan National Research Technical University named after A.N.Tupolev-KAI16112022281-246542812202228122022Copyright © 2022, Velikanov P.G., Artyukhin Y.P.2022<p>Modern mechanical engineering sets the tasks of calculating thin-walled structures that simultaneously combine sometimes mutually exclusive properties: lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of orthotropic materials and plastics seems quite justified.</p>
<p>The article demonstrates the complex representation method of the equations of the orthotropic shells general theory, which allowed in a complex form to significantly reduce the number of unknowns and the order of the system of differential equations. A feature of the proposed technique for orthotropic shells is the appearance of complex conjugate unknown functions. Despite this, the proposed technique allows for a more compact representation of the equations, and in some cases it is even possible to calculate a complex conjugate function. In the case of axisymmetric deformation, this function vanishes, and in other cases the influence of the complex conjugate function can be neglected.</p>
<p>Verification of the correctness of the proposed technique was demonstrated on a shallow orthotropic spherical shell of rotation under the action of a distributed load. In the limiting case, results were obtained for an isotropic shell as well.</p>механикадифференциальные уравненияортотропные пластинки и оболочкипологие оболочки вращенияосесимметричная деформацияуравнение и функции Бесселяфункция Ломмелягипергеометрические функцииmechanicsdifferential equationsorthotropic plates and shellsshallow shells of rotationaxisymmetric deformationBessel equation and functionsLommel functionhypergeometric functions[Novozhilov V.V. Theory of thin shells. Moscow: Sudpromgiz, 1962, 431 p. Available at: https://bookree.org/reader?file=661745. (in Russ.)][Artyukhin Y.P. Calculation of single-layer and multilayer orthotropic shells for local loads. Research on the theory of plates and shells. Kazan: Izd.-vo KGU, 1966, issue 4, pp. 91–110. Available at: http://mi.mathnet.ru/kutpo593. (in Russ.)][Artyukhin Y.P., Velikanov P.G. Effect of local loads on orthotropic spherical and conical shells of rotation. In: Analytical mechanics, stability and motion control: materials of the All-Russian seminar. Kazan: Izd.-vo KGU, 2008, pp. 22–23. Available at: https://repository.kpfu.ru/?p_id=9408. (In Russ.)][Ambartsumyan S.A. General theory of anisotropic shells. Moscow: Fizmatgiz, 1961, 384 p. Available at: https://bookree.org/reader?file=438699. (In Russ.)][Stanescu K., Vissarion V. Static-geometric analogy for thin elastic shells with orthotropy of the material and its application to the calculation of flat shells and cylindrical shells of circular cross-section. Revue de Mechanique Appliquee (RPR), 1958, vol. 3, no. 1. (In Russ.)][Artyukhin Yu.P., Guryanov N.G., Kotlyar L.M. The Mathematics 4.0 system and its applications in mechanics: textbook. Kazan: Kazanskoe matematicheskoe obshchestvo. Izd-vo KamPI, 2002, 415 p. Available at: https://repository.kpfu.ru/?p_id=53958. (In Russ.)][Velikanov P.G. Fundamentals of work in the Mathematics system: laboratory workshop. Kazan: Izd-vo Kazanskogo gos. tekhn. un-ta, 2010, 40 p. Available at: https://elibs.kai.ru/_docs_file/806166/HTML/. (In Russ.)][Gradstein I.S., Ryzhik I.M. Tables of integrals, sums of series and products. Moscow: Nauka, 1971, 1108 p. Available at: http://www.vixri.ru/?p=991. (In Russ.)][Guryanov N.G., Tyuleneva O.N. Orthotropic plates and flat shells. Theory, methods of solving boundary value problems. Kazan: KGU, 2002, 112 p. (In Russ.)][Matthews F., Rollings R. Composite materials. Mechanics and technology. Moscow: Tekhnosfera, 2004, 408 p. Available at: https://www.technosphera.ru/lib/book/89?read=1. (In Russ.)][Kornishin M.S., Isanbayeva F.S. Flexible plates and panels. Moscow: Nauka, 1968, 260 p. (In Russ.)]