Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Karpov V. V., Bakusov P. A., Maslennikov A. M., Semenov A. A. Simulation models and research algorithms of thin shell structures deformation Part I. Shell deformation models. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 3, pp. 370-410. DOI: 10.18500/1816-9791-2023-23-3-370-410, EDN: YSOXDU

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2023
Full text:
(downloads: 794)
Language: 
Russian
Heading: 
Article type: 
Review
UDC: 
539.3
EDN: 
YSOXDU

Simulation models and research algorithms of thin shell structures deformation Part I. Shell deformation models

Autors: 
Karpov Vladimir Vasil'evich, Saint Petersburg State University of Architecture and Civil Engineering
Bakusov Pavel Anatol`evich, Saint Petersburg State University of Architecture and Civil Engineering
Maslennikov Alexander M., Saint Petersburg State University of Architecture and Civil Engineering
Semenov Alexey Aleksandrovich, Saint Petersburg State University of Architecture and Civil Engineering
Abstract: 
In the article the development of thin shell construction theory is considered according to the contribution of researchers, chronology, including the most accurate and simplified solutions. The review part of the article consists only of those publications which are related to the development of shell theory. The statement is based on the works of famous Russian researchers (V. V. Novozhilov, A. I. Lurie, A. L. Goldenweiser, H. M. Mushtari, V. Z. Vlasov), who developed the specified theory the most. The paper also mentions the researchers who improved the theory, calculation methods in aspects of strength, sustainability and vibrations of thin elastic shell constructions. Separately the application of the models for ribbed shells constructions is shown. It is reporting the basic principles of nonlinear thin shell construction theory development, including the nonlinear relations for deformations. In the article it is shown that if median surface of the shell is referred to the orthogonal coordinate system, then the expressions for deformations, obtained by different authors, practically correspond. The case in which the median surface of the shell is referred to an oblique-angled coordinate system was developed by A. L. Goldenweiser. For static problem, the functional of the total potential energy of deformation, representing the difference between the potential energy and the work of external forces, is used. The equilibrium equations and natural boundary conditions are derived from the minimum condition of this functional. In case of dynamic problem, the functional of the total deformation energy of the shell is described in which it is necessary to consider the kinetic energy of shell deformation. It is necessary to underline that the condition for minimum of the specified functional lets to derive the movement equations and natural boundary and initial conditions. Also, in the article the results of contemporary research of thin shell theory are presented.
References: 
  1. Aron H. Das Gleichgewicht und die Bewegung einer unendlich dunnen, beliebig gekrummten elastischen Schale. Journal fur die reine und angewandte Mathematik / ed. C. W. Borchardt. Berlin, Boston, De Gruyter, 1874, vol. 78, pp. 136–174 (in German). https://doi.org/10.1515/9783112389843-010
  2. Love A. E. H. XVI. The small free vibrations and deformation of a thin elastic shell. Philosophical Transactions of the Royal Society of London A, 1888, vol. 179, pp. 491–546. https://doi.org/10.1098/rsta.1888.0016
  3. Reissner H. Formanderung und Spannungen einer dunnwandigen, an den Randern frei aufliegenden, beliebig belasteten Zylinderschale. Eine Erweiterung der Navierschen Integrationsmethode. ZAMM, 1933, vol. 13, iss. 2, pp. 133–138 (in German). https://doi.org/10.1002/zamm.19330130219
  4. Donell L. H. Stability of Thin-Walled Tubes Under Torsion. NASA, 1933, Rep. no. 479. Available at: https://ntrs.nasa.gov/citations/19930091553 (accessed November 16, 2022).
  5. Galerkin B. G. On the theory of elastic cylindrical shell. Doklady Akademii Nauk SSSR, 1934, vol. 4, no. 5–6, pp. 270–275 (in Russian).
  6. Feinberg S. On the construction of the moment theory of cylindrical shells. Proekt i standart [Project and Standard], 1936, iss. 12, pp. 7–11 (in Russian).
  7. Lurie A. I. Research on the theory of elastic shells. Trudy Leningradskogo industrial’nogo instituta [Proceedings of the Leningrad Industrial Institute], 1937, no. 6, iss. 3, pp. 37–52 (in Russian).
  8. Mushtari H. M. Some generalizations of the theory of thin shells. Izvestiya fiziko-matematicheskogo obshchestva pri Kazanskom universitete. Seriya 8 [Proceedings of the Physics and Mathematics Society at Kazan University. Series 8], 1938, vol. 11, pp. 71–150 (in Russian).
  9. Goldenweiser A. L. Equations of shell theory. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1940, vol. 4, iss. 2, pp. 35–42 (in Russian).
  10. Novozhilov V. V. Teoriya tonkikh obolochek [Theory of Thin Shells]. Leningrad, Oborongiz, 1941. 431 p. (in Russian).
  11. Vlasov V. Z. Basic differential equations of the general theory of elastic shells. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1944, vol. 8, iss. 2, pp. 109–140 (in Russian).
  12. Robotnov Yu. N. Basic equations of shell theory. Doklady Akademii Nauk SSSR, 1945, vol. 47, iss. 2, pp. 90–93 (in Russian).
  13. Vekua I. N. On the theory of thin flat elastic shells. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1948, vol. 12, iss. 1, pp. 69–74 (in Russian).
  14. Ambartsumyan S. A. On the theory of anisotropic flat shells. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1948, vol. 12, iss. 1, pp. 75–80 (in Russian).
  15. Alumae N. A. Differential equations of equilibrium states of thin-walled elastic shells in the post-critical stage. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1949, vol. 13, iss. 1, pp. 95–106 (in Russian).
  16. Krauss F. Uber die Grundgeichunden der Elastizitatstheorie schwach deformierter Schalen. Mathematische Annalen, 1929, vol. 101, iss. 1, pp. 61–92 (in German). https://doi.org/10.1007/BF01454824
  17. Kilchevsky N. A. Generalization of the modern theory of shells. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1939, vol. 2, iss. 4, pp. 427–438 (in Russian).
  18. Grigolyuk E. I., Kabanov V. V. Ustoychivost’ obolochek [Shell Stability]. Moscow, Nauka, 1978. 359 p. (in Russian).
  19. Tovstik P. E. Ustoychivost’ tonkikh obolochek [Stability of Thin Shells]. Moscow, Nauka. Fizmatlit, 1995. 320 p. (in Russian).
  20. Bubnov I. G. Stroitel’naya mekhanika korablya [Ship Construction Mechanics]. Pt. 1–2. Sankt-Peterburg, tip. Mor. m-va, 1912–1914. Pt. 1, 1912. 330 p. ; Pt. 2, 1914. 647 p. (in Russian).
  21. Karman Th. V. Festigkeitsprobleme im Maschinenbau. Encyklopadie der mathematischen Wissenschaften. Leipzig, 1910, vol. 4, pp. 311–385 (in German). https://doi.org/10.1007/978-3-663-16028-1_5
  22. Feodos’ev V. I. Uprugie elementy tochnogo priborostroeniya: Teoriya i raschet [Elastic Elements of Precision Instrumentation: Theory and Calculation]. Moscow, Oborongiz, 1949. 344 p. (in Russian).
  23. Vorovich I. I. On the existence of solutions in nonlinear shell theory. Izvestiya Akademii nauk SSSR. Seriya matematicheskaya, 1955, vol. 19, iss. 4, pp. 173–186 (in Russian).
  24. Donell L. N. A new theory for the buckling of thin cylinders under axial compression and bending. Transactions of the American Society of Mechanical Engineers, 1934, vol. 56, iss. 11, pp. 795–806. https://doi.org/10.1115/1.4019867
  25. Karman Th. V, Tsien H.-S. The buckling of spherical shells by external pressure. Journal of the Aeronautical Sciences, 1939, vol. 7, iss. 2, pp. 43–50. https://doi.org/10.2514/8.1019
  26. Marguerre K. Zur Teorie der gekremmten Platte grosser Formanderung. Jahzbuch 1939 deutseher Luftfahrtsforchung. Bd. 1. Berlin, Ablershof Buecherei, 1939 (in German).
  27. Petrov V. V. To the calculation of flat shells with finite deflections. Nauchnye doklady vysshey shkoly. Stroitel’stvo [Scientific Reports of the Higher School. Construction], 1959, iss. 1, pp. 27–35 (in Russian).
  28. Lurie A. I. Obshchie uravneniya obolochki, podkreplennoy rebrami zhestkosti [General Equations of a Shell Supported by Stiffeners]. Leningrad, 1948. 28 p. (in Russian).
  29. Vlasov V. Z. Contact problems in the theory of shells and thin-walled rods. Izvestiya Akademii nauk SSSR. Otdelenie tekhnicheskikh nauk, 1949, iss. 6, pp. 819–939 (in Russian).
  30. Amiro I. Ya., Zarutskiy V. A., Polyakov P. S. Rebristye tsilindricheskie obolochki [Ribbed Cylindrical Shells]. Kiev, Naukova dumka, 1973. 248 p. (in Russian).
  31. Greben’ E. S. The main relations of the technical theory of ribbed shells. Izvestiya Akademii nauk SSSR. Mekhanika, 1965, iss. 3, pp. 81–92 (in Russian).
  32. Mikhaylov B. K. Plastiny i obolochki s razryvnymi parametrami [Plates and Shells with Discontinuous Parameters]. Leningrad, Leningrad University Publ., 1980. 196 p. (in Russian).
  33. Rassudov V. M. Deformation of flat shells supported by stiffening ribs. Uchenye zapiski Saratovskogo universiteta, 1956, vol. 52, pp. 51–91 (in Russian).
  34. Belostochnyy G. N. Analytical methods for integrating differential equations of thermoelasticity of geometrically irregular shells. Doklady Akademii voennykh nauk. Povolzhskoe regional’noe otdelenie [Doklady of the Academy of Military Sciences. Volga Region Regional Office], 1999, iss. 1, pp. 14–26 (in Russian).
  35. Terebushko O. I. Stability and supercritical deformation of shells supported by sparsely spaced ribs. Raschet prostranstvennykh konstruktsiy [Calculation of Spatial Structures]. Moscow, Mashstroyizdat, 1964, iss. 9, pp. 131–160 (in Russian).
  36. Timashev S. A. Ustoychivost’ podkreplennykh obolochek [Stability of Reinforced Shells]. Moscow, Stroyizdat, 1974. 256 p. (in Russian).
  37. Mileykovskiy I. E., Grechaninov I. P. Stability of rectangular flat shells in terms of. Raschet prostranstvennykh konstruktsiy [Calculation of Spatial Structures]. Moscow, Mashstroyizdat, 1969, iss. 12, pp. 168–176 (in Russian).
  38. Burmistrov E. F. Symmetric deformation of a shell that differs little from a cylindrical one. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1949, vol. 13, iss. 4, pp. 401–412 (in Russian).
  39. Zhilin P. A. General theory of ribbed shells. Prochnost’ gidroturbin: Trudy TsKTI [Strength of Hydraulic Turbines: Proceedings of the CCTI], 1971, iss. 88, pp. 46–70 (in Russian).
  40. Endzhievskiy L. V. Nelineynye deformatsii rebristykh oboloche [Nonlinear Deformations of Ribbed Shells]. Krasnoyarsk, Krasnoyarsk University Publ., 1982. 295 p. (in Russian).
  41. Preobrazhenskiy I. N. Ustoychivost’ i kolebaniya plastinok i obolochek s otverstiyami [Stability and Vibrations of Plates and Shells with Holes]. Moscow, Mashinostroenie, 1981. 191 p. (in Russian).
  42. Il’in V. P., Karpov V. V. Ustoychivost’ rebristykh obolochek pri bol’shikh peremeshcheniyakh [Stability of Ribbed Shells at Large Displacements]. Leningrad, Stroyizdat, 1986. 168 p. (in Russian).
  43. Karpov V. V. Models of the shells having ribs, reinforcement plates and cutouts. International Journal of Solids and Structures, 2018, vol. 146, pp. 117–135. https://doi.org/10.1016/j.ijsolstr.2018.03.024
  44. Rikards R. B., Teters G. A. Ustoychivost’ obolochek iz kompozitnykh materialov [Stability of Shells Made of Composite Materials]. Riga, Zinatne, 1974. 310 p. (in Russian).
  45. Karpov V. V., Semenov A. A. Refined model of stiffened shells. International Journal of Solids and Structures, 2020, vol. 199, pp. 43–56. https://doi.org/10.1016/j.ijsolstr.2020.03.019
  46. Semenov A. A. Mathematical model of deformation of orthotropic shell structures under dynamic loading with transverse shears. Computers & Structures, 2019, vol. 221, pp. 65–73. https://doi.org/10.1016/j.compstruc.2019.05.017
  47. Semenov A. A. Strength and stability of geometrically nonlinear orthotropic shell structures. Thin-Walled Structures, 2016, vol. 106, pp. 428–436. https://doi.org/10.1016/j.tws.2016.05.018
  48. Vol’mir A. S. Gibkie plastiny i obolochki [Flexible Plates and Shells]. Moscow, Gostekhizdat, 1956. 419 p. (in Russian).
  49. Vol’mir A. S. Ustoychivost’ deformirovannykh system [Stability of Deformed Systems]. Moscow, Nauka, 1956. 984 p. (in Russian).
  50. Vol’mir A. S. Nelineynaya dinamika plastinok i obolochek [Nonlinear Dynamics of Plates and Shells]. Moscow, Nauka, 1972. 432 p. (in Russian).
  51. Chernykh K. F. Theory of thin shells of elastomers — rubber-like materials. Uspekhi mekhaniki [Advances in Mechanics], 1983, vol. 6, iss. 1–2, pp. 111–147 (in Russian).
  52. Chernykh K. F., Kabrits S. A., Mikhaylovskiy E. I., Tovstik P. E., Shamina V. A. Obshchaya nelineynaya teoriya uprugikh obolochek [General Nonlinear Theory of Elastic Shells]. St. Petersburg, St. Petersburg State University Publ., 2002. 388 p. (in Russian).
  53. Chernykh K. F. Lineynaya teoriya obolochek. T. 2. Nekotorye voprosy teorii [Linear Theory of Shells. Vol. 2. Some Questions of Theory]. Leningrad, Leningrad State University Publ., 1964. 396 p. (in Russian).
  54. Petrov V. V. Metod posledovatel’nykh nagruzheniy v nelineynoy teorii plastinok i obolochek [Sequential Loading Method in the Nonlinear Theory of Plates and Shells]. Saratov, Saratov University Publ., 1975. 119 p. (in Russian).
  55. Petrov V. V., Inozemtsev V. K., Sineva N. F. Teoriya navedennoy neodnorodnosti i ee prilozheniya k probleme ustoychivosti plastin i obolochek [Theory of Induced Inhomogeneity and its Applications to the Problem of Stability of Plates and Shells]. Saratov, State Technical University of Saratov Publ., 1996. 312 p. (in Russian).
  56. Kossovich L. Yu. Nestatsionarnye zadachi teorii uprugikh tonkikh obolochek [Nonstationary Problems of the Theory of Elastic Thin Shells]. Saratov, Saratov University Publ., 1986. 176 p. (in Russian).
  57. Kossovich L. Yu. Asymptotic integration of nonlinear equations of elasticity theory for a cylindrical shell. Mekhanika deformiruemykh sred [Mechanics of Deformable Media]. Saratov, Saratov University Publ., 1977, iss. 3, pp. 86–96 (in Russian).
  58. Aksel’rad E. L. Gibkie obolochki [Flexible Shells]. Moscow, Nauka, 1976. 376 p. (in Russian).
  59. Mushtari Kh. M., Galimov K. Z. Nelineynaya teoriya uprugikh obolochek [Nonlinear Theory of Elastic Shells]. Kazan, Tatknigoizdat, 1957. 431 p. (in Russian).
  60. Paimushin V. N. Static and dynamic beam forms of loss of stability of a long orthotropic cylindrical shell under external pressure. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 2008, vol. 72, iss. 6, pp. 1014–1027 (in Russian).
  61. Pshenichnov G. I. Teoriya tonkikh uprugikh setchatykh obolochek i plastin [Theory of Thin Elastic Mesh Shells and Plates]. Moscow, Nauka, 1982. 352 p. (in Russian).
  62. Maksimyuk V. A., Storozhuk E. A., Chernyshenko I. S. Variational finite-difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review). International Applied Mechanics, 2012, vol. 48, pp. 613–687. https://doi.org/10.1007/s10778-012-0544-8
  63. Mileykovskiy I. E., Trushin S. I. Raschet tonkostennykh konstruktsiy [Calculation of Thin-Walled Structures]. Moscow, Stroyizdat, 1989. 200 p. (in Russian).
  64. Guz’ A. N., Chernyshenko I. S., Chekhov V. N., Shnerenko K. N. Tsilindricheskie obolochki, oslablennye otverstiyami [Cylindrical Shells Weakened by Holes]. Kiev, Naukova dumka, 1974. 272 p. (in Russian).
  65. Balabukh L. I., Alfutov N. A., Usyukin V. I. Stroitel’naya mekhanika raket [Rocket Construction Mechanics]. Moscow, Vysshaya shkola, 1984. 391 p. (in Russian).
  66. Shalashilin V. N., Kuznetsov E. B. Metody prodolzheniya resheniya po parametru i nailuchshaya parametrizatsiya [Methods of Continuation of the Solution by Parameter and the Best Parameterization]. Moscow, Editorial URSS, 1999. 224 p. (in Russian).
  67. Gavryushin S. S., Nikolaeva A. S. Method of change of the subspace of control parameters and its application to problems of synthesis of nonlinearly deformable axisymmetric thin-walled structures. Mechanics of Solids, 2016, vol. 51, pp. 339–348. https://doi.org/10.3103/S0025654416030110
  68. Valishvili N. V. Metody rascheta obolochek vrashcheniya na ETsVM [Methods of Calculation of Shells of Rotation on ECM]. Moscow, Mashinostroenie, 1976. 278 p. (in Russian).
  69. Kovalenko A. D. Osnovy termouprugosti [Fundamentals of Thermoelasticity]. Kiev, Naukova dumka, 1970. 306 p. (in Russian).
  70. Abovskiy N. P., Chernyshov V. N., Pavlov A. S. Gibkie rebristye pologie obolochki [Flexible Ribbed flat Shells]. Krasnoyarsk, 1975. 128 p. (in Russian).
  71. Alfutov N. A. Stability of a cylindrical shell supported by a transverse force set and loaded with an external uniform pressure. Inzhenernyy sbornik [Engineering Collection], 1956, vol. 23, pp. 36–46 (in Russian).
  72. Kantor B. Ya. Nelineynye zadachi teorii neodnorodnykh pologikh obolochek [Nonlinear Problems of the Theory of Inhomogeneous Flat Shells]. Kiev, Naukova dumka, 1971. 136 p. (in Russian).
  73. Karmishin A. V., Lyaskovets V. A., Myachenkov V. I., Frolov A. N. Statika i dinamika tonkostennykh obolochechnykh konstruktsiy [Statics and Dynamics of Thin-Walled Shell Structures]. Moscow, Mashinostroenie, 1975. 376 p. (in Russian).
  74. Klimanov V. I., Timashev S. A. Nelineynye zadachi podkreplennykh obolochek [Nonlinear Problems of Reinforced Shells]. Sverdlovsk, UNTs AN SSSR, 1985. 291 p. (in Russian).
  75. Teregulov I. G. Izgib i ustoychivost’ tonkikh plastin i obolochek pri polzuchesti [Bending and Stability of Thin Plates and Shells Under Creep]. Moscow, Nauka, 1969. 206 p. (in Russian).
  76. Krys’ko V. A. Nelineynaya statika i dinamika neodnorodnykh obolochek [Nonlinear Statics and Dynamics of Inhomogeneous Shells]. Saratov, Saratov University Publ., 1976. 216 p. (in Russian).
  77. Pertsev A. K., Platonov E. G. Dinamika obolochek i plastin [Dynamics of Shells and Plates]. Leningrad, Sudostroenie, 1987. 316 p. (in Russian).
  78. Filin A. P. Elementy teorii obolochek [Elements of Shell Theory]. Leningrad, Stroyizdat, 1987. 384 p. (in Russian).
  79. Kornishin M. S. Nelineynye zadachi teorii plastin i obolochek i metody ikh resheniya [Nonlinear Problems of the Theory of Plates and Shells and Methods of Their Solution]. Moscow, Nauka, 1964. 192 p. (in Russian).
  80. Krivoshapko S. N. About the possibilities of shell structures in modern architecture and construction. Stroitel’naya mekhanika inzhenernykh konstruktsiy i sooruzheniy, 2013, iss. 1, pp. 51–56 (in Russian).
  81. Meissner E. Das Elastizitatsproblem fur dunne Schalen von Ringflachen, Kugel- und Kegelform. Phisikalische Zeitschrift, 1913, vol. 14, pp. 343–349 (in German).
  82. Yakushev V. L. Nelineynye deformatsii i ustoychivost’ tonkikh obolochek [Nonlinear Deformations and Stability of Thin Shells]. Moscow, Nauka, 2004. 276 p. (in Russian).
  83. Andreev L. V., Obodan N. I., Lebedev A. G. Ustoychivost’ obolochek pri neosesimmetrichnoy deformatsii [Stability of Shells under Non-axisymmetric Deformation]. Moscow, Nauka, 1988. 208 p. (in Russian).
  84. Karpov V. V. Prochnost’ i ustoychivost’ podkreplennykh obolochek vrashcheniya. Ch. 1. Modeli i algoritmy issledovaniya prochnosti i ustoychivosti podkreplennykh obolochek vrashcheniya [The Strength and Stability of the Reinforced Shells of Rotation. Part 1. Models and Algorithms for Studying the Strength and Stability of Reinforced Shells of Rotation]. Moscow, Fizmatlit, 2010. 288 p. (in Russian).
  85. Lurie A. I. General theory of elastic thin shells. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics], 1940, vol. 4, iss. 2, pp. 7–34 (in Russian).
  86. Vlasov V. Z. Obshchaya teoriya obolochek i ee prilozhenie v tekhnike [General Theory of Shells and its Application in Engineering]. Moscow, Leningrad, Gostekhizdat, 1949. 784 p. (in Russian).
  87. Gol’denveyzer A. L. Teoriya tonkikh uprugikh obolochek [Theory of thin Elastic Shells]. Moscow, GITTL, 1953. 544 p. (in Russian).
  88. Mileykovskiy I. E., Kupar A. K. Gipary. Raschet i proektirovanie pologikh obolochek pokrytiy v forme giperbolicheskikh paraboloidov [Hypars. Calculation and Design of flat Shells of Coatings in the Form of Hyperbolic Paraboloids]. Moscow, Stroyizdat, 1978. 223 p. (in Russian).
  89. Dykhovichnyy Yu. A., Zhukovskiy E. Z. Prostranstvennye sostavnye konstruktsii [Spatial Composite Constructions]. Moscow, Vysshaya shkola, 1989. 288 p. (in Russian).
  90. Krivoshapko S. N., Ivanov V. N., Khalabi S. M. Analiticheskie poverkhnosti: materialy po geometrii 500 poverkhnostey i informatsiya k raschetu na prochnost’ tonkikh obolochek [Analytical Surfaces: Materials on the Geometry of 500 Surfaces and Information for Calculating the Strength of Thin Shells]. Moscow, Nauka, 2006. 544 p. (in Russian).
  91. Zhilin P. A. Prikladnaya mekhanika. Osnovy teorii obolochek [Applied Mechanics. Fundamentals of Shell Theory]. St. Petersburg, St. Petersburg Politechnic University Publ., 2006. 167 p. (in Russian).
  92. Mikhaylova E. Yu., Tarlakovskiy D. V., Fedotenkov G. V. Obshchaya teoriya uprugikh obolochek [General Theory of Elastic Shells]. Moscow, MAI Publ., 2018. 112 p. (in Russian).
  93. Mikhailova E. Yu., Tarlakovsky D. V., Fedotenkov G. V. Generalized linear model of dynamics of thin elastic shells. Uchenye zapiski Kazanskogo universiteta. Seriya Fiziko-matematicheskie nauki [Scientific Notes of Kazan University. Series of Physical and Mathematical Sciences], 2018, vol. 160, book 3, pp. 561–577 (in Russian). EDN: YZSUDR
  94. Pogorelov A. V. Geometricheskie metody v nelineynoy teorii uprugikh obolochek [Geometric Methods in the Nonlinear Theory of Elastic Shells]. Moscow, Nauka, 1967. 280 p. (in Russian).
  95. Pogorelov A. V. Izgibanie vypuklykh poverkhnostey [Bending of Convex Surfaces]. Moscow, Leningrad, GITTL, 1951. 183 p. (in Russian).
  96. Ivochkina N. M., Filimonenkova N. V. Differential geometry in the theory of Hessian operators. Available at: https://arxiv.org/pdf/1904.04157.pdf (accessed July 8, 2021).
Received: 
16.11.2022
Accepted: 
16.01.2023
Published: 
31.08.2023