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

V. V. Karpov ; P. A. Bakusov ; A. M. Maslennikov ; A. A. Semenov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 3, pp. 370-410

Résumé

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.