Abstract
The paper deals with the study of solvability to geometrically nonlinear boundary value problem for elastic inhomogeneous isotropic shallow shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation relative to deflection of shell in Sobolev space. Solvability of equation is proved with the use of contracting mappings principle.
Similar content being viewed by others
REFERENCES
I. I. Vorovich, Mathematical Problems of Nonlinear Theory of Shallow Shells (Nauka, Moscow, 1989) [in Russian].
N. F. Morozov, Selected Two-Dimensional Problems of Elasticity Theory (LGU, Leningrad, 1978) [in Russian].
M. M. Karchevskii, ‘‘Solvability of variational problems in the nonlinear theory of shallow shells,’’ Differ. Equat. 27, 1196–1203 (1991).
M. M. Karchevskii, ‘‘Investigation of solvability of the nonlinear equilibrium problem of shallow unfixed shell,’’ Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 155, 105–110 (2013).
S. N. Timergaliev, Existence Theorems in Nonlinear Theory of Thin Elastic Shells (Kazan Univ. Press, Kazan, 2011) [in Russian].
S. N. Timergaliev, ‘‘Proof of the solvability of a system of partial differential equations in the nonlinear theory of shallow shells of Timoshenko type,’’ Differ. Equat. 48, 450–454 (2012).
S. N. Timergaliev, ‘‘On existence of solutions to geometrically nonlinear problems for shallow shells of the Timoshenko Type with free edges,’’ Russ. Math. 58 (3), 31–46 (2014).
S. N. Timergaliev, ‘‘On the existence of solutions of a nonlinear boundary value problems for the system of partial differential equations of the theory of Timoshenko type shallow shells with free edges,’’ Differ. Equat. 51, 373–386 (2015).
S. N. Timergaliev, A. N. Uglov and L. S. Kharasova, ‘‘Solvability of geometrically nonlinear boundary value problems for shallow shells of Timoshenko type with pivotally supported edges,’’ Russ. Math. 59 (5), 41–51 (2015).
S. N. Timergaliev and L. S. Kharasova, ‘‘Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type,’’ Differ. Equat. 52, 651–664 (2016).
S. N. Timergaliev, ‘‘A method of integral equations in nonlinear boundary value problems for flat shells of the Timoshenko type with free edges,’’ Russ. Math. 61 (4), 49–64 (2017).
S. N. Timergaliev, ‘‘On the solvability problem for nonlinear equilibrium problems of shallow shells of the Timoshenko type,’’ J. Appl. Math. Mech. 82, 98–113 (2018).
S. N. Timergaliev and A. N. Uglov, ‘‘Application of Riemann–Hilbert problem solutions to a study of nonlinear boundary value problems for Timoshenko type inhomogeneous shells with free edges,’’ Lobachevskii J. Math. 39 (6), 855–865 (2018).
S. N. Timergaliev, ‘‘Method of integral equations for studying the solvability of boundary value problems for the system of nonlinear differential equations of the theory of Timoshenko type shallow inhomogeneous shells,’’ Differ. Equat. 55, 243–259 (2019).
S. N. Timergaliev, ‘‘On existence of solutions of nonlinear equilibrium problems on shallow inhomogeneous anisotropic shells of the Timoshenko type,’’ Russ. Math. 63 (8), 38–53 (2019).
K. Z. Galimov, Principles of the Nonlinear Theory of Thin Shells (Kazan Univ. Press, Kazan, 1975) [in Russian].
I. N. Vekua, Generalized Analytic Function (Nauka, Moscow, 1988) [in Russian].
M. A. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968) [in Russian].
F. D. Gakhov, Boundary-Value Problems (Fizmatgiz, Moscow, 1963) [in Russian].
M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (Gostekhizdat, Moscow, 1956) [in Russian].
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. M. Elizarov)
Rights and permissions
About this article
Cite this article
Akhmadiev, M.G., Timergaliev, S.N., Uglov, A.N. et al. On the Existence of Solutions of Nonlinear Boundary Value Problems for Inhomogeneous Isotropic Shallow Shells of the Timoshenko Type with Free Edges. Lobachevskii J Math 42, 30–43 (2021). https://doi.org/10.1134/S1995080221010054
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080221010054