Equilibrium problem for a Kirchhoff–Love plate contacting with an inclined and lateral obstacles
Keywords:
variational problem, inclined obstacle, plate, non-penetration condition, contact problem
Abstract
A nonlinear mathematical model of the equilibrium of a plate contacting with two obstacles is investigated. The first non-deformable obstacle is defined by inclined generatrices, and the second one restricts the plate displacements on the side face. In this case, the plate can contact both along the side edge and at the points of the curve corresponding to the intersection of the front surface of the plate and the side cylindrical surface of the plate. These circumstances lead to the fact that boundary conditions are imposed in the form of three inequalities fulfilled on the same curve. Along with the model of a homogeneous plate, the case of a nonhomogeneous plate in which a rigid inclusion is located near the contact boundary is also considered. The unique solvability of the problems for both models is proven. Under the condition of additional smoothness of the solutions to these problems, optimality conditions are found in the form of boundary conditions, as well as the corresponding equivalent differential formulations.
References
[1] Fichera G., Boundary Value Problems of Elasticity with Unilateral Constraints, Handbook der Physik, Band 6a/2, Springer, Berlin; Heidelberg; New York (1972).
[2] Dal Maso G. and Paderni G., “Variational inequalities for the biharmonic operator with variable obstacles,” Ann. Mat. Pura Appl., 153, 203–227 (1988).
[3] Kovtunenko V. A., Itou H., Khludnev A. M., and Rudoy E. M., “Non-smooth variational problems and applications,” Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 380, article ID 20210364 (2022).
[4] Baiocchi C. and Capello A., Variational and Quasivariational Inequalities: Application to Free Boundary Problems, Wiley, New York (1984).
[5] Khludnev A. M. and Kovtunenko V. A., Analysis of Cracks in Solids, WIT-Press, Southampton (2000).
[6] Pelekh B. L., Theory of Shells with Finite Shear Modulus [in Russian], Nauk. Dumka, Kiev (1973).
[7] Lazarev N. P., Itou H., and Neustroeva N. V., “Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle,” Japan J. Ind. Appl. Math., 33, No. 1, 63–80 (2016).
[8] Lazarev N. P., Nikiforov D. Y., and Romanova N. A., “Equilibrium problem for a Timoshenko plate contacting by the side and face surfaces,” Chelyab. Fiz.-Mat. Zhurn., 8, No. 4, 528–541 (2023).
[9] Lazarev N. P., “Fictitious domain method in the equilibrium problem for a Timoshenko-type plate contacting with a rigid obstacle,” J. Math. Sci., 203, No. 4, 527–539 (2014).
[10] Rudoi E. M. and Khludnev A. M., “Unilateral contact of a plate with a thin elastic obstacle,” J. Appl. Ind. Math., 4, 389–398 (2010).
[11] Furtsev A. I., “The unilateral contact problem for a Timoshenko plate and a thin elastic obstacle,” Sib. Electron. Math. Rep., 17, 364–379 (2020).
[12] Furtsev A. I., “On contact between a thin obstacle and a plate containing a thin inclusion,” J. Math. Sci., 237, No. 4, 530–545 (2019).
[13] Popova T. S., “A contact problem for a viscoelastic plate and an elastic beam,” J. Appl. Ind. Math., 10, No. 3, 404–416 (2016).
[14] Pyatkina E. V., “A Contact of two elastic plates connected along a thin rigid inclusion,” Sib. Electron. Math. Rep., 17, 1797–1815 (2020).
[15] Khludnev A. M., “The contact between two plates, one of which contains a crack,” J. Appl. Math. Mech., 61, No. 5, 851–862 (1997).
[16] Khludnev A. M., “On unilateral contact of two plates aligned at an angle to each other,” J. Appl. Mech. Tech. Phys., 49, 553–567 (2008).
[17] Lazarev N. P., Semenova G. M., and Fedotov E. D., “An equilibrium problem for a Kirchhoff– Love plate, contacting an obstacle by top and bottom edges,” Lobachevskii J. Math., 44, No. 2, 614–619 (2023).
[18] Nikolaeva N. A., “Method of fictitious domains for Signorini’s problem in Kirchhoff–Love theory of plates,” J. Math. Sci., 221, No. 6, 872–882 (2017).
[19] Khludnev A. M., “Problem of a crack on the boundary of a rigid inclusion in an elastic plate,” Mech. Solids, 45, No. 5, 733–742 (2010).
[20] Rotanova T. A., “Contact problem for plates with rigid inclusions intersecting the boundary,” Vestn. Tomsk. Gos. Univ., Mat., Mekh., No. 3, 99–107 (2011).
[21] Rudoy E. and Shcherbakov V., “First-order shape derivative of the energy for elastic plates with rigid inclusions and interfacial cracks,” Appl. Math. Optimization, 84, No. 3, 2775–2802 (2021).
[22] Rudoy E. M., “Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body,” Z. Angew. Math. Phys., 66, 1923–1937 (2015).
[23] Lazarev N. P., Semenova G. M., and Romanova N. A., “On a limiting passage as the thickness of a rigid inclusions in an equilibrium problem for a Kirchhoff-Love plate with a crack,” J. Sib. Fed. Univ., Math. Phys., 14, No. 1, 28–41 (2021).
[24] Furtsev A. I., “Problem of equilibrium for hyperelastic body with rigid inclusion and non- penetrating crack,” Sib. Electron. Math. Rep., 21, No. 1, 17–40 (2024).
[25] Rudoy E. M. and Shcherbakov V. V., “Domain decomposition method for a membrane with a delaminated thin rigid inclusion,” Sib. Electron. Math. Rep., 13, No. 1, 395–410 (2016).
[26] Namm R. V. and Tsoy G. I., “Solution of a contact elasticity problem with a rigid inclusion,” Comput. Math. Math. Phys., 59, 659–666 (2019).
[27] Khludnev A. M., Elasticity Problems in Nonsmooth Domains [in Russian], Fizmatlit, Moscow (2010).
[28] Khludnev A. M., “Contact problems for elastic bodies with rigid inclusions,” Q. Appl. Math., 70, 269–284 (2012).
Received
09-07-2024
How to Cite
Lazarev, N., Semenova, G. and Nikulin, A. (2024) “Equilibrium problem for a Kirchhoff–Love plate contacting with an inclined and lateral obstacles”, Mathematical notes of NEFU, 31(2), pp. 14-30. doi: https://doi.org/10.25587/2411-9326-2024-2-14-30.
Section
Mathematics