О равновесии упругой пластины с включением и жёсткими перемычками: некоэрцитивные задачи

О равновесии упругой пластины с включением и жёсткими перемычками: некоэрцитивные задачи

Хлуднев А. М., Лазарев Н. П.
Сибирский журнал индустриальной математики, 28, 4(104), 189-204 (2025)
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УДК 539.3:517.9 
DOI: 10.33048/SIBJIM.2025.28.413

Аннотация:

В работе исследуются краевые задачи о равновесии упругой пластины, содержащей объёмное упругое включение и тонкие жёсткие перемычки. Включение отслаивается от пластины, образуя межфазную трещину. Рассматриваемые в работе краевые условия на внешней границе пластины и на границе упругого включения соответствуют некоэрцитивным краевым задачам. Найдены необходимые и достаточные условия существования решения рассматриваемых задач и доказано существование решений. Дано обоснование возможности предельного перехода по параметру жёсткости включения при стремлении параметра к бесконечности и к нулю. Проведён анализ предельных задач, описывающих равновесие пластины с объёмным жёстким включением и с полостью. Анализируются как вариационные, так и дифференциальные постановки рассматриваемых задач.

Литература:
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Работа выполнена при поддержке Математического Центра в Академгородке, соглашение № 075-15-2025-349 с Министерством науки и высшего образования РФ (разд. 1 и 2) и при поддержке Минобрнауки РФ, соглашение от 11.03.2025 № 075-02-2025-1792 (разд. 3). Других источников финансирования проведения или руководства данным конкретным исследованием не было.

А. М. Хлуднев
  1. Институт гидродинамики им. М. А. Лаврентьева СО РАН, 
    просп. Акад. Лаврентьева, 15, Новосибирск 630090, Россия
  2. Новосибирский государственный университет, 
    ул. Пирогова, 1, г. Новосибирск 630090, Россия

E-mail: khlud@hydro.nsc.ru 

Н. П. Лазарев
  1. Научно-исследовательский институт математики Северо-Восточного федерального университета им. М. К. Аммосова, 
    ул. Белинского, 58, г. Якутск 677000, Россия

E-mail: nyurgunlazarev@yandex.ru 

Статья поступила 14.05.2025 г.
После доработки — 06.01.2026 г.
Принята к публикации 20.01.2026 г.

Abstract:

The paper investigates boundary value problems on the equilibrium of an elastic plate containing a volume elastic inclusion and thin rigid bridges.The inclusion peels off the plate, forming an interfacial crack.The boundary conditions considered in this paper on the outer boundary of the plate and on the boundary of the elastic inclusion correspond to non-coercive boundary value problems. Necessary and sufficient conditions for the existence of solutions to the problems under consideration are found and the existence of solutions is proved. A justification is given for the possibility of a passage to limit in terms of the inclusion stiffness parameter when the parameter tends to infinity and to zero. The analysis of limit problems describing the equilibrium of the plate with a volume rigid inclusion and with a cavity is carried out. Both variational and differential formulations of the problems under consideration are analyzed.

References:
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  2. El Jarroudi M. Homogenization of an elastic material reinforced with thin rigid von Karman ribbons. Math. Mech. Solids, 2018, Vol. 24, No. 7, pp. 1965–1991; DOI: 10.1177/1081286518810757
     
  3. Bessoud A.-L., Krasucki F., Michaille G. Multi-materials with strong interface: Variational modelings. Asymptotic Analysis, 2009, Vol. 61, No. 1, pp. 1–19; DOI: 10.3233/ASY-2008-0903
     
  4. Rudoy E. Asymptotic justification of models of plates containing inside hard thin inclusions. Technologies, 2020, Vol. 8, No. 4, Article number 59; DOI: 10.3390/technologies8040059
     
  5. Fankina I. V., Furtsev A. I., Rudoy E. M., Sazhenkov S. A. Asymptotic modeling of curvilinear narrow inclusions with rough boundaries in elastic bodies: case of a soft inclusion. Sib. Electron. Math. Reports, 2022, Vol. 19, No. 2, pp. 935–948; DOI: 0.33048/semi.2022.19.078
     
  6. Fankina I. V., Furtsev A. I., Rudoy E. M., Sazhenkov S. A. Multiscale analysis of stationary thermoelastic vibrations of a composite material. Phil. Trans. R. Soc. A, 2022, Vol. 380, 20210354; DOI: 10.1098/rsta.2021.0354 
     
  7. Gaudiello A, Sili A. Limit models for thin heterogeneous structures with high contrast. J. Diff. Equat., 2021, Vol. 302, pp. 37–63; DOI: 10.1016/j.jde.2021.08.032
     
  8. De Maio U., Gaudiello A., Sili A. An uncoupled limit model for a high-contrast problem in a thin multistructure. Rendiconti Lincei: Matematica e Applicazioni, 2022, Vol. 33, No. 1, pp. 39–64; DOI: 10.4171/RLM/963
     
  9. Caillerie D., Nedelec J. C. The effect of a thin inclusion of high rigidity in an elastic body. Math. Meth. Appl. Sci., 1980, Vol. 2, pp.251–270; DOI: 10.1002/mma.1670020302
     
  10. Itou H., Kovtunenko V. A., Rajagopal K. R. Well-posedness of the problem of non-penetrating cracks in elastic bodies whose material moduli depend on the mean normal stress. Internat. J. Engrg. Sci., 2019, Vol. 136, 17-25; DOI: 10.1016/j.ijengsci.2018.12.005
     
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  13. Khludnev А. М. On the crossing bridge between two Kirchhoff–Love plates. Axioms, 2023, Vol. 12, No. 2, 120; DOI: 10.3390/axioms12020120
     
  14. Shcherbakov V. V. Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks. Nonlinear Analysis: Real World Applications, 2022, Vol. 65, 103505; DOI: 10.1016/j.nonrwa.2021.103505
     
  15. Shcherbakov V. V. The Griffith formula and J-integral for elastic bodies with Timoshenko inclusions. Z. Angew. Math. Mech., 2016, Vol. 96, pp. 1306–1317; DOI: 10.1002/zamm.201500145
     
  16. Shcherbakov V. V. Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions. Z. Angew. Math. Phys., 2017, 68:26; DOI: 10.1007/s00033-017-0769-9
     
  17. Kovtunenko V. A., Kunisch K. Shape derivative for penalty-constrained nonsmooth-nonconvex optimization: cohesive crack problem. J. Opt. Theory Appl., 2022, Vol. 194, pp. 597–635; DOI: 10.1007/s10957-022-02041-y 
     
  18. Pyatkina E. V. Optimal control problem for two-layer elastic body with crack. J. Math. Sci., 2018, Vol. 230, No. 1, pp. 159–166; DOI: 10.1007/s10958-018-3735-y
     
  19. Lazarev N., Itou H. Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff-Love plates with a crack. Math. Mech. Solids, 2019, Vol. 24, pp. 3743–3752; DOI: 10.1177/1081286519850608 
     
  20. Kovtunenko V. A., Leugering G. A shape-topological control problem for nonlinear crack – defect interaction: the anti-plane variational model. SIAM J. Control Optim., 2016, Vol. 54, pp. 1329–1351; DOI: 10.1137/151003209
     
  21. Saccomandi G., Beatty M. F. Universal relations for fiber-reinforced elastic materials. Math. Mech. Solids, 2002, Vol. 7, pp. 99–110; DOI: 10.1177/108128650200700
     
  22. Rudoy E. M. On numerical solving a rigid inclusions in 2D elasticity. Z. Angew. Math. Phys., 2017, Vol. 68, No. 1, 19; DOI:10.1007/s00033-016-0764-6
     
  23. Kozlov V. A., Mazya V. G., Movchan A. B. Asymptotic Analysis of Fields in a Multi-Structure. Oxford Math. Monogr. N.-Y.: Oxford University Press, 1999.
     
  24. Panasenko G. Multi-Scale Modelling for Structures and Composites. N.-Y.: Springer, 2005.