О равновесии упругих тел со слабо искривлённой перемычкой

О равновесии упругих тел со слабо искривлённой перемычкой

Хлуднев А. М.

УДК 517.958:539.3 
DOI: 10.33048/SIBJIM.2023.26.312


Аннотация:

Работа посвящена анализу краевой задачи с неизвестной областью контакта, описывающей равновесие двумерных упругих тел с тонкой слабо искривлённой перемычкой. Перемычка отслаивается от упругих тел, образуя тем самым межфазные трещины. На берегах трещин задаются нелинейные краевые условия вида неравенств, исключающие взаимное проникание берегов. Установлена однозначная разрешимость краевой задачи. Проведён анализ предельных переходов по параметру жёсткости тонкой перемычки при стремлении параметра жёсткости к бесконечности и к нулю и исследованы предельные модели.

Литература:
  1. Caillerie D., Nedelec J. C. The effect of a thin inclusion of high rigidity in an elastic body // Math. Meth. Appl. Sci. 1980. V. 2. P. 251–270; https://doi.org/10.1002/mma.1670020302
     
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  22. Хлуднев А. М. Асимптотика анизотропных слабо искривлённых включений в упругом теле // Сиб. журнал индустр. математики. 2017. Т. 20, № 1. С. 93–104; https://doi.org/10.17377/sibjim.2017.20.110
     
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  24. Khludnev А. М. Asymptotics of solutions for two elastic plates with thin junction // Sib. Electron. Math. Rep. 2022. V. 19, N 2. P. 484–501; DOI: 10.33048/semi.2022.19.041
     
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А. М. Хлуднев
  1. Институт гидродинамики им. М. А. Лаврентьева СО РАН, 
    просп. Акад. Лаврентьева, 15, г. Новосибирск 630090, Россия

E-mail: khlud@hydro.nsc.ru

Статья поступила 10.04.2023 г.
После доработки — 19.04.2023 г.
Принята к публикации 27.04.2023 г.

Abstract:

The work is addressed to the analysis of a boundary value problem with an unknown contact area, which describes equilibrium of two-dimensional elastic bodies with a thin weakly curved junction. It is assumed that the junction exfoliates from the elastic bodies, thereby forming interfacial cracks. Nonlinear boundary conditions of the inequality form are set on the crack faces, excluding the mutual penetration. The unique solvability of the boundary value problem is established. The analysis of limit transitions in terms of the junction stiffness parameter is provided as the parameter tends to infinity and to zero, and limiting models are investigated.

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. 1–27; https://doi.org/10.1177/1081286518810757
     
  3. Lazarev N. Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion. Z. Angew. Math. Phys., 2015, Vol. 66, pp. 2025-2040; https://doi.org/10.1007/s00033-014-0488-4
     
  4. Rudoy E. Asymptotic justification of models of plates containing inside hard thin inclusions. Technologies, 2020, Vol. 8, No. 4, article 59; https://doi.org/10.3390/technologies8040059
     
  5. Bessoud A.-L., Krasucki F., Michaille G. Multi-materials with strong interface: Variational modelings. Asympt. Analysis, 2009, Vol. 61, No. 1, pp. 1–19; DOI:10.3233/ASY-2008-0903
     
  6. Khludnev A. M., Fankina I. V. Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary. Z. Angew. Math. Phys., 2021, Vol. 72, No. 121; https://doi.org/10.1007/s00033-021-01553-3
     
  7. Lazarev N. P., Rudoy E. M. Optimal location of a finite set of rigid inclusions in contact problems for inhomogeneous two-dimensional bodies J. Comp. Appl. Math. 2022. V. 403. Article 113710; DOI:10.1016/j.cam.2021.113710
     
  8. 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; https://doi.org/10.1177/1081286519850608
     
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  11. Shcherbakov V. V. Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions. Z. Angew. Math. Phys., 2017, Vol. 68, No. 26; https://doi.org/10.1007/s00033-017-0769-9
     
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  13. Furtsev A. I. On contact between a thin obstacle and a plate containing a thin inclusion. J. Math. Sci., 2019, Vol. 237, No. 4, pp. 530–545; https://doi.org/10.17377/PAM.2017.17.9
     
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  25. Khludnev А. М. On the crossing bridge between two Kirchhoff —Love plates. Axioms, 2023, Vol. 12, No. 2, article 120; https://doi.org/10.3390/axioms12020120