О равновесии упругих тел со слабо искривлённой перемычкой
О равновесии упругих тел со слабо искривлённой перемычкой
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:
- 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
- El Jarroudi M. Homogenization of an elastic material reinforced with thin rigid von Karman ribbons // Math. Mech. Solids. 2018. V. 24, N 7. P. 1–27; https://doi.org/10.1177/1081286518810757
- 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. V. 66. P. 2025-2040; https://doi.org/10.1007/s00033-014-0488-4
- Rudoy E. Asymptotic justification of models of plates containing inside hard thin inclusions // Technologies. 2020. V. 8, N 4. Article 59; https://doi.org/10.3390/technologies8040059
- Bessoud A.-L., Krasucki F., Michaille G. Multi-materials with strong interface: Variational modelings // Asymptot. Anal. 2009. V. 61, N 1. P. 1–19; DOI:10.3233/ASY-2008-
- Khludnev A. M., Fankina I. V. Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary // Z. Angew. Math. Phys. 2021. V. 72, N 121; https://doi.org/10.1007/s00033-021-01553-3
- 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
- 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. V. 24. P. 3743–3752; https://doi.org/10.1177/1081286519850608
- Рудой Е. М., Итоу Х., Лазарев Н. П. Асимптотическое обоснование моделей тонких включений в упругом теле в рамках антиплоского сдвига // Сиб. журн. индустр. математики. 2021. Т. 24. № 1. С. 103–119; DOI: 10.33048/SIBJIM.2021.24.108
- 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. Rep. 2022. V. 19, N 2. P. 935–948; DOI: 0.33048/semi.2022.19.078
- Shcherbakov V. V. Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions // Z. Angew. Math. Phys. 2017. V. 68, N 26; https://doi.org/10.1007/s00033-017-0769-9
- Khludnev A. M. T-shape inclusion in elastic body with a damage parameter // J. Comp. Appl. Math. 2021. V. 393. Article 113540; https://doi.org/10.1016/j.cam.2021.113540
- Furtsev A. I. On contact between a thin obstacle and a plate containing a thin inclusion // J. Math. Sci. 2019. V. 237, N 4. P. 530–545; https://doi.org/10.17377/PAM.2017.17.9
- Bellieud M., Bouchitte G. Homogenization of a soft elastic material reinforced by fibers // Asymptotic Anal. 2002. V. 32. P. 153–183.
- Fankina I. V., Furtsev A. I., Rudoy E. M., Sazhenkov S. A. Multiscale analysis of stationary thermoelastic vibrations of a composite material // Philos. Trans. R. Soc. Ser. A. 2022. V. 380. Article 20210354; https://doi.org/10.1098/rsta.2021.0354
- Gaudiello A., Sili A. Limit models for thin heterogeneous structures with high contrast// J. Differ. Equ. 2021. V. 302. P. 37–63; https://doi.org/10.1016/j.jde.2021.08.032
- 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. V. 54. P. 1329–1351; https://doi.org/10.1137/151003209
- Khludnev A. M., Kovtunenko V. A. Analysis of Cracks in Solids. Southampton; Boston: WIT Press, 2000.
- Хлуднев А. М. Задачи теории упругости в негладких областях. М.: Физматлит, 2010.
- Вольмир А. С. Нелинейная динамика пластинок и оболочек. М.: Наука, 1972.
- Хлуднев A. M. Слабо искривлённое включение в упругом теле при наличии отслоения // Изв. РАН. 2015. № 5. С. 131–144.
- 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
- Хлуднев А. М., Попова Т. С. О задаче сопряжения двух слабо искривлённых включений в упругом теле // Сиб. мат. журн. 2020. Т. 61, № 4. С. 932–945; DOI: 10.33048/smzh.2020.61.414
- 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
- Khludnev А. М. On the crossing bridge between two Kirchhoff —Love plates // Axioms. 2023. V. 12, N 2. Article 120; https://doi.org/10.3390/axioms12020120
А. М. Хлуднев
- 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:
- 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; https://doi.org/10.1002/mma.1670020302
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Rudoj E. M., Itou H., Lazarev N. P. Asimptoticheskoe obosnovanie modelej tonkih vkljuchenij v uprugom tele v ramkah antiploskogo sdviga [Asymptotic substantiation of models of thin inclusions in an elastic body within the framework of an antiplane shift]. Sib. Zhurn. Indust. Mat., 2021, Vol. 24, No. 1, pp. 103– 119 (in Russian); DOI: 10.33048/SIBJIM.2021.24.108
- 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. Rep., 2022, Vol. 19, No. 2, pp. 935–948; DOI: 0.33048/semi.2022.19.078
- 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
- Khludnev A. M. T-shape inclusion in elastic body with a damage parameter. J. Comp. Appl. Math., 2021, Vol. 393, article 113540; https://doi.org/10.1016/j.cam.2021.113540
- 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
- Bellieud M., Bouchitte G. Homogenization of a soft elastic material reinforced by fibers. Asymptotic Anal., 2002, Vol. 32, pp. 153–183.
- Fankina I. V., Furtsev A. I., Rudoy E. M., Sazhenkov S. A. Multiscale analysis of stationary thermoelastic vibrations of a composite material. Philos. Trans. R. Soc. Ser. A, 2022, Vol. 380, article 20210354; https://doi.org/10.1098/rsta.2021.0354
- Gaudiello A., Sili A. Limit models for thin heterogeneous structures with high contrast. J. Differ. Equ., 2021, Vol. 302, pp. 37–63; https://doi.org/10.1016/j.jde.2021.08.032
- 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; https://doi.org/10.1137/151003209
- Khludnev A. M., Kovtunenko V. A. Analysis of Cracks in Solids. Southampton; Boston: WIT Press, 2000.
- Khludnev A. M. Zadachi teorii uprugosti v negladkih oblastjah [Problems of elasticity theory in nonsmooth domains]. Moscow: Fizmatlit, 2010 (in Russian).
- Vol’mir A. S. Nelinejnaja dinamika plastinok i obolochek [Nonlinear dynamics of plates and shells]. Moscow: Nauka, 1972 (in Russian).
- Khludnev A. M. A weakly curved inclusion in an elastic body with separation. Mechanics of Solids, 2015, Vol. 50, No. 5, pp. 591–601; DOI: https://doi.org/10.3103/S0025654415050106
- Khludnev A. M. Asimptotika anizotropnyh slabo iskrivljonnyh vkljuchenij v uprugom tele [Asymptotics of anisotropic weakly curved inclusions in an elastic body]. Sib. Zhurn. Indust. Mat., 2017, Vol. 20, No. 1, pp. 93–104 (in Russian); https://doi.org/10.17377/sibjim.2017.20.110
- Khludnev A. M., Popova T. C. The junction problem for two weakly curved inclusions in an elastic body. Sib. Math. J., 2020, Vol. 61, No. 4, pp. 743–754.
- Khludnev А. М. Asymptotics of solutions for two elastic plates with thin junction. Sib. Electron. Math. Rep., 2022, Vol. 19, No 2, pp. 484–501; DOI: 10.33048/semi.2022.19.041
- 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