Geodesic
On~the~equilibrium of~elastic bodies with~weakly curved junction
Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 3, pp. 154-168.

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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.
Keywords: boundary value problem, nonlinear boundary conditions, elastic body, thin junction, crack. 
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A. M. Khludnev. On~the~equilibrium of~elastic bodies with~weakly curved junction. Sibirskij žurnal industrialʹnoj matematiki, Tome 26 (2023) no. 3, pp. 154-168. http://geodesic.mathdoc.fr/item/SJIM_2023_26_3_a11/

[1] Caillerie D., Nedelec J.C., “The effect of a thin inclusion of high rigidity in an elastic body”, Math. Meth. Appl. Sci., 2 (1980), 251–270 | DOI | MR | Zbl

[2] El Jarroudi M., “Homogenization of an elastic material reinforced with thin rigid von Karman ribbons”, Math. Mech. Solids, 24:7 (2018), 1–27 | DOI | MR

[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., 66 (2015), 2025–2040 | DOI | MR | Zbl

[4] Rudoy E., “Asymptotic justification of models of plates containing inside hard thin inclusions”, Technologies, 8:4 (2020), 59 | DOI

[5] Bessoud A.-L., Krasucki F., Michaille G., “Multi-materials with strong interface: Variational modelings”, Asymptot. Anal., 61:1 (2009), 1–19 | DOI | MR | Zbl

[6] Khludnev A.M., Fankina I.V., “Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary”, Z. Angew. Math. Phys., 72:121 (2021) | DOI | MR | Zbl

[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., 403 (2022), 113710 | DOI | MR | Zbl

[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, 24 (2019), 3743–3752 | DOI | MR

[9] Rudoj E.M., Itou H., Lazarev N.P., “Asymptotic substantiation of models of thin inclusions in an elastic body within the framework of an antiplane shift”, Sib. Zhurn. Indust. Mat., 24:1 (2021), 103–119 (in Russian) | DOI | MR | Zbl

[10] 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., 19:2 (2022), 935–948 | DOI | MR

[11] Shcherbakov V.V., “Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions”, Z. Angew. Math. Phys., 68:26 (2017) | DOI | MR | Zbl

[12] Khludnev A.M., “T-shape inclusion in elastic body with a damage parameter”, J. Comp. Appl. Math., 393 (2021), 113540 | DOI | MR | Zbl

[13] Furtsev A.I., “On contact between a thin obstacle and a plate containing a thin inclusion”, J. Math. Sci., 237:4 (2019), 530–545 | DOI | MR

[14] Bellieud M., Bouchitte G., “Homogenization of a soft elastic material reinforced by fibers”, Asymptotic Anal., 32 (2002), 153–183 | MR | Zbl

[15] 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, 380 (2022), 20210354 | DOI | MR

[16] Gaudiello A., Sili A., “Limit models for thin heterogeneous structures with high contrast”, J. Differ. Equyu, 302 (2021), 37–63 | DOI | MR | Zbl

[17] Kovtunenko V.A., Leugering G., “A shape-topological control problem for nonlinear crack — defect interaction: the anti-plane variational model”, SIAM J. Control Optimyu, 54 (2016), 1329–1351 | DOI | MR | Zbl

[18] Khludnev A.M., Kovtunenko V.A., Analysis of Cracks in Solids, WIT Press, Southampton–Boston, 2000

[19] Khludnev A.M., Problems of elasticity theory in non-smooth domains, Fizmatlit, M., 2010 (in Russian)

[20] Vol'mir A.S., Nonlinear dynamics of plates and shells, Nauka, M., 1972 (in Russian)

[21] Khludnev A.M., “A weakly curved inclusion in an elastic body with separation”, Mechanics of Solids, 50:5 (2015), 591–601 | DOI | MR

[22] Khludnev A.M., “Asymptotics of anisotropic weakly curved inclusions in an elastic body”, Sib. Zhurn. Indust. Mat., 20:1 (2017), 93–104 (in Russian) | DOI | Zbl

[23] Khludnev A.M., Popova T.C., “The junction problem for two weakly curved inclusions in an elastic body”, Sib. Math. J., 61:4 (2020), 743–754 | DOI | DOI | MR | Zbl

[24] Khludnev A.M., “Asymptotics of solutions for two elastic plates with thin junction”, Sib. Electron. Math. Rep., 19:2 (2022), 484–501 | DOI | MR

[25] Khludnev A.M., “On the crossing bridge between two Kirchhoff —Love plates”, Axioms, 12:2 (2023), 120 | DOI