Geodesic
A problem of glueing of two Kirchhoff--Love plates
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1351-1374.

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An equilibrium problem of two parallel elastic plates is considered in the paper. The plates are located without a gap and have the same size and shape. They are clamped at their edges and joined to each other along a straight interval. The deflections of the plates satisfy the nonpenetration condition. The case is considered when at the contact surface of the plates not only the lateral load from another plate, but also additional elastic force acts. It is assumed that this elastic force acts both in contact plane and orthogonal to it, and its value is characterized by a so-called damage parameter. Two extreme cases are studied when the parameter equals zero or tends to infinity. The first case corresponds to contact without friction of two plates. The second one corresponds to equilibrium of two-layer plate. The strong convergence of the solutions sequence of equilibrium problem of two plates with elastic force acting at the contact surface to the solutions of extreme problem is proved when the damage parameter tends to zero or to infinity. For the case of the contact without friction singular solution is found near the tip of the interval along which the plates are glued.
Keywords: Kirchhoff–Love plate, contact problem, nonpenetration condition, variational inequality. 
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E. V. Pyatkina. A problem of glueing of two Kirchhoff--Love plates. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1351-1374. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a98/

[1] A.M. Khludnev, “On modelling elastic bodies with defects”, Sib. Electr. Math. Rep., 15 (2018), 153–166 | MR | Zbl

[2] V.V. Sil'vestrov, I.A. Ivanov, “A packet of elastic plates joined along a periodic system of segments”, J. Appl. Mech. Tech. Phys., 42:4 (2001), 731–736 | Zbl

[3] M. Bach, A.M. Khludnev, V.V. Kovtunenko, “Derivatives of the energy functional for 2D-problems with a crack under Singorini and friction condition”, Math. Meth. Appl. Sci., 23:6 (2000), 515–534 | MR | Zbl

[4] H. Itou, A.M. Khludnev, E.M. Rudoy, A. Tani, “Asymptotic behaviour at a tip of a rigid line inclusion in linearized elasticity”, Z.A.M.M., 92:9 (2012), 716–730 | MR | Zbl

[5] E.M. Rudoy, “On numerical solving a rigid inclusions problem in 2D elasticity”, Z.A.M.P., 68:19 (2017), 518–534 | MR | Zbl

[6] E.M. Rudoy, N.A. Kazarinov, V. Yu. Slesarenko, “Numerical simulation of equilibrium of an elastic two-layer structure with a through crack”, Numer. Analys. Appl., 10:1 (2017), 63–73 | MR

[7] Y. Beneveniste, T. Miloh, “Imperfect soft and stiff interfaces in two-dimensional elasticity”, Mech. of Materials, 33 (2001), 309–323

[8] Yu. I. Dimitrienko, “Asymptotic theory of multylayer thin plates”, Vestnik Moskov.Gos. Univ. im. Baumana Ser. Estestv Nauki, 3 (2012), 86–99 (in Russian)

[9] Yu. I. Dimitrienko, E.A. Gubareva, Yu.V. Yurin, “Variational equations of asymptotic theory of multylayer thin plates”, Vestnik Moskov.Gos. Univ. im. Baumana Ser. Estestv. Nauki, 4 (2015), 67–87 (in Russian)

[10] A.M. Khludnev, “Problem on the contact of two elastic plates”, J. Appl. Math. Mech., 47:1 (1983), 110–115 | MR

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

[12] A.M. Khludnev, Elasticity Theory Problems in Nonsmooth Domains, Fizmatlit, M., 2010 (in Russian)

[13] E.I. Grigolyuk, V.M. Tolkachev, Contact Problems of Theory of Plates and Shells, Mashinostroenie, M., 1980 (in Russian)

[14] P.A. Zhilin, Applied Mechanics. Basis of Shell Theory, Politech. Gos. Univ., St. Petersburg, 2006 (in Russian)

[15] S.N. Dmitriev, “Analytical solution to the Hertzian contact problem of elastic deformations of a thin plate positioned in a cylindrical cavity”, Nauka i obrazovanie, 2013, no. 1 (in Russian) | DOI | MR

[16] V.V. Bolotin, Yu.N. Novichkov, Mechanics of Multilayered Structures, Mashinostroenie, M., 1980 (in Russian)

[17] A.R. Rzhanitsyn, Built-up Bars and Plates, Stroiizdat, M., 1986 (in Russian)

[18] E.V. Pyatkina, “A contact problem for two plates of the same shape glued along one edge of a crack”, J. Appl. Ind. Mech., 12:2 (2018), 334–346 | MR | Zbl

[19] A.M. Khludnev, “On unilateral contact of two plates aligned at an angle to each other”, J. Appl.Mech. Tech. Phys., 49:4 (2008), 553–567 | MR | Zbl

[20] N.V. Neustroeva, “Unilateral contact of elastic plates to rigid inclusion”, Vestn. Novosib. Gos. Univ. Ser. Math. Mech. Inform., 9:4 (2009), 51–64 (in Russian) | Zbl

[21] T.A. Rotanova, “On the statements and solvability of the problems on the contact of two plates containing rigid inclusions”, Sibirskii zhurnal industrial'noi matematiki, 15:2 (2012), 107–118 (in Russian) | MR | Zbl

[22] A.M. Il'in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Amer. Math. Soc., Providence, 1989 | MR | Zbl