Geodesic
A contact problem for a~viscoelastic plate and an elastic beam
Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 3, pp. 41-54.

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We consider the problem of contact of a viscoelastic plate with an elastic beam. For characterizing the viscoelastic deformation of the plate, we use hereditary integrals. We present a differential statement of the problem with conditions having the form of a system of equalities and inequalities in the domain of possible contact and prove its equivalence to a variational inequality. We establish the unique solvability and the existence of the derivative of a solution with respect to time. The limit problem is considered with the parameter of bending rigidity of the plate tending to infinity.
Keywords: viscoelasticity, beam, plate, hereditary integral, variational inequality, nonpenetration condition. 
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T. S. Popova. A contact problem for a~viscoelastic plate and an elastic beam. Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 3, pp. 41-54. http://geodesic.mathdoc.fr/item/SJIM_2016_19_3_a3/

[1] Rzhanitsyn A. R., Nekotorye voprosy mekhaniki sistem, deformiruyuschikhsya vo vremeni, Gostekhizdat, M., 1949

[2] Verichev S. N., Metrikin A. V., “Dinamicheskaya zhestkost balki v dvizhuschemsya kontakte”, Prikl. mekhanika i tekhn. fizika, 41:6 (2000), 170–177 | Zbl

[3] Li H., Dempsey J., “Unbonded contact of finite Timoshenko beam on elastic layer”, J. Engrg. Mech., 114:8 (1988), 1265–1284 | DOI

[4] Yakovleva T. V., “O pamyati nelineinykh differentsialnykh sistem v teorii plastin”, Vestn. Nizhegorodskogo gos. un-ta, 4:2 (2011), 21–23

[5] Nobili A., Tarantino A., “Unilateral contact problem for aging viscoelastic beams”, J. Engrg. Mech., 131:12 (2005), 1229–1238 | DOI

[6] Khludnev A. M., Zadachi teorii uprugosti v negladkikh oblastyakh, Fizmatlit, M., 2010

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

[8] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[9] Khludnev A. M., Khoffmann K.-Kh., Botkin N. D., “Variatsionnaya zadacha o kontakte uprugikh ob'ektov raznykh razmernostei”, Sib. mat. zhurn., 47:3 (2006), 707–717 | MR | Zbl

[10] Stekina T. A., “Variatsionnaya zadacha ob odnostoronnem kontakte uprugoi plastiny s balkoi”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 9:1 (2009), 45–56 | Zbl

[11] Rudoi E. M., Khludnev A. M., “Odnostoronnii kontakt plastiny s tonkim uprugim prepyatstviem”, Sib. zhurn. industr. matematiki, 12:2 (2009), 120–130 | MR | Zbl

[12] Neustroeva N. V., “A rigid inclusion in the contact problem for elastic plates”, J. Appl. Industr. Math., 4:4 (2010), 526–538 | DOI | MR

[13] Neustroeva N. V., “Kontaktnaya zadacha dlya uprugikh tel raznykh razmernostei”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 8:4 (2008), 60–75 | Zbl

[14] Scherbakov V. V., Krivorotko O. I., “Optimalnye formy treschin v vyazkouprugomtele”, Tr. IMM UrO RAN, 21, no. 1, 2015, 294–304 | MR

[15] Lazarev N. P., “Zadacha o ravnovesii plastiny Timoshenko s naklonnoi treschinoi”, Prikl. mekhanika i tekhn. fizika, 54:4 (2013), 171–181 | MR | Zbl

[16] Eck C., Jarušek J., “Existence of solutions for the dynamic frictional contact problemof isotropic viscoelastic bodies”, Nonlinear Anal., 53 (2003), 157–181 | DOI | MR | Zbl

[17] Bock I., Jarušek J., “Unilateral dynamic contact problem for viscoelastic Reissner–Mindlin plates”, Nonlinear Anal., 74 (2011), 4192–4202 | DOI | MR | Zbl

[18] Khludnev A. M., “On equilibrium problem for a plate having a crack under the creep condition”, Control and Cybernetics, 25:5 (1996), 1015–1030 | MR

[19] Popova T. S., “Zhestkoe vklyuchenie v zadache o vyazkouprugom tele s treschinoi”, Mat. zametki YaGU, 20:1 (2013), 87–106

[20] Popova T. S., “Zadacha o ravnovesii vyazkouprugogo tela s tonkim zhestkim vklyucheniem”, Mat. zametki SVFU, 21:1 (2014), 47–55 | Zbl

[21] Popova T. S., “Zadacha o ravnovesii vyazkouprugogo tela s treschinoi i tonkim zhestkim vklyucheniem”, Mat. zametki SVFU, 21:2 (2014), 94–105 | Zbl

[22] Li Y., Liu Z., “A quasistatic contact problem for viscoelastic materials with friction and damage”, Nonlinear Anal., 73 (2010), 2221–2229 | DOI | MR | Zbl

[23] Yao S.-S., Huang N.-J., “A class of quasistatic contact problems for viscoelastic materials with nonlocal coulomb friction and time-delay”, Math. Modelling Anal., 19:4 (2014), 491–508 | DOI | MR