Applications of Mathematics, Vol. 65, No. 1, pp. 43-65, 2020
Solvability of a dynamic rational contact with limited interpenetration for viscoelastic plates
Jiří Jarušek
Received August 27, 2019. Published online January 31, 2020.
Abstract: Solvability of the rational contact with limited interpenetration of different kind of viscolastic plates is proved. The biharmonic plates, von Kármán plates, Reissner-Mindlin plates, and full von Kármán systems are treated. The viscoelasticity can have the classical ("short memory") form or the form of a certain singular memory. For all models some convergence of the solutions to the solutions of the Signorini contact is proved provided the thickness of the interpenetration tends to zero.
Keywords: dynamic contact problem; limited interpenetration; viscoelastic plate; existence of solution
References: [1] I. Bock, J. Jarušek: Unilateral dynamic contact of viscoelastic von Kármán plates. Adv. Math. Sci. Appl. 16 (2006), 175-187. MR 2253231 | Zbl 1110.35049
[2] I. Bock, J. Jarušek: Unilateral dynamic contact of von Kármán plates with singular memory. Appl. Math., Praha 52 (2007), 515-527. DOI 10.1007/s10492-007-0030-5 | MR 2357578 | Zbl 1164.35447
[3] I. Bock, J. Jarušek: Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system. Z. Angew. Math. Phys. 61 (2010), 865-876. DOI 10.1007/s00033-010-0066-3 | MR 2726632 | Zbl 1273.74352
[4] I. Bock, J. Jarušek: Unilateral dynamic contact problem for viscoelastic Reissner-Mindlin plates. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 4192-4202. DOI 10.1016/j.na.2011.03.054 | MR 2803022 | Zbl 1402.74068
[5] J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 20, Springer, New York (2005). DOI 10.1007/0-387-28271-8 | MR 2144010 | Zbl 1076.49001
[6] C. Eck, J. Jarušek, M. Krbec: Unilateral Contact Problems. Variational Methods and Existence Theorems. Pure and Applied Mathematics (Boca Raton) 270, Chapman & Hall/CRC, Boca Raton (2005). DOI 10.1201/9781420027365 | MR 2128865 | Zbl 1079.74003
[7] C. Eck, J. Jarušek, J. Stará: Normal compliance contact models with finite interpenetration. Arch. Ration. Mech. Anal. 208 (2013), 25-57. DOI 10.1007/s00205-012-0602-8 | MR 3021543 | Zbl 1320.74083
[8] J. Jarušek: Static semicoercive normal compliance contact problem with limited interpenetration. Z. Angew. Math. Phys. 66 (2015), 2161-2172. DOI 10.1007/s00033-015-0539-5 | MR 3412294 | Zbl 1327.35364
[9] J. Jarušek, J. Stará: Solvability of a rational contact model with limited interpenetration in viscoelastodynamics. Math. Mech. Solids 23 (2018), 1040-1048. DOI 10.1177/1081286517703262 | MR 3825900 | Zbl 1401.74218
[10] H. Koch, A. Stachel: Global existence of classical solutions to the dynamical von Kármán equations. Math. Methods Appl. Sci. 16 (1993), 581-586. DOI 10.1002/mma.1670160806 | MR 1233041 | Zbl 0778.73029
[11] J. E. Lagnese: Boundary Stabilization of Thin Plates. SIAM Studies in Applied Mathematics 10, Society for Industrial and Applied Mathematics, Philadelphia (1989). DOI 10.1137/1.9781611970821 | MR 1061153 | Zbl 0696.73034
[12] A. Signorini: Sopra alcune questioni di statica dei sistemi continui. Ann. Sc. Norm. Super. Pisa, II. Ser. 2 (1933), 231-251. (In Italian.) MR 1556704 | JFM 59.0738.01
[13] A. Signorini: Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. Appl., V. Ser. 18 (1959), 95-139. (In Italian.) MR 0118021 | Zbl 0091.38006
Affiliations: Jiří Jarušek, Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic, e-mail: jarusek@math.cas.cz
