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Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. (English) Zbl 1081.74036

Summary: We examine an evolution problem which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and by a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First, we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.

MSC:

74M15 Contact in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74G65 Energy minimization in equilibrium problems in solid mechanics
49J40 Variational inequalities
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[1] Aubin JP, Differential Inclusions. Set-Valued Maps and Viability Theory (1984)
[2] DOI: 10.1137/0317040 · Zbl 0439.49018 · doi:10.1137/0317040
[3] Awbi B, Annales Polonici Mathematici 75 pp 233– (2000)
[4] DOI: 10.1016/0362-546X(95)00131-E · Zbl 0894.34055 · doi:10.1016/0362-546X(95)00131-E
[5] DOI: 10.1016/0022-1236(72)90070-5 · Zbl 0249.47044 · doi:10.1016/0022-1236(72)90070-5
[6] DOI: 10.1016/0022-247X(81)90095-0 · Zbl 0487.49027 · doi:10.1016/0022-247X(81)90095-0
[7] DOI: 10.1023/A:1014501802247 · Zbl 1008.74058 · doi:10.1023/A:1014501802247
[8] Clarke FH, Optimization and Nonsmooth Analysis (1983)
[9] Duvaut G, Les Inéquations en Mécanique et en Physique (1972)
[10] Gasiński L, Jagiellonian University pp p. 61– (2000)
[11] DOI: 10.1002/1522-2616(200207)242:1<79::AID-MANA79>3.0.CO;2-S · Zbl 1003.49006 · doi:10.1002/1522-2616(200207)242:1<79::AID-MANA79>3.0.CO;2-S
[12] DOI: 10.1023/A:1021783924105 · Zbl 0973.90060 · doi:10.1023/A:1021783924105
[13] Goeleven D, From Convexity to Nonconvexity pp pp. 111–122– (2001)
[14] Haslinger J, Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications (1999)
[15] Jarusek J, Czechoslovak Mathematical Journal 46 pp 475– (1996)
[16] DOI: 10.1016/0362-546X(95)00170-Z · Zbl 0865.73054 · doi:10.1016/0362-546X(95)00170-Z
[17] DOI: 10.1142/S0219199799000067 · Zbl 0959.34049 · doi:10.1142/S0219199799000067
[18] Lions JL, Quelques Méthodes de Résolution des Problémes Aux Limites non Linéaires (1969)
[19] Miettinen M, University of Jyväskylä (1993)
[20] Migórski S, Topological Methods Nonlinear Anal. 16 pp 125– (2000) · Zbl 0979.34051 · doi:10.12775/TMNA.2000.034
[21] DOI: 10.1016/S0362-546X(01)00160-2 · Zbl 1042.49515 · doi:10.1016/S0362-546X(01)00160-2
[22] Migórski S, Industrial Mathematics and Statistics pp pp. 248–279– (2003)
[23] Migórski S, Journal of Global Optimization · Zbl 0927.49019
[24] Motreanu D, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications (1999) · doi:10.1007/978-1-4615-4064-9
[25] Naniewicz Z, Mathematical Theory of Hemivariational Inequalities and Applications (1995)
[26] Ne[cbreve]as J, Les Méthodes Directes en Théorie des Équations Elliptiques (1967)
[27] Ne[cbreve]as J, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction (1981)
[28] Ochal A, Jagiellonian University pp p. 63– (2001)
[29] Panagiotopoulos PD, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions (1985) · doi:10.1007/978-1-4612-5152-1
[30] DOI: 10.1016/0362-546X(91)90224-O · Zbl 0733.49012 · doi:10.1016/0362-546X(91)90224-O
[31] DOI: 10.1016/0377-0427(95)00050-X · Zbl 0847.49016 · doi:10.1016/0377-0427(95)00050-X
[32] Panagiotopoulos PD, Atti del Seminario Matematico e Fisico dell’Universita di Modena pp 159– (1995)
[33] Panagiotopoulos PD, Hemivariational Inequalities, Applications in Mechanics and Engineering (1993) · doi:10.1007/978-3-642-51677-1
[34] DOI: 10.1515/JAA.1999.95 · Zbl 0929.34049 · doi:10.1515/JAA.1999.95
[35] DOI: 10.1007/BF02857308 · Zbl 0931.34043 · doi:10.1007/BF02857308
[36] DOI: 10.1080/00036819808840639 · Zbl 0904.73055 · doi:10.1080/00036819808840639
[37] Sofonea M, Comm. Appl. Anal. 5 pp 135– (2001)
[38] DOI: 10.1155/S0161171201003994 · Zbl 0983.34054 · doi:10.1155/S0161171201003994
[39] Zeidler E, Nonlinear Functional Analysis and Applications II A/B (1990) · doi:10.1007/978-1-4612-0985-0
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