zbMATH — the first resource for mathematics

A unified approach to dynamic contact problems in viscoelasticity. (English) Zbl 1138.74375
Summary: In this paper we consider mathematical models describing dynamic viscoelastic contact problems with the Kelvin-Voigt constitutive law and subdifferential boundary conditions. We treat evolution hemivariational inequalities which are weak formulations of contact problems. We establish the existence of solutions to hemivariational inequalities with different types of non-monotone multivalued boundary relations. These results are consequences of an existence theorem for second order evolution inclusions. In a particular case we deliver sufficient conditions under which the solution to a hemivariational inequality is unique. Finally, applications to several unilateral and bilateral problems in contact mechanics are provided.

74M15 Contact in solid mechanics
74D05 Linear constitutive equations for materials with memory
74M25 Micromechanics of solids
49J40 Variational inequalities
Full Text: DOI
[1] A. Amassad and C. Fabre, On the analysis of viscoplastic contact problem with time dependent Tresca’s friction law. Electron. J. Math. Phys. Sci. 1(1) (2002) 47–71. · Zbl 1154.74325
[2] A. Amassad and M. Sofonea, Analysis of a quasistatic viscoplastic problem involving Tresca friction law. Discrete Contin. Dyn. Syst. 4 (1998) 55–72. · Zbl 0972.74050
[3] A. Amassad and M. Sofonea, Analysis of some nonlinear evolution systems arising in rate-type viscoplasticity, In: W. Chen and S. Hu (eds.), Dynamical System and Differential Equations, an added volume to Discrete and Cont. Dyn. Syst. (1998) 58–71. · Zbl 1304.35671
[4] L.-E. Anderson, A global existence result for a quasistatic contact problem with friction. Adv. Math. Sci. Appl. 5 (1995) 249–286. · Zbl 0832.35056
[5] B. Awbi, E.H. Essoufi and M. Sofonea, A viscoelastic contact problem with normal damped response and friction. Ann. Pol. Math. 75 (2000) 233–246. · Zbl 0994.74051
[6] K.C. Chang, Variational methods for nondifferentiable functionals and applications to partial differential equations. J. Math. Anal. Appl. 80 (1981) 102–129. · Zbl 0487.49027 · doi:10.1016/0022-247X(81)90095-0
[7] O. Chau, W. Han and M. Sofonea, A dynamic frictional contact problem with normal damped response. Acta Appl. Math. 71 (2002) 159–178. · Zbl 1008.74058 · doi:10.1023/A:1014501802247
[8] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York (1983). · Zbl 0582.49001
[9] Z. Denkowski and S. Migórski, A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact. Nonlinear Anal. 60 (2005) 1415–1441. · Zbl 1190.74019 · doi:10.1016/j.na.2004.11.004
[10] Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory and Applications. Kluwer/Plenum, Boston, Dordrecht, London, New York (2003). · Zbl 1030.35106
[11] Y. Dumont, D. Goeleven, M. Rochdi, K.L. Kuttler and M. Shillor, A dynamic model with friction and adhesion with applications to rocks. J. Math. Anal. Appl. 247 (2000) 87–109. · Zbl 0976.74043 · doi:10.1006/jmaa.2000.6828
[12] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer, Berlin Heidelberg New York (1976). · Zbl 0331.35002
[13] D. Goeleven, M. Miettinen and P.D. Panagiotopoulos, Dynamic hemivariational inequalities and their applications. J. Optim. Theory Appl. 103(3) (1999) 567–601. · Zbl 0973.90060 · doi:10.1023/A:1021783924105
[14] D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications. Kluwer, Boston, Dordrecht, London (2003). · Zbl 1259.49002
[15] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, International Press (2002). · Zbl 1013.74001
[16] J. Jarušek, Dynamic contact problems with given friction for viscoelastic bodies. Czechoslov. Math. J. 46 (1996) 475–487. · Zbl 0879.73022
[17] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). · Zbl 0685.73002
[18] A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26 (1988) 811–832. · Zbl 0662.73079 · doi:10.1016/0020-7225(88)90032-8
[19] K.L. Kuttler, Dynamic friction contact problem with general normal and friction laws. Nonlinear Anal. 28 (1997) 559–575. · Zbl 0865.73054 · doi:10.1016/0362-546X(95)00170-Z
[20] K.L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions. Comm. Contemporary Math. 1(1) (1999) 87–123. · Zbl 0959.34049 · doi:10.1142/S0219199799000067
[21] K.L. Kuttler and M. Shillor, Dynamic bilateral contact with discontinuous friction coefficient. Nonlinear Anal. 45 (2001) 309–327. · Zbl 1046.74037 · doi:10.1016/S0362-546X(99)00345-4
[22] K.L. Kuttler and M. Shillor, Dynamic contact with normal compliance wear and discontinuous friction coefficient. SIAM J. Math. Anal. 34(1) (2002) 1–27. · Zbl 1029.74033 · doi:10.1137/S0036141001391184
[23] S. Migórski, Existence and convergence results for evolution hemivariational inequalities. Topol. Methods Nonlinear Anal. 16 (2000) 125–144. · Zbl 0979.34051
[24] S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84 (2005) 669–699. · Zbl 1081.74036 · doi:10.1080/00036810500048129
[25] S. Migórski and A. Ochal, Boundary hemivariational inequality of parabolic type. Nonlinear Anal. 57(4) (2004) 579–596. · Zbl 1050.35043 · doi:10.1016/j.na.2004.03.004
[26] S. Migórski and A. Ochal, Hemivariational inequality for viscoelastic contact problem with slip dependent friction. Nonlinear Anal. 61 (2005) 135–161. · Zbl 1190.74020 · doi:10.1016/j.na.2004.11.018
[27] S. Migórski and A. Ochal, Existence of solutions for second order evolution inclusions and their applications to mechanical contact problems, Optimization, (2006), in press. · Zbl 1104.34045
[28] E.S. Mistakidis and P.D. Panagiotopoulos, The search for substationarity points in the unilateral contact problems with nonmonotone friction. Math. Comput. Model. 28(4–8) (1998) 341–358. · Zbl 1126.74481 · doi:10.1016/S0895-7177(98)00126-5
[29] Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York, Basel, Hong Kong (1995). · Zbl 0968.49008
[30] A. Ochal, Existence results for evolution hemivariational inequalities of second order. Nonlinear Anal. 60 (2005) 1369–1391. · Zbl 1082.34052 · doi:10.1016/j.na.2004.10.021
[31] P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985). · Zbl 0579.73014
[32] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer, Berlin Heidelberg New York (1993). · Zbl 0826.73002
[33] M. Rochdi, M. Shillor and M. Sofonea, A quasistatic contact problem with directional friction and damped response. Appl. Anal. 68 (1998) 409–422. · Zbl 0904.73055 · doi:10.1080/00036819808840639
[34] M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51 (1998) 105–126. · Zbl 0921.73231 · doi:10.1023/A:1007413119583
[35] M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci. 38 (2000) 1517–1533. · Zbl 1210.74132 · doi:10.1016/S0020-7225(99)00126-3
[36] E. Zeidler, Nonlinear Functional Analysis and Applications II A/B. Springer, Berlin Heidelberg New York (1990). · Zbl 0684.47028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.