Abstract
We consider two initial boundary value problems which describe the evolution of a viscoelastic and viscoplastic body, respectively, in contact with a piston or a device. In both problems the contact process is assumed to be dynamic and the friction is described with a subdifferential boundary condition. Both the constitutive laws and the contact conditions we use involve memory terms. For each problem we derive a variational formulation which is in the form of a system coupling a nonlinear integral equation with a history-dependent hemivariational inequality. Then, we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proofs are based on results for history-dependent hemivariational inequalities presented in Chap. 2
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Banks, H.T., Pinter, G.A., Potter, L.K., Munoz, B.C., Yanyo, L.C.: Estimation and control related issues in smart material structure and fluids. In: Caccetta, L., et al. (eds.) Optimization Techniques and Applications, pp. 19–34. Curtin University Press, Perth (1998)
Banks, H.T., Hu, S., Kenz, Z.R.: A brief review of elasticity and viscoelasticity for solids. Adv. Appl. Math. Mech. 3, 1–51 (2011)
Banks, H.T., Pinter, G.A., Potter, L.K., Gaitens, J.M., Yanyo, L.C.: Modeling of quasistatic and dynamic load responses of filled viesoelastic materials, Chapter 11. In: Cumberbatch, E., Fitt, A. (eds.) Mathematical Modeling: Case Studies from Industry, pp. 229–252. Cambridge University Press, Cambridge (2011)
Cristescu, N., Suliciu, I.: Viscoplasticity. Martinus Nijhoff Publishers, Editura Tehnica, Bucharest (1982)
Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum, Boston (2003)
Drozdov, A.D.: Finite Elasticity and Viscoelasticity—A Course in the Nonlinear Mechanics of Solids. World Scientific, Singapore (1996)
Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics, vol. 270. Chapman/CRC Press, New York (2005)
Farcaş, A., Pătrulescu, F., Sofonea, M.: A history-dependent contact problem with unilateral constraint. Math. Appl. 2, 105–111 (2012)
Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics, vol. 30. American Mathematical Society/International Press, Providence/Somerville (2002)
Ionescu, I.R., Sofonea, M.: Functional and Numerical Methods in Viscoplasticity. Oxford University Press, Oxford (1993)
Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
Klarbring, A., Mikelič, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)
Klarbring, A., Mikelič, A., Shillor, M.: On friction problems with normal compliance. Nonlinear Anal. 13, 935–955 (1989)
Martins, J.A.C., Oden, J.T.: Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. Theory Methods Appl. 11, 407–428 (1987)
Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elast. 83, 247–275 (2006)
Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)
Oden, J.T., Martins, J.A.C.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527–634 (1985)
Piotrowski, J.: Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons. Veh. Syst. Dyn. 48, 675–703 (2010)
Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Springer, Berlin (2004)
Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)
Sofonea, M., Pătrulescu, F.: Analysis of a history-dependent frictionless contact problem. Math. Mech. Solids 18, 409–430 (2013)
Sofonea, M., Shillor, M.: A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Commun. Pure Appl. Anal. 13, 371–387 (2014)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators. Springer, New York (1990)
Acknowledgements
This research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the National Science Center of Poland under grant no. N N201 604640, the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant no. W111/7.PR/2012, the National Science Center of Poland under Maestro Advanced Project no. DEC-2012/06/A/ST1/00262, and the project Polonium “Mathematical and Numerical Analysis for Contact Problems with Friction” 2014/15 between the Jagiellonian University in Krakow and Université de Perpignan Via Domitia.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Sofonea, M., Migórski, S., Ochal, A. (2015). Two History-Dependent Contact Problems. In: Han, W., Migórski, S., Sofonea, M. (eds) Advances in Variational and Hemivariational Inequalities. Advances in Mechanics and Mathematics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-14490-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-14490-0_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14489-4
Online ISBN: 978-3-319-14490-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)