A class of subdifferential inclusions for elastic unilateral contact problems. (English) Zbl 06584642
Summary: We consider a class of stationary subdifferential inclusions in a reflexive Banach space. We reformulate the problem in terms of a variational inequality with multivalued term and prove an existence result using the Kakutani-Fan-Glicksberg fixed point theorem. This approach allows to consider, in a natural way, a dual variational formulation of the problem. Next, we study the link between the primal and dual formulations and provide an equivalence result. Then, we consider a new mathematical model which describes the contact of an elastic body with a foundation. We apply the abstract formalism to derive the primal and the dual variational formulations of the problem, in terms of displacement and stress, respectively. Finally, we present existence and equivalence results in the study of this contact model.
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47J22 | Variational and other types of inclusions |
| 74M10 | Friction in solid mechanics |
| 74M15 | Contact in solid mechanics |
| 74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |
Keywords:
subdifferential inclusion; multivalued operator; variational inequality; Clarke subdifferential; fixed point; primal and dual formulation; elastic material; frictional contact; normal compliance; unilateral constraint; weak solutionReferences:
| [1] | Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, A Hitchhiker’s Guide, Third Edition. Springer, Berlin (2006) · Zbl 1156.46001 |
| [2] | Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984) · Zbl 0538.34007 · doi:10.1007/978-3-642-69512-4 |
| [3] | Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems. John Wiley, Chichester (1984) · Zbl 0551.49007 |
| [4] | Barboteu, M., Cheng, X.L., Sofonea, M.: Analysis of a contact problem with unilateral constraint and slip-dependent friction. Mathematics and Mechanics of Solids, published online on June 13, 2014. doi:10.1177/1081286514537289 · Zbl 1370.74115 |
| [5] | Brézis, H.: Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115-175 (1968) · Zbl 0169.18602 · doi:10.5802/aif.280 |
| [6] | Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1-168 (1972) · Zbl 0237.35001 |
| [7] | Clarke, F.H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247-262 (1975) · Zbl 0307.26012 · doi:10.1090/S0002-9947-1975-0367131-6 |
| [8] | Clarke, F.H.: Generalized gradients of Lipschitz functionals. Adv. Math. 40, 52-67 (1981) · Zbl 0463.49017 · doi:10.1016/0001-8708(81)90032-3 |
| [9] | Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983) · Zbl 0582.49001 |
| [10] | Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003) · Zbl 1040.46001 · doi:10.1007/978-1-4419-9156-0 |
| [11] | Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003) · Zbl 1054.47001 · doi:10.1007/978-1-4419-9156-0 |
| [12] | 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) · Zbl 1079.74003 · doi:10.1201/9781420027365 |
| [13] | Friedman, A.: Variational Principles and Free-Boundary Problems. John Wiley, New York (1982) · Zbl 0564.49002 |
| [14] | Han, W., Migórski, S., Sofonea, M.: A class of variational-hemivariational inequalities with applications to elastic contact problems. SIAM J. Math. Anal. 46, 3891-3912 (2014) · Zbl 1309.47068 · doi:10.1137/140963248 |
| [15] | Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis, second edition. Springer (2013) · Zbl 1258.74002 |
| [16] | Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity vol. 30, Studies in Advanced Mathematics, Americal Mathematical Society, Providence. RI-International Press, Somerville (2002) · Zbl 1013.74001 |
| [17] | Haslinger, J., Hlaváček, I., Nečas, J.: Numerical methods for unilateral problems in solid mechanics. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. IV, pp 313-485, North-Holland, Amsterdam (1996) · Zbl 0873.73079 |
| [18] | Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer Academic Publishers, Boston (1999) · Zbl 0949.65069 · doi:10.1007/978-1-4757-5233-5 |
| [19] | Hlaváček, I., Haslinger, J., Necǎs, J.: and J. Lovíšek, Solution of Variational Inequalities in Mechanics. Springer, New York (1988) · Zbl 0654.73019 · doi:10.1007/978-1-4612-1048-1 |
| [20] | Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988) · Zbl 0685.73002 · doi:10.1137/1.9781611970845 |
| [21] | Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics, vol. 31. SIAM, Philadelphia (2000) · Zbl 0988.49003 · doi:10.1137/1.9780898719451 |
| [22] | Lions, J.-L.: Quelques méthodes de resolution des problémes aux limites non linéaires. Dunod, Paris (1969) · Zbl 0189.40603 |
| [23] | Migórski, S., Ochal, A., Sofonea, M.: History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 12, 3384-3396 (2011) · Zbl 1231.74065 · doi:10.1016/j.nonrwa.2011.06.002 |
| [24] | 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) · Zbl 1262.49001 |
| [25] | Migórski, S., Ochal, A., Sofonea, M.: History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 22, 604-618 (2015) · Zbl 1326.74101 · doi:10.1016/j.nonrwa.2014.09.021 |
| [26] | Naniewicz, Z., Panagiotopoulos, P. D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York (1995) · Zbl 0968.49008 |
| [27] | Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985) · Zbl 0579.73014 · doi:10.1007/978-1-4612-5152-1 |
| [28] | Panagiotopoulos, P. D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993) · Zbl 0826.73002 · doi:10.1007/978-3-642-51677-1 |
| [29] | Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact, Lect. Notes Phys, vol. 655. Springer, Berlin Heidelberg (2004) · Zbl 1069.74001 · doi:10.1007/b99799 |
| [30] | Sofonea, M., Danan, D., Zheng, C.: Primal and dual variational formulation of a frictional contact problem. Mediterr. J. Math. (2015). doi:10.1007/s00009-014-0504-0. to appear · Zbl 1335.74048 |
| [31] | Sofonea, M., Han, W., Migórski, S.: Numerical analysis of history-dependent variational inequalities with applications to contact problems. Eur. J. Appl. Math. 26, 427-452 (2015) · Zbl 1439.74230 · doi:10.1017/S095679251500011X |
| [32] | Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012) · Zbl 1255.49002 · doi:10.1017/CBO9781139104166 |
| [33] | Zeidler, E.: Nonlinear Functional Analysis and Applications II B. Springer, New York (1990) · Zbl 0684.47029 · doi:10.1007/978-1-4612-0985-0 |
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