Dynamic thermoviscoelastic problem with friction and damage. (English) Zbl 1334.74062
Summary: We present a model of dynamic frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a temperature effect described by the parabolic equation with the subdifferential boundary condition and a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition. Contact is modeled with bilateral condition and is associated to a subdifferential frictional law. The variational formulation of the problem leads to a system of hyperbolic hemivariational inequality for the displacement, parabolic hemivariational inequality for the temperature and parabolic variational inequality for the damage. The existence of a unique weak solution is proved by using recent results from the theory of hemivariational inequalities, variational inequalities, and a fixed point argument.
MSC:
| 74M10 | Friction in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74F05 | Thermal effects in solid mechanics |
| 74R05 | Brittle damage |
| 49J40 | Variational inequalities |
Keywords:
nonlinear thermoviscoelastic material; subdifferential frictional condition; damage; temperature; variational inequality; hemivariational inequalityReferences:
| [1] | Frémond, M.; Nedjar, B., Damage in concrete: the unilateral phenomenon, Nucl. Eng. Des., 156, 323-335 (1995) |
| [2] | Frémond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual power, Internat J. Solids Struct., 33, 1083-1103 (1996) · Zbl 0910.73051 |
| [3] | Han, W.; Shillor, M.; Sofonea, M., Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Anal. Appl., 137, 377-398 (2001) · Zbl 0999.74087 |
| [4] | Denkowski, Z.; Migórski, S., A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact, Nonlinear Anal., 60, 1415-1441 (2005) · Zbl 1190.74019 |
| [5] | Denkowski, Z.; Migórski, S.; Ochal, A., Optimal control for a class of mechanical thermoviscoelastic frictional contact problems, Control Cybernet., 36, 611-632 (2007) · Zbl 1166.49009 |
| [6] | Migórski, S.; Szafraniec, P., A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity, Nonlinear Anal. RWA, 15, 158-171 (2014) · Zbl 1302.74114 |
| [7] | Campo, M.; Fernández, J. R.; Rodríguez-Arós, Á., A quasistatic contact problem with normal compliance and damage involving viscoelastic materials with long memory, Appl. Numer. Math., 58, 1274-1290 (2008) · Zbl 1142.74031 |
| [8] | Li, Y.; Liu, Z., A quasistatic contact problem for viscoelastic materials with friction and damage, Nonlinear Anal., 73, 2221-2229 (2010) · Zbl 1197.35155 |
| [9] | Campo, M.; Fernández, J. R.; Han, W.; Sofonea, M., A dynamic viscoelastic contact problem with normal compliance and damage, Finite Elem. Anal. Des., 42, 1-24 (2005) |
| [10] | Campo, M.; Fernández, J. R.; Kuttler, K. L., Analysis of a dynamic frictional contact problem with damage, Finite Elem. Anal. Des., 45, 659-674 (2009) |
| [11] | Casal, G.; Fernández, J. R., A thermoviscoelastic problem with damage, Finite Elem. Anal. Des., 50, 255-265 (2012) |
| [12] | Chau, O.; Fernández-García, J. R.; Han, W.; Sofonea, M., A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage, Comput. Methods Appl. Mech. Engrg., 191, 5007-5026 (2002) · Zbl 1042.74039 |
| [13] | Bonetti, E.; Bonfanti, G., Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. Inst. H. Poincaré, 25, 1187-1208 (2008) · Zbl 1152.35505 |
| [14] | Strömberg, N.; Johansson, L.; Klarbring, A., Derivation and analysis of a generalized standard model for contact, friction and wear, Internat. J. Solids Structures, 33, 1817-1836 (1996) · Zbl 0926.74012 |
| [15] | Barboteu, M.; Bartosz, K.; Kalita, P.; Ramadan, A., Analysis of a contact problem with normal compliance, finite penetration and nonmonotone slip dependent friction, Commun. Contemp. Math., 16, 1350016 (2014) · Zbl 1292.74026 |
| [16] | Bartosz, K.; Kalita, P., Optimal control for a class of dynamic viscoelastic contact problems with adhesion, Dynam. Systems Appl., 21, 269-292 (2012) · Zbl 1262.35173 |
| [17] | Bartosz, K., Hemivariational inequalities modeling dynamic contact problems with adhesion, Nonlinear Anal., 71, 1747-1762 (2009) · Zbl 1165.74035 |
| [18] | Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley: Wiley New York · Zbl 0582.49001 |
| [19] | Denkowski, Z.; Migórski, S.; Papageorgiou, N. S., An Introduction to Nonlinear Analysis: Applications (2003), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers Boston, Dordrecht, London, New York · Zbl 1030.35106 |
| [20] | Migórski, S.; Ochal, A.; Sofonea, M., (Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26 (2013), Springer: Springer New York) · Zbl 1262.49001 |
| [21] | Migórski, S.; Ochal, A., Boundary hemivariational inequality of parabolic type, Nonlinear Anal., 57, 579-596 (2004) · Zbl 1050.35043 |
| [22] | Barbu, V., Optimal Control of Variational Inequalities (1984), Pitman: Pitman Boston · Zbl 0574.49005 |
| [23] | Sofonea, M.; Han, W.; Shillor, M., (Analysis and Approximation of Contact Problems with Adhesion or Damage. Analysis and Approximation of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics, vol. 276 (2006), Chapman-Hall/CRC Press) · Zbl 1089.74004 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
