×

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. (English) Zbl 1159.74026

Summary: We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described by a linearly viscoelastic constitutive law, and friction is modeled by a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second-order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.

MSC:

74M15 Contact in solid mechanics
74M10 Friction in solid mechanics
74D05 Linear constitutive equations for materials with memory
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
49J40 Variational inequalities
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andreu, F.; Mazón, J. M.; Sofonea, M., Entropy solutions in the study of antiplane shear deformations for elastic solids, Math. Models Methods Appl. Sci., 10, 96-126 (2000) · Zbl 1077.74583
[2] Borelli, A.; Horgan, C. O.; Patria, M. C., Saint-Venant’s principle for antiplane shear deformations of linear piezoelectric materials, SIAM J. Appl. Math., 62, 2027-2044 (2002) · Zbl 1047.74019
[3] Campillo, M.; Dascalu, C.; Ionescu, I. R., Instability of a periodic system of faults, Geophys. Int. J., 159, 212-222 (2004)
[4] Campillo, M.; Ionescu, I. R., Initiation of antiplane shear instability under slip dependent friction, J. Geophys. Res., 102 B9, 363-371 (1997)
[5] Chang, K. C., Variational methods for nondifferentiable functionals and applications to partial differential equations, J. Math. Anal. Appl., 80, 102-129 (1981) · Zbl 0487.49027
[6] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley, Interscience: Wiley, Interscience New York · Zbl 0727.90045
[7] Dalah, M.; Sofonea, M., Antiplane frictional contact of electro-viscoelastic cylinders, Electron. J. Differential Equations, 161, 1-14 (2007) · Zbl 1139.74039
[8] Denkowski, Z.; Migórski, S.; Papageorgiou, N. S., An Introduction to Nonlinear Analysis: Theory (2003), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers Boston, Dordrecht, London, New York · Zbl 1040.46001
[9] 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
[10] Horgan, C. O., Anti-plane shear deformation in linear and nonlinear solid mechanics, SIAM Rev., 37, 53-81 (1995) · Zbl 0824.73018
[11] Horgan, C. O.; Miller, K. L., Anti-plane shear deformation for homogeneous and inhomogeneous anisotropic linearly elastic solids, J. Appl. Mech., 61, 23-29 (1994) · Zbl 0809.73016
[12] Ionescu, I. R.; Dascalu, C.; Campillo, M., Slip-weakening friction on a periodic system of faults: Spectral analysis, Z. Angew. Math. Phys. (ZAMP), 53, 980-995 (2002) · Zbl 1014.35068
[13] Ionescu, I. R.; Wolf, S., Interaction of faults under slip dependent friction. Nonlinear eigenvalue analysis, Math. Methods Appl. Sci. \((M^2 A S), 28, 77-100 (2005)\) · Zbl 1062.86006
[14] Migórski, S., Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems, Comput. Math. Appl., 52, 677-698 (2006) · Zbl 1121.74047
[15] Migórski, S.; Ochal, A., Hemivariational inequality for viscoelastic contact problem with slip-dependent friction, Nonlinear Anal., 61, 135-161 (2005) · Zbl 1190.74020
[16] S. Migórski, A. Ochal, M. Sofonea, Weak solvability of antiplane frictional contact problems for elastic cylinders (submitted for publication); S. Migórski, A. Ochal, M. Sofonea, Weak solvability of antiplane frictional contact problems for elastic cylinders (submitted for publication) · Zbl 1241.74029
[17] Panagiotopoulos, P. D., Hemivariational Inequalities, Applications in Mechanics and Engineering (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0826.73002
[18] Rabinowicz, E., Friction and Wear of Materials (1995), Wiley: Wiley New York
[19] Shillor, M.; Sofonea, M.; Telega, J. J., (Models and Analysis of Quasistatic Contact. Models and Analysis of Quasistatic Contact, Lect. Notes Phys., vol. 655 (2004), Springer: Springer Berlin, Heidelberg) · Zbl 1180.74046
[20] Sofonea, M.; Dalah, M.; Ayadi, A., Analysis of an antiplane electro-elastic contact problem, Adv. Math. Sci. Appl., 17, 385-400 (2007) · Zbl 1131.74036
[21] M. Sofonea, A. Matei, Variational inequalities with applications, in: A Study of Antiplane Frictional Contact Problems, Springer, New York (in press); M. Sofonea, A. Matei, Variational inequalities with applications, in: A Study of Antiplane Frictional Contact Problems, Springer, New York (in press) · Zbl 1195.49002
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.