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Analysis of a piezoelectric contact problem with subdifferential boundary condition

Published online by Cambridge University Press:  03 October 2014

Stanisław Migórski ,
Anna Ochal  and
Mircea Sofonea
Stanisław Migórski
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow, Poland, ([email protected])
Anna Ochal
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow, Poland, ([email protected])
Mircea Sofonea
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France
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Abstract

We consider a mathematical model which describes the frictionless contact between a piezoelectric body and a foundation. The contact process is quasi-static and the foundation is assumed to be insulated. The novelty of the model consists in the fact that the material behaviour is described with an electro-elastic–visco-plastic constitutive law and the contact is modelled with a subdifferential boundary condition. We derive a variational formulation of the problem which is in the form of a system coupling two nonlinear integral equations with a history-dependent hemivariational inequality and a time-dependent linear equation. We prove the existence of a weak solution to the problem and, under additional assumptions, its uniqueness. The proof is based on a recent result on history-dependent hemivariational inequalities obtained by Migórski, Ochal and Sofonea in 2011.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 5 , October 2014 , pp. 1007 - 1025
Copyright
Copyright © Royal Society of Edinburgh 2014 

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References

1Batra, R. C. and Yang, J. S.. Saint-Venant's principle in linear piezoelectricity. J. Elasticity 38 (1995), 209–218.Google Scholar
2Bisegna, P., Lebon, F. and Maceri, F.. The unilateral frictional contact of a piezoelectric body with a rigid support. In Contact mechanics (ed. Martins, J. A. C. and Marques, M. D. P. Monteiro), pp. 347–354 (Dordrecht: Kluwer, 2002).Google Scholar
3Boureanu, M., Matei, A. and Sofonea, M.. Analysis of a contact problem for electro-elastic- visco-plastic materials. Commun. Pure Appl. Analysis 11 (2012), 1185–1203.Google Scholar
4Buchukuri, T. and Gegelia, T.. Some dynamic problems of the theory of electroelasticity. Mem. Diff. Eqns Math. Phys. 10 (1997), 1–53.Google Scholar
5Clarke, F. H.. Optimization and nonsmooth analysis (Wiley, 1983).Google Scholar
6Cristescu, N. and Suliciu, I.. Viscoplasticity (Bucharest: Martinus Nijhoff, 1982).Google Scholar
7Denkowski, Z., Migorski, S. and Papageorgiou, N. S.. An introduction to nonlinear analysis: theory (Dordrecht: Kluwer Academic/Plenum, 2003).Google Scholar
8Denkowski, Z., Migórski, S. and Papageorgiou, N. S.. An introduction to nonlinear analysis: applications (Kluwer Academic/Plenum, 2003).Google Scholar
9Han, W. and Sofonea, M.. Quasistatic contact problems in viscoelasticity and viscoplasticity. Studies in Advanced Mathematics, vol. 30 (American Mathematical Society/International Press, 2002).Google Scholar
10Han, W., Sofonea, M. and Kazmi, K.. A frictionless contact problem for electro-elastic-visco-plastic materials. Computat. Meth. Appl. Mech. Engng 196 (2007), 3915–3926.CrossRefGoogle Scholar
11Ionescu, I. R. and Sofonea, M.. Functional and numerical methods in viscoplasticity (Oxford University Press, 1993).CrossRefGoogle Scholar
12Maceri, F. and Bisegna, P.. The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Modelling 28 (1998), 19–28.Google Scholar
13Migórski, S., Ochal, A. and Sofonea, M.. History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics. Nonlin. Analysis 12 (2011), 3384–3396.Google Scholar
14Migórski, S., Ochal, A. and Sofonea, M.. Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems. Advances in Mechanics and Mathematics, vol. 26 (Springer, 2013).Google Scholar
15Mindlin, R. D.. Polarisation gradient in elastic dielectrics. Int. J. Solids Struct. 4 (1968), 637–663.Google Scholar
16Mindlin, R. D.. Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Int. J. Solids Struct. 4 (1969), 1197–1213.CrossRefGoogle Scholar
17Mindlin, R. D.. Elasticity, piezoelasticity and crystal lattice dynamics. J. Elasticity 4 (1972), 217–280.CrossRefGoogle Scholar
18Panagiotopoulos, P. D.. Inequality problems in mechanics and applications (Birkhauser, 1985).Google Scholar
19Panagiotopoulos, P. D.. Hemivariational inequalities, applications in mechanics and engineering (Springer, 1993).CrossRefGoogle Scholar
20Shillor, M., Sofonea, M. and Telega, J. J.. Models and analysis of quasistatic contact (Springer, 2004).Google Scholar
21Sofonea, M. and Essoufi, El H.. A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Analysis 9 (2004), 229–242.Google Scholar
22Sofonea, M. and Essoufi, El H.. Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004), 613–631.Google Scholar
23Solymar, L. and Au, L. B.. Solutions manual for lectures on the electrical properties of materials, 5th edn (Oxford University Press, 1993).Google Scholar
24Toupin, R. A.. The elastic dielectrics. J. Ration. Mech. Analysis 5 (1956), 849–915.Google Scholar
25Toupin, R. A.. A dynamical theory of elastic dielectrics. Int. J. Engng Sci. 1 (1963), 101–126.CrossRefGoogle Scholar
26Voigt, W.. Lehrbuch der Kristall-Physik (Leipzig: Teubner, 1910).Google Scholar
27Yang, J. (ed.). Special topics in the theory of piezoelectricity (Springer, 2009).Google Scholar
28Yang, J. and Yang, J. S.. An introduction to the theory of piezoelectricity (Springer, 2005).Google Scholar
29Zgurovsky, M., Mel'nik, V. and Kasyanov, P.. Evolution inclusions and variation inequalities for Earth data processing. II. Differential-operator inclusions and evolution variation inequalities for Earth data processing. Advances in Mechanics and Mathematics, vol. 25 (Springer, 2011).Google Scholar
30Zgurovsky, M., Kasyanov, P., Kapustyan, O., Valero, J. and Zadoianchuk, N.. Evolution inclusions and variation inequalities for Earth data processing. III. Long-time behavior of evolution inclusions solutions in Earth data analysis. Advances in Mechanics and Mathematics, vol. 27 (Springer, 2012).Google Scholar