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Frictional contact of an anisotropic piezoelectric plate

Published online by Cambridge University Press:  23 January 2009

Isabel N. Figueiredo  and
Georg Stadler
Isabel N. Figueiredo
Affiliation:
Centro de Matemática da Universidade de Coimbra (CMUC), Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal. isabelf@mat.uc.pt
Georg Stadler
Affiliation:
Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, USA. georgst@ices.utexas.edu
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Abstract

The purpose of this paper is to derive and study a new asymptoticmodel for the equilibrium state of a thin anisotropicpiezoelectric plate in frictional contact with a rigid obstacle.In the asymptotic process, the thickness of the piezoelectricplate is driven to zero and the convergence of the unknowns isstudied. This leads to two-dimensional Kirchhoff-Love plateequations, in which mechanical displacement and electric potentialare partly decoupled. Based on this model numerical examples arepresented that illustrate the mutual interaction between themechanical displacement and the electric potential. We observethat, compared to purely elastic materials, piezoelectric bodiesyield a significantly different contact behavior.

Keywords

Contactfrictionasymptotic analysisanisotropicmaterialpiezoelectricityplate
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009 , pp. 149 - 172
Copyright
© EDP Sciences, SMAI, 2008

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References

Bernadou, M. and Haenel, C., Modelization and numerical approximation of piezoelectric thin shells. I. The continuous problems. Comput. Methods Appl. Mech. Engrg. 192 (2003) 4003–4043. CrossRef
P. Bisegna, F. Lebon and F. Maceri, The unilateral frictional contact of a piezoelectric body with a rigid support, in Contact mechanics (Praia da Consolação, 2001), Solid Mech. Appl., Kluwer Acad. Publ., Dordrecht (2002) 347–354.
P.G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates, Studies in Mathematics and its Applications 27. North-Holland Publishing Co., Amsterdam (1997).
P.G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells, Studies in Mathematics and its Applications 29. North-Holland Publishing Co., Amsterdam (2000).
Ciarlet, P.G. and Destuynder, P., Une justification d'un modèle non linéaire en théorie des plaques. C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A33–A36.
Ciarlet, P.G. and Destuynder, P., A justification of the two-dimensional linear plate model. J. Mécanique 18 (1979) 315–344.
Collard, C. and Miara, B., Two-dimensional models for geometrically nonlinear thin piezoelectric shells. Asymptotic Anal. 31 (2002) 113–151.
Costa, L., Figueiredo, I., Leal, R., Oliveira, P. and Stadler, G., Modeling and numerical study of actuator and sensor effects for a laminated piezoelectric plate. Comput. Struct. 85 (2007) 385–403. CrossRef
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der mathematischen Wissenschaften 219. Springer-Verlag, Berlin (1976).
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28. SIAM, Philadelphia (1999).
Figueiredo, I. and Leal, C., A piezoelectric anisotropic plate model. Asymptotic Anal. 44 (2005) 327–346.
Figueiredo, I. and Leal, C., A generalized piezoelectric Bernoulli-Navier anisotropic rod model. J. Elasticity 85 (2006) 85–106. CrossRef
R. Glowinski, Numerical Methods for Nonlinear Variational Inequalities. Springer-Verlag, New York (1984).
J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities, Nonconvex Optimization and its Applications 35. Kluwer Academic Publishers, Dordrecht (1999).
Hüeber, S., Matei, A. and Wohlmuth, B.I., A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48 (2005) 209–232.
Hüeber, S., Stadler, G. and Wohlmuth, B., A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J. Sci. Comp. 30 (2008) 572–596.
T. Ikeda, Fundamentals of Piezoelectricity. Oxford University Press (1990).
N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988).
Klinkel, S. and Wagner, W., A geometrically non-linear piezoelectric solid shell element based on a mixed multi-field variational formulation. Int. J. Numer. Meth. Engng. 65 (2005) 349–382. CrossRef
Léger, A. and Miara, B., Mathematical justification of the obstacle problem in the case of a shallow shell. J. Elasticity 9 (2008) 241–257. CrossRef
J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics 323 . Springer-Verlag, Berlin (1973).
Maceri, F. and Bisegna, P., The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Model. 28 (1998) 19–28. CrossRef
Maugin, G.A. and Attou, D., An asymptotic theory of thin piezoelectric plates. Quart. J. Mech. Appl. Math. 43 (1990) 347–362. CrossRef
Miara, B., Justification of the asymptotic analysis of elastic plates. I. The linear case. Asymptotic Anal. 9 (1994) 47–60.
Rahmoune, M., Benjeddou, A. and Ohayon, R., New thin piezoelectric plate models. J. Int. Mat. Sys. Struct. 9 (1998) 1017–1029. CrossRef
Raoult, A. and Sène, A., Modelling of piezoelectric plates including magnetic effects. Asymptotic Anal. 34 (2003) 1–40.
Sabu, N., Vibrations of thin piezoelectric flexural shells: Two-dimensional approximation. J. Elast. 68 (2002) 145–165. CrossRef
Sene, A., Modelling of piezoelectric static thin plates. Asymptotic Anal. 25 (2001) 1–20.
R.C. Smith, Smart Material Systems: Model Development, Frontiers in Applied Mathematics 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005).
Sofonea, M. and Essoufi, El-H., A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9 (2004) 229–242.
Sofonea, M. and Essoufi, El-H., Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004) 613–631.
L. Trabucho and J.M. Viaño, Mathematical modelling of rods, in Handbook of Numerical Analysis IV , P.G. Ciarlet and J.-L. Lions Eds., Elsevier, Amsterdam, North-Holland (1996) 487–974.
Weller, T. and Licht, C., Analyse asymptotique de plaques minces linéairement piézoélectriques. C. R. Math. Acad. Sci. Paris 335 (2002) 309–314. CrossRef