# Frictional contact of an anisotropic piezoelectric plate

Isabel N. Figueiredo; Georg Stadler

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 1, page 149-172
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topFigueiredo, Isabel N., and Stadler, Georg. "Frictional contact of an anisotropic piezoelectric plate." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 149-172. <http://eudml.org/doc/250573>.

@article{Figueiredo2009,

abstract = {
The purpose of this paper is to derive and study a new asymptotic
model for the equilibrium state of a thin anisotropic
piezoelectric plate in frictional contact with a rigid obstacle.
In the asymptotic process, the thickness of the piezoelectric
plate is driven to zero and the convergence of the unknowns is
studied. This leads to two-dimensional Kirchhoff-Love plate
equations, in which mechanical displacement and electric potential
are partly decoupled. Based on this model numerical examples are
presented that illustrate the mutual interaction between the
mechanical displacement and the electric potential. We observe
that, compared to purely elastic materials, piezoelectric bodies
yield a significantly different contact behavior.
},

author = {Figueiredo, Isabel N., Stadler, Georg},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Contact; friction; asymptotic analysis; anisotropic
material; piezoelectricity; plate; convergence; Kirchhoff-Love plate},

language = {eng},

month = {1},

number = {1},

pages = {149-172},

publisher = {EDP Sciences},

title = {Frictional contact of an anisotropic piezoelectric plate},

url = {http://eudml.org/doc/250573},

volume = {15},

year = {2009},

}

TY - JOUR

AU - Figueiredo, Isabel N.

AU - Stadler, Georg

TI - Frictional contact of an anisotropic piezoelectric plate

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2009/1//

PB - EDP Sciences

VL - 15

IS - 1

SP - 149

EP - 172

AB -
The purpose of this paper is to derive and study a new asymptotic
model for the equilibrium state of a thin anisotropic
piezoelectric plate in frictional contact with a rigid obstacle.
In the asymptotic process, the thickness of the piezoelectric
plate is driven to zero and the convergence of the unknowns is
studied. This leads to two-dimensional Kirchhoff-Love plate
equations, in which mechanical displacement and electric potential
are partly decoupled. Based on this model numerical examples are
presented that illustrate the mutual interaction between the
mechanical displacement and the electric potential. We observe
that, compared to purely elastic materials, piezoelectric bodies
yield a significantly different contact behavior.

LA - eng

KW - Contact; friction; asymptotic analysis; anisotropic
material; piezoelectricity; plate; convergence; Kirchhoff-Love plate

UR - http://eudml.org/doc/250573

ER -

## References

top- M. Bernadou and C. Haenel, Modelization and numerical approximation of piezoelectric thin shells. I. The continuous problems. Comput. Methods Appl. Mech. Engrg.192 (2003) 4003–4043. Zbl1052.74035
- 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. Zbl1053.74583
- P.G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates, Studies in Mathematics and its Applications27. North-Holland Publishing Co., Amsterdam (1997). Zbl0888.73001
- P.G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells, Studies in Mathematics and its Applications29. North-Holland Publishing Co., Amsterdam (2000).
- P.G. Ciarlet and P. Destuynder, Une justification d'un modèle non linéaire en théorie des plaques. C. R. Acad. Sci. Paris Sér. A-B287 (1978) A33–A36. Zbl0382.73012
- P.G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model. J. Mécanique18 (1979) 315–344. Zbl0415.73072
- C. Collard and B. Miara, Two-dimensional models for geometrically nonlinear thin piezoelectric shells. Asymptotic Anal.31 (2002) 113–151. Zbl1045.74034
- L. Costa, I. Figueiredo, R. Leal, P. Oliveira and G. Stadler, Modeling and numerical study of actuator and sensor effects for a laminated piezoelectric plate. Comput. Struct.85 (2007) 385–403.
- G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der mathematischen Wissenschaften219. Springer-Verlag, Berlin (1976).
- I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics28. SIAM, Philadelphia (1999). Zbl0939.49002
- I. Figueiredo and C. Leal, A piezoelectric anisotropic plate model. Asymptotic Anal.44 (2005) 327–346. Zbl1086.35108
- I. Figueiredo and C. Leal, A generalized piezoelectric Bernoulli-Navier anisotropic rod model. J. Elasticity85 (2006) 85–106. Zbl1104.74028
- R. Glowinski, Numerical Methods for Nonlinear Variational Inequalities. Springer-Verlag, New York (1984). Zbl0536.65054
- J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities, Nonconvex Optimization and its Applications35. Kluwer Academic Publishers, Dordrecht (1999). Zbl0949.65069
- S. Hüeber, A. Matei and B.I. Wohlmuth, 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. Zbl1105.74028
- S. Hüeber, G. Stadler and B. Wohlmuth, A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J. Sci. Comp.30 (2008) 572–596. Zbl1158.74045
- 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 Mathematics8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988). Zbl0685.73002
- S. Klinkel and W. Wagner, 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. Zbl1146.74052
- A. Léger and B. Miara, Mathematical justification of the obstacle problem in the case of a shallow shell. J. Elasticity9 (2008) 241–257. Zbl1133.74033
- J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics 323. Springer-Verlag, Berlin (1973).
- F. Maceri and P. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Model.28 (1998) 19–28. Zbl1126.74392
- G.A. Maugin and D. Attou, An asymptotic theory of thin piezoelectric plates. Quart. J. Mech. Appl. Math.43 (1990) 347–362. Zbl0704.73087
- B. Miara, Justification of the asymptotic analysis of elastic plates. I. The linear case. Asymptotic Anal.9 (1994) 47–60. Zbl0806.73029
- M. Rahmoune, A. Benjeddou and R. Ohayon, New thin piezoelectric plate models. J. Int. Mat. Sys. Struct.9 (1998) 1017–1029.
- A. Raoult and A. Sène, Modelling of piezoelectric plates including magnetic effects. Asymptotic Anal.34 (2003) 1–40. Zbl1050.74029
- N. Sabu, Vibrations of thin piezoelectric flexural shells: Two-dimensional approximation. J. Elast.68 (2002) 145–165. Zbl1046.74032
- A. Sene, Modelling of piezoelectric static thin plates. Asymptotic Anal.25 (2001) 1–20. Zbl0995.74038
- R.C. Smith, Smart Material Systems: Model Development, Frontiers in Applied Mathematics32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005). Zbl1086.74002
- M. Sofonea and El-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal.9 (2004) 229–242. Zbl1092.74029
- M. Sofonea and El-H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl.14 (2004) 613–631. Zbl1078.74036
- 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. Zbl0873.73041
- T. Weller and C. Licht, Analyse asymptotique de plaques minces linéairement piézoélectriques. C. R. Math. Acad. Sci. Paris335 (2002) 309–314.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.