Frictional contact of an anisotropic piezoelectric plate
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 149-172.

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.

DOI : https://doi.org/10.1051/cocv:2008022
Classification : 74K20,  78M35,  74M15,  74M10,  74F15
Mots clés : contact, friction, asymptotic analysis, anisotropic material, piezoelectricity, plate
@article{COCV_2009__15_1_149_0,
author = {Figueiredo, Isabel N. and Stadler, Georg},
title = {Frictional contact of an anisotropic piezoelectric plate},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {149--172},
publisher = {EDP-Sciences},
volume = {15},
number = {1},
year = {2009},
doi = {10.1051/cocv:2008022},
zbl = {1155.74031},
mrnumber = {2488573},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_1_149_0/}
}
Figueiredo, Isabel N.; Stadler, Georg. Frictional contact of an anisotropic piezoelectric plate. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 149-172. doi : 10.1051/cocv:2008022. http://www.numdam.org/item/COCV_2009__15_1_149_0/

[1] 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. | MR 1999616 | Zbl 1052.74035

[2] 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. | MR 1968676 | Zbl 1053.74583

[3] P.G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates, Studies in Mathematics and its Applications 27. North-Holland Publishing Co., Amsterdam (1997). | MR 1477663

[4] P.G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells, Studies in Mathematics and its Applications 29. North-Holland Publishing Co., Amsterdam (2000). | MR 1757535 | Zbl 0953.74004

[5] P.G. Ciarlet and P. Destuynder, Une justification d'un modèle non linéaire en théorie des plaques. C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A33-A36. | MR 495644 | Zbl 0382.73012

[6] P.G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model. J. Mécanique 18 (1979) 315-344. | MR 533827 | Zbl 0415.73072

[7] C. Collard and B. Miara, Two-dimensional models for geometrically nonlinear thin piezoelectric shells. Asymptotic Anal. 31 (2002) 113-151. | MR 1938601 | Zbl 1045.74034

[8] 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. | MR 2303505

[9] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der mathematischen Wissenschaften 219. Springer-Verlag, Berlin (1976). | MR 521262 | Zbl 0331.35002

[10] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28. SIAM, Philadelphia (1999). | MR 1727362 | Zbl 0939.49002

[11] I. Figueiredo and C. Leal, A piezoelectric anisotropic plate model. Asymptotic Anal. 44 (2005) 327-346. | MR 2176277 | Zbl 1086.35108

[12] I. Figueiredo and C. Leal, A generalized piezoelectric Bernoulli-Navier anisotropic rod model. J. Elasticity 85 (2006) 85-106. | MR 2265722 | Zbl 1104.74028

[13] R. Glowinski, Numerical Methods for Nonlinear Variational Inequalities. Springer-Verlag, New York (1984). | MR 737005 | Zbl 0536.65054

[14] J. Haslinger, M. Miettinen and P. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities, Nonconvex Optimization and its Applications 35. Kluwer Academic Publishers, Dordrecht (1999). | MR 1784436 | Zbl 0949.65069

[15] 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. | MR 2153041 | Zbl 1105.74028

[16] 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. | MR 2385876 | Zbl 1158.74045

[17] T. Ikeda, Fundamentals of Piezoelectricity. Oxford University Press (1990).

[18] 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). | MR 961258 | Zbl 0685.73002

[19] 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. | Zbl 1146.74052

[20] A. Léger and B. Miara, Mathematical justification of the obstacle problem in the case of a shallow shell. J. Elasticity 9 (2008) 241-257. | MR 2387957 | Zbl 1133.74033

[21] J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics 323. Springer-Verlag, Berlin (1973). | MR 600331 | Zbl 0268.49001

[22] F. Maceri and P. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Model. 28 (1998) 19-28. | MR 1616376 | Zbl 1126.74392

[23] G.A. Maugin and D. Attou, An asymptotic theory of thin piezoelectric plates. Quart. J. Mech. Appl. Math. 43 (1990) 347-362. | MR 1070961 | Zbl 0704.73087

[24] B. Miara, Justification of the asymptotic analysis of elastic plates. I. The linear case. Asymptotic Anal. 9 (1994) 47-60. | MR 1285016 | Zbl 0806.73029

[25] M. Rahmoune, A. Benjeddou and R. Ohayon, New thin piezoelectric plate models. J. Int. Mat. Sys. Struct. 9 (1998) 1017-1029.

[26] A. Raoult and A. Sène, Modelling of piezoelectric plates including magnetic effects. Asymptotic Anal. 34 (2003) 1-40. | MR 1981723 | Zbl 1050.74029

[27] N. Sabu, Vibrations of thin piezoelectric flexural shells: Two-dimensional approximation. J. Elast. 68 (2002) 145-165. | MR 2024309 | Zbl 1046.74032

[28] A. Sene, Modelling of piezoelectric static thin plates. Asymptotic Anal. 25 (2001) 1-20. | MR 1814987 | Zbl 0995.74038

[29] R.C. Smith, Smart Material Systems: Model Development, Frontiers in Applied Mathematics 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005). | MR 2132740 | Zbl 1086.74002

[30] M. Sofonea and El-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9 (2004) 229-242. | MR 2099952 | Zbl 1092.74029

[31] M. Sofonea and El-H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004) 613-631. | MR 2111832 | Zbl 1078.74036

[32] 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. | MR 1422507 | Zbl 0873.73041

[33] T. Weller and C. Licht, Analyse asymptotique de plaques minces linéairement piézoélectriques. C. R. Math. Acad. Sci. Paris 335 (2002) 309-314. | MR 1933680 | Zbl 1019.74024