Numerical analysis of a frictionless viscoelastic piezoelectric contact problem
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 667-682.

In this work, we consider the quasistatic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelastic constitutive law is employed to model the piezoelectric material and the normal compliance condition is used to model the contact. The variational formulation is derived in a form of a coupled system for the displacement and electric potential fields. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.

DOI : 10.1051/m2an:2008022
Classification : 65N15, 65N30, 74D10, 74M15, 74S05, 74S20
Mots-clés : piezoelectricity, viscoelasticity, normal compliance, error estimates, numerical simulations 
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     title = {Numerical analysis of a frictionless viscoelastic piezoelectric contact problem},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {667--682},
     publisher = {EDP-Sciences},
     volume = {42},
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     doi = {10.1051/m2an:2008022},
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     zbl = {1142.74029},
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     url = {http://www.numdam.org/articles/10.1051/m2an:2008022/}
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Barboteu, Mikael; Fernández, Jose Ramon; Ouafik, Youssef. Numerical analysis of a frictionless viscoelastic piezoelectric contact problem. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 667-682. doi : 10.1051/m2an:2008022. http://www.numdam.org/articles/10.1051/m2an:2008022/

[1] P. Alart, M. Barboteu and F. Lebon, Solution of frictional contact problems by an EBE preconditioner. Comput. Mech. 20 (1997) 370-378. | Zbl

[2] F. Auricchio, P. Bisegna and C. Lovadina, Finite element approximation of piezoelectric plates. Internat. J. Numer. Methods Engrg. 50 (2001) 1469-1499. | MR | Zbl

[3] M. Barboteu, J.R. Fernández and Y. Ouafik, Numerical analysis of two frictionless elastic-piezoelectric contact problems. J. Math. Anal. Appl. 339 (2008) 905-917. | MR | Zbl

[4] R.C. Batra and J.S. Yang, Saint-Venant's principle in linear piezoelectricity. J. Elasticity 38 (1995) 209-218. | MR | Zbl

[5] 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. 103, Kluwer Acad. Publ., Dordrecht (2002) 347-354. | MR | Zbl

[6] P.G. Ciarlet, The finite element method for elliptic problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991) 17-352. | MR | Zbl

[7] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer Verlag, Berlin (1976). | MR | Zbl

[8] J.R. Fernández, M. Sofonea and J.M. Viaño, A frictionless contact problem for elastic-viscoplastic materials with normal compliance: Numerical analysis and computational experiments. Numer. Math. 90 (2002) 689-719. | MR | Zbl

[9] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). | MR | Zbl

[10] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society-International Press (2002). | MR | Zbl

[11] W. Han, M. Shillor and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage. J. Comput. Appl. Math. 137 (2001) 377-398. | MR | Zbl

[12] 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 48 (2005) 209-232. | MR | Zbl

[13] T. Ideka, Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990).

[14] A. Klarbring, A. Mikelić and M. Shillor, Frictional contact problems with normal compliance. Internat. J. Engrg. Sci. 26 (1988) 811-832. | MR | Zbl

[15] F. Maceri and B. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Modelling 28 (1998) 19-28. | MR | Zbl

[16] J.A.C. Martins and J.T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11 (1987) 407-428. | MR | Zbl

[17] R.D. Mindlin, Polarisation gradient in elastic dielectrics. Internat. J. Solids Structures 4 (1968) 637-663. | Zbl

[18] R.D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Internat. J. Solids Structures 5 (1969) 1197-1213.

[19] R.D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics. J. Elasticity 4 (1972) 217-280.

[20] A. Morro and B. Straughan, A uniqueness theorem in the dynamical theory of piezoelectricity. Math. Methods Appl. Sci. 14 (1991) 295-299. | MR | Zbl

[21] Y. Ouafik, A piezoelectric body in frictional contact. Bull. Math. Soc. Sci. Math. Roumanie 48 (2005) 233-242. | MR | Zbl

[22] M. Sofonea and E.-H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004) 25-40. | MR | Zbl

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

[24] M. Sofonea and Y. Ouafik, A piezoelectric contact problem with normal compliance. Appl. Math. 32 (2005) 425-442. | EuDML | MR | Zbl

[25] R.A. Toupin, The elastic dielectrics. J. Rational Mech. Anal. 5 (1956) 849-915. | MR | Zbl

[26] R.A. Toupin, Stress tensors in elastic dielectrics. Arch. Rational Mech. Anal. 5 (1960) 440-452. | MR | Zbl

[27] R.A. Toupin, A dynamical theory of elastic dielectrics. Internat. J. Engrg. Sci. 1 (1963) 101-126. | MR

[28] N. Turbé and G.A. Maugin, On the linear piezoelectricity of composite materials. Math. Methods Appl. Sci. 14 (1991) 403-412. | MR | Zbl

[29] P. Wriggers, Computational Contact Mechanics. Wiley-Verlag (2002). | Zbl

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