Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-24T16:11:57.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysisof a frictionless viscoelastic piezoelectric contact problem

Published online by Cambridge University Press:  05 June 2008

Mikael Barboteu ,
Jose Ramon Fernández  and
Youssef Ouafik
Mikael Barboteu
Affiliation:
Laboratoire de Mathématiques et Physique pour les Systèmes (MEPS), Bâtiment B3, case courrier 12, 52 Avenue Paul Alduy, 66860 Perpignan, France. [email protected]; [email protected]
Jose Ramon Fernández
Affiliation:
Departamento de Matemática Aplicada, Facultade de Matemáticas, Campus Sur s/n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. [email protected]
Youssef Ouafik
Affiliation:
Laboratoire de Mathématiques et Physique pour les Systèmes (MEPS), Bâtiment B3, case courrier 12, 52 Avenue Paul Alduy, 66860 Perpignan, France. [email protected]; [email protected]
Get access

Abstract

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

Keywords

Piezoelectricityviscoelasticitynormal complianceerror estimatesnumerical simulations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008 , pp. 667 - 682
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alart, P., Barboteu, M. and Lebon, F., Solution of frictional contact problems by an EBE preconditioner. Comput. Mech. 20 (1997) 370–378. CrossRef
Auricchio, F., Bisegna, P. and Lovadina, C., Finite element approximation of piezoelectric plates. Internat. J. Numer. Methods Engrg. 50 (2001) 1469–1499. 3.0.CO;2-I>CrossRef
Barboteu, M., Fernández, J.R. and Ouafik, Y., Numerical analysis of two frictionless elastic-piezoelectric contact problems. J. Math. Anal. Appl. 339 (2008) 905–917. CrossRef
Batra, R.C. and Yang, J.S., Saint-Venant's principle in linear piezoelectricity. J. Elasticity 38 (1995) 209–218. 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. 103, Kluwer Acad. Publ., Dordrecht (2002) 347–354.
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.
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer Verlag, Berlin (1976).
Fernández, J.R., Sofonea, M. and Viaño, J.M., A frictionless contact problem for elastic-viscoplastic materials with normal compliance: Numerical analysis and computational experiments. Numer. Math. 90 (2002) 689–719.
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984).
W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society-International Press (2002).
Han, W., Shillor, M. and Sofonea, M., Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage. J. Comput. Appl. Math. 137 (2001) 377–398. CrossRef
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 48 (2005) 209–232.
T. Ideka, Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990).
Klarbring, A., Mikelić, A. and Shillor, M., Frictional contact problems with normal compliance. Internat. J. Engrg. Sci. 26 (1988) 811–832. CrossRef
Maceri, F. and Bisegna, B., The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Modelling 28 (1998) 19–28. CrossRef
Martins, J.A.C. and Existence, J.T. Oden and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11 (1987) 407–428. CrossRef
Mindlin, R.D., Polarisation gradient in elastic dielectrics. Internat. J. Solids Structures 4 (1968) 637–663. CrossRef
Mindlin, R.D., Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Internat. J. Solids Structures 5 (1969) 1197–1213. CrossRef
Mindlin, R.D., Elasticity, piezoelasticity and crystal lattice dynamics. J. Elasticity 4 (1972) 217–280. CrossRef
Morro, A. and Straughan, B., A uniqueness theorem in the dynamical theory of piezoelectricity. Math. Methods Appl. Sci. 14 (1991) 295–299. CrossRef
Ouafik, Y., A piezoelectric body in frictional contact. Bull. Math. Soc. Sci. Math. Roumanie 48 (2005) 233–242.
Sofonea, M. and Essoufi, E.-H., Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004) 25–40.
Sofonea, M. and Essoufi, E.-H., A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9 (2004) 229–242.
Sofonea, M. and Ouafik, Y., A piezoelectric contact problem with normal compliance. Appl. Math. 32 (2005) 425–442.
Toupin, R.A., The elastic dielectrics. J. Rational Mech. Anal. 5 (1956) 849–915.
Toupin, R.A., Stress tensors in elastic dielectrics. Arch. Rational Mech. Anal. 5 (1960) 440–452. CrossRef
Toupin, R.A., A dynamical theory of elastic dielectrics. Internat. J. Engrg. Sci. 1 (1963) 101–126. CrossRef
Turbé, N. and Maugin, G.A., On the linear piezoelectricity of composite materials. Math. Methods Appl. Sci. 14 (1991) 403–412. CrossRef
P. Wriggers, Computational Contact Mechanics. Wiley-Verlag (2002).