### Mostafa Kabbaj and El-H. Essoufi

Département de Mathématiques, Faculté des Sciences et Techniques, Université Moulay Ismail, B.P. 509-Boutalamine 52000, Errachidia, Maroc
e-mail: kabbaj.mostafa@gmail.com

Département de Mathématiques, Faculté des Sciences et Techniques, Universitée Hassan Premier, Km 3, route Casablanca, B.P. 577, Settat, Maroc
e-mail: essoufi@gmail.com

Abstract.   The dynamic evolution with frictional contact of a electroelastic body is considered. In modelling the contact, the Tresca model is used. We derive a variational formulation for the model in a form of a coupled system involving the displacement and the electric potential fields. We provide existence and uniqueness result. The proof is based on a regularization method, Galerkin method, compactness and lower semicontinuity arguments. Such a result extend the result obtained by Duvaut and Lions, where the analysis of friction in dynamic elasticity materials was provided. The novelty of this paper consists in the fact that here we take into account the piezoelectric properties of the materials.

2000 Mathematics Subject Classification.   35J20, 37L65, 46B50, 49J40, 65F22, 74H20, 74H25.

Key words and phrases.   Dynamic electroelasticity, second-order hyperbolic variational inequality, regularization method, Faedo-Galerkin method, compactness method, existence, uniqueness.

Full text (PDF) (free access)

DOI: 10.3336/gm.43.1.10

References:

1. R. C. Batra and J. S. Yang, Saint-Venant's principle in linear piezoelectricity, Journal of Elasticity 38 (1995), 209-218.
MathSciNet     CrossRef

2. P. Bisenga, F. Lebon and F. Maceri, The unilateral frictional contact of a piezoelectric body with a rigid support, in: Contact Mechanics, J. A. C. Martins and Manuel D. P. Monteiro Marques (Eds.), Kluwer, Dordrecht, 2002, p. 347-354.
MathSciNet

3. G. Duvaut, J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
MathSciNet

4. C. Eck, J. Jarusek and M. Krbec, Unilateral Contact Problems, Pure and Applied Mathematics 270, Chapman & Hall/CRC Press, Boca Raton, Florida, 2005.
MathSciNet

5. El H. Essoufi and M. Kabbaj, Existence of solutions of a dynamic Signorini's Problem with non local friction for viscoelastic Piezoelectric Materials, Bull. Math. Soc. SC. Math. Roumanie 48 (2005), 181-195.
MathSciNet

6. El H. Essoufi and M. Sofonea, A Piezoelectric Contact Problem with Slip Dependent Coefficient of friction, Mathematical Modelling and Analysis 9 (2004), 229-242.
MathSciNet

7. El H. Essoufi and M. Sofonea, Quasistatic Frictional Contact of a Viscoelastic Piezoelectric Body, Advances in Mathematical Sciences and Applications 14 (2004), 613-631.
MathSciNet

8. W. Han and M. Sofonea, Evolutionary Variational inequalities arising in viscoelastic contact problems, SIAM Journal of Numerical Analysis 38 (2000), 556-579.
MathSciNet     CrossRef

9. W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, Americal Mathematical Society - Intl. Press, 2002.
MathSciNet

10. T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1990.

11. I. R. Ionescu, Viscosity solutions for dynamic problems with slip-rate dependent friction, Quart. Appl. Math. I.X 3 (2002), 461-476.
MathSciNet

12. I. R. Ionescu, Q-L. Nguyen, S. Wolf, Slip-dependent friction in dynamic elasticity, Nonlinear Analysis 53 (2003), 375-390.
MathSciNet     CrossRef

13. K. L. Kuttler and M. Shillor, Dynamic contact with Signorini's condition and slip rate dependent friction, Electronic J. Diff. Eqns. 83 (2004), 1-21.
MathSciNet

14. K. L. Kuttler and M. Shillor, Dynamic contact with normal compliance wear and discontinuous friction coefficient, SIAM J. Math. Anal. 34 (2002), 1-27.
MathSciNet     CrossRef

15. F. Maceri and P. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support, Math. Comp. Modelling 28 (1998), 19-28.
MathSciNet     CrossRef

16. R. D. Mindlin, Polarisation gradient in elastic dielectrics, Int. J. Solids Structures 4 (1968), 637-663.
CrossRef

17. R. D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films, Int. J. Solids Structures 4 (1969), 1197-1213.
CrossRef

18. R. D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics, Journal of Elasticity 4 (1972), 217-280.

19. J. Necas and I. Hlavácek, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier Scientific Publishing Company, Amsterdam, Oxford, New York, 1981.
MathSciNet

20. B. Tengiz and G. Tengiz, Some Dynamic Problems of the Theory of electroelasticity, Memoirs on Differential Equations and Mathematical Physics 10 (1997), 1-53.
MathSciNet

21. R. A. Toupin, The elastic dielectrics, J. Rat. Mech. Analysis 5 (1956), 849-915.
MathSciNet

22. R. A. Toupin, A dynamical theory of elastic dielectrics, Int. J. Engrg. Sci. 1 (1963), 101-126.
MathSciNet     CrossRef