Glasnik Matematicki, Vol. 45, No.1 (2010), 125-137.

SLIP-DEPENDENT FRICTION IN DYNAMIC ELECTROVISCOELASTICITY

Mostafa Kabbaj and El-Hassan Essoufi

Department of Mathematics, Faculty of Sciences and Techniques Settat, University Hassan Premier, Km 3, route Casablanca, B.P. 577, Settat, Morocco
e-mail: essoufi@gmail.com

Abstract.   Existence and uniqueness of a weak solution for dynamical frictional contact between an electro-viscoelastic body and a rigid electrically non-conductive obstacle is established. The contact is modelled with a simplified version of Coulomb's law of dry friction in which the coefficient of friction depends on the slip. The proof is based on the regularization method, Faedo-Galerkin method, compactness and lower semicontinuity arguments.

2000 Mathematics Subject Classification.   37L65, 49J40, 74A55, 74D05, 74H20, 74H25.

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

DOI: 10.3336/gm.45.1.10

References:

1. R. C. Batra and J. S. Yang, Saint-Venant's principle in linear piezoelectricity, J. 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, 347-354.
   MathSciNet

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

4. 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. Sci. Math. Roumanie (N.S.)} 48(96) (2005), 181-195.
   MathSciNet

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

6. I. R. Ionescu and Q-L. Nguyen, Dynamic contact problems with slip-dependent friction in viscoelasticity, Int. J. Math. Comput. Sci. 12 (2002), 71-80.
   MathSciNet

7. M. Kabbaj and El-H. Essoufi, Frictional contact problem in dynamic electroelasticity, Glas. Mat. Ser. III 43(63) (2008), 137-158.
   MathSciNet     CrossRef

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

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

10. R. D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films, Int. J. Solids Structures 4 (1969), 1197-1208.
    CrossRef

11. R. D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics, J. Elasticity 4 (1972), 217-280.
    CrossRef

12. M. Sofonea and El-H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction, Math. Model. Anal. 9 (2004), 229-242.
    MathSciNet

13. M. Sofonea and El-H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body, Adv. Math. Sci. Appl. 14 (2004), 613-631.
    MathSciNet

14. B. Tengiz and G. Tengiz, Some dynamic problems of the theory of electroelasticity, Mem. Differential Equations Math. Phys. 10 (1997), 1-53.
    MathSciNet

15. R. A. Toupin, The elastic dielectrics, J. Rational Mech. Anal. 5 (1956), 849-915.
    MathSciNet

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