VARIATIONAL ANALYSIS OF AN ELECTRO-VISCOELASTIC CONTACT PROBLEM WITH FRICTION AND ADHESION

- Journal title : Journal of the Korean Mathematical Society
- Volume 53, Issue 1, 2016, pp.161-185
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2016.53.1.161

Title & Authors

VARIATIONAL ANALYSIS OF AN ELECTRO-VISCOELASTIC CONTACT PROBLEM WITH FRICTION AND ADHESION

CHOUGUI, NADHIR; DRABLA, SALAH; HEMICI, NACERDINNE;

CHOUGUI, NADHIR; DRABLA, SALAH; HEMICI, NACERDINNE;

Abstract

We consider a mathematical model which describes the quasistatic frictional contact between a piezoelectric body and an electrically conductive obstacle, the so-called foundation. A nonlinear electro-viscoelastic constitutive law is used to model the piezoelectric material. Contact is described with Signorini's conditions and a version of Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equation. We derive a variational formulation for the model, in the form of a system for the displacements, the electric potential and the adhesion. Under a smallness assumption which involves only the electrical data of the problem, we prove the existence of a unique weak solution of the model. The proof is based on arguments of time-dependent quasi-variational inequalities, differential equations and Banach's fixed point theorem.

Keywords

Piezoelectric material;electro-viscoelastic;frictional contact;nonlocal Coulomb's law;adhesion;quasi-variational inequality;weak solution;fixed point theorem;

Language

English

References

1.

K. T. Andrews, L. Chapman, J. R. Fernandez, M. Fisackerly, M. Shillor, L. Vanerian, and T. Van Houten, A membrane in adhesive contact, SIAM J. Appl. Math. 64 (2003), no. 1, 152-169.

2.

K. T. Andrews and M. Shillor, Dynamic adhesive contact of a membrane, Adv. Math. Sci. Appl. 13 (2003), no. 1, 343-356.

3.

R. C. Batra and J. S. Yang, Saint-Venant's principle in linear piezoelectricity, J. Elas- ticity 38 (1995), no. 2, 209-218.

4.

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

5.

T. Buchukuri and T. Gegelia, Some dynamic problems of the theory of electro-elasticity, Mem. Differential Equations Math. Phys. 10 (1997), 1-53.

6.

O. Chau, J. R. Fernandez, M. Shillor, and M. Sofonea, Variational and numerical anal- ysis of a quasistatic viscoelastic contact problem with adhesion, J. Comput. Appl. Math. 159 (2003), no. 2, 431-465.

7.

O. Chau, M. Shillor, and M. Sofonea, Dynamic frictionless contact with adhesion, J. Appl. Math. Phys. (ZAMP) 55 (2004), no. 1, 32-47.

8.

M. Cocu, Existence of solution of Signorini problem with friction, Internat. J. Engrg. Sci. 39 (1984), no. 5, 567-575.

9.

A. Curnier and C. Talon, A model of adhesion added to contact with friction, in Contact Mechanics, 161-168, JAC Martins and MDP Monteiro Marques (Eds.), Kluwer, Dordrecht, 2002.

10.

S. Drabla and M. Sofonea, Analysis of a Signorini's problem with friction, IMA J. Appl. Math. 63 (1999), no. 2, 113-130.

11.

S. Drabla and Z. Zellagui, Analyse of a electro-elastic contact problem with friction and adhesion, Studia Universitatis. "Babes-Bolyai", Mathematica, LIV(1) March, 2009.

12.

S. Drabla, Variational analysis and the convergence of the finite element approximation of an electro-elastic contact problem with adhesion, Arab J. Sci. Eng. 36 (2011), no. 8, 1501-1515.

13.

M. Fremond, Equilibre des structures qui adherent a leur support, C. R. Acad. Sci. Paris Ser. II Mec. Phys. Chim. Sci. Univers Sci. Terre 295 (1982), no. 11, 913-916.

14.

M. Fremond, Adherence des Solides, J. Mec. Theor. Appl. 6 (1987), no. 3, 383-407.

15.

M. Fremond, Non-Smooth Thermomechanics, Springer, Berlin, 2002.

16.

W. Han, K. L. Kuttler, M. Shillor, and M. Sofonea, Elastic beam in adhesive contact, Internat. J. Solids Structures 39 (2002), no. 5, 1145-1164.

17.

W. Han and M. Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM J. Numer. Anal. 38 (2000), no. 2, 556-579.

18.

W. Han, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, American Mathematical Society, Providence, RI - Intl.Press, Sommerville, MA, 2002.

19.

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

20.

Z. Lerguet, M. Shillor, and M. Sofonea, A frictional contact problem for an electro-viscoelastic body, Electron. J. Differential Equations 2007 (2007), no. 170, 1-16.

21.

Z. Lerguet, M. Sofonea, and S. Drabla, Analysis of frictional contact problem with adhesion, Acta Math. Univ. Comenian. 77 (2008), no. 2, 181-198.

22.

F. Maceri and P. Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support, Math. Comput. Modelling 28 (1998), no. 4-8, 19-28.

23.

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

24.

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

25.

V. Z. Patron and B. A. Kudryavtsev, Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids, Gordon & Breach, London, 1988.

26.

M. Raous, L. Cangemi, and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact, Comput. Methods Appl. Mech. Engrg. 177 (1999), no. 3-4, 383- 399.

27.

J. Rojek and J. J. Telega, Contact problems with friction, adhesion and wear in ortho- pedic biomechanics. I: general developments, J. Theoretical Appl. Mechanics 39 (2001), no. 3, 655-677.

28.

J. Rojek, J. J. Telega, and S. Stupkiewicz, Contact problems with friction, adhesion and wear in orthopedic biomechanics, II: numerical implementation and application to implanted knee joints, J. Theoretical Appl. Mechanics 39 (2001), 679-706.

29.

M. Shillor, M. Sofonea, and J. J. Telega, Models and Analysis of Quasistatic Contact. Variational Methods, Lect. Notes Phys., vol. 655, Springer, Berlin, 2004.

30.

M. Sofonea, W. Han, and M. Shillor, Analysis and approximation of contact problems with adhesion or damage, Pure and Applied Mathematics (Boca Raton), vol. 276, Chapman & Hall/CRC Press, Florida, 2006.

31.

M. Sofonea and El H. Essoufi, A Piezoelectric contact problem with slip dependent coefficient of friction, Math. Model. Anal. 9 (2004), no. 3, 229-242.

32.

M. Sofonea, Quasistatic frictional contact of a viscoelastic piezoelectric body, Adv. Math. Sci. Appl. 14 (2004), no. 2, 613-631.

33.

M. Sofonea and A. Matei, Variational inequalities with applications, Advances in Mecanics and Mathematics Springer, 2009.