A quasi-static contact problem with friction in electro viscoelasticity with long-term memory body with damage and thermal effects.

Document Type : Research Paper


1 Laboratory of Operator Theory and PDE: Foundations and Applications, Faculty of Exact Sciences, University of El Oued, 39000, El Oued, Algeria

2 Department of Mathematics, University of El Oued, 39000 El Oued, Algeria


In this paper, we consider a mathematical model that describes the quasi-static process of contact between a piezoelectric body and a deformable foundation. A nonlinear thermo-electro-viscoelastic constitutive law with long term memory and damage is used and the contact is described with the normal compliance condition and a version of Coulomb’s law of friction. We derive variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field, the temperature field and the damage field, existence and uniqueness of a weak solution of the problem is proved. The proof is based on arguments of time-dependent variational inequalities, parabolic inequalities, differential equations and fixed points.


[1] M.S. Aamur, T.H. Ammar and L. Maiza, Analysis of a frictional contact problem for viscoelastic piezoelectric
materials, Aust. J. Math. Anal. Appl. 17 (2020), no. 1, Article 6.
[2] R.C. Batra and J.S. Yang. Saint-Venant’s principle in linear piezoelectricity, J. Elastic. 38 (1995), no. 2, 209–218.
[3] V.L. Berdichevsky, Variational principles, Variational Principles of Continuum Mechanics Springer, Berlin, Heidelberg, 2009.
[4] H. Brezis, Equations et in´equations non lin´eaires dans les espaces vectoriels en dualit´e, Ann. Inst. Fourier 18
(1968), 115–175.
[5] H. L. Dai and X. Wang, Thermo-electro-elastic transient responses in piezoelectric hollow structures, Inter. J. Sol.
Struct. 42 (2005), 1151—1171.
[6] M. Fremond and B. Nedjar, Damage in concrete: the unilateral phenomenon, Nucl. Eng. Des. 156 (1995), 323–335.
[7] M. Fremond and B. Nedjar, Damage, gradient of damage and principle of virtual work, Internat. J. Solids Struct.
33 (1996), 1083–1103.
[8] M. Fremond, KL. Kuttler, B. Nedjar and M. Shillor, One-dimensional models of damage, Adv. Math. Sci. Appl.
8 (1998), 541–570.
[9] A.C. FisherCripps Introduction to Contact Mechanics, Mechanical Engineering Series, Springer, 2000.
[10] I. Figueiredo and L. Trabucho, A class of contact and friction dynamic problems in thermoelasticity and in
thermoviscoelasticity, Internat. J. Engrg. Sci. 33 (1995), no. 1, 45—66.
[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.
[12] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced
Mathematics, Americal Mathematical Society and International Press, 2002.
[13] I.R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press,
Oxford, 1993.
[14] J.L. Lions and E. Magenes, Probl`emes aux limites non homog`enes et applications, vol. 1, Dunod, Paris, 1968.
[15] Z. Lerguet, M. Shillor and M. Sofonea, A frictional contact problem for an electro-viscoelastic body, Electron. J.
Diferential Equ. 2007 (2007), Article ID 170.
[16] A.D. Muradova and G.E. Stavroulakis, A unilateral contact model with buckling in von K´arm´an plates, Nonlinear
Anal. 8 (2007), no. 4, 1261–1271.
[17] F. Messelmi and B. Merouani, Quasi-static evolution of damage in thermo-viscoplastic materials, An. Univ.
Oradea Fasc. Mat, Tom XVII (2010), no. 2, 133–148.
[18] A. Merouani and F. Messelmi, Dynamic evolution of damage in elasticthermo-viscoplastic materials, Electron. J.
Differential Equ. 2010 (2010), no. 129, pp. 1–15.
[19] R.D. Mindlin, Polarisation gradient in elastic dielectrics, Int. J. Solids Struct. 4 (1968), 637–663.
[20] R.D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin
dielectric films, Int. J. Solids 4 (1969), 1197–1213.
[21] R.D. Mindlin, Elasticity, Piezoelectricity and Cristal lattice dynamics, J. Elastic. 4 (1972), 217–280.
[22] J. Necas and I. Hlav´acek, Mathematical Theory of Elastic and Elastoplastic Bodies: An Introduction, Elsevier,
Amsterdam, 1981.[23] M. Rochdi, M. Shillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, J.
Elast. 51 (1998), 105–126.
[24] A. Rodr´ıguez-Ar´os, J.M. Via˜no and M. Sofonea, A class of evolutionary variational inequalities with Volterra-type
term, Math. Models Methods Appl. Sci. 14 (2004), no. 4, 557-–577.
[25] M. Sofonea, W. Han and M. Shillor, Analysis and Approximations of Contact Problems with Adhesion Or Damage,
Pure and Applied Mathematics Chapman and Hall/CRC Press, Boca Raton, Florida, 2006.
[26] M. Sofonea, A. Rodr´ıguez-Ar´os, J.M. Via˜no, A class of integro-differential variational inequalities with applications
to viscoelastic contact, Math. Comput. Modell. 41 (2005), no. 11–12, 1355—1369.
Volume 13, Issue 2
July 2022
Pages 205-220
  • Receive Date: 12 August 2021
  • Revise Date: 07 December 2021
  • Accept Date: 25 January 2022
  • First Publish Date: 12 February 2022