Document Type : Original Article


Laboratory of Operator Theory and PDE, Foundations and Applications, 39000, El Oued, Algeria


We consider a quasistatic frictional contact problem with a subdifferential boundary condition for general thermo-electro-elastic-viscoplastic materials. The frictional contact is modeled by a general velocity-dependent dissipation function. We derive a weak formulation of the system and then prove the existence of a unique weak solution to the problem. The proof is based on arguments of evolutionary variational inequalities, parabolic equations, the variational equation, differential equations, and the fixed-point theorem. Finally, we describe a number of concrete contact and friction conditions to which our results apply.


[1] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Application to Free Boundary Problems, Wiley-Interscience, Chichester-New York, 1984.
[2] S. Boutechebak, A dynamic problem of frictionless contact for elastic-thermoviscoplastic materials with damage, Int. J. Pure Appl. Math., 86 (2013), 173–197.
[3] H. Brezis, Equations et inequations non lineaires dans les espaces vectoriels en dualit´e, Ann. Inst. Fourier 18 (1968), 115–175.
[4] O. Chau, D. Motreanu, and M. Sofonea, Quasistatic frictional problems for elastic and viscoelastic materials, Appl. Math., 47(4) (2002), 341-360.
[5] H. L. Dai and X. Wang, Thermo-electro-elastic transient responses in piezoelectric hollow structures, Int. J. Sol. Struct., 42 (2005), 1151–1171.
[6] T. Haje Ammar, S. Drabla, and B. Benabderrahmane, Analysis and approximation of frictionless contact problems between two piezoelectric bodies with adhesion, Georgian Math. J., 44 (2014), 1-15.
[7] T. Haje Ammar, A. Saidi, and A. Azeb Ahmed, Dynamic contact problem with adhesion and damage between thermo-electro-elasto-viscoplastic bodies, C. R. Mecani., 345 (2017), 329–336.
[8] A. Hamidat and A. Aissaoui, A quasistatic frictional contact problem with normal damped response for thermo-electro-elastic-viscoelastic bodies, Adv. Math.: Sci. J. 10(12) (2021).
[9] A. Hamidat and A. Aissaoui, A quasi-static contact problem with friction in electro viscoelasticity
with long-term memory body with damage and thermal effects, Int. J. Nonlinear Anal. Appl., 13(2) (2022), 205-220.
[10] W. Han, M. Sofonea, and K. Kazmi, Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials, Compu. Methods. Appl. Mech. Engrg., 196 (2007), 3915–3926.
[11] Z. Lerguet, M. Shillor, and M. Sofonea, A frictional contact problem for an electroviscoelastic body, Electron. J. Differ. Equat., 2007(170) (2007), 1–16.
[12] J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites Non Lineaires, Dunod, 1969.
[13] J. L. Lions and E. Mag´enes, Problemes aux limites non homogenes et applications, vol. 1 et 2, Dunod, Paris, 1968.
[14] F. Messelmi, B. Merouani, and M. Meflah, Nonlinear thermoelasticity problem, Anal. Univer. Oradea, Fasc. Math. Tome 15 (2008), 207–217.
[15] R. D. Mindlin, Polarisation gradient in elastic dielectrics, Int. J. Solids Struct., 4(6) (1968), 637–642.
[16] J. Necas and J. Kratochvil, On existence of the solution boundary value problems for elastic inelastic solids, Comment. Math. Univ. Carolina, 14 (1973), 755–760.
[17] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhauser, Basel, 1985.
[18] M. Sofonea, Functional Methods in Thermo-ElastoVisco-Plasticity, Ph. D. Thesis, Univ of Bucharest, 1988. [in Romanian]
[19] M. Sofonea, Quasistatic processes for elastic-viscoplastic materials with internal state variables, Ann. Sci. Univer. Clermont. Math., 94(25) (1989), 47-60.
[20] M. Sofonea and M. Shillor, Variational analysis of quasistatic viscoplastic contact problems with friction, Comm. Appl. Anal., 5 (2001), 135–151.
[21] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, Cambridge University Press, 2012.