Sliding Contact problem with wear for thermoviscoelastic material
Résumé
We study a quasistatic sliding contact problem with wear between a viscoelastic with long memory and a rigid moving foundation. We model the wear with a version of Archard's law. Thermal effects is taken into account. We establish a variational formulation of the model and we prove the existence and uniqueness of the weak solution. The proofs are based on the nonlinear equations involving the monotone operators, the classical result of nonlinear first order evolution inequalities, and the fixed-point arguments. We also establish the dependent results with respect to certain data.
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Références
1 : S. Adly, O. Chau and M. Rochdi, Solvability of a class of thermal dynamical contact problems with subdifferential conditions, Numerical algebra, Control and optimization, 2012, 2 (1), pp. 91-104. 10. 3934/naco.2012.2.91. hal-00683058.
2 : Amassad, A. kuttler, K. L., Rochdi, M., and Shillor, M. Quasistatic thermoviscoelastic contact problem with slip dependent friction coefficient. Math. Comput. Model., 36(7-8), 839-854 (2002).
3 : M. Campo and J. R. Fernandez, Numerical analysis of a quasistatic thermoviscoelastic frictional contact problem, Computational mechanics, 2005, 35 (6), pp,459-469. 10.1007/s00466-004-0635-4. hal-04912121.
4 : Chau, O. R. Oujja and M. Rochdi, A mathematical analysis of dynamical frictional contact model in thermoviscoelasticity, Discrete and continuous dynamical systems series s Volume 1, Number 1, March 2008.
5 : Chau, O. and Awbi, B. Quasistatic thermoviscoelastic frictional contact problem with damped response. Appl. Anal., 83(6), 635-648 (2004)/
6 : J. Chen, W. Han and M. Sofonea, Numerical analysis of a quasistatic problem of sliding frictional contact with wear, Methods and applications of analysis, 2000.
7 : C. Ciulcu, T. V. Hoarau-Mante and M. Sofonea, Viscoelastic sliding contact problems with wear, Mathematical and computer modelling 36 (2002) 861-874.
8 : A. D. Rodrigue, J. M. Viano and M. Sofonea, A class of variational inequalities with Volterra-type term*, Mathematical Models and Methods in Applied Sciences, Vol. 14, No. 04, pp. 557-577 (2004)
9 : L. Selmani and M. Selmani, Analysis of a viscoelastic contact problem with normal damped response and damage, Bull. Belg. Math. Soc. 13 (2006), 207-220.
10 : M. Selmani, Frictional contact problem with wear for electro-viscoelastic materials with long memory, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 461-479.
11 : M. Selmani and M. Sofonea, Viscoelastic frictionless contact problem with adhesion, Hindawi Publishing Corporation, Journal of Inequalities and Applications, Volume 2006, Article ID 36130, Pages 1--22, DOI 10.1155/JIA/2006/36130.
12 : M. Selmani. A dynamic problem with adhesion and damage in electro-viscoelasticity with long-term memory. JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only], 10(1):1-19,2009.
13 : M. Selmani. A frictional contact problem involving piezoelectric materials with long memory. Mediterranean Journal of Mathematics 12(3):1177-1197, 2015. https://doi.org/10.1007/s00009-014-0430-1.
14 : P. Shi and M. Shillor, Uniqueness and stability of the solution to a thermoelastic contact problem, Eur. J. Appl. Math. 1 (1990), 371-387.
15 : P. Shi, M. Shillor, and X. L. Zou, Numerical solution to the one-dimensional problems of thermoelastic contact, Comput. Math. Appl. 22 (1991), 65-78.
16 : Shillor. M, Sofonea. M and Telega. J.J, Models and Analysis of Quasistatic Contact. Lecturre Notes in Physics Springer, Berlin Heidelberg, 2004. https://doi.org/10.1007/b99799.
17 : Sofonea. M, Han. W and Shillor. M, Analysis and Approximation of Contact Problems with Adhesion or Damage, Chapman-Hall/ CRC, New York (2006).
18 : Sofonea. M and Matei. A, Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398, Cambridge University Press, Cambridge, 2012.
19 : M. Sofonea and F. Patrulescu, A viscoelastic contact problem with adhesion and surface memory effects, Mathematical modeling and analysis, Volume 19 Number 5, November 2014, 607-626.
20 : M. Sofonea, A. Rodreguez-Aros and J. M. Viano, A class of integro-differential variational inequalities with applications to viscoelastic contact, Mathematical and computer modelling 41 (2005) 1355-1369.
21 : M. Sofonea and S. Migorski, A class of history-dependent variational-hemivariational inequalities, Nonlinear Differ. Equ. Appl. (2016), Springer International Publishing, 1021-9722/16/030001-23.
2 : Amassad, A. kuttler, K. L., Rochdi, M., and Shillor, M. Quasistatic thermoviscoelastic contact problem with slip dependent friction coefficient. Math. Comput. Model., 36(7-8), 839-854 (2002).
3 : M. Campo and J. R. Fernandez, Numerical analysis of a quasistatic thermoviscoelastic frictional contact problem, Computational mechanics, 2005, 35 (6), pp,459-469. 10.1007/s00466-004-0635-4. hal-04912121.
4 : Chau, O. R. Oujja and M. Rochdi, A mathematical analysis of dynamical frictional contact model in thermoviscoelasticity, Discrete and continuous dynamical systems series s Volume 1, Number 1, March 2008.
5 : Chau, O. and Awbi, B. Quasistatic thermoviscoelastic frictional contact problem with damped response. Appl. Anal., 83(6), 635-648 (2004)/
6 : J. Chen, W. Han and M. Sofonea, Numerical analysis of a quasistatic problem of sliding frictional contact with wear, Methods and applications of analysis, 2000.
7 : C. Ciulcu, T. V. Hoarau-Mante and M. Sofonea, Viscoelastic sliding contact problems with wear, Mathematical and computer modelling 36 (2002) 861-874.
8 : A. D. Rodrigue, J. M. Viano and M. Sofonea, A class of variational inequalities with Volterra-type term*, Mathematical Models and Methods in Applied Sciences, Vol. 14, No. 04, pp. 557-577 (2004)
9 : L. Selmani and M. Selmani, Analysis of a viscoelastic contact problem with normal damped response and damage, Bull. Belg. Math. Soc. 13 (2006), 207-220.
10 : M. Selmani, Frictional contact problem with wear for electro-viscoelastic materials with long memory, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 461-479.
11 : M. Selmani and M. Sofonea, Viscoelastic frictionless contact problem with adhesion, Hindawi Publishing Corporation, Journal of Inequalities and Applications, Volume 2006, Article ID 36130, Pages 1--22, DOI 10.1155/JIA/2006/36130.
12 : M. Selmani. A dynamic problem with adhesion and damage in electro-viscoelasticity with long-term memory. JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only], 10(1):1-19,2009.
13 : M. Selmani. A frictional contact problem involving piezoelectric materials with long memory. Mediterranean Journal of Mathematics 12(3):1177-1197, 2015. https://doi.org/10.1007/s00009-014-0430-1.
14 : P. Shi and M. Shillor, Uniqueness and stability of the solution to a thermoelastic contact problem, Eur. J. Appl. Math. 1 (1990), 371-387.
15 : P. Shi, M. Shillor, and X. L. Zou, Numerical solution to the one-dimensional problems of thermoelastic contact, Comput. Math. Appl. 22 (1991), 65-78.
16 : Shillor. M, Sofonea. M and Telega. J.J, Models and Analysis of Quasistatic Contact. Lecturre Notes in Physics Springer, Berlin Heidelberg, 2004. https://doi.org/10.1007/b99799.
17 : Sofonea. M, Han. W and Shillor. M, Analysis and Approximation of Contact Problems with Adhesion or Damage, Chapman-Hall/ CRC, New York (2006).
18 : Sofonea. M and Matei. A, Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398, Cambridge University Press, Cambridge, 2012.
19 : M. Sofonea and F. Patrulescu, A viscoelastic contact problem with adhesion and surface memory effects, Mathematical modeling and analysis, Volume 19 Number 5, November 2014, 607-626.
20 : M. Sofonea, A. Rodreguez-Aros and J. M. Viano, A class of integro-differential variational inequalities with applications to viscoelastic contact, Mathematical and computer modelling 41 (2005) 1355-1369.
21 : M. Sofonea and S. Migorski, A class of history-dependent variational-hemivariational inequalities, Nonlinear Differ. Equ. Appl. (2016), Springer International Publishing, 1021-9722/16/030001-23.
Publiée
2025-12-20
Rubrique
Advances in Nonlinear Analysis and Applications
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