A history-dependent frictional contact problem with wear for thermoviscoelastic materials
Abstract
In this manuscript we study a contact problem between a deformable viscoelastic body and a rigid foundation. Thermal effects, wear and friction between surfaces are taken into account. A variational formulation of the problem is supplied and an existence and uniqueness result is proved. The idea of the proof rested on a recent result on history-dependent quasivariational inequalities. Finally, a perturbation of the data is initiated and a convergence result is demonstrated when the perturbation parameter converges to zero.
Keyword : viscoelastic material, thermal effects, friction, history-dependent quasivariational inequality, convergence result
How to Cite
Chouchane, L., & Selmani, L. (2019). A history-dependent frictional contact problem with wear for thermoviscoelastic materials. Mathematical Modelling and Analysis, 24(3), 351-371. https://doi.org/10.3846/mma.2019.022
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Shillor, M. Sofonea and J.J. Telega. Models and Analysis of Quasistatic Contact. Lecture Notes in Physics Springer, Berlin Heidelberg, 2004. https://doi.org/10.1007/b99799.
M. Sofonea, W. Han and M. Barboteu. Analysis of a viscoelastic contact problem with multivalued normal compliance and unilateral constraint. Comput. Methods Appl. Mech. Eng., 264:12–22, 2013. https://doi.org/10.1016/j.cma.2013.05.006.
M. Sofonea, W. Han and M. Shillor. Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman & Hall/ CRC, New York, 2006.
M. Sofonea, K. Kazmi, M. Barboteu and W. Han. Analysis and numerical solution of a piezoelectric frictional contact problem. Appl. Math. Model., 36(9):4483–4501, 2012. https://doi.org/10.1016/j.apm.2011.11.077.
M. Sofonea and A. Matei. History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math., 22(5):471–491, 2011. https://doi.org/10.1017/S0956792511000192.
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398, Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139104166.
M. Sofonea and F. Pătrulescu. A viscoelastic contact problem with adhesion and surface memory effects. Math. Model. Anal., 19(5):607–626, 2014. https://doi.org/10.3846/13926292.2014.979334.
M. Sofonea, F. Pătrulescu and A. Farca¸s. A viscoplastic contact problem with normal compliance, unilateral constraint and memory term. Appl. Math. Opt., 69(2):175–198, 2014. https://doi.org/10.1007/s00245-013-9216-2.
M. Sofonea and Y. Souleiman. A viscoelastic sliding contact problem with normal compliance, unilateral constraint and memory term. Mediterr. J. Math., 13(5):2863–2886, 2016. https://doi.org/10.1007/s00009-015-0661-9.
N. Strőmberg. Continuum Thermodynamics of Contact, Friction and Wear. Ph.D. Theisis, Linőkping University, Sweeden, 1995.
A. Capatina. Variational Inequalities and Frictional Contact Problems-Advances in Mechanics and Mathematics. Springer, New York, 2014.
O. Chau. A class of thermal sub-differential contact problems. AIMS Mathematics, 2(4):658–681, 2017. https://doi.org/10.3934/Math.2017.4.658.
J. Chen, W. Han and M. Sofonea. Numerical analysis of a quasistatic problem of sliding frictional contact with wear. Methods Appl. Anal., 7(4):687–704, 2000. https://doi.org/10.4310/MAA.2000.v7.n4.a5.
C. Ciulcu, T.V. Hoarau-Mante and M. Sofonea. Viscoelastic sliding contact problems with wear. Math. Comput. Modelling., 36(7–8):861–874, 2002. https://doi.org/10.1016/S0895-7177(02)00233-9.
C. Corduneanu. Problémes globaux dans la théorie des équations intégrales de Volterra. Ann. Math. Pures Appl., 67(1):349–363, 1965. https://doi.org/10.1007/BF02410815.
C. Eck, J. Jarušek and M. Krbeč. Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure Applied Mathematics, Chapman/CRC Press, New York, 2005.
I. Halvácek, J. Haslinger, J. Nečas and J. Lovišek. Solution of Variational Inequalities in Mechanics. Springer, New York, 1988. https://doi.org/10.1007/978- 1-4612-1048-1.
W. Han and M. Sofonea. Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity- Studies in Advanced Mathematics, volume 30. American Mathematical Society-International Press, 2002. https://doi.org/10.1090/amsip/030.
K. Kazmi, M. Barboteu, W. Han and M. Sofonea. Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics. Esaim Math. Model. Numer. Anal., 48(3):919–942, 2014. https://doi.org/10.1051/m2an/2013127.
J. J. Massera and J. J. Schãffer. Linear Differential Equations and Function Spaces. Academic Press, New York, 1966.
S. Migórski, A. Ochal and M. Sofonea. Non Linear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26, Springer, New York, 2013.
S. Migórski, M. Shillor and M. Sofonea. Special section on contact mechanics. Nonlinear Anal. Real World Appl, 22:435–436, 2015. https://doi.org/10.1016/j.nonrwa.2014.10.005.
P.D. Panagitopoulos. Inequality Problems in Mechanics and Applications. Birkhãuser, Boston, 1985. https://doi.org/10.1007/978-1-4612-5152-1.
M. Selmani and L. Selmani. Frictional contact problem for elastic-viscoplastic materials with thermal effects. Appl. Math. Mech. -Engl. Ed., 34(6):761–776, 2013. https://doi.org/10.1007/s10483-013-1705-7.
M. Shillor, M. Sofonea and J.J. Telega. Models and Analysis of Quasistatic Contact. Lecture Notes in Physics Springer, Berlin Heidelberg, 2004. https://doi.org/10.1007/b99799.
M. Sofonea, W. Han and M. Barboteu. Analysis of a viscoelastic contact problem with multivalued normal compliance and unilateral constraint. Comput. Methods Appl. Mech. Eng., 264:12–22, 2013. https://doi.org/10.1016/j.cma.2013.05.006.
M. Sofonea, W. Han and M. Shillor. Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman & Hall/ CRC, New York, 2006.
M. Sofonea, K. Kazmi, M. Barboteu and W. Han. Analysis and numerical solution of a piezoelectric frictional contact problem. Appl. Math. Model., 36(9):4483–4501, 2012. https://doi.org/10.1016/j.apm.2011.11.077.
M. Sofonea and A. Matei. History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math., 22(5):471–491, 2011. https://doi.org/10.1017/S0956792511000192.
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398, Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139104166.
M. Sofonea and F. Pătrulescu. A viscoelastic contact problem with adhesion and surface memory effects. Math. Model. Anal., 19(5):607–626, 2014. https://doi.org/10.3846/13926292.2014.979334.
M. Sofonea, F. Pătrulescu and A. Farca¸s. A viscoplastic contact problem with normal compliance, unilateral constraint and memory term. Appl. Math. Opt., 69(2):175–198, 2014. https://doi.org/10.1007/s00245-013-9216-2.
M. Sofonea and Y. Souleiman. A viscoelastic sliding contact problem with normal compliance, unilateral constraint and memory term. Mediterr. J. Math., 13(5):2863–2886, 2016. https://doi.org/10.1007/s00009-015-0661-9.
N. Strőmberg. Continuum Thermodynamics of Contact, Friction and Wear. Ph.D. Theisis, Linőkping University, Sweeden, 1995.