# American Institute of Mathematical Sciences

January  2018, 23(1): 387-400. doi: 10.3934/dcdsb.2018027

## Numerical treatment of contact problems with thermal effect

 Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Lojasiewicza 6, 30-348 Krakow, Poland

* Corresponding author: Anna Ochal

Received  December 2016 Revised  May 2017 Published  January 2018

Fund Project: The research was supported by the National Center Science of Poland under Maestro Advanced Project no DEC-2012/06/A/ST1/00262.

The paper deals with the formulation and the finite element approximation of a quasi-static thermoviscoelastic problem which describes frictional contact between a deformable body and a rigid foundation. The contact is modeled by normal damped response condition whereas the friction is described by the Coulomb law of dry friction. The weak formulation of the model consists of a coupled system of the variational inequality for the displacement and the parabolic equation for the temperature. The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.

Citation: Anna Ochal, Michal Jureczka. Numerical treatment of contact problems with thermal effect. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 387-400. doi: 10.3934/dcdsb.2018027
##### References:
 [1] K. Bartosz, D. Danan and P. Szafraniec, Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law, Computers & Mathematics with Applications, 73 (2017), 727-746. doi: 10.1016/j.camwa.2016.12.026. Google Scholar [2] O. Chau and R. Oujja, Numerical treatment of a class of thermal contact problems, Mathematics and Computers in Simulation, 118 (2015), 163-176. doi: 10.1016/j.matcom.2014.12.007. Google Scholar [3] L. Gasinki, A. Ochal and M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Journal of Analysis and its Applications (ZAA), 34 (2015), 251-275. doi: 10.4171/ZAA/1538. Google Scholar [4] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society and International Press, 2002. [5] S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems Advances in Mechanics and Mathematics, vol. 26, Springer, 2013. doi: 10.1007/978-1-4614-4232-5. Google Scholar [6] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, 1993. doi: 10.1007/978-3-642-51677-1. [7] M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Springer-Verlag, 2004. doi: 10.1007/b99799. [8] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics Lecture Note Series, vol. 398, Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. Google Scholar [9] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

##### References:
 [1] K. Bartosz, D. Danan and P. Szafraniec, Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law, Computers & Mathematics with Applications, 73 (2017), 727-746. doi: 10.1016/j.camwa.2016.12.026. Google Scholar [2] O. Chau and R. Oujja, Numerical treatment of a class of thermal contact problems, Mathematics and Computers in Simulation, 118 (2015), 163-176. doi: 10.1016/j.matcom.2014.12.007. Google Scholar [3] L. Gasinki, A. Ochal and M. Shillor, Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage, Journal of Analysis and its Applications (ZAA), 34 (2015), 251-275. doi: 10.4171/ZAA/1538. Google Scholar [4] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society and International Press, 2002. [5] S. Migorski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems Advances in Mechanics and Mathematics, vol. 26, Springer, 2013. doi: 10.1007/978-1-4614-4232-5. Google Scholar [6] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, 1993. doi: 10.1007/978-3-642-51677-1. [7] M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Springer-Verlag, 2004. doi: 10.1007/b99799. [8] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics Lecture Note Series, vol. 398, Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. Google Scholar [9] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
Body's shape and temperature at t = 0:25
Body's shape and temperature at t = 0:5
Body's shape and temperature at t = 0:75
Body's shape and temperature at t = 1
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