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April  2021, 20(4): 1521-1543. doi: 10.3934/cpaa.2021031

Numerical analysis of a thermal frictional contact problem with long memory

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author

Received  October 2020 Revised  January 2021 Published  March 2021

Fund Project: This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grand Agreement No. 823731 CONMECH and Major Scientifc Research Project of Zhejiang Lab No. 2019DB0ZX01

The objective in this paper is to study a thermal frictional contact model. The deformable body consists of a viscoelastic material and the process is assumed to be dynamic. It is assumed that the material behaves in accordance with Kelvin-Voigt constitutive law and the thermal effect is added. The variational formulation of the model leads to a coupled system including a history-dependent hemivariational inequality for the displacement field and an evolution equation for the temperature field. In study of this system, we first consider a fully discrete scheme of it and then focus on deriving error estimates for numerical solutions. Under appropriate assumptions of solution regularity, an optimal order error estimate is obtained. At the end of this manuscript, we report some numerical simulation results for the contact problem so as to verify the theoretical results.

Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1521-1543. doi: 10.3934/cpaa.2021031
References:
[1]

K. T. AndrewsM. ShillorS. Wright and A. Klarbring, A dynamic thermviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci., 35 (1997), 1291-1309.  doi: 10.1016/S0020-7225(97)87426-5.  Google Scholar

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K. E. Atkinson and W. Han, Theortical Numerical Analysis: A Functional Analysis Framework, 3rd edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar

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C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-boundary Problems, John Wiley, Chichester, 1984.  Google Scholar

[4]

K. BartoszX. ChengP. YuY. J. Kalita and C Zheng, Rothe method for parabolic variational–hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862.  doi: 10.1016/j.jmaa.2014.09.078.  Google Scholar

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M. CampoJ. R. Fernández and T. V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math., 196 (2006), 180-197.  doi: 10.1016/j.cam.2005.08.025.  Google Scholar

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A. Capatina, Variational Inequalities Frictional Contact Problems. vol. 31, Advances in Mechanics and Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-10163-7.  Google Scholar

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O. Chau, Numerical analysis of a thermal contact problem with adhesion, Comp. Appl. Math., 37 (2018), 5424-5455.  doi: 10.1007/s40314-018-0642-2.  Google Scholar

[9]

X. ChengS. MigórskiA. Ochal and M. Sofonea, Anaysis of two quasistatic history-dependent contact models, Discrete Contin. Dyn. Ser. Ser. B, 19 (2014), 2425-2445.  doi: 10.3934/dcdsb.2014.19.2425.  Google Scholar

[10]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.  Google Scholar

[11]

W. HanS. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar

[12] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, RI-International Press, Somerville, 2002.  doi: 10.1090/amsip/030.  Google Scholar
[13]

W. HanM. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085.  Google Scholar

[14]

L. JianuM. Shillor and M. Sofonea, A viscoelastic frictioness contact problem with adhesion, Appl. Anal., 80 (2001), 233-255.  doi: 10.1080/00036810108840990.  Google Scholar

[15]

A. Matei and M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009.  Google Scholar

[16]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[17]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, M. Dekker, New york, 1995.  Google Scholar

[18]

J. Nedoma and J. Stehlik, Mathematical and computational methods and algorithms in biomechanics, in Human Skeletal Systems, John Wiley and Sons, 2011. Google Scholar

[19]

J. Ogorzaly, Dynamic contact problem with thermal effect, Georgian Math. J, 2016. doi: 10.1515/gmj-2016-0025.  Google Scholar

[20]

M. Rochdi and M. Shillor, Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity, Quart. Appl. Math, 58 (2000), 543-560.  doi: 10.1090/qam/1770654.  Google Scholar

[21]

M. SofoneaF. Pǎtrulescu and Y. Souleiman, Analysis of contact problem with wear and unilateral constraint, Appl. Anal., 95 (2017), 2590-2607.  doi: 10.1080/00036811.2015.1102892.  Google Scholar

[22]

W. XuZ. HuangW. HanW. Chen and C. Wang, Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration, Comput. Math. Appl., 77 (2019), 2596-2607.  doi: 10.1016/j.camwa.2018.12.038.  Google Scholar

[23]

H. XuanX. ChengW. Han and Q. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math. Theory Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.  Google Scholar

show all references

References:
[1]

K. T. AndrewsM. ShillorS. Wright and A. Klarbring, A dynamic thermviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci., 35 (1997), 1291-1309.  doi: 10.1016/S0020-7225(97)87426-5.  Google Scholar

[2]

K. E. Atkinson and W. Han, Theortical Numerical Analysis: A Functional Analysis Framework, 3rd edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar

[3]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-boundary Problems, John Wiley, Chichester, 1984.  Google Scholar

[4]

K. BartoszX. ChengP. YuY. J. Kalita and C Zheng, Rothe method for parabolic variational–hemivariational inequalities, J. Math. Anal. Appl., 423 (2015), 841-862.  doi: 10.1016/j.jmaa.2014.09.078.  Google Scholar

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Elements Methods, 3rd edition, Spring-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[6]

M. CampoJ. R. Fernández and T. V. Hoarau-Mantel, Analysis of two frictional viscoplastic contact problems with damage, J. Comput. Appl. Math., 196 (2006), 180-197.  doi: 10.1016/j.cam.2005.08.025.  Google Scholar

[7]

A. Capatina, Variational Inequalities Frictional Contact Problems. vol. 31, Advances in Mechanics and Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-10163-7.  Google Scholar

[8]

O. Chau, Numerical analysis of a thermal contact problem with adhesion, Comp. Appl. Math., 37 (2018), 5424-5455.  doi: 10.1007/s40314-018-0642-2.  Google Scholar

[9]

X. ChengS. MigórskiA. Ochal and M. Sofonea, Anaysis of two quasistatic history-dependent contact models, Discrete Contin. Dyn. Ser. Ser. B, 19 (2014), 2425-2445.  doi: 10.3934/dcdsb.2014.19.2425.  Google Scholar

[10]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.  Google Scholar

[11]

W. HanS. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM J. Math. Anal., 46 (2014), 3891-3912.  doi: 10.1137/140963248.  Google Scholar

[12] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, RI-International Press, Somerville, 2002.  doi: 10.1090/amsip/030.  Google Scholar
[13]

W. HanM. Sofonea and M. Barboteu, Numerical analysis of elliptic hemivariational inequalities, SIAM J. Numer. Anal., 55 (2017), 640-663.  doi: 10.1137/16M1072085.  Google Scholar

[14]

L. JianuM. Shillor and M. Sofonea, A viscoelastic frictioness contact problem with adhesion, Appl. Anal., 80 (2001), 233-255.  doi: 10.1080/00036810108840990.  Google Scholar

[15]

A. Matei and M. Sofonea, Variational Inequalities with Applications: A Study of Antiplane Frictional Contact Problems, Springer-Verlag, New York, 2009.  Google Scholar

[16]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[17]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, M. Dekker, New york, 1995.  Google Scholar

[18]

J. Nedoma and J. Stehlik, Mathematical and computational methods and algorithms in biomechanics, in Human Skeletal Systems, John Wiley and Sons, 2011. Google Scholar

[19]

J. Ogorzaly, Dynamic contact problem with thermal effect, Georgian Math. J, 2016. doi: 10.1515/gmj-2016-0025.  Google Scholar

[20]

M. Rochdi and M. Shillor, Existence and uniqueness for a quasistatic frictional bilateral contact problem in thermoviscoelasticity, Quart. Appl. Math, 58 (2000), 543-560.  doi: 10.1090/qam/1770654.  Google Scholar

[21]

M. SofoneaF. Pǎtrulescu and Y. Souleiman, Analysis of contact problem with wear and unilateral constraint, Appl. Anal., 95 (2017), 2590-2607.  doi: 10.1080/00036811.2015.1102892.  Google Scholar

[22]

W. XuZ. HuangW. HanW. Chen and C. Wang, Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration, Comput. Math. Appl., 77 (2019), 2596-2607.  doi: 10.1016/j.camwa.2018.12.038.  Google Scholar

[23]

H. XuanX. ChengW. Han and Q. Xiao, Numerical analysis of a dynamic contact problems with history-dependent operators, Numer. Math. Theory Methods Appl., 13 (2020), 569-594.  doi: 10.4208/nmtma.oa-2019-0130.  Google Scholar

Figure 1.  Reference configuration of the two-dimensional example
Figure 2.  the deformed configuration at several times
Figure 3.  the deformed configuration at $ c_1 = c_2 = a = 0.1 $ and $ c_1 = c_2 = a = 1 $
Figure 4.  the temperature field at $ \theta_R = 0 $ and $ \theta_R = 8 $
Figure 5.  the deformed configuration at t = 0.5s and t = 1s
Figure 6.  the deformed configuration at $ \theta_R = 0 $ and $ \theta_R = 10 $
Figure 7.  the deformed configuration at $ \theta_R = 0 $ and $ \theta_R = 8 $
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