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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

Hailing Xuan , and  Xiaoliang Cheng

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

* Corresponding author

Received  October 2020 Revised  January 2021 Published  April 2021 Early access  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.

Keywords: hemivariational inequality, history-dependent operators, numerical approximation, optimal order error estimate, thermal.
Mathematics Subject Classification: Primary: 65M15, 74G30, 65N22; Secondary: 74M15, 47J20.
Citation: Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1521-1543. doi: 10.3934/cpaa.2021031
References:
[1]

K. T. Andrews, M. Shillor, S. 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.

[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.

[3]

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

[4]

K. Bartosz, X. Cheng, P. Yu, Y. 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.

[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.

[6]

M. Campo, J. 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.

[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.

[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.

[9]

X. Cheng, S. Migórski, A. 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.

[10]

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

[11]

W. Han, S. 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.

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

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

[14]

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

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[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.

[21]

M. Sofonea, F. 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.

[22]

W. Xu, Z. Huang, W. Han, W. 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.

[23]

H. Xuan, X. Cheng, W. 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.

show all references

References:
[1]

K. T. Andrews, M. Shillor, S. 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.

[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.

[3]

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

[4]

K. Bartosz, X. Cheng, P. Yu, Y. 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.

[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.

[6]

M. Campo, J. 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.

[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.

[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.

[9]

X. Cheng, S. Migórski, A. 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.

[10]

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

[11]

W. Han, S. 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.

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

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

[14]

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

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[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.

[21]

M. Sofonea, F. 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.

[22]

W. Xu, Z. Huang, W. Han, W. 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.

[23]

H. Xuan, X. Cheng, W. 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.

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