doi: 10.3934/eect.2021057
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Impulsive hemivariational inequality for a class of history-dependent quasistatic frictional contact problems

1. 

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

2. 

Department of Mathematics and Statistics, Shanxi Datong University, Datong, Shanxi 037009, China

3. 

Department of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China

* Corresponding author: Jinrong Wang

Received  February 2021 Revised  July 2021 Early access November 2021

This paper deals with a class of history-dependent frictional contact problem with the surface traction affected by the impulsive differential equation. The weak formulation of the contact problem is a history-dependent hemivariational inequality with the impulsive differential equation. By virtue of the surjectivity of multivalued pseudomonotone operator theorem and the Rothe method, existence and uniqueness results on the abstract impulsive differential hemivariational inequalities is established. In addition, we consider the stability of the solution to impulsive differential hemivariational inequalities in relation to perturbation data. Finally, the existence and uniqueness of weak solution to the contact problem is proved by means of abstract results.

Citation: Furi Guo, Jinrong Wang, Jiangfeng Han. Impulsive hemivariational inequality for a class of history-dependent quasistatic frictional contact problems. Evolution Equations and Control Theory, doi: 10.3934/eect.2021057
References:
[1]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, New York, 1984.

[2]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[3]

C. Carstensen and J. Gwinner, A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems, Ann. Mat. Pura Appl., 177 (1999), 363-394.  doi: 10.1007/BF02505918.

[4]

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

[5]

Z. Denkowski and S. Migórski, Hemivariational inequalities in thermoviscoelasticity, Nonlinear Anal., 63 (2005), 87-97.  doi: 10.1016/j.na.2005.01.011.

[6]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Plenum Publishers, Boston, 2003. doi: 10.1007/978-1-4419-9158-4.

[7]

A. D. Drozdov, Finite Elasticity and Viscoelasticity: A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore, 1996. doi: 10.1142/2905.

[8]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.

[9]

A. FarcasF. Patrulescu and M. Sofonea, A history-dependent contact problem with unilateral constraint, Ann. Acad. Rom. Sci. Ser. Math. Appl., 4 (2012), 90-96. 

[10]

M. Frigon and D. O'Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl., 193 (1995), 96-113.  doi: 10.1006/jmaa.1995.1224.

[11]

J. HanY. Li and S. Migórski, Analysis of an adhesive contact problem for viscoelastic materials with long memory, J. Math. Anal. Appl., 427 (2015), 646-668.  doi: 10.1016/j.jmaa.2015.02.055.

[12]

W. Han, S. Migórski and M. Sofonea, Advances in Variational and Hemivariational Inequalities with Applications: Theory, Numerical Analysis, and Applications, Advances in Mechanics and Mathematics, Springer, 2015. doi: 10.1007/978-3-319-14490-0.

[13]

W. HanS. Migórski and M. Sofonea, Analysis of a general dynamic history-dependent variational hemivariational inequality, Nonlinear Anal. Real World Appl., 36 (2017), 69-88.  doi: 10.1016/j.nonrwa.2016.12.007.

[14]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, vol. 30. Americal Mathematical Society, Providence, International Press, Somerville, 2002. doi: 10.1090/amsip/030.

[15]

S. Migórski, Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems, Comput. Math. Appl., 52 (2006), 677-698.  doi: 10.1016/j.camwa.2006.10.007.

[16]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.

[17]

S. Migórski and P. Gamorski, A new class of quasistatic frictional contact problems governed by a variational-hemivariational inequality, Nonlinear Anal. Real World Appl., 50 (2019), 583-602.  doi: 10.1016/j.nonrwa.2019.05.014.

[18]

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435.  doi: 10.1137/080733231.

[19]

S. MigórskiA. Ochal and M. Sofonea, Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact, Math. Models Methods Appl. Sci., 18 (2008), 271-290.  doi: 10.1142/S021820250800267X.

[20]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, NewYork, 2013. doi: 10.1007/978-1-4614-4232-5.

[21]

S. MigórskiA. Ochal and M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 22 (2015), 604-618.  doi: 10.1016/j.nonrwa.2014.09.021.

[22]

S. MigórskiA. Ochal and M. Sofonea, Analysis of frictional contact problem for viscoelastic materials with long memory, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 687-705.  doi: 10.3934/dcdsb.2011.15.687.

[23]

S. Migórski and S. Zeng, Mixed variational inequalities driven by fractional evolutionary equations, Acta Math. Sci. Ser. B, 39 (2019), 461-468.  doi: 10.1007/s10473-019-0211-9.

[24]

S. Migórski and S. Zeng, A class of differential hemivariational inequalities in Banach spaces, J. Global Optim., 72 (2018), 761-779.  doi: 10.1007/s10898-018-0667-5.

[25]

S. Migórski and S. Zeng, Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics, Numer. Algorithms, 82 (2019), 423-450.  doi: 10.1007/s11075-019-00667-0.

[26]

A. C. Pipkin, Lectures on Viscoelasticity Theory, Applied Mathematical Sciences, Springer, New York, 1972. doi: 10.1007/978-1-4615-9970-8.

[27]

S. ShenF. LiuJ. ChenI. Turner and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218 (2012), 10861-10870.  doi: 10.1016/j.amc.2012.04.047.

[28]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods, Springer, Berlin, 2004. doi: 10.1007/b99799.

[29]

M. Sofonea and A. Farcaş, Analysis of a history-dependent frictional contact problem, Appl. Anal., 93 (2014), 428-444.  doi: 10.1080/00036811.2013.778981.

[30]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics: Preliminaries on Functional Analysis, London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166.

[31]

M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics, European J. Appl. Math., 22 (2011), 471-491.  doi: 10.1017/S0956792511000192.

[32]

M. Sofonea and F. Pătrulescu, Analysis of a history-dependent frictionless contact problem, Math. Mech. Solids, 18 (2012), 409-430.  doi: 10.1177/1081286512440004.

[33]

M. SofoneaF. Pătrulescu and A. Farcas, A viscoplastic contact problem with normal compliance, unilateral constraint and memory term, Appl. Math. Optim., 69 (2014), 175-198.  doi: 10.1007/s00245-013-9216-2.

[34]

G. XueF. Lin and B. Qin, Solvability and optimal control of fractional differential hemivariational inequalities, Optimization, 3 (2020), 1-32.  doi: 10.1080/02331934.2020.1786089.

[35]

S. Zeng and S. Migórski, A class of time-fractional hemivariational inequalities with application to frictional contact problem, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 34-48.  doi: 10.1016/j.cnsns.2017.07.016.

[36]

S. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072.

show all references

References:
[1]

C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, New York, 1984.

[2]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[3]

C. Carstensen and J. Gwinner, A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems, Ann. Mat. Pura Appl., 177 (1999), 363-394.  doi: 10.1007/BF02505918.

[4]

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

[5]

Z. Denkowski and S. Migórski, Hemivariational inequalities in thermoviscoelasticity, Nonlinear Anal., 63 (2005), 87-97.  doi: 10.1016/j.na.2005.01.011.

[6]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Plenum Publishers, Boston, 2003. doi: 10.1007/978-1-4419-9158-4.

[7]

A. D. Drozdov, Finite Elasticity and Viscoelasticity: A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore, 1996. doi: 10.1142/2905.

[8]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.

[9]

A. FarcasF. Patrulescu and M. Sofonea, A history-dependent contact problem with unilateral constraint, Ann. Acad. Rom. Sci. Ser. Math. Appl., 4 (2012), 90-96. 

[10]

M. Frigon and D. O'Regan, Existence results for first-order impulsive differential equations, J. Math. Anal. Appl., 193 (1995), 96-113.  doi: 10.1006/jmaa.1995.1224.

[11]

J. HanY. Li and S. Migórski, Analysis of an adhesive contact problem for viscoelastic materials with long memory, J. Math. Anal. Appl., 427 (2015), 646-668.  doi: 10.1016/j.jmaa.2015.02.055.

[12]

W. Han, S. Migórski and M. Sofonea, Advances in Variational and Hemivariational Inequalities with Applications: Theory, Numerical Analysis, and Applications, Advances in Mechanics and Mathematics, Springer, 2015. doi: 10.1007/978-3-319-14490-0.

[13]

W. HanS. Migórski and M. Sofonea, Analysis of a general dynamic history-dependent variational hemivariational inequality, Nonlinear Anal. Real World Appl., 36 (2017), 69-88.  doi: 10.1016/j.nonrwa.2016.12.007.

[14]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, vol. 30. Americal Mathematical Society, Providence, International Press, Somerville, 2002. doi: 10.1090/amsip/030.

[15]

S. Migórski, Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems, Comput. Math. Appl., 52 (2006), 677-698.  doi: 10.1016/j.camwa.2006.10.007.

[16]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.

[17]

S. Migórski and P. Gamorski, A new class of quasistatic frictional contact problems governed by a variational-hemivariational inequality, Nonlinear Anal. Real World Appl., 50 (2019), 583-602.  doi: 10.1016/j.nonrwa.2019.05.014.

[18]

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435.  doi: 10.1137/080733231.

[19]

S. MigórskiA. Ochal and M. Sofonea, Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact, Math. Models Methods Appl. Sci., 18 (2008), 271-290.  doi: 10.1142/S021820250800267X.

[20]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, NewYork, 2013. doi: 10.1007/978-1-4614-4232-5.

[21]

S. MigórskiA. Ochal and M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 22 (2015), 604-618.  doi: 10.1016/j.nonrwa.2014.09.021.

[22]

S. MigórskiA. Ochal and M. Sofonea, Analysis of frictional contact problem for viscoelastic materials with long memory, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 687-705.  doi: 10.3934/dcdsb.2011.15.687.

[23]

S. Migórski and S. Zeng, Mixed variational inequalities driven by fractional evolutionary equations, Acta Math. Sci. Ser. B, 39 (2019), 461-468.  doi: 10.1007/s10473-019-0211-9.

[24]

S. Migórski and S. Zeng, A class of differential hemivariational inequalities in Banach spaces, J. Global Optim., 72 (2018), 761-779.  doi: 10.1007/s10898-018-0667-5.

[25]

S. Migórski and S. Zeng, Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics, Numer. Algorithms, 82 (2019), 423-450.  doi: 10.1007/s11075-019-00667-0.

[26]

A. C. Pipkin, Lectures on Viscoelasticity Theory, Applied Mathematical Sciences, Springer, New York, 1972. doi: 10.1007/978-1-4615-9970-8.

[27]

S. ShenF. LiuJ. ChenI. Turner and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218 (2012), 10861-10870.  doi: 10.1016/j.amc.2012.04.047.

[28]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods, Springer, Berlin, 2004. doi: 10.1007/b99799.

[29]

M. Sofonea and A. Farcaş, Analysis of a history-dependent frictional contact problem, Appl. Anal., 93 (2014), 428-444.  doi: 10.1080/00036811.2013.778981.

[30]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics: Preliminaries on Functional Analysis, London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166.

[31]

M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics, European J. Appl. Math., 22 (2011), 471-491.  doi: 10.1017/S0956792511000192.

[32]

M. Sofonea and F. Pătrulescu, Analysis of a history-dependent frictionless contact problem, Math. Mech. Solids, 18 (2012), 409-430.  doi: 10.1177/1081286512440004.

[33]

M. SofoneaF. Pătrulescu and A. Farcas, A viscoplastic contact problem with normal compliance, unilateral constraint and memory term, Appl. Math. Optim., 69 (2014), 175-198.  doi: 10.1007/s00245-013-9216-2.

[34]

G. XueF. Lin and B. Qin, Solvability and optimal control of fractional differential hemivariational inequalities, Optimization, 3 (2020), 1-32.  doi: 10.1080/02331934.2020.1786089.

[35]

S. Zeng and S. Migórski, A class of time-fractional hemivariational inequalities with application to frictional contact problem, Commun. Nonlinear Sci. Numer. Simul., 56 (2018), 34-48.  doi: 10.1016/j.cnsns.2017.07.016.

[36]

S. Zeng and S. Migórski, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072.

Figure 1.  A deformable body in contact with a foundation
Figure 2.  The surface traction $ \boldsymbol f_N $ with impact influence
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