A NEW CLASS OF THE DYNAMIC VISCOPLASTIC FRICTIONAL CONTACT PROBLEM WITH ADHESION

  • Furi GUO1 ,
  • * ,
  • Jinrong WANG2
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  • 1. Department of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China;
    2. Department of Mathematics, Guizhou University, Guiyang 550025, China
Jinrong WANG, E-mail: jrwang@gzu.edu.cn

Received date: 2024-08-22

  Revised date: 2024-11-15

  Online published: 2026-05-22

Supported by

The first author was supported by the NSF of Shanxi (202303021221168), and the Industry-university-research project of Shanxi Datong University (2022CXY10, 2022CXY13).

Abstract

In this paper, our main goal is to study a new mathematical model which describes the frictional contact between a foundation and a deformable body which is composed of viscoplastic materials and where the process is considered dynamic. The contact condition on the normal plane is modeled by a unilateral constraint condition for a version of normal velocity in which the memory effect and the adhesion are considered. On the tangential plane a frictional contact condition is governed by the Clarke subdifferential of a locally Lipschitz function, and the evolution of the bonding field is governed by an ordinary differential equation. We formulate this problem as coupled system that consists of two ordinary differential equations and a variational-hemivariational inequality. Then, the existence, uniqueness and continuous dependence of the solution on the data results concerning the abstract system are established. Finally, we use the abstract results to show the existence and uniqueness of the solution to the contact problem.

Cite this article

Furi GUO1 , * , Jinrong WANG2 . A NEW CLASS OF THE DYNAMIC VISCOPLASTIC FRICTIONAL CONTACT PROBLEM WITH ADHESION[J]. Acta mathematica scientia, Series B, 2026 , 46(1) : 69 -98 . DOI: 10.1007/s10473-026-0105-6

References

[1] Zeng S D, Migórski S. Noncoercive hyperbolic variational inequalities with applications to contact mechanics. J Math Anal Appl, 2017, $\textbf{455}$: 619-637
[2] Han J F, Zeng H D. Variational analysis and optimal control of dynamic unilateral contact models with friction. J Math Anal Appl, 2019, $\textbf{473}$: 712-748
[3] Migórski S. Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl Anal, 2005, $\textbf{84}$: 669-699
[4] Chau O, Fernández-García J R, Han W, Sofonea M. A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage. Comput Methods Appl Mech Eng, 2002, $\textbf{191}$: 5007-5026
[5] Amassad A, Fabre C, Sofonea M. A quasistatic viscoplastic contact problem with normal compliance and friction. IMA J Appl Math, 2004, $\textbf{69}$: 463-482
[6] Sofonea M, Shillor M. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Commun Pure Appl Anal, 2014, $\textbf{13}$: 371-387
[7] Sofonea M, Pătrulescu F, Farcas A. A viscoplastic contact problem with normal compliance, unilateral constraint,memory term. Appl Math Optim, 2014, $\textbf{69}$: 175-198
[8] Kulig A. Variational-hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage. Nonlinear Anal: Real World Appl, 2018, $\textbf{44}$: 401-416
[9] Kulig A. A quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term and friction. Nonlinear Anal: Real World Appl, 2017, $\textbf{33}$: 226-236
[10] Han W, Sofonea M. Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer, 2019, $\textbf{28}$: 175-286
[11] Han W, Sofonea M, Barboteu M. Numerical analysis of elliptic hemivariational inequalities. SIAM J Numer Anal, 2017, $\textbf{55}$: 640-663
[12] Migórski S. Optimal control of history-dependent evolution inclusions with applications to frictional contact. J Optim Theory Appl, 2020, $\textbf{185}$: 574-596.
[13] Panagiotopoulos P D.Inequality Problems in Mechanics and Applications. Boston: Birkhäuser, 1985
[14] Panagiotopoulos P D. Hemivariational Inequalities, Applications in Mechanics and Engineering. Berlin: Springer-Verlag, 1993
[15] Raous M, Jean M, Moreau J J. Contact Mechanics.New York: Plenum Press, 1995
[16] Shillor M, Sofonea M, Telega J J.Models and Analysis of Quasistatic Contact. Berlin: Springer, 2004
[17] Migórski S, Liu Z H, Zeng S D. A class of history-dependent differential variational inequalities with application to contact problems. Optimization, 2020, $\textbf{69}$: 743-775
[18] Han W M, Sofonea M.Quasistatic Contact Problems in Viscoelastictity and Viscoplasticity. Studies in Advanced Mathematics $\textbf{30}$. Providence: Americal Mathematical Society, 2002
[19] Zhao J, Gan C M, Liu Z H. Differential evolution hemivariational inequalities with anti-periodic conditions. Acta Math Sin, 2024, $\textbf{40}$: 1143-1160
[20] Liu Y J, Liu Z H, Papageorgiou N S. Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities. J Differential Equations, 2023, $\textbf{342}$: 559-595
[21] Li X W, Liu Z H, Papageorgiou N S. Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications. Nonlinearity, 2023, $\textbf{36}$: 1323-1348
[22] Migórski S. Well-posedness of constrained evolutionary differential variational-hemivariational inequalities with applications. Nonlinear Anal: Real World Appl, 2022, $\textbf{67}$: 1-22
[23] Anh N T V, Ke T D. On the differential variational inequalities of parabolic-parabolic type. Acta Appl Math, 2017, $\textbf{176}$: 1-25
[24] Zeng S D, Migórski S. A class of time-fractional hemivariational inequalities with application to frictional contact problem. Commun Nonlinear Sci Numer Simul, 2018, $\textbf{56}$: 34-48
[25] Liu Z H, Motreanu D, Zeng S D. Generalized penalty and regularization method for differential variational-hemivariational inequalities. SIAM J Optim, 2021, $\textbf{31}$: 1158-1183
[26] Jiang C J, Zeng B. Continuous dependence and optimal control for a class of variational-hemivariational inequalities. Appl Math Optim, 2020, $\textbf{82}$: 637-656
[27] Migórski S, Bai Y R, Zeng S D. A new class of history-dependent quasi variational-hemivariational inequalities with constraints. Commun Nonlinear Sci Numer Simul, 2022, $\textbf{114}$: 1-15
[28] Denkowski Z, Migórski S, Papageorgiou N S.An Introduction to Nonlinear Analysis: Applications. New York: Kluwer Academic/Plenum Publishers, 2003
[29] Zeidler E.Nonlinear Functional Analysis and Applications II A/B. New York: Springer, 1990
[30] Migórski S, Zeng B. A new class of history-dependent evolutionary variational-hemivariational inequalities with unilateral constraints. Appl Math Optim, 2021, $\textbf{84}$: 2671-2697
[31] Han W, Migórski S, Sofonea M. A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J Math Anal, 2014, $\textbf{46}$: 3891-3912
[32] Raous M, Canǵemi L, Cocu M. A consistent model coupling adhesion, friction,unilateral contact. Comput Methods Appl Mech Eng, 1999, $\textbf{177}$: 383-399
[33] Han J F, Li Y, Migórski S. Analysis of an adhesive contact problem for viscoelastic materials with long memory. J Math Anal Appl, 2015, $\textbf{427}$: 646-668
[34] Dumont Y, Goeleven D, Rochdi M, et al. A dynamic model with friction and adhesion with applications to rocks. J Math Anal Appl, 2000, $\textbf{247}$: 87-109
[35] Paczka D. Adhesive contact problem for viscoplastic materials. Nonlinear Anal: Real World Appl, 2020, $\textbf{51}$: 1-24
[36] Roubíček T. Adhesive contact of viscoelastic bodies and defect measures arising by vanishing viscosity. SIAM J Math Anal, 2013, $\textbf{45}$: 101-126
[37] Cocu M, Rocca R. Existence results for unilateral quasistatic contact problems with friction and adhesion. ESAIM Math Model Numer Anal, 2000, $\textbf{34}$: 981-1001
[38] Roubíček T, Panagiotopoulos C G, Mantič V. Quasistatic adhesive contact of viscoelastic bodies and its numerical treatment for very small viscosity. J Math Anal Appl, 2013, $\textbf{93}$: 823-840
[39] Migórski S, Ochal A. Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Anal: Theory Methods Appl, 2008, $\textbf{69}$: 495-509
[40] Migórski S, Zeng S D. Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model. Nonlinear Anal: Real World Appl, 2018, $\textbf{43}$: 121-143
[41] Sofonea M, Han W, Shillor M.Analysis and Approximation of Contact Problems with Adhesion or Damage. Florida: CRC Press, 2005.
[42] Chau O, Shillor M, Sofonea M. Dynamic frictionless contact with adhesion. Z Angew Math Phys, 2004, $\textbf{55}$: 32-47
[43] Chau O, Fernández-García J R, Han W, Sofonea M. Variational and numerical analysis of a dynamic frictionless contact problem with adhesion. J Comput Appl Math, 2003, $\textbf{156}$: 127-157
[44] Bartosz K. Hemivariational inequalities modeling dynamic contact problems with adhesion. Nonlinear Anal Theory Methods Appl, 2009, $\textbf{71}$: 1747-1762
[45] Migórski S, Ochal A, Sofonea M.Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics $\textbf{26}$. New York: Springer, 2013.
[46] Sofonea M, Migórski S.Variational-Hemivariational Inequalities with Applications. New York: Chapman and Hall/CRC, 2017
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