American Institute of Mathematical Sciences

Advanced
October  2014, 19(8): 2425-2445. doi: 10.3934/dcdsb.2014.19.2425

Analysis of two quasistatic history-dependent contact models

Xiaoliang Cheng 1, , Stanisław Migórski 2, , Anna Ochal 2, and  Mircea Sofonea 3,

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2. 

Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow
3. 

Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan

Received  October 2013 Revised  January 2014 Published  August 2014

We consider two mathematical models which describe the evolution of a viscoelastic and viscoplastic body, respectively, in contact with a piston or a device, the so-called obstacle or foundation. In both models the contact process is assumed to be quasistatic and the friction is described with a Clarke subdifferential boundary condition. The novelty of the models consists in the constitutive laws as well as in the contact conditions we use, which involve a memory term. We derive a variational formulation of the problems which is in the form of a system coupling a nonlinear integral equation with a history--dependent hemivariational inequality. Then, we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proof is based on a result on history--dependent hemivariational inequalities obtained in [18].
Keywords: Viscoelastic material, memory term, history-dependent hemivariational inequality, normal compliance, frictional contact, weak solution., viscoplastic material, Clarke subdifferential.
Mathematics Subject Classification: Primary: 74M15, 74M10, 74G25, 74G30, 74C10, 49J40; Secondary: 47J20, 35Q74, 35J8.
Citation: Xiaoliang Cheng, Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of two quasistatic history-dependent contact models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2425-2445. doi: 10.3934/dcdsb.2014.19.2425
References:
[1]

H. T. Banks, S. Hu and Z. R. Kenz, A brief review of elasticity and viscoelasticity for solids,, Adv. Appl. Math. Mech., 3 (2011), 1.   Google Scholar

[2]

H. T. Banks, G. A. Pinter, L. K. Potter, J. M. Gaitens and L. C. Yanyo, Modeling of quasistatic and dynamic load responses of filled viesoelastic materials,, Chapter 11 in Mathematical Modeling: Case Studies from Industry (eds. E. Cumberbatch and A. Fitt), (2011), 229.   Google Scholar

[3]

H. T. Banks, G. A. Pinter, L. K. Potter, B. C. Munoz and L. C. Yanyo, Estimation and control related issues in smart material structure and fluids,, Optimization Techniques and Applications, (1998), 19.   Google Scholar

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).   Google Scholar

[5]

N. Cristescu and I. Suliciu, Viscoplasticity,, Martinus Nijhoff Publishers, (1982).   Google Scholar

[6]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory,, Kluwer Academic/Plenum Publishers, (2003).   Google Scholar

[7]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications,, Kluwer Academic/Plenum Publishers, (2003).   Google Scholar

[8]

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

[9]

C. Eck, J. Jarušek and M. Krbeč, Unilateral Contact Problems: Variational Methods and Existence Theorems,, Pure and Applied Mathematics, (2005).  doi: 10.1201/9781420027365.  Google Scholar

[10]

A. Farcaş, F. Pătrulescu and M. Sofonea, A history-dependent contact problem with unilateral constraint,, Mathematics and its Applications, 2 (2012), 105.   Google Scholar

[11]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,, Studies in Advanced Mathematics, (2002).   Google Scholar

[12]

I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity,, Oxford University Press, (1993).   Google Scholar

[13]

A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance,, Int. J. Engng. Sci., 26 (1988), 811.  doi: 10.1016/0020-7225(88)90032-8.  Google Scholar

[14]

A. Kulig and S. Migórski, Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator,, Nonlinear Analysis, 75 (2012), 4729.  doi: 10.1016/j.na.2012.03.023.  Google Scholar

[15]

J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws,, Nonlinear Analysis TMA, 11 (1987), 407.  doi: 10.1016/0362-546X(87)90055-1.  Google Scholar

[16]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction,, Applicable Analysis, 84 (2005), 669.  doi: 10.1080/00036810500048129.  Google Scholar

[17]

S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity,, Journal of Elasticity, 83 (2006), 247.  doi: 10.1007/s10659-005-9034-0.  Google Scholar

[18]

S. Migórski, A. Ochal and M. Sofonea, History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics,, Nonlinear Anal. Real World Appl., 12 (2011), 3384.  doi: 10.1016/j.nonrwa.2011.06.002.  Google Scholar

[19]

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

[20]

J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena,, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527.  doi: 10.1016/0045-7825(85)90009-X.  Google Scholar

[21]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications,, Birkhäuser, (1985).  doi: 10.1007/978-1-4612-5152-1.  Google Scholar

[22]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering,, Springer-Verlag, (1993).  doi: 10.1007/978-3-642-51677-1.  Google Scholar

[23]

J. Piotrowski, Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons,, Vehicle System Dynamics, 48 (2010), 675.  doi: 10.1080/00423110903126478.  Google Scholar

[24]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact,, Springer, (2004).  doi: 10.1007/b99799.  Google Scholar

[25]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics,, London Mathematical Society Lecture Note Series, (2012).  doi: 10.1017/CBO9781139104166.  Google Scholar

[26]

M. Sofonea and F. Pätrulescu, Analysis of a history-dependent frictionless contact problem,, Mathematics and Mechanics of Solids, 18 (2013), 409.  doi: 10.1177/1081286512440004.  Google Scholar

[27]

M. Sofonea and M. Shillor, A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient,, Communications in Pure and Appled Analysis, 13 (2014), 371.  doi: 10.3934/cpaa.2014.13.371.  Google Scholar

show all references

References:
[1]

H. T. Banks, S. Hu and Z. R. Kenz, A brief review of elasticity and viscoelasticity for solids,, Adv. Appl. Math. Mech., 3 (2011), 1.   Google Scholar

[2]

H. T. Banks, G. A. Pinter, L. K. Potter, J. M. Gaitens and L. C. Yanyo, Modeling of quasistatic and dynamic load responses of filled viesoelastic materials,, Chapter 11 in Mathematical Modeling: Case Studies from Industry (eds. E. Cumberbatch and A. Fitt), (2011), 229.   Google Scholar

[3]

H. T. Banks, G. A. Pinter, L. K. Potter, B. C. Munoz and L. C. Yanyo, Estimation and control related issues in smart material structure and fluids,, Optimization Techniques and Applications, (1998), 19.   Google Scholar

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).   Google Scholar

[5]

N. Cristescu and I. Suliciu, Viscoplasticity,, Martinus Nijhoff Publishers, (1982).   Google Scholar

[6]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory,, Kluwer Academic/Plenum Publishers, (2003).   Google Scholar

[7]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications,, Kluwer Academic/Plenum Publishers, (2003).   Google Scholar

[8]

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

[9]

C. Eck, J. Jarušek and M. Krbeč, Unilateral Contact Problems: Variational Methods and Existence Theorems,, Pure and Applied Mathematics, (2005).  doi: 10.1201/9781420027365.  Google Scholar

[10]

A. Farcaş, F. Pătrulescu and M. Sofonea, A history-dependent contact problem with unilateral constraint,, Mathematics and its Applications, 2 (2012), 105.   Google Scholar

[11]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,, Studies in Advanced Mathematics, (2002).   Google Scholar

[12]

I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity,, Oxford University Press, (1993).   Google Scholar

[13]

A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance,, Int. J. Engng. Sci., 26 (1988), 811.  doi: 10.1016/0020-7225(88)90032-8.  Google Scholar

[14]

A. Kulig and S. Migórski, Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator,, Nonlinear Analysis, 75 (2012), 4729.  doi: 10.1016/j.na.2012.03.023.  Google Scholar

[15]

J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws,, Nonlinear Analysis TMA, 11 (1987), 407.  doi: 10.1016/0362-546X(87)90055-1.  Google Scholar

[16]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction,, Applicable Analysis, 84 (2005), 669.  doi: 10.1080/00036810500048129.  Google Scholar

[17]

S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity,, Journal of Elasticity, 83 (2006), 247.  doi: 10.1007/s10659-005-9034-0.  Google Scholar

[18]

S. Migórski, A. Ochal and M. Sofonea, History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics,, Nonlinear Anal. Real World Appl., 12 (2011), 3384.  doi: 10.1016/j.nonrwa.2011.06.002.  Google Scholar

[19]

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

[20]

J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena,, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527.  doi: 10.1016/0045-7825(85)90009-X.  Google Scholar

[21]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications,, Birkhäuser, (1985).  doi: 10.1007/978-1-4612-5152-1.  Google Scholar

[22]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering,, Springer-Verlag, (1993).  doi: 10.1007/978-3-642-51677-1.  Google Scholar

[23]

J. Piotrowski, Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons,, Vehicle System Dynamics, 48 (2010), 675.  doi: 10.1080/00423110903126478.  Google Scholar

[24]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact,, Springer, (2004).  doi: 10.1007/b99799.  Google Scholar

[25]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics,, London Mathematical Society Lecture Note Series, (2012).  doi: 10.1017/CBO9781139104166.  Google Scholar

[26]

M. Sofonea and F. Pätrulescu, Analysis of a history-dependent frictionless contact problem,, Mathematics and Mechanics of Solids, 18 (2013), 409.  doi: 10.1177/1081286512440004.  Google Scholar

[27]

M. Sofonea and M. Shillor, A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient,, Communications in Pure and Appled Analysis, 13 (2014), 371.  doi: 10.3934/cpaa.2014.13.371.  Google Scholar

[1]

Mircea Sofonea, Meir Shillor. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Communications on Pure & Applied Analysis, 2014, 13 (1) : 371-387. doi: 10.3934/cpaa.2014.13.371
[2]

Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339
[3]

Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
[4]

Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553
[5]

Stanislaw Migórski. A class of hemivariational inequalities for electroelastic contact problems with slip dependent friction. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 117-126. doi: 10.3934/dcdss.2008.1.117
[6]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[7]

Simone Göttlich, Sebastian Kühn, Jan Peter Ohst, Stefan Ruzika, Markus Thiemann. Evacuation dynamics influenced by spreading hazardous material. Networks & Heterogeneous Media, 2011, 6 (3) : 443-464. doi: 10.3934/nhm.2011.6.443
[8]

María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201
[9]

Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587
[10]

Oanh Chau, R. Oujja, Mohamed Rochdi. A mathematical analysis of a dynamical frictional contact model in thermoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 61-70. doi: 10.3934/dcdss.2008.1.61
[11]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
[12]

Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084
[13]

Rejeb Hadiji, Ken Shirakawa. Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1345-1361. doi: 10.3934/cpaa.2010.9.1345
[14]

Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014
[15]

Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041
[16]

Rainer Picard. On a comprehensive class of linear material laws in classical mathematical physics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 339-349. doi: 10.3934/dcdss.2010.3.339
[17]

Michela Eleuteri, Jana Kopfová, Pavel Krejčí. A new phase field model for material fatigue in an oscillating elastoplastic beam. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2465-2495. doi: 10.3934/dcds.2015.35.2465
[18]

Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415
[19]

Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071
[20]

Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences