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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), 1-51.  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), Cambridge University Press, 2011, 229-252. 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, Curtin University Press, Perth, Western Australia (eds. L. Caccetta, et al.), 1998, 19-34. Google Scholar

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, Interscience, New York, 1983.  Google Scholar

[5]

N. Cristescu and I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982.  Google Scholar

[6]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.  Google Scholar

[7]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.  Google Scholar

[8]

A. D. Drozdov, Finite Elasticity and Viscoelasticity-A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore, 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, 270, Chapman/CRC Press, New York, 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), 105-111. Google Scholar

[11]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI, International Press, Somerville, MA, 2002.  Google Scholar

[12]

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

[13]

A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811-832. 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), 4729-4746. 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), 407-428. 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), 669-699. 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), 247-275. 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), 3384-3396. 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, 26, Springer, New York, 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), 527-634. doi: 10.1016/0045-7825(85)90009-X.  Google Scholar

[21]

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

[22]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 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), 675-703. doi: 10.1080/00423110903126478.  Google Scholar

[24]

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

[25]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge, 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), 409-430. 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), 371-387. 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), 1-51.  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), Cambridge University Press, 2011, 229-252. 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, Curtin University Press, Perth, Western Australia (eds. L. Caccetta, et al.), 1998, 19-34. Google Scholar

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, Interscience, New York, 1983.  Google Scholar

[5]

N. Cristescu and I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982.  Google Scholar

[6]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.  Google Scholar

[7]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.  Google Scholar

[8]

A. D. Drozdov, Finite Elasticity and Viscoelasticity-A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore, 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, 270, Chapman/CRC Press, New York, 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), 105-111. Google Scholar

[11]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI, International Press, Somerville, MA, 2002.  Google Scholar

[12]

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

[13]

A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811-832. 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), 4729-4746. 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), 407-428. 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), 669-699. 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), 247-275. 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), 3384-3396. 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, 26, Springer, New York, 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), 527-634. doi: 10.1016/0045-7825(85)90009-X.  Google Scholar

[21]

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

[22]

P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 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), 675-703. doi: 10.1080/00423110903126478.  Google Scholar

[24]

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

[25]

M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge, 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), 409-430. 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), 371-387. doi: 10.3934/cpaa.2014.13.371.  Google Scholar

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