Analysis of two quasistatic history-dependent contact models

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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

Abstract
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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 and 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.
[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.
[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.
[4]	F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, Interscience, New York, 1983.
[5]	N. Cristescu and I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982.
[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.
[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.
[8]	A. D. Drozdov, Finite Elasticity and Viscoelasticity-A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore, 1996. doi: 10.1142/2905.
[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.
[10]	A. Farcaş, F. Pătrulescu and M. Sofonea, A history-dependent contact problem with unilateral constraint, Mathematics and its Applications, 2 (2012), 105-111.
[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.
[12]	I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford, 1993.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985. doi: 10.1007/978-1-4612-5152-1.
[22]	P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.
[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.
[24]	M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Springer, Berlin, 2004. doi: 10.1007/b99799.
[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.
[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.
[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.

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.
[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.
[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.
[4]	F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, Interscience, New York, 1983.
[5]	N. Cristescu and I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982.
[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.
[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.
[8]	A. D. Drozdov, Finite Elasticity and Viscoelasticity-A Course in the Nonlinear Mechanics of Solids, World Scientific, Singapore, 1996. doi: 10.1142/2905.
[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.
[10]	A. Farcaş, F. Pătrulescu and M. Sofonea, A history-dependent contact problem with unilateral constraint, Mathematics and its Applications, 2 (2012), 105-111.
[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.
[12]	I. R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford, 1993.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985. doi: 10.1007/978-1-4612-5152-1.
[22]	P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.
[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.
[24]	M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Springer, Berlin, 2004. doi: 10.1007/b99799.
[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.
[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.
[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.

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