Dual formulation of a viscoplastic contact problem with unilateral constraint

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December 2013, 6(6): 1587-1598. doi: 10.3934/dcdss.2013.6.1587

Dual formulation of a viscoplastic contact problem with unilateral constraint

Andaluzia Matei ^1, and Mircea Sofonea ^2,

1.	Departement of Mathematics, University of Craiov, A.I. Cuza Street 13, 200585, Craiova
2.	Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan

Received June 2012 Revised September 2012 Published April 2013

Abstract
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We consider a mathematical model which describes the contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the contact is frictionless and is modelled with unilateral constraint. We derive a variational formulation of the model which leads to a history-dependent quasivariational inequality for stress field, associated to a time-dependent convex. Then we prove the unique weak solvability of the model. The proof is based on an abstract existence and uniqueness result obtained in [11].

Keywords: frictionless contact, quasivariational inequality., dual formulation, unilateral constraint, Viscoplastic material.

Mathematics Subject Classification: Primary: 74M15, 74G25; Secondary: 49J4.

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

References:

[1]	M. Anders, "Dual-Dual Formulations for Frictional Contact Problems in Mechanics," Ph.D thesis, Leibniz Universität, Hannover, 2011.
[2]	B. Awbi, M. Shillor and M. Sofonea, Dual formulation of a quasistatic viscoelastic contact problem with Tresca's friction law, Applicable Analysis, 79 (2001), 1-20. doi: 10.1080/00036810108840949.
[3]	M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Q. J. Mechanics Appl. Math., 65 (2012), 555-579. doi: 10.1093/qjmam/hbs016.
[4]	N. Cristescu and I. Suliciu, "Viscoplasticity," Translated from the Romanian, Mechanics of Plastic Solids, 5, Martinus Nijhoff Publishers, The Hague, 1982.
[5]	W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," AMS/IP Studies in Advanced Mathematics, 30, Americal Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.
[6]	I. Hlaváček, J. Haslinger, J. Nečas and J. Lovášek, "Solution of Variational Inequalities in Mechanics," Translated from the Slovak by J. Jarník, Applied Mathematical Sciences, 66, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1.
[7]	I. R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.
[8]	N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, SIAM, Philadelphia, PA, 1988.
[9]	M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact," Lecture Notes in Physics, 655, Springer, Berlin, 2004. doi: 10.1007/b99799.
[10]	M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems, Communications on Pure and Applied Analysis, 7 (2008), 645-658. doi: 10.3934/cpaa.2008.7.645.
[11]	M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics, European Journal of Applied Mathematics, 22 (2011), 471-491. doi: 10.1017/S0956792511000192.
[12]	J. J. Telega, Topics on unilateral contact problems of elasticity and inelasticity, in "Nonsmooth Mechanics and Applications" (eds. J.-J. Moreau, P. D. Panagiotopoulos and G. Strang), Birkhäuser Verlag, Basel, (1988), 340-461.

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References:

[1]	M. Anders, "Dual-Dual Formulations for Frictional Contact Problems in Mechanics," Ph.D thesis, Leibniz Universität, Hannover, 2011.
[2]	B. Awbi, M. Shillor and M. Sofonea, Dual formulation of a quasistatic viscoelastic contact problem with Tresca's friction law, Applicable Analysis, 79 (2001), 1-20. doi: 10.1080/00036810108840949.
[3]	M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Q. J. Mechanics Appl. Math., 65 (2012), 555-579. doi: 10.1093/qjmam/hbs016.
[4]	N. Cristescu and I. Suliciu, "Viscoplasticity," Translated from the Romanian, Mechanics of Plastic Solids, 5, Martinus Nijhoff Publishers, The Hague, 1982.
[5]	W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," AMS/IP Studies in Advanced Mathematics, 30, Americal Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.
[6]	I. Hlaváček, J. Haslinger, J. Nečas and J. Lovášek, "Solution of Variational Inequalities in Mechanics," Translated from the Slovak by J. Jarník, Applied Mathematical Sciences, 66, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1.
[7]	I. R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.
[8]	N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods," SIAM Studies in Applied Mathematics, 8, SIAM, Philadelphia, PA, 1988.
[9]	M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact," Lecture Notes in Physics, 655, Springer, Berlin, 2004. doi: 10.1007/b99799.
[10]	M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems, Communications on Pure and Applied Analysis, 7 (2008), 645-658. doi: 10.3934/cpaa.2008.7.645.
[11]	M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics, European Journal of Applied Mathematics, 22 (2011), 471-491. doi: 10.1017/S0956792511000192.
[12]	J. J. Telega, Topics on unilateral contact problems of elasticity and inelasticity, in "Nonsmooth Mechanics and Applications" (eds. J.-J. Moreau, P. D. Panagiotopoulos and G. Strang), Birkhäuser Verlag, Basel, (1988), 340-461.

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