# American Institute of Mathematical Sciences

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

## Dual formulation of a viscoplastic contact problem with unilateral constraint

 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

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].
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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