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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

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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

N. Cristescu and I. Suliciu, "Viscoplasticity," Translated from the Romanian, Mechanics of Plastic Solids, 5, Martinus Nijhoff Publishers, The Hague, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Anders, "Dual-Dual Formulations for Frictional Contact Problems in Mechanics," Ph.D thesis, Leibniz Universität, Hannover, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

N. Cristescu and I. Suliciu, "Viscoplasticity," Translated from the Romanian, Mechanics of Plastic Solids, 5, Martinus Nijhoff Publishers, The Hague, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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