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Dual formulation of a viscoplastic contact problem with unilateral constraint

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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].
    Mathematics Subject Classification: Primary: 74M15, 74G25; Secondary: 49J40.


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  • [1]

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    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.


    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.


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


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    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.


    I. R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity," Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.


    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.


    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.


    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.


    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.


    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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