A viscoplastic contact problem with normal compliance, unilateral constraint and memory term


We consider a mathematical model which describes the quasistatic contact between a viscoplastic body and a foundation. The material’s behavior is modelled with a rate-type constitutive law with internal state variable. The contact is frictionless and is modelled with normal compliance, unilateral constraint and memory term.

We present the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. Then we prove its unique weak solvability. The proof is based on arguments of history-dependent quasivariational inequalities.

We also study the dependence of the solution with respect to the data and prove a convergence result.


Mircea Sofonea
(Laboratoire de Mathématiques et Physique, Université de Perpignan)

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Anca Farcaş
(Babeş-Bolyai University Faculty of Mathematics and Computer Sciences)


viscoplastic material; frictionless contact; normal compliance; unilateral constraint; memory term; history-dependent variational inequality, weak solution; Fréchet space

Cite this paper as:

M. Sofonea, F. Pătrulescu, A. Farcaş, A viscoplastic contact problem with normal compliance, unilateral constraint and memory term, Appl. Math. Opt., vol. 62 (2014), pp. 175-198


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Applied Mathematics and Optimization

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Springer US, New York, NY

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[1] M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quarterly of Mechanics and Applied Mathematics, 65 (2012), 555-579.
[2] C. Corduneanu, Problemes globaux dans la theorie des equations Integrales de Volterra, Ann. Math. Pure Appl., 67 (1965), 349-363.
[3] N. Cristescu, I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, (1982).
[4] A. Farcas, F. Patrulescu, M. Sofonea, A history-dependent contact problem with unilateral constraint, Mathematics and its Applications, 2 (2012), 105-111.
[5] J.R. Fernandez-Garcia, W. Han, M. Sofonea, J.M. Viano, Variational and numerical analysis of a frictionless contact problem for elastic-viscoplastic materials with internal state variable, Quarterly of Mechanics and Applied Mathematics, 54 (2001), 501-522.
[6] W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, 30, American Mathematical Society–International Press, Sommerville, MA (2002).
[7] I.R. Ionescu, M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford (1993).
[8] J. Jarusek, M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, Zeitschrift fur Angewandte Matematik und Mechanik (ZAMM), 88 (2008), 3-22.
[9] N. Kikuchi, J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988.
[10] A. Klarbring, A. Mikelic, M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811-832.
[11] A. Klarbring, A. Mikelic, M. Shillor, On friction problems with normal compliance, Nonlinear Analysis, 13 (1989), 935-955.
[12] J.J. Massera, J.J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York-London (1966).
[13] J.A.C.Martins, 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.
[14] J.T. Oden, J.A.C. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527-634.
[15] M. Shillor, M. Sofonea, J.J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, 655, Springer, Berlin (2004).
[16] M. Sofonea, C. Avramescu, 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.
[17] M. Sofonea, A. Matei. History-dependent quasivariational inequalities arising in Contact Mechanics, Eur. J. Appl. Math., 22 (2011), 471-491.
[18] M. Sofonea, A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge (2012).



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