History-dependent contact models for viscoplastic materials


We consider two mathematical models which describe the frictionless process of contact between a rate-type viscoplastic body and a foundation. The contact is modelled with normal compliance and memory term such that penetration is not restricted in the first problem, but is restricted with unilateral constraint in the second one.

For each problem, we derive a variational formulation in terms of displacements, which is in a form of a history-dependent variational equation and a history-dependent variational inequality. Then we prove the unique weak solvability of each model. Next, we prove the convergence of the weak solution of the first problem and the weak solution of the second problem, as the stiffness coefficient of the foundation converges to infinity.

Finally, we provide numerical simulations which illustrate this convergence result.


Mikael Barboteu
(Laboratoire de Mathématiques et Physique, Université de Perpignan)

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

Ahmad Ramadan
(Laboratoire de Mathématiques et Physique, Université de Perpignan)

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


viscoplastic material; frictionless contact; normal compliance; unilateral constraint; memory term; history-dependent variational inequality; weak solution; numerical simulations

Cite this paper as:

M. Barboteu, F. Pătrulescu, A. Ramadan, M. Sofonea, History-dependent contact models for viscoplastic materials, IMA J. Appl. Math., 79 (2014) no. 6, pp. 1180-1200.


About this paper

Publisher Name

Oxford University Press, Oxford

Print ISSN


Online ISSN






[1] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods, Computer Methods in Applied Mechanics 26 and Engineering 92 (1991), 353–375.
[2] M. Barboteu, A. Matei, M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quarterly Journal of Mechanics and Applied Mathematics, 65 (2012), 555–579.
[3] N. Cristescu and I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982.
[4] W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, American Mathematical Society–International Press, 2002.
[5] I.R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford, 1993.
[6] H. B. Khenous, P. Laborde and Y. Renard, On the discretization of contact problems in elastodynamics, Lecture Notes in Applied Computational Mechanics 27 (2006), 31–38.
[7] H. B. Khenous, J. Pommier, Y. Renard, Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers, Applied Numerical Mathematics 56 (2006), 163–192.
[8] A. Klarbring, A. Mikeliˇc and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci. 26 (1988), 811–832.
[9] A. Klarbring, A. Mikeliˇc and M. Shillor, On friction problems with normal compliance, Nonlinear Analysis 13 (1989), 935–955.
[10] T. Laursen, Computational Contact and Impact Mechanics, Springer, Berlin, 2002.
[11] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Analysis : Theory, Methods and Applications 11 (1987), 407–428.
[12] J.T. Oden and J.A.C. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in Applied Mechanics and Engineering 52 (1985), 527–634.
[13] M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, 655, Springer, Berlin, 2004.
[14] 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.
[15] M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in Contact Mechanics, European Journal of Applied Mathematics, 22 (2011), 471–491.
[16] M. Sofonea and F. P˘atrulescu, Analysis of a history-dependent frictionless contact problem, Mathematics and Mechanics of Solids, 18, no. 4 (2013), 409–430.
[17] P. Wriggers, Computational Contact Mechanics, Wiley, Chichester, 2002.



Related Posts