Convergence results for contact problems with memory term


In this paper, we consider two quasistatic contact problems. The material’s behavior is modelled with an elastic constitutive law for the first problem and a viscoplastic constitutive law for the second problem. The novelty arises in the fact that the contact is frictionless and is modelled with a condition which involves normal compliance and memory term. Moreover, for the second problem we consider a condition with unilateral constraint. For each problem we derive a variational formulation of the model and prove its unique solvability. Also, we analyze the dependence of the solution with respect to the data.


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

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


convergence result; memory term; history-dependent variational inequality; weak solution; Fréchet space; Gronwall inequality

Cite this paper as

F. Pătrulescu, A. Ramadan, Convergence results for contact problems with memory term, Math. Rep., vol. 17 (67), no. 1 (2015), pp. 24-41


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Publishing House of the Romanian Academy (Editura Academiei Române), Bucharest

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[1] C. Corduneanu, Problemes globaux dans la theorie des equations Integrales de Volterra, Ann. Math. Pure Appl., 67 (1965), 349-363.
[2] M. Barboteu, A. Matei, Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quart. J. of Mech. and Appl. Math., 65 (2012), 555-579.
[3] M. Barboteu, F. Patrulescu, A. Ramadan,M. Sofonea, History-dependent contact models for viscoplastic materials, IMA J. Appl. Math., 79, no. 6 (2014), 1180-1200.
[4] W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, 30, American Mathematical Society–International Press, Sommerville, MA (2002).
[5] J.J. Massera, J.J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York-London (1966).
[6] M. Sofonea, A. Matei. History-dependent quasivariational inequalities arising in Contact Mechanics, Eur. J. Appl. Math., 22 (2011), 471-491.
[7] M. Sofonea, A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge (2012).
[8] M. Sofonea, F. Patrulescu, Analysis of a history-dependent frictionless contact problem, Math. and Mech. of Solids,18 (2013), 409-430.

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