Geodesic
A frictional contact problem with damage in viscoplasticity
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 255-281.

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In this paper, we study a quasistatic contact problem with damage between a viscoplastic body and an obstacle the so-called foundation. The contact is modelled with a general normal compliance condition and the associated version of Coulomb's law of dry friction. We provide a variational formulation of the mechanical problem for which we establish an existence theorem of a weak solution including a regularity result.
Keywords: viscoplastic material, damage, Coulomb's law of dry friction, normal compliance, Rothe method, variational inequalities.
Mots-clés : quasistatic 
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A. Kasri. A frictional contact problem with damage in viscoplasticity. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 255-281. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a26/

[1] V. Barbu, Optimal control of variational inequalities, Research Notes in Mathematics, 100, Pitman, Boston etc., 1984 | MR | Zbl

[2] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011 | MR | Zbl

[3] M. Campo, J.R. Fernández, T.-V. Hoarau-Mantel, “Analysis of two frictional viscoplastic contact problems with damage”, J. Comput. Appl. Math., 196:1 (2006), 180–197 | DOI | MR | Zbl

[4] N. Cristescu, I. Suliciu, Viscoplasticity, Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982 | MR | Zbl

[5] G. Duvaut, J.L. Lions, Inequalities in mechanics and physics, Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin etc., 1976 | DOI | MR | Zbl

[6] E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems, Preprint No 637, TU Berlin, Fachbereich Mathematik, 1999

[7] J.R. Fernández, M. Sofonea, “Variational and numerical analysis of the Signorini's contact problem in viscoplasticity with damage”, J. Appl. Math., 2003:2 (2003), 87–114 | DOI | MR | Zbl

[8] M. Frémond, B. Nedjar, “Damage in concrete: the unilateral phenomenon”, Nuclear Engng. Design, 156:1-2 (1995), 323–335 | DOI

[9] M. Frémond, B. Nedjar, “Damage, gradient of damage and principle of virtual power”, Int. J. Solids Struct., 33:8 (1996), 1083–1103 | DOI | MR | Zbl

[10] J. Han, S. Migórski, “A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage”, J. Math. Anal. Appl., 443:1 (2016), 57–80 | DOI | MR | Zbl

[11] W. Han, M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS, Providence, 2002 | MR | Zbl

[12] A. Kasri, “A viscoplastic contact problem with friction and adhesion”, Sib. Electron. Math. Izv., 17 (2020), 540–565 | DOI | MR | Zbl

[13] A. Kasri, A. Touzaline, “A quasistatic frictional contact problem for viscoelastic materials with long memory”, Georgian Math. J., 27:2 (2020), 249–264 | DOI | MR | Zbl

[14] A. Kasri, A. Touzaline, “Analysis of a dynamic contact problem with friction, damage and adhesion”, Appl. Math., 46:1 (2019), 127–153 | MR | Zbl

[15] A. Klarbring, A. Mikelić, M. Shillor, “Frictional contact problems with normal compliance”, Int. J. Eng. Sci., 26:8 (1988), 811–832 | DOI | MR | Zbl

[16] Y. Li, S. Migórski, J. Han, “A quasistatic frictional contact problem with damage involving viscoelastic materials with short memory”, Math. Mech. Solids, 21:10 (2016), 1167–1183 | DOI | MR | Zbl

[17] J.A.C. Martins, J.T. Oden, “Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface law”, Nonlinear Anal., Theory Methods Appl., 11:3 (1987), 407–428 | DOI | MR | Zbl

[18] J. Nec̃as, I. Hlavac̃ek, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Studies in Applied Mechanics, 3, Elsevier, Amsterdam etc., 1981 | MR | Zbl

[19] P.D. Panagiotopoulos, Inequality problems in mechanics and applications. Convex and nonconvex energy functions, Birkhäuser, Boston etc., 1985 | MR | Zbl

[20] E. Roth, “Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben”, Math. Ann., 102 (1930), 650–670 | DOI | MR | Zbl

[21] M. Rudd, K. Schmitt, “Variational inequalities of elliptic and parabolic type”, Taiwanese J. Math., 6:3 (2002), 287–322 | DOI | MR | Zbl

[22] M. Shillor, M. Sofonea, J.J. Telega, Models and analysis of quasistatic contact. Variational methods, Lecture Notes in Physics, 655, Springer, Berlin, 2004 | DOI | Zbl

[23] M. Sofonea, M. Shillor, “Variational analysis of a quasistatic viscoplastic contact problem with friction”, Commun. Appl. Anal., 5:1 (2001), 135–151 | MR | Zbl