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Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics

Published online by Cambridge University Press:  24 April 2014

Kamran Kazmi ,
Mikael Barboteu ,
Weimin Han  and
Mircea Sofonea
Kamran Kazmi
Affiliation:
Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI 54901, USA. kazmis@uwosh.edu
Mikael Barboteu
Affiliation:
Laboratoire de Mathématiques et Physique, University of Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France; barboteu@univ-perp.fr; sofonea@univ-perp.fr
Weimin Han
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA; weimin-han@uiowa.edu
Mircea Sofonea
Affiliation:
Laboratoire de Mathématiques et Physique, University of Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France; barboteu@univ-perp.fr; sofonea@univ-perp.fr
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Abstract

A new class of history-dependent quasivariational inequalities was recently studied in [M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491]. Existence, uniqueness and regularity results were proved and used in the study of several mathematical models which describe the contact between a deformable body and an obstacle. The aim of this paper is to provide numerical analysis of the quasivariational inequalities introduced in the aforementioned paper. To this end we introduce temporally semi-discrete and fully discrete schemes for the numerical approximation of the inequalities, show their unique solvability, and derive error estimates. We then apply these results to a quasistatic frictional contact problem in which the material’s behavior is modeled with a viscoelastic constitutive law, the contact is bilateral, and friction is described with a slip-rate version of Coulomb’s law. We discuss implementation of the corresponding fully-discrete scheme and present numerical simulation results on a two-dimensional example.

Keywords

Quasivariational inequalitynumerical analysisfinite element methoderror estimatesquasistatic frictional contact problemviscoelastic constitutive lawCoulomb’s lawnumerical simulations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 919 - 942
Copyright
© EDP Sciences, SMAI, 2014

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References

Alart, P. and Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92 (1991) 353–375. Google Scholar
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems. John Wiley, Chichester (1984).
Barboteu, M. and Sofonea, M., Modelling and analysis of the unilateral contact of a piezoelectric body with a conductive support. J. Math. Anal. Appl. 358 (2009) 110–124. Google Scholar
Barboteu, M. and Sofonea, M., Analysis and numerical approach of a piezoelectric contact problem. Annals of the Academy of Romanian Scientists, Series on Mathematics and its Applications 1 (2009) 7–31. Google Scholar
D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007).
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008).
Brezis, H., Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18 (1968) 115–175. Google Scholar
P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, vol. II, edited by P.G. Ciarlet and J.-L. Lions. North-Holland, Amsterdam (1991) 17–351.
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976).
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York (1984).
C. Eck, J. Jarušek and M. Krbeč, Unilateral Contact Problems: Variational Methods and Existence Theorems, vol. 270, Pure Appl. Math. Chapman/CRC Press, New York (2005).
Han, W. and Reddy, B.D., Computational plasticity: the variational basis and numerical analysis. Comput. Mech. Adv. 2 (1995) 283–400. Google Scholar
W. Han and B.D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, 2nd edn. Springer-Verlag, New York (2013).
W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. In vol. 30, Stud. Adv. Math. American Mathematical Society, Providence, RI-International Press, Sommerville, MA (2002).
J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities. Theory, Methods Appl. Kluwer Academic Publishers, Boston, Dordrecht, London (1999).
I. Hlaváček, J. Haslinger, J. Necǎs and J. Lovíšek, Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York (1988).
Khenous, H.B., Laborde, P., and Renard, Y., On the discretization of contact problems in elastodynamics. Lect. Notes Appl. Comput. Mech. 27 (2006) 31–38. Google Scholar
Khenous, H.B., Pommier, J. and Renard, Y., Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers. Appl. Numer. Math. 56 (2006) 163–192. Google Scholar
Kikuchi, N. and Oden, J.T., Theory of variational inequalities with applications to problems of flow through porous media. Int. J. Engng. Sci. 18 (1980) 1173–1284. Google Scholar
N. Kikuchi and T.J. Oden, Contact Problems in Elasticity. SIAM, Philadelphia (1988).
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications. In vol. 31, Classics Appl. Math. SIAM, Philadelphia (2000).
T. Laursen, Computational contact and impact mechanics. Springer, Berlin (2002).
J.A.C. Martins and M.D.P. Monteiro Marques, eds., Contact Mechanics. Kluwer, Dordrecht (2002).
Mistakidis, E.S. and Panagiotopulos, P.D., Numerical treatment of problems involving nonmonotone boundary or stress-strain laws. Comput. Structures 64 (1997) 553–565. Google Scholar
Mistakidis, E.S. and Panagiotopulos, P.D., The search for substationary points in the unilateral contact problems with nonmonotone friction. Math. Comput. Modelling 28 (1998) 341–358. Google Scholar
P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Birkhäuser, Boston, 1985.
M. Raous, M. Jean and J.J. Moreau, Contact Mechanics. Plenum Press, New York (1995).
M. Shillor, ed., Recent advances in contact mechanics, Special issue of Math. Comput. Modelling 28 (4–8) (1998).
M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact. Variational Methods. In vol. 655, Lect. Notes Phys. Springer, Berlin (2004).
Sofonea, M., Avramescu, C. and Matei, A., A fixed point result with applications in the study of viscoplastic frictionless contact problems. Commun. Pure Appl. Anal. 7 (2008) 645–658. Google Scholar
M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage. Chapman & Hall/CRC, New York (2006).
M. Sofonea and A. Matei, Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. vol. 18, Adv. Mech. Math. Springer, New York (2009).
Sofonea, M. and Matei, A., History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491. Google Scholar
C.H. Scholz, The Mechanics of Earthquakes and Faulting. Cambridge University Press (1990).
Tzaferopoulos, M.A., Mistakidis, E.S., Bisbos, C.D., and Panagiotopulos, P.D., Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms. Comput. Struct. 57 (1995) 959–971. Google Scholar
P. Wriggers and U. Nackenhorst, eds., Analysis and Simulation of Contact Problems. In vol. 27, Lect. Notes Appl. Comput. Mech. Springer, Berlin (2006).
P. Wriggers, Computational Contact Mechanics. Wiley, Chichester (2002).