Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T20:57:02.919Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

Part of: Special kinds of problems Elliptic equations and systems

Published online by Cambridge University Press:  03 August 2011

A. MATEI  and
R. CIURCEA
A. MATEI*
Affiliation:
Department of Mathematics, University of Craiova, A.I. Cuza 13, 200585 Craiova, Romania (email: andaluziamatei2000@yahoo.com, miciloi@yahoo.com)
R. CIURCEA
Affiliation:
Department of Mathematics, University of Craiova, A.I. Cuza 13, 200585 Craiova, Romania (email: andaluziamatei2000@yahoo.com, miciloi@yahoo.com)
*
✉For correspondence; e-mail: andaluziamatei2000@yahoo.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

Keywords

contact modelsnonlinearly elastic materialsdual Lagrange multipliersweak solutions

MSC classification

Secondary: 74M10: Friction 35J50: Variational methods for elliptic systems 35J66: Nonlinear boundary value problems for nonlinear elliptic equations
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 2 , October 2010 , pp. 160 - 178
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Braess, D., Finite elements (Cambridge University Press, Cambridge, 2001).Google Scholar
[2]Brezzi, F. and Fortin, M., Mixed and hybrid finite element methods (Springer, New York, 1991).CrossRefGoogle Scholar
[3]Burguera, M., Fernandez-Garcia, J. R. and Viano, J. M., “A 3d-fem simulation of highest stress lines in mandible fractures by elastic impact”, Comput. Methods Biomech. Biomed. Eng. 3 (2000) 273–285.Google Scholar
[4]Campillo, M., Dascǎlu, C. and Ionescu, I. R., “Instability of a periodic system of faults”, Geophys. J. Int. 159 (2004) 212–222, doi:10.1111/j.1365-246X.2004.02365.x.CrossRefGoogle Scholar
[5]Campillo, M. and Ionescu, I. R., “Initiation of antiplane shear instability under slip dependent friction”, J. Geophys. Res. 102 (1997) 20363–20371, doi:10.1029/97JB01508.CrossRefGoogle Scholar
[6]Ekeland, I. and Temam, R., Convex analysis and variational problems, Volume 1 of Studies in Mathematics and its Applications (North-Holland, Amsterdam, 1976).Google Scholar
[7]Fernandez, J. R., Gallas, M., Burguera, M. and Viano, J. M., “A three-dimensional numerical simulation of mandible fracture reduction with screwed miniplates”, J. Biomech. 36 (2003) 329–337, doi:10.1016/S0021-9290(02)00416-5.CrossRefGoogle ScholarPubMed
[8]Han, W. and Sofonea, M., Quasistatic contact problems in viscoelasticity and viscoplasticity, Studies in Advanced Mathematics (American Mathematical Society, Providence, RI, 2002).CrossRefGoogle Scholar
[9]Haslinger, J., Hlavác̆ek, I. and Nec̆as, J., “Numerical methods for unilateral problems in solid mechanics”, in: Handbook of numerical analysis, Volume IV (ed. Ciarlet, P. G.), (North-Holland, Amsterdam, 1996) 313–485.Google Scholar
[10]Hüeber, S., Mair, M. and Wohlmuth, B., “A priori error estimates and an inexact primal–dual active set strategy for linear and quadratic finite elements applied to multibody contact problems”, Appl. Numer. Math. 54 (2005) 555–576, doi:10.1016/j.apnum.2004.09.019.CrossRefGoogle Scholar
[11]Hüeber, S., Matei, A. and Wohlmuth, B., “A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity”, Bull. Math. Soc. Math. Roumanie 48 (2005) 209–232.Google Scholar
[12]Hüeber, S., Matei, A. and Wohlmuth, B., “Efficient algorithms for problems with friction”, SIAM J. Sci. Comput. 29 (2007) 70–92, doi:10.1137/050634141.CrossRefGoogle Scholar
[13]Hüeber, S. and Wohlmuth, B., “An optimal a priori error estimate for nonlinear multibody contact problems”, SIAM J. Numer. Anal. 43 (2005) 157–173, doi:10.1137/S0036142903436678.CrossRefGoogle Scholar
[14]Hüeber, S. and Wohlmuth, B., “A primal–dual active set strategy for nonlinear multibody contact problems”, Comput. Methods Appl. Mech. Engrg. 194 (2005) 3147–3166, doi:10.1016/j.cma.2004.08.006.CrossRefGoogle Scholar
[15]Laursen, T., Computational contact and impact mechanics (Springer, Berlin, 2002).Google Scholar
[16]Matei, A., “A variational approach for an electro-elastic unilateral contact problem”, Math. Model. Anal. 14 (2009) 323–334, doi:10.3846/1392-6292.2009.14.323-334.CrossRefGoogle Scholar
[17]Shillor, M., Sofonea, M. and Telega, J., Models and variational analysis of quasistatic contact, Volume 655 of Lecture Notes in Physics (Springer, Berlin, 2004).CrossRefGoogle Scholar
[18]Sofonea, M., Han, W. and Shillor, M., Analysis and approximation of contact problems with adhesion or damage, Volume 276 of Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, FL, 2006).Google Scholar
[19]Sofonea, M. and Matei, A., Variational inequalities with applications: a study of antiplane frictional contact problems, Volume 18 of Advances in Mechanics and Mathematics (Springer, New York, 2009).Google Scholar
[20]Wohlmuth, B., “A mortar finite element method using dual spaces for the Lagrange multiplier”, SIAM J. Numer. Anal. 38 (2000) 989–1012, doi:10.1137/S0036142999350929.CrossRefGoogle Scholar
[21]Wohlmuth, B., Discretization methods and iterative solvers based on domain decomposition, Volume 17 of Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2001).CrossRefGoogle Scholar
[22]Wriggers, P., Computational contact mechanics, 2nd edn (Springer, Berlin, 2006).CrossRefGoogle Scholar