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Keywords:
unilateral contact and friction; solution-dependent coefficient of friction
unilateral contact and friction; solution-dependent coefficient of friction
Summary:
The paper presents the analysis, approximation and numerical realization of 3D contact problems for an elastic body unilaterally supported by a rigid half space taking into account friction on the common surface. Friction obeys the simplest Tresca model (a slip bound is given a priori) but with a coefficient of friction $\Cal F$ which depends on a solution. It is shown that a solution exists for a large class of $\Cal F$ and is unique provided that $\Cal F$ is Lipschitz continuous with a sufficiently small modulus of the Lipschitz continuity. The problem is discretized by finite elements, and convergence of discrete solutions is established. Finally, methods for numerical realization are described and several model examples illustrate the efficiency of the proposed approach.
References:
[1] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, Vol. 4. North-Holland Amsterdam-New York-Oxford (1978). MR 0520174
[2] Clement, P.: Approximation by finite element functions using local regularization. Rev. Franc. Automat. Inform. Rech. Operat. 9, R-2 (1975), 77-84. MR 0400739 | Zbl 0368.65008
[3] Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften, Band 219. Springer Berlin-Heidelberg-New York (1976). MR 0521262
[4] Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer Series in Computational Physics. Springer New York (1984). MR 0737005
[5] Haslinger, J., Hlaváček, I., Nečas, J.: Numerical methods for unilateral problems in solid mechanics. Handbook of Numerical Analysis, Vol. IV P. G. Ciarlet et al. North-Holland Amsterdam (1995), 313-485. MR 1422506
[6] Haslinger, J., Vlach, O.: Signorini problem with a solution dependent coefficient of friction (model with given friction): Approximation and numerical realization. Appl. Math. 50, (2005), 153-171. DOI 10.1007/s10492-005-0010-6 | MR 2125156 | Zbl 1099.65109
[7] Hlaváček, I.: Finite element analysis of a static contact problem with Coulomb friction. Appl. Math. 45 (2000), 357-379. DOI 10.1023/A:1022220711369 | MR 1777018
[8] Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Applied Mathematical Sciences, Vol. 66. Springer New York (1988). DOI 10.1007/978-1-4612-1048-1 | MR 0952855
[9] Kikuchi, N., Oden, J. T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM Studies in Applied Mathematics, Vol. 8. SIAM Philadelphia (1988). MR 0961258
[10] Kučera, R.: Convergence rate of an optimization algorithm for minimizing quadratic functions with separable convex constraints. SIAM J. Optim. 19 (2008), 846-862. DOI 10.1137/060670456 | MR 2448917 | Zbl 1168.65028
[11] Ligurský, T.: Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution. Diploma thesis MFF UK, 2007 ( http://artax.karlin.mff.cuni.cz/ {ligut2am/tl21.pdf}).
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