# Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution

• Volume: 54, Issue: 5, page 391-416
• ISSN: 0862-7940

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## Abstract

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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 $ℱ$ which depends on a solution. It is shown that a solution exists for a large class of $ℱ$ and is unique provided that $ℱ$ 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.

## How to cite

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Haslinger, Jaroslav, and Ligurský, Tomáš. "Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution." Applications of Mathematics 54.5 (2009): 391-416. <http://eudml.org/doc/37829>.

@article{Haslinger2009,
abstract = {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 $\mathcal \{F\}$ which depends on a solution. It is shown that a solution exists for a large class of $\mathcal \{F\}$ and is unique provided that $\mathcal \{F\}$ 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.},
author = {Haslinger, Jaroslav, Ligurský, Tomáš},
journal = {Applications of Mathematics},
keywords = {unilateral contact and friction; solution-dependent coefficient of friction; unilateral contact and friction; solution-dependent coefficient of friction},
language = {eng},
number = {5},
pages = {391-416},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution},
url = {http://eudml.org/doc/37829},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Haslinger, Jaroslav
AU - Ligurský, Tomáš
TI - Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 5
SP - 391
EP - 416
AB - 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 $\mathcal {F}$ which depends on a solution. It is shown that a solution exists for a large class of $\mathcal {F}$ and is unique provided that $\mathcal {F}$ 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.
LA - eng
KW - unilateral contact and friction; solution-dependent coefficient of friction; unilateral contact and friction; solution-dependent coefficient of friction
UR - http://eudml.org/doc/37829
ER -

## References

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8. Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J., 10.1007/978-1-4612-1048-1, Springer New York (1988). (1988) MR0952855DOI10.1007/978-1-4612-1048-1
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). (1988) MR0961258
10. Kučera, R., 10.1137/060670456, SIAM J. Optim. 19 (2008), 846-862. (2008) Zbl1168.65028MR2448917DOI10.1137/060670456
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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