Article
MSC:
74G20,
74G55,
74M10,
74M15,
74S05 | MR 2984603 | Zbl 1265.74069 | DOI: 10.1007/s10492-012-0016-9
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
unilateral contact; Coulomb friction; local uniqueness; qualitative behaviour
unilateral contact; Coulomb friction; local uniqueness; qualitative behaviour
Summary:
A discrete model of the two-dimensional Signorini problem with Coulomb friction and a coefficient of friction $\mathcal {F}$ depending on the spatial variable is analysed. It is shown that a solution exists for any $\mathcal {F}$ and is globally unique if $\mathcal {F}$ is sufficiently small. The Lipschitz continuity of this unique solution as a function of $\mathcal {F}$ as well as a function of the load vector $\boldsymbol {f}$ is obtained. Furthermore, local uniqueness of solutions for arbitrary $\mathcal {F} > 0$ is studied. The question of existence of locally Lipschitz-continuous branches of solutions with respect to the coefficient $\mathcal {F}$ is converted to the question of existence of locally Lipschitz-continuous branches of solutions with respect to the load vector $\boldsymbol {f}$. A condition guaranteeing the existence of locally Lipschitz-continuous branches of solutions in the latter case and results for determining their directional derivatives are given. Finally, the general approach is illustrated on an elementary example, whose solutions are calculated exactly.
References:
[1] Beremlijski, P., Haslinger, J., Kočvara, M., Kučera, R., Outrata, J. V.: Shape optimization in three-dimensional contact problems with Coulomb friction. SIAM J. Optim. 20 (2009), 416-444. DOI 10.1137/080714427 | MR 2507130 | Zbl 1186.49028
[2] Beremlijski, P., Haslinger, J., Kočvara, M., Outrata, J. V.: Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13 (2002), 561-587. DOI 10.1137/S1052623401395061 | MR 1951035 | Zbl 1025.49026
[3] Dontchev, A. L., Hager, W. W.: Implicit functions, Lipschitz maps, and stability in optimization. Math. Oper. Res. 19 (1994), 753-768. DOI 10.1287/moor.19.3.753 | MR 1288898 | Zbl 0835.49019
[4] Eck, C., Jarušek, J.: Existence results for the static contact problem with Coulomb friction. Math. Models Methods Appl. Sci. 8 (1998), 445-468. DOI 10.1142/S0218202598000196 | MR 1624879 | Zbl 0907.73052
[5] Ekeland, I., Temam, R.: Analyse convexe et problèmes variationnels. Études mathématiques. Dunod/Gauthier-Villars Paris/Bruxelles-Montréal (1974), French. MR 0463993
[6] Glowinski, R.: Numerical methods for nonlinear variational problems. Springer Series in Computational Physics. Springer New York (1984). MR 0737005
[7] Haslinger, J.: Approximation of the Signorini problem with friction, obeying the Coulomb law. Math. Methods Appl. Sci. 5 (1983), 422-437. DOI 10.1002/mma.1670050127 | MR 0716664 | Zbl 0525.73130
[8] 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 (1996), 313-485. DOI 10.1016/S1570-8659(96)80005-6 | MR 1422506
[9] Hild, P.: An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction. C. R. Math., Acad. Sci. Paris 337 (2003), 685-688. DOI 10.1016/j.crma.2003.10.010 | MR 2030112 | Zbl 1035.35090
[10] Hild, P.: Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity. Q. J. Mech. Appl. Math. 57 (2004), 225-235. DOI 10.1093/qjmam/57.2.225 | MR 2068404 | Zbl 1059.74042
[11] Hild, P., Renard, Y.: Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics. Quart. Appl. Math. 63 (2005), 553-573 \MR 2169034. DOI 10.1090/S0033-569X-05-00974-0 | MR 2169034
[12] Janovský, V.: Catastrophic features of Coulomb friction model. The mathematics of finite elements and applications IV, MAFELAP 1981, Proc. Conf., Uxbridge/Middlesex 1981 (1982), 259-264. Zbl 0504.73077
[13] Nečas, J., Jarušek, J., Haslinger, J.: On the solution of the variational inequality to the Signorini problem with small friction. Boll. Unione Mat. Ital., V. Ser., B 17 (1980), 796-811. MR 0580559
[14] Renard, Y.: A uniqueness criterion for the Signorini problem with Coulomb friction. SIAM J. Math. Anal. 38 (2006), 452-467. DOI 10.1137/050635936 | MR 2237156 | Zbl 1194.74225
[15] Robinson, S. M.: Strongly regular generalized equations. Math. Oper. Res. 5 (1980), 43-62. DOI 10.1287/moor.5.1.43 | MR 0561153 | Zbl 0437.90094
[16] Scholtes, S.: Introduction to piecewise differentiable equations. Preprint No. 53/1994. Institut für Statistik und Mathematische Wirtschaftstheorie Universität Karlsruhe (1994).
Search
Partner of
