×
Itou, H.; Kovtunenko, V. A.; Tani, A.

The interface crack with Coulomb friction between two bonded dissimilar elastic media. (English) Zbl 1224.35054

Appl. Math., Praha 56, No. 1, 69-97 (2011).
Summary: We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.

Cited in 14 Documents

MSC:

35C20 Asymptotic expansions of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
49J40 Variational inequalities
74B05 Classical linear elasticity
74M10 Friction in solid mechanics

Keywords:

linearized elasticity; singularity at the crack tip; interfacial crack; non-penetration condition; Coulomb friction
PDFBibTeX XMLCite
\textit{H. Itou} et al., Appl. Math., Praha 56, No. 1, 69--97 (2011; Zbl 1224.35054)
Full Text: DOI EuDML Link

References:

[1] L.-E. Andersson: Existence results for quasistatic contact problems with Coulomb friction. Appl. Math. Optim. 42 (2000), 169–202. · Zbl 0972.35058 · doi:10.1007/s002450010009
[2] B. Audoly: Asymptotic study of the interfacial crack with friction. J. Mech. Phys. Solids 48 (2000), 1851–1864. · Zbl 0963.74048 · doi:10.1016/S0022-5096(99)00098-8
[3] M. Bach, A.M. Khludnev, V.A. Kovtunenko: Derivatives of the energy functional for 2D-problems with a crack under Signorini and friction conditions. Math. Methods Appl. Sci. 23 (2000), 515–534. · Zbl 0954.35076 · doi:10.1002/(SICI)1099-1476(200004)23:6<515::AID-MMA122>3.0.CO;2-S
[4] H.D. Bui, A. Oueslati: The sliding interface crack with friction between elastic and rigid bodies. J. Mech. Phys. Solids 53 (2005), 1397–1421. · Zbl 1120.74744 · doi:10.1016/j.jmps.2004.12.007
[5] M. Comninou: An overview of interface crack. Eng. Fract. Mech. 37 (1990), 197–208. · doi:10.1016/0013-7944(90)90343-F
[6] M. Comninou, J. Dundurs: Effect of friction on the interface crack loaded in shear. J. Elasticity 10 (1980), 203–212. · Zbl 0457.73098 · doi:10.1007/BF00044504
[7] J. Dundurs, M. Comninou: Some consequences of the inequality conditions in contact and crack problems. J. Elasticity 9 (1979), 71–82. · Zbl 0393.73117 · doi:10.1007/BF00040981
[8] Ch. Eck, J. Jarušek, M. Krbec: Unilateral Contact Problems. Chapman&amp;Hall/CRC, Boca Raton, 2005.
[9] A.H. England: Complex Variable Methods in Elasticity. John Wiley &amp; Sons, London, 1971. · Zbl 0222.73017
[10] G. Fichera: Existence theorems in elasticity. Mechanics of Solids Vol. II (C. Truesdell, ed.). Springer, Berlin, 1984, pp. 347–389.
[11] J. Haslinger, J. Kučera, O. Vlach: Bifurcations in contact problems with local Coulomb friction. Num. Math. Adv. Appl. (K. Kunisch, G. Of, O. Steinbach, eds.). Springer, Berlin, 2008, pp. 811–818. · Zbl 1155.74032
[12] P. Hild: Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity. Q. J. Mech. Appl. Math. 57 (2004), 225–235. · Zbl 1059.74042 · doi:10.1093/qjmam/57.2.225
[13] M. Hintermüller, V.A. Kovtunenko, K. Kunisch: Obstacle problems with cohesion: A hemi-variational inequality approach and its efficient numerical solution. MATHEON Report 687. DFG-Forschungszentrum, TU-Berlin, Berlin, 2010. · Zbl 1228.49012
[14] S. Hüeber, G. Stadler, B. I. Wohlmuth: A primal-dual active set algorithm for threedimensional contact problems with Coulomb friction. SIAM J. Sci. Comput. 30 (2008), 572–596. · Zbl 1158.74045 · doi:10.1137/060671061
[15] M. Ikehata, H. Itou: Reconstruction of a linear crack in an isotropic elastic body from a single set of measured data. Inverse Probl. 23 (2007), 589–607. · Zbl 1115.35149 · doi:10.1088/0266-5611/23/2/008
[16] M. Ikehata, H. Itou: Extracting the support function of a cavity in an isotropic elastic body from a single set of boundary data. Article ID 105005. Inverse Probl. 25 (2009), 1–21. · Zbl 1180.35564
[17] H. Itou, A. Tani: A boundary value problem for an infinite elastic strip with a semi-infinite crack. J. Elasticity 66 (2002), 193–206. · Zbl 1018.74033 · doi:10.1023/A:1021903404039
[18] Y. Kato: Signorini’s problem with friction in linear elasticity. Japan J. Appl. Math. 4 (1987), 237–268. · Zbl 0627.73098 · doi:10.1007/BF03167776
[19] A.M. Khludnev, V.A. Kovtunenko: Analysis of Cracks in Solids. WIT-Press, Southampton, Boston, 2000. · Zbl 0954.35076
[20] A.M. Khludnev, V.A. Kovtunenko, A. Tani: Evolution of a crack with kink and non-penetration. J. Math. Soc. Japan 60 (2008), 1219–1253. · Zbl 1153.49040 · doi:10.2969/jmsj/06041219
[21] A.M. Khludnev, V.A. Kovtunenko, A. Tani: On the topological derivative due to kink of a crack with non-penetration. J. Math. Pures Appl. 94 (2010), 571–596. · Zbl 1203.49035
[22] A.M. Khludnev, V.A. Kozlov: Asymptotics of solutions near crack tips for Poisson equation with inequality type boundary conditions. Z. Angew. Math. Phys. 59 (2008), 264–280. · Zbl 1138.74043 · doi:10.1007/s00033-007-6032-z
[23] N. Kikuchi, J.T. Oden: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, 1988. · Zbl 0685.73002
[24] V.A. Kovtunenko: Crack in a solid under Coulomb friction law. Appl. Math. 45 (2000), 265–290. · Zbl 1058.74064 · doi:10.1023/A:1022319428441
[25] A. S. Kravchuk: Variational and Quasivariational Inequations in Mechanics. MGAPI, Moscow, 1997. (In Russian.)
[26] V. Maz’ya, S. Nazarov, B. Plamenevskii: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. II. Birkhäuser, Basel, 2000. · Zbl 1127.35301
[27] N. I. Muskhelishvili: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, 1963.
[28] J. Nečas, J. Jarušek, J. Haslinger: On the solution of the variational inequality to the Signorini problem with small friction. Boll. Unione Mat. Ital. 17-B (1980), 796–811.
[29] Y. Renard: A uniqueness criterion for the Signorini problem with Coulomb friction. SIAM J. Math. Anal. 38 (2006), 452–467. · Zbl 1155.35383 · doi:10.1137/050635936
[30] J.R. Rice: Elastic fracture mechanics concepts for interfacial cracks. J. Appl. Mech. 55 (1988), 98–103. · doi:10.1115/1.3173668
[31] M. Shillor, M. Sofonea, J. Telega: Models and Analysis of Quasistatic Contact. Springer, Berlin, 2004. · Zbl 1069.74001
[32] R.A. Toupin: Saint-Venant’s principle. Arch. Ration. Mech. Anal. 18 (1965), 83–96. · Zbl 0203.26803 · doi:10.1007/BF00282253
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.