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

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.

### 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
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### 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&Hall/CRC, Boca Raton, 2005. [9] A.H. England: Complex Variable Methods in Elasticity. John Wiley & 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
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