The interface crack with Coulomb friction between two bonded dissimilar elastic media

Hiromichi Itou; Victor A. Kovtunenko; Atusi Tani

Applications of Mathematics (2011)

  • Volume: 56, Issue: 1, page 69-97
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Itou, Hiromichi, Kovtunenko, Victor A., and Tani, Atusi. "The interface crack with Coulomb friction between two bonded dissimilar elastic media." Applications of Mathematics 56.1 (2011): 69-97. <http://eudml.org/doc/116505>.

@article{Itou2011,
abstract = {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.},
author = {Itou, Hiromichi, Kovtunenko, Victor A., Tani, Atusi},
journal = {Applications of Mathematics},
keywords = {linearized elasticity; singularities at the crack tip; interfacial crack; non-penetration condition; Coulomb friction; linearized elasticity; singularity at the crack tip; interfacial crack; non-penetration condition; Coulomb friction},
language = {eng},
number = {1},
pages = {69-97},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The interface crack with Coulomb friction between two bonded dissimilar elastic media},
url = {http://eudml.org/doc/116505},
volume = {56},
year = {2011},
}

TY - JOUR
AU - Itou, Hiromichi
AU - Kovtunenko, Victor A.
AU - Tani, Atusi
TI - The interface crack with Coulomb friction between two bonded dissimilar elastic media
JO - Applications of Mathematics
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 69
EP - 97
AB - 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.
LA - eng
KW - linearized elasticity; singularities at the crack tip; interfacial crack; non-penetration condition; Coulomb friction; linearized elasticity; singularity at the crack tip; interfacial crack; non-penetration condition; Coulomb friction
UR - http://eudml.org/doc/116505
ER -

References

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  1. Andersson, L.-E., 10.1007/s002450010009, Appl. Math. Optim. 42 (2000), 169-202. (2000) Zbl0972.35058MR1784173DOI10.1007/s002450010009
  2. Audoly, B., 10.1016/S0022-5096(99)00098-8, J. Mech. Phys. Solids 48 (2000), 1851-1864. (2000) Zbl0963.74048DOI10.1016/S0022-5096(99)00098-8
  3. Bach, M., Khludnev, A. M., Kovtunenko, V. A., 10.1002/(SICI)1099-1476(200004)23:6<515::AID-MMA122>3.0.CO;2-S, Math. Methods Appl. Sci. 23 (2000), 515-534. (2000) Zbl0954.35076MR1748320DOI10.1002/(SICI)1099-1476(200004)23:6<515::AID-MMA122>3.0.CO;2-S
  4. Bui, H. D., Oueslati, A., 10.1016/j.jmps.2004.12.007, J. Mech. Phys. Solids 53 (2005), 1397-1421. (2005) Zbl1120.74744MR2137068DOI10.1016/j.jmps.2004.12.007
  5. Comninou, M., 10.1016/0013-7944(90)90343-F, Eng. Fract. Mech. 37 (1990), 197-208. (1990) DOI10.1016/0013-7944(90)90343-F
  6. Comninou, M., Dundurs, J., 10.1007/BF00044504, J. Elasticity 10 (1980), 203-212. (1980) Zbl0457.73098MR0576168DOI10.1007/BF00044504
  7. Dundurs, J., Comninou, M., 10.1007/BF00040981, J. Elasticity 9 (1979), 71-82. (1979) Zbl0393.73117DOI10.1007/BF00040981
  8. Eck, Ch., Jarušek, J., Krbec, M., Unilateral Contact Problems, Chapman&Hall/CRC Boca Raton (2005). (2005) Zbl1079.74003MR2128865
  9. England, A. H., Complex Variable Methods in Elasticity, John Wiley & Sons London (1971). (1971) Zbl0222.73017MR0464824
  10. Fichera, G., Existence theorems in elasticity, Mechanics of Solids Vol. II C. Truesdell Springer Berlin (1984), 347-389. (1984) 
  11. Haslinger, J., Kučera, J., Vlach, O., 10.1007/978-3-540-69777-0_97, Num. Math. Adv. Appl. K. Kunisch, G. Of, O. Steinbach Springer Berlin (2008), 811-818. (2008) Zbl1155.74032MR3615958DOI10.1007/978-3-540-69777-0_97
  12. Hild, P., 10.1093/qjmam/57.2.225, Q. J. Mech. Appl. Math. 57 (2004), 225-235. (2004) Zbl1059.74042MR2068404DOI10.1093/qjmam/57.2.225
  13. Hintermüller, M., Kovtunenko, V. A., Kunisch, K., Obstacle problems with cohesion: A hemi-variational inequality approach and its efficient numerical solution, MATHEON Report 687 DFG-Forschungszentrum, TU-Berlin Berlin (2010). (2010) MR2817476
  14. Hüeber, S., Stadler, G., Wohlmuth, B. I., 10.1137/060671061, SIAM J. Sci. Comput. 30 (2008), 572-596. (2008) Zbl1158.74045MR2385876DOI10.1137/060671061
  15. Ikehata, M., Itou, H., 10.1088/0266-5611/23/2/008, Inverse Probl. 23 (2007), 589-607. (2007) Zbl1115.35149MR2309665DOI10.1088/0266-5611/23/2/008
  16. Ikehata, M., Itou, H., 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. (2009) MR2545974
  17. Itou, H., Tani, A., 10.1023/A:1021903404039, J. Elasticity 66 (2002), 193-206. (2002) Zbl1018.74033MR1956323DOI10.1023/A:1021903404039
  18. Kato, Y., 10.1007/BF03167776, Japan J. Appl. Math. 4 (1987), 237-268. (1987) Zbl0627.73098MR0899912DOI10.1007/BF03167776
  19. Khludnev, A. M., Kovtunenko, V. A., Analysis of Cracks in Solids, WIT-Press Southampton, Boston (2000). (2000) 
  20. Khludnev, A. M., Kovtunenko, V. A., Tani, A., 10.2969/jmsj/06041219, J. Math. Soc. Japan 60 (2008), 1219-1253. (2008) Zbl1153.49040MR2467876DOI10.2969/jmsj/06041219
  21. Khludnev, A. M., Kovtunenko, V. A., Tani, A., 10.1016/j.matpur.2010.06.002, J. Math. Pures Appl. 94 (2010), 571-596. (2010) Zbl1203.49035MR2737389DOI10.1016/j.matpur.2010.06.002
  22. Khludnev, A. M., Kozlov, V. A., 10.1007/s00033-007-6032-z, Z. Angew. Math. Phys. 59 (2008), 264-280. (2008) Zbl1138.74043MR2400558DOI10.1007/s00033-007-6032-z
  23. Kikuchi, N., Oden, J. T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Philadelphia (1988). (1988) Zbl0685.73002MR0961258
  24. Kovtunenko, V. A., 10.1023/A:1022319428441, Appl. Math. 45 (2000), 265-290. (2000) Zbl1058.74064MR1763172DOI10.1023/A:1022319428441
  25. Kravchuk, A. S., Variational and Quasivariational Inequations in Mechanics, MGAPI Moscow (1997), Russian. (1997) 
  26. Maz'ya, V., Nazarov, S., Plamenevskii, B., Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. II, Birkhäuser Basel (2000). (2000) Zbl1127.35301MR1779978
  27. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff Groningen (1963). (1963) Zbl0124.17404MR0176648
  28. Nečas, J., Jarušek, J., Haslinger, J., On the solution of the variational inequality to the Signorini problem with small friction, Boll. Unione Mat. Ital. 17-B (1980), 796-811. (1980) MR0580559
  29. Renard, Y., 10.1137/050635936, SIAM J. Math. Anal. 38 (2006), 452-467. (2006) Zbl1194.74225MR2237156DOI10.1137/050635936
  30. Rice, J. R., 10.1115/1.3173668, J. Appl. Mech. 55 (1988), 98-103. (1988) DOI10.1115/1.3173668
  31. Shillor, M., Sofonea, M., Telega, J., Models and Analysis of Quasistatic Contact, Springer Berlin (2004). (2004) Zbl1069.74001
  32. Toupin, R. A., 10.1007/BF00282253, Arch. Ration. Mech. Anal. 18 (1965), 83-96. (1965) Zbl0203.26803MR0172506DOI10.1007/BF00282253

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