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Depinning of a Discrete Elastic String from a Random Array of Weak Pinning Points with Finite Dimensions

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Abstract

In possible connection with dislocation pinning by foreign atoms in alloys and vortex pinning in type II superconductors, we compute the external force required to drag an elastic string along a discrete two-dimensional random array with finite dimensions. The obstacles, with a maximum pinning force f m are distributed randomly on a rectangular lattice with square symmetry. The system dimensions are fixed by the total course of the elastic string L x and the string length L y . Our study shows that Larkin’s length is larger than L y when f m is less than a certain bound depending on the system size as well as on the obstacle density c s . Below such a bound an analytical theory is developed to compute the depinning threshold. Some numerical simulations allow us to demonstrate the accuracy of the theory for an obstacle density ranging from 1 to 50% and for different geometries.

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References

  1. Arsenault, R., Esterling, D.: Metall. Trans. A 20, 1411 (1989)

    Article  Google Scholar 

  2. Auslaender, O., Luan, L., Straver, E., Hoffman, J., Koshnick, N., Zeldov, E., Bonn, D., Liang, R., Hardy, W., Moler, K.: Nature 5, 35 (2009)

    Google Scholar 

  3. Barabasi, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  4. Blanter, Y.M., Vinokur, V.: Phys. Rev. B 66, 132101 (2002)

    Article  ADS  Google Scholar 

  5. Blatter, G., Feigelman, M., Geshkenbein, V., Larkin, A., Vinokur, V.: Rev. Mod. Phys. 66, 1125 (1994)

    Article  ADS  Google Scholar 

  6. Bouchaud, E.: J. Phys.: Condens. Matter 9, 4319 (1997)

    Article  ADS  Google Scholar 

  7. Brazovskii, S., Nattermann, T.: Adv. Phys. 53, 177 (2004)

    Article  ADS  Google Scholar 

  8. Chauve, P., Giamarchi, T., LeDoussal, P.: Phys. Rev. B 62, 6241 (2000)

    Article  ADS  Google Scholar 

  9. Curtin, W., Olmsted, D., Hector, L.: Nat. Mater. 5, 875 (2006)

    Article  ADS  Google Scholar 

  10. D’Anna, G., Benoit, W., Vinokur, V.: J. Appl. Phys. 82, 5983 (1997)

    Article  ADS  Google Scholar 

  11. Didenko, D.A., Pustovalov, V.: J. Low Temp. Phys. 11, 65 (1973)

    Article  ADS  Google Scholar 

  12. Fisher, D.: Phys. Rev. Lett. 56, 1964 (1986)

    Article  ADS  Google Scholar 

  13. Fleischer, R.: Acta Metall. 11, 203 (1963)

    Article  Google Scholar 

  14. Fleischer, R.: The Strengthening of Metals. Reinold Press Edition (1964), p. 93

  15. Fleischer, R., Hibbard, W.: The Relation Between Structure and Mechanical Properties of Metals, vol. 1. H.M.S.O., London (1963), p. 262

    Google Scholar 

  16. Foreman, A., Makin, M.: Philos. Mag. 14, 911 (1966)

    Article  ADS  Google Scholar 

  17. Friedel, J.: Dislocations. Addison-Wesley, New York (1964), p. 224

    MATH  Google Scholar 

  18. Gennes, P.D., Matricon, J.: Rev. Mod. Phys. 36, 45 (1964)

    Article  ADS  Google Scholar 

  19. Hiratani, M., Bulatov, V.: Philos. Mag. Lett. 84, 461 (2004)

    Article  ADS  Google Scholar 

  20. Hirth, J., Lothe, J.: Theory of Dislocations. Wiley Interscience, New York (1982), p. 180

    Google Scholar 

  21. Joanny, J., de Gennes, P.: J. Chem. Phys. 81, 552 (1984)

    Article  ADS  Google Scholar 

  22. Kierfeld, J., Vinokur, V.: Phys. Rev. Lett. 96, 175502 (2006)

    Article  ADS  Google Scholar 

  23. Labusch, R.: Phys. Status Solidi 19, 715 (1967)

    Article  Google Scholar 

  24. Labusch, R.: Phys. Status Solidi 41, 659 (1970)

    Article  Google Scholar 

  25. Labusch, R.: Acta Metall. 20, 917 (1972)

    Article  Google Scholar 

  26. Larkin, A.: Sov. Phys. JETP 31, 784 (1970)

    ADS  Google Scholar 

  27. Larkin, A., Ovchinnikov, Y.N.: J. Low Temp. Phys. 34, 409 (1979)

    Article  ADS  Google Scholar 

  28. Marian, J., Caro, A.: Phys. Rev. B 74, 024113 (2006)

    Article  ADS  Google Scholar 

  29. Moretti, P., Carmen-Miguel, M., Zaiser, M., Zapperi, S.: Phys. Rev. B 69, 214103 (2004)

    Article  ADS  Google Scholar 

  30. Mott, N.: Imperfections in Nearly Perfect Crystals. Wiley, New York (1952), p. 173

    Google Scholar 

  31. Mott, N., Nabarro, F.: Report on the Strength of Solids. Physical Society, London (1948), pp. 1–19

    Google Scholar 

  32. Nabarro, F.: Dislocations and Properties of Real Materials. The Institute of Metals, London (1985), p. 152

    Google Scholar 

  33. Narayan, O., Fisher, D.: Phys. Rev. B 48, 7030 (1993)

    Article  ADS  Google Scholar 

  34. Nogaret, T., Rodney, D.: Phys. Rev. B 74, 134110 (2006)

    Article  ADS  Google Scholar 

  35. Olmsted, D., Hector, L., Curtin, W., Clifton, R.: Model. Simul. Mater. Sci. Eng. 13, 371 (2005)

    Article  ADS  Google Scholar 

  36. Park, G., Cunningham, C., Cabrera, B., Huber, M.: Phys. Rev. B 68, 1920 (1992)

    ADS  Google Scholar 

  37. Patinet, S., Proville, L.: Phys. Rev. B 78, 104109 (2008)

    Article  ADS  Google Scholar 

  38. Proville, L.: arXiv:0904.3357

  39. Proville, L., Rodney, D., Bréchet, Y., Martin, G.: Philos. Mag. 86, 3893 (2006)

    Article  ADS  Google Scholar 

  40. Rodary, E., Rodney, D., Proville, L., Bréchet, Y., Martin, G.: Phys. Rev. B 70, 054111 (2004)

    Article  ADS  Google Scholar 

  41. Rosso, A., LeDoussal, P., Wiese, K.: Phys. Rev. B 75, 220201 (2007)

    Article  ADS  Google Scholar 

  42. Rusakov, V., Vasiliev, S., Chabanenko, V., Yurov, A., Nabialek, A., Piechota, S., Szymczak, H.: Acta Phys. Pol. A 109, 641 (2006)

    Google Scholar 

  43. Suzuki, T., Takeuchi, S., Yoshinaga, H.: Dislocation Dynamics and Plasticity. Springer, Berlin (1991), p. 32

    Google Scholar 

  44. Tanguy, A., Vettorel, T.: Eur. Phys. J. B 38, 71 (2004)

    ADS  Google Scholar 

  45. Tapasa, K., Bacon, D., Osetsky, Y.N.: Model. Simul. Mater. Sci. Eng. 14, 1153 (2006)

    Article  ADS  Google Scholar 

  46. Tapasa, K., Bacon, D., Osetsky, Y.N.: Acta Mater. 55, 93 (2007)

    Article  Google Scholar 

  47. Wiese, K.: Acta Phys. Slovaca 52, 341 (2002)

    Google Scholar 

  48. Xu, Z., Picu, R.: Phys. Rev. B 76, 94112 (2007)

    Article  ADS  Google Scholar 

  49. Zapperi, S., Ciseau, P., Durin, G., Stanley, E.: Phys. Rev. B 58, 6353 (1998)

    Article  ADS  Google Scholar 

  50. Zapperi, S., Zaiser, M.: Mater. Sci. Eng. A 309–310, 348 (2001)

    Article  Google Scholar 

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  1. CEA, DEN Service de Recherches de Métallurgie Physique, 91191, Gif-sur-Yvette, France

    Laurent Proville

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  1. Laurent Proville
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Correspondence to Laurent Proville.

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Proville, L. Depinning of a Discrete Elastic String from a Random Array of Weak Pinning Points with Finite Dimensions. J Stat Phys 137, 717–727 (2009). https://doi.org/10.1007/s10955-009-9860-8

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  • DOI: https://doi.org/10.1007/s10955-009-9860-8

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