Abstract.
Forecasting failure events is one of the most important problems in fracture mechanics and related sciences. In this paper, we use the Molchan scheme to investigate the error diagrams in a fracture model which has the notable advantage of displaying two completely different regimes according to the heterogeneity of the system. In one regime, a characteristic event is observed while for the second regime a power-law spectrum of avalanches is obtained reminiscent of self-organized criticality. We find that both regimes are different when predicting large avalanches and that, in the second regime, there are non-trivial temporal correlations associated to clustering of large events. Finally, we extend the discussion to seismology, where both kinds of avalanche size distributions can be seen.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H.J. Herrmann, S. Roux, Statistical Models for the Fracture of Disordered Media (North Holland, Amsterdam, 1990), and references therein
D.L. Turcotte, Fractals and Chaos in Geology and Geophysics. 2nd edn (Cambridge University Press, Cambridge, New York, 1997)
D. Sornette, Critical Phenomena in Natural Sciences, (Springer-Verlag, New York, 2000)
L. de Arcangelis, S. Redner, H.J. Herrmann, J. Phys. (Paris) 46, L585 (1985)
S. Roux, E. Guyon, J. Phys. Lett. France 46, L999 (1985)
R. Burridgeq, L. Knopoff, Bull. Seismol. Soc. Am. 57, 341 (1967)
Z. Olami, H.J. Feder, K. Christensen, Phys. Rev. Lett. 68, 1244 (1992)
K. Christensen, Z. Olami, J. Geophys. Res. 97, 8729 (1992)
H.E. Daniels, Proc. R. Soc. London, Ser. A 183, 404 (1945)
W.I. Newman, D.L. Turcotte, A.M. Gabrielov, Phys. Rev. E 52, 4827 (1995)
P.C. Hemmer, A. Hansen, J. Appl. Mech. 59, 909 (1992)
Y. Moreno, J.B. Gómez, A.F. Pacheco, Phys. Rev. Lett. 85, 2865 (2000)
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A. 38, 364 (1988)
G. Gutenberg, C.F. Richter, Ann. Geophys. 9, 1 (1956)
P. Bak, C. Tang, J. Geophys. Res. 94, 15635 (1989)
D. Sornette, A. Sornette, Europhys. Lett. 9, 197 (1989)
K. Ito, M. Matsuzaki, J. Geophys. Res. 95, 6853 (1990)
V. Keilis-Borok, Annu. Rev. Earth Planet. Sci. 30, 1 (2002)
Y. Moreno, J.B. Gómez, A.F. Pacheco, Physica A 274, 400 (1999)
S.G. Wesnousky, Bull Seismol. Soc. Am. 84, 1940 (1994)
K. Sieh, Proc. Natl. Acad. Sci. USA 93, 3764 (1996)
G.M. Molchan, Pure Appl. Geophys. 149, 233 (1997)
This choice may seem arbitrary. While the equal load sharing is the simplest load transfer rule one may consider, it also gives more accurate results when modeling elastic forces than other load transfer schemes, at least within the FBM context. However, one can in principle consider general load transfer schemes as the one introduced in [25]. There, it is shown that the behavior of the system can be regarded as global provided that the interactions follow a power law of the form \(r^{-\gamma}<Emphasis Type="Italic">with</Emphasis>\gamma\le 2\). On the other hand, similar results have been obtained when using local models [26]
R.C. Hidalgo, Y. Moreno, F. Kun, H.J. Herrmann, Phys. Rev. E 65, 046148 (2002)
Z. Olami, H.J. Feder, K. Christensen, Phys. Rev. Lett. 68, 1244 (1992)
W.I. Newman, D.L. Turcotte, Nonlin. Proc. Geophys. 9, 1 (2002)
A. Helmstetter, D. Sornette, e-print cond-mat/0208597, (2002)
M. Vázquez-Prada, Á. González, J.B. Gómez, A.F. Pacheco, (unpublished) 2003
A.C. Johnston, S.J. Nava, J. Geophys. Res. 90, 6737 (1985)
Y.Y. Kagan, D.D. Jackson, Geophys. J. Int. 104, 117 (1991)
M.S. Mega, P. Allegrini, P. Grigolini, V. Latora, L. Palatella, A. Rapisarda, S. Vinciguerra, Phys. Rev. Lett. (in press) also e-print cond-mat/0208597 (2002)
M. Vázquez-Prada, Á. González, J.B. Gómez, A.F. Pacheco, Nonlin. Proc. Geophys. 9, 513 (2002)
Models able to pass from the GR to a characteristic earthquake behavior are not found very often in the literature. One of such models was proposed in [35], where the concept of configurational entropy is used to argue the existence of the two regimes. Our model also suggests that these two behaviors can be recognized when the degree of heterogeneity of the faults under study is different
K. Dahmen, D. Ertas, Y. Ben-Zion, Phys. Rev. E 58, 1494 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: 16 April 2003, Published online: 9 September 2003
PACS:
46.50.+a Fracture mechanics, brittleness, fracture and cracks - 91.45.Vz Fracture and faults - 62.20.Mk Fatigue, brittleness, fracture, and cracks
Rights and permissions
About this article
Cite this article
Moreno, Y., Vázquez-Prada, M., Gómez, J.B. et al. Error diagrams and temporal correlations in a fracture model with characteristic and power-law distributed avalanches. Eur. Phys. J. B 34, 489–494 (2003). https://doi.org/10.1140/epjb/e2003-00248-6
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2003-00248-6