- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Universal avalanche statistics and triggering close to failure in a mean-field model of rheological fracture

Jordi Baró* and Jörn Davidsen

  • Department of Physics and Astronomy University of Calgary, 2500 University Drive, NW Calgary, Alberta, Canada T2N 1N4

  • *jordi.barourbea@ucalgary.ca
PDF

Phys. Rev. E 97, 033002 – Published 29 March, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.033002

Abstract

The hypothesis of critical failure relates the presence of an ultimate stability point in the structural constitutive equation of materials to a divergence of characteristic scales in the microscopic dynamics responsible for deformation. Avalanche models involving critical failure have determined common universality classes for stick-slip processes and fracture. However, not all empirical failure processes exhibit the trademarks of criticality. The rheological properties of materials introduce dissipation, usually reproduced in conceptual models as a hardening of the coarse grained elements of the system. Here, we investigate the effects of transient hardening on (i) the activity rate and (ii) the statistical properties of avalanches. We find the explicit representation of transient hardening in the presence of generalized viscoelasticity and solve the corresponding mean-field model of fracture. In the quasistatic limit, the accelerated energy release is invariant with respect to rheology and the avalanche propagation can be reinterpreted in terms of a stochastic counting process. A single universality class can be defined from such analogy, and all statistical properties depend only on the distance to criticality. We also prove that interevent correlations emerge due to the hardening—even in the quasistatic limit—that can be interpreted as “aftershocks” and “foreshocks.”

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (107)

  1. J. Li, K. J. Van Vliet, T. Zhu, S. Yip, and S. Suresh, Atomistic mechanisms governing elastic limit and incipient plasticity in crystals, Nature (London) 418, 307 (2002).
  2. N. Friedman, A. T. Jennings, G. Tsekenis, J.-Y. Kim, M. Tao, J. T. Uhl, J. R. Greer, and K. A. Dahmen, Statistics of Dislocation Slip Avalanches in Nanosized Single Crystals Show Tuned Critical Behavior Predicted by a Simple Mean Field Model, Phys. Rev. Lett. 109, 095507 (2012).
  3. J. Antonaglia, X. Xie, G. Schwarz, M. Wraith, J. Qiao, Y. Zhang, P. K. Liaw, J. T. Uhl, and K. A. Dahmen, Tuned critical avalanche scaling in bulk metallic glasses, Sci. Rep. 4, 4382 (2014).
  4. K. M. Salerno and M. O. Robbins, Effect of inertia on sheared disordered solids: Critical scaling of avalanches in two and three dimensions, Phys. Rev. E 88, 062206 (2013).
  5. R. C. Hidalgo, C. U. Grosse, F. Kun, H. W. Reinhardt, and H. J. Herrmann, Evolution of Percolating Force Chains in Compressed Granular Media, Phys. Rev. Lett. 89, 205501 (2002).
  6. D. V. Denisov, K. A. Lőrincz, W. J. Wright, T. C. Hufnagel, A. Nawano, X. Gu, J. T. Uhl, K. A. Dahmen, and P. Schall, Universal slip dynamics in metallic glasses and granular matter–linking frictional weakening with inertial effects, Sci. Rep. 7, 43376 (2017).
  7. D. Bonamy and E. Bouchaud, Failure of heterogeneous materials: A dynamic phase transition? Phys. Rep. 498, 1 (2011).
  8. K. Mogi, The influence of the dimensions of specimens on the fracture strength of rocks: Comparison between the strength of rock specimens and that of the earth's crust, Bulletin of the Earthquake Research Institute 40, 175 (1962).
  9. R. Burridge and L. Knopoff, Model and theoretical seismicity, Bull. Seismol. Soc. Am. 57, 341 (1967).
  10. Z. Olami, H. J. S. Feder, and K. Christensen, Self-Organized Criticality in a Continuous, Nonconservative Cellular Automaton Modeling Earthquakes, Phys. Rev. Lett. 68, 1244 (1992).
  11. M. J. Alava, P. K. V. V. Nukala, and S. Zapperi, Statistical models of fracture, Adv. Phys. 55, 349 (2006).
  12. Y. Ben-Zion, K. A. Dahmen, and J. T. Uhl, A unifying phase diagram for the dynamics of sheared solids and granular materials, Pure Appl. Geophys. 168, 2221 (2011).
  13. L. Girard, D. Amitrano, and J. Weiss, Failure as a critical phenomenon in a progressive damage model, J. Stat. Mech.: Theor. Exp. (2010) P01013.
  14. A. Shekhawat, S. Zapperi, and J. P. Sethna, From Damage Percolation to Crack Nucleation Through Finite Size Criticality, Phys. Rev. Lett. 110, 185505 (2013).
  15. L. De Arcangelis, S. Redner, and H. J. Herrmann, A random fuse model for breaking processes, J. Phys. Lett. 46, 585 (1985).
  16. P. M. Duxbury, P. L. Leath, and P. D. Beale, Breakdown properties of quenched random systems: The random-fuse network, Phys. Rev. B 36, 367 (1987).
  17. S. Zapperi, P. Ray, H. E. Stanley, and A. Vespignani, First-Order Transition in the Breakdown of Disordered Media, Phys. Rev. Lett. 78, 1408 (1997).
  18. F. F. Csikor, C. Motz, D. Weygand, M. Zaiser, and S. Zapperi, Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale, Science 318, 251 (2007).
  19. D. Amitrano, Variability in the power-law distributions of rupture events, Eur. Phys. J.: Spec. Top. 205, 199 (2012).
  20. Y. Moreno, J. B. Gomez, and A. F. Pacheco, Fracture and Second-Order Phase Transitions, Phys. Rev. Lett. 85, 2865 (2000).
  21. F. Kun, S. Zapperi, and H. J. Herrmann, Damage in fiber bundle models, Eur. Phys. J. B 17, 269 (2000).
  22. J. P. Sethna, K. A. Dahmen, and C. R. Myers, Crackling noise, Nature (London) 410, 242 (2001).
  23. Y. Ben-Zion, Collective behavior of earthquakes and faults: Continuum-discrete transitions, progressive evolutionary changes, and different dynamic regimes, Rev. Geophys. 46, RG4006 (2008).
  24. H. Kanamori and E. E. Brodsky, The physics of earthquakes, Rep. Prog. Phys. 67, 1429 (2004).
  25. C. B. Scruby, An introduction to acoustic emission, J. Phys. E 20, 946 (1987).
  26. D. Lockner, The role of acoustic emission in the study of rock fracture, Int. J. Rock Mech. Min. Sci. 30, 883 (1993).
  27. I. G. Main, P. G. Meredith, and C. Jones, A reinterpretation of the precursory seismic b-value anomaly from fracture mechanics, Geophys. J. Int. 96, 131 (1989).
  28. C. H. Scholz, The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes, Bull. Seismol. Soc. Am. 58, 399 (1968).
  29. P. G. Meredith and B. K. Atkinson, Stress corrosion and acoustic emission during tensile crack propagation in whin sill dolerite and other basic rocks, Geophys. J. Int. 75, 1 (1983).
  30. J.-C. Anifrani, C. Le Floc'h, D. Sornette, and B. Souillard, Universal log-periodic correction to renormalization group scaling for rupture stress prediction from acoustic emissions, J. de Physique I, EDP Sciences 5, 631 (1995).
  31. J. Davidsen, S. Stanchits, and G. Dresen, Scaling and Universality in Rock Fracture, Phys. Rev. Lett. 98, 125502 (2007).
  32. G. Michlmayr, D. Cohen, and D. Or, Sources and characteristics of acoustic emissions from mechanically stressed geologic granular media—a review, Earth-Sci. Rev. 112, 97 (2012).
  33. T. H. W. Goebel, D. Schorlemmer, T. W. Becker, G. Dresen, and C. G. Sammis, Acoustic emissions document stress changes over many seismic cycles in stick-slip experiments, Geophys. Res. Lett. 40, 2049 (2013).
  34. J. Baró, Á. Corral, X. Illa, A. Planes, E. K. H. Salje, W. Schranz, D. E. Soto-Parra, and E. Vives, Statistical Similarity Between the Compression of a Porous Material and Earthquakes, Phys. Rev. Lett. 110, 088702 (2013).
  35. H. V. Ribeiro, L. S. Costa, L. G. A. Alves, P. A. Santoro, S. Picoli, E. K. Lenzi, and R. S. Mendes, Analogies Between the Cracking Noise of Ethanol-Dampened Charcoal and Earthquakes, Phys. Rev. Lett. 115, 025503 (2015).
  36. L. S. Costa, E. K. Lenzi, R. S. Mendes, and H. V. Ribeiro, Extensive characterization of seismic laws in acoustic emissions of crumpled plastic sheets, Europhys. Lett. 114, 59002 (2016).
  37. J. Davidsen, G. Kwiatek, E.-M. Charalampidou, T. Goebel, S. Stanchits, M. Rück, and G. Dresen, Triggering Processes in Rock Fracture, Phys. Rev. Lett. 119, 068501 (2017).
  38. C. U. Grosse and M. Ohtsu, Acoustic Emission Testing (Springer-Verlag, Berlin, Heidelberg, 2008).
  39. C. U. Grosse, H. W. Reinhardt, and F. Finck, Signal-based acoustic emission techniques in civil engineering, J. Mater. Civ. Eng. 15, 274 (2003).
  40. S. C. Jaumé and L. R. Sykes, Evolving towards a critical point: A review of accelerating seismic moment/energy release prior to large and great earthquakes, in Seismicity Patterns, their Statistical Significance and Physical Meaning (Springer, Basel AG, 1999), pp. 279–305.
  41. Y. Ben-Zion and V. Lyakhovsky, Accelerated seismic release and related aspects of seismicity patterns on earthquake faults, Earthquake Processes: Physical Modelling, Numerical Simulation and Data Analysis Part II (Springer, Basel AG, 2002), pp. 2385–2412.
  42. H.-h. Zhang, X.-c. Yin, N.-g. Liang, H.-z. Yu, S.-y. Li, Y. C. Wang, C. Yin, V. Kukshenko, N. Tomiline, and S. Elizarov, Acoustic emission experiments of rock failure under load simulating the hypocenter condition, Pure Appl. Geophys. 163, 2389 (2006).
  43. L. Wang, S. Ma, and L. Ma, Accelerating moment release of acoustic emission during rock deformation in the laboratory, Pure Appl. Geophys. 165, 181 (2008).
  44. D. Vere-Jones, R. Robinson, and W. Yang, Remarks on the accelerated moment release model: Problems of model formulation, simulation and estimation, Geophys. J. Int. 144, 517 (2001).
  45. A. Mignan, Retrospective on the accelerating seismic release (ASR) hypothesis: Controversy and new horizons, Tectonophysics 505, 1 (2011).
  46. R. Robinson, S. Zhou, S. Johnston, and D. Vere-Jones, Precursory accelerating seismic moment release (AMR) in a synthetic seismicity catalog: A preliminary study, Geophys. Res. Lett. 32, L07309 (2005).
  47. D. Sornette and C. G. Sammis, Complex critical exponents from renormalization group theory of earthquakes: Implications for earthquake predictions, J. Phys. I France 5, 607 (1995).
  48. D. Amitrano, J. R. Grasso, and G. Senfaute, Seismic precursory patterns before a cliff collapse and critical point phenomena, Geophys. Res. Lett. 32, L08314 (2005).
  49. K. A. Dahmen, Y. Ben-Zion, and J. T. Uhl, A simple analytic theory for the statistics of avalanches in sheared granular materials, Nat. Phys. 7, 554 (2011).
  50. J. Vasseur, F. B. Wadsworth, Y. Lavallée, A. F. Bell, I. G. Main, and D. B. Dingwell, Heterogeneity: The key to failure forecasting, Sci. Rep. 5, 13259 (2015).
  51. J. L. Hardebeck, K. R. Felzer, and A. J. Michael, Improved tests reveal that the accelerating moment release hypothesis is statistically insignificant, J. Geophys. Res. 113, B08310 (2008).
  52. O. Ramos, Criticality in earthquakes, good or bad for prediction? Tectonophysics 485, 321 (2010).
  53. A. Vespignani and S. Zapperi, How self-organized criticality works: A unified mean-field picture, Phys. Rev. E 57, 6345 (1998).
  54. K. Bækgaard Lauritsen, S. Zapperi, and H. E. Stanley, Self-organized branching processes: Avalanche models with dissipation, Phys. Rev. E 54, 2483 (1996).
  55. A. P. Mehta, K. A. Dahmen, and Y. Ben-Zion, Universal mean moment rate profiles of earthquake ruptures, Phys. Rev. E 73, 056104 (2006).
  56. E. A. Jagla and A. B. Kolton, A mechanism for spatial and temporal earthquake clustering, J. Geophys. Res. 115, B05312 (2010).
  57. F. Liakopoulou-Morris, I. G. Main, B. R. Crawford, and B. G. D. Smart, Microseismic properties of a homogeneous sandstone during fault nucleation and frictional sliding, Geophys. J. Int. 119, 219 (1994).
  58. S. Lennartz-Sassinek, I. G. Main, M. Zaiser, and C. C. Graham, Acceleration and localization of subcritical crack growth in a natural composite material, Phys. Rev. E 90, 052401 (2014).
  59. T. Utsu, Y. Ogata, and R. S. Matsu'ura, The centenary of the omori formula for a decay law of aftershock activity, J. Phys. Earth 43, 1 (1995).
  60. I. O. Ojala, I. G. Main, and B. T. Ngwenya, Strain rate and temperature dependence of Omori law scaling constants of AE data: Implications for earthquake foreshock-aftershock sequences, Geophys. Res. Lett. 31, L24617 (2004).
  61. M. Kloster, A. Hansen, and P. Christian Hemmer, Burst avalanches in solvable models of fibrous materials, Phys. Rev. E 56, 2615 (1997).
  62. K. Mogi, Earthquakes and fractures, Tectonophysics 5, 35 (1967).
  63. J. Dieterich, A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res.: Solid Earth 99, 2601 (1994).
  64. Y. Ben-Zion, and V. Lyakhovsky, Analysis of aftershocks in a lithospheric model with seismogenic zone governed by damage rheology, Geophys. J. Int. 165, 197 (2006).
  65. S. Hainzl, G. Zöller, and J. Kurths, Similar power laws for foreshock and aftershock sequences in a spring-block model for earthquakes, J. Geophys. Res.: Solid Earth 104, 7243 (1999).
  66. H. Nakanishi, Earthquake dynamics driven by a viscous fluid, Phys. Rev. A 46, 4689 (1992).
  67. R. Shcherbakov and D. L. Turcotte, A damage mechanics model for aftershocks, Pure Appl. Geophys. 161, 2379 (2004).
  68. Y. Kawada and H. Nagahama, Cumulative Benioff strain-release, modified Omori's law and transient behaviour of rocks, Tectonophysics 424, 157 (2006).
  69. E. A. Jagla, F. P. Landes, and A. Rosso, Viscoelastic Effects in Avalanche Dynamics: A Key to Earthquake Statistics, Phys. Rev. Lett. 112, 174301 (2014).
  70. E. Lippiello, F. Giacco, W. Marzocchi, C. Godano, and L. De Arcangelis, Mechanical origin of aftershocks, Sci. Rep. 5, 15560 (2015).
  71. Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes, J. Am. Stat. Assoc. 83, 9 (1988).
  72. C. Gu, A. Y. Schumann, M. Baiesi, and J. Davidsen, Triggering cascades and statistical properties of aftershocks, J. Geophys. Res.: Solid Earth 118, 4278 (2013).
  73. I. Zaliapin and Y. Ben-Zion, Earthquake clusters in southern california i: Identification and stability, J. Geophys. Res.: Solid Earth 118, 2847 (2013).
  74. J. Baró and J. Davidsen, Are triggering rates of labquakes universal? Inferring triggering rates from incomplete information, Eur. Phys. J.: Spec. Top. 226, 3211 (2017).
  75. S. Zapperi, K. B. Lauritsen, and H. E. Stanley, Self-Organized Branching Processes: Mean-Field Theory for Avalanches, Phys. Rev. Lett. 75, 4071 (1995).
  76. S. Pradhan and B. K. Chakrabarti, Precursors of catastrophe in the Bak-Tang-Wiesenfeld, Manna, and random-fiber-bundle models of failure, Phys. Rev. E 65, 016113 (2001).
  77. F. Kun, F. Raischel, R. C. Hidalgo, and H. J. Herrmann, Extensions of fibre bundle models, in Modelling Critical and Catastrophic Phenomena in Geoscience (Springer-Verlag, Berlin, Heidelberg, 2006), pp. 57–92.
  78. R. C. Hidalgo, F. Kun, and H. J. Herrmann, Bursts in a fiber bundle model with continuous damage, Phys. Rev. E 64, 066122 (2001).
  79. Z. Halász and F. Kun, Fiber bundle model with stick-slip dynamics, Phys. Rev. E 80, 027102 (2009).
  80. S. Hergarten and H. J. Neugebauer, Foreshocks and Aftershocks in the Olami-Feder-Christensen Model, Phys. Rev. Lett. 88, 238501 (2002).
  81. S. Hergarten and R. Krenn, Synchronization and desynchronization in the Olami-Feder-Christensen earthquake model and potential implications for real seismicity, Nonlin. Processes Geophys. 18, 635 (2011).
  82. M. Paczuski, S. Boettcher, and M. Baiesi, Interoccurrence Times in the Bak-Tang-Wiesenfeld Sandpile Model: A Comparison with the Observed Statistics of Solar Flares, Phys. Rev. Lett. 95, 181102 (2005).
  83. A. Deluca, N. R. Moloney, and Á. Corral, Data-driven prediction of thresholded time series of rainfall and self-organized criticality models, Phys. Rev. E 91, 052808 (2015).
  84. S. Janićević, L. Laurson, K. Jørgen Måløy, S. Santucci, and M. J. Alava, Interevent Correlations from Avalanches Hiding Below the Detection Threshold, Phys. Rev. Lett. 117, 230601 (2016).
  85. H. Benioff, Earthquakes and rock creep: (Part I: Creep characteristics of rocks and the origin of aftershocks), Bull. Seismol. Soc. Am. 41, 31 (1951).
  86. V. B. Smirnov and A. V. Ponomarev, Seismic regime relaxation properties from in situ and laboratory data, Izv., Acad. Sci., USSR, Phys. Solid Earth (Engl. Transl.) 40, 807 (2004).
  87. J. Davidsen and M. Baiesi, Self-similar aftershock rates, Phys. Rev. E 94, 022314 (2016).
  88. T. Hirata, Omori's power law aftershock sequences of microfracturing in rock fracture experiment, J. Geophys. Res.: Solid Earth 92, 6215 (1987).
  89. G. F. Nataf, P. O. Castillo-Villa, J. Baró, X. Illa, E. Vives, A. Planes, and E. K. H. Salje, Avalanches in compressed porous sio2-based materials, Phys. Rev. E 90, 022405 (2014).
  90. W. G. Glöckle and Th. F. Nonnenmacher, Fractional relaxation and the time-temperature superposition principle, Rheol. Acta 33, 337 (1994).
  91. N. Heymans and J.-C. Bauwens, Fractal rheological models and fractional differential equations for viscoelastic behavior, Rheol. Acta 33, 210 (1994).
  92. S. Hellinckx, N. Heymans, and J.-C. Bauwens, Analytical and fractal descriptions of non-linear mechanical behavior of polymers, J. Non-Cryst. Solids 172, 1058 (1994).
  93. H. Schiessel, R. Metzler, A. Blumen, and T. F. Nonnenmacher, Generalized viscoelastic models: Their fractional equations with solutions, J. Phys. A 28, 6567 (1995).
  94. C. H. R. Friedrich, Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol. Acta 30, 151 (1991).
  95. G. W. Scott Blair, The role of psychophysics in rheology, J. Colloid Sci. 2, 21 (1947).
  96. A. Jaishankar and G. H. McKinley, Power-law rheology in the bulk and at the interface: Quasi-properties and fractional constitutive equations, Proc. R. Soc. A 469, 20120284 (2013).
  97. F. Mainardi and G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology, Eur. Phys. J.: Spec. Top. 193, 133 (2011).
  98. X. Zhang and R. Shcherbakov, Power-law rheology controls aftershock triggering and decay, Sci. Rep. 6, 36668 (2016).
  99. M. Baiesi and M. Paczuski, Scale-free networks of earthquakes and aftershocks, Phys. Rev. E 69, 066106 (2004).
  100. S. Hainzl, A. Christophersen, D. Rhoades, and D. Harte, Statistical estimation of the duration of aftershock sequences, Geophys. J. Int. 205, 1180 (2016).
  101. D. L. Turcotte, J. R. Holliday, and J. B. Rundle, BASS, an alternative to ETAS, Geophys. Res. Lett. 34, L12303 (2007).
  102. K. A. Dahmen, Mean field theory of slip statistics, In Avalanches in Functional Materials and Geophysics (Springer International Publishing, Cham, 2017), pp. 19–30.
  103. A. Corral, Local distributions and rate fluctuations in a unified scaling law for earthquakes, Phys. Rev. E 68, 035102 (2003).
  104. P. O. Castillo-Villa, J. Baró, A. Planes, E. K. H. Salje, P. Sellappan, W. M. Kriven, and E. Vives, Crackling noise during failure of alumina under compression: The effect of porosity, J. Phys.: Condens. Matter 25, 292202 (2013).
  105. M. Kachanov, Elastic solids with many cracks and related problems, Adv. Appl. Mech. 30, 259 (1993).
  106. T. Yamashita and L. Knopoff, Models of aftershock occurrence, Geophys. J. Int. 91, 13 (1987).
  107. R. Maaß, M. Wraith, J. T. Uhl, J. R. Greer, and K. A. Dahmen, Slip statistics of dislocation avalanches under different loading modes, Phys. Rev. E 91, 042403 (2015).

Outline

Information

Phys. Rev. E 97, 033002– Published 29 March, 2018
  • Received 5 January 2018
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation