International Journal of Fracture

, Volume 204, Issue 2, pp 225–237 | Cite as

A fiber-bundle model for the continuum deformation of brittle material

  • K. Z. Nanjo
Original Paper


The deformation of brittle material is primarily accompanied by micro-cracking and faulting. However, it has often been found that continuum fluid models, usually based on a non-Newtonian viscosity, are applicable. To explain this rheology, we use a fiber-bundle model, which is a model of damage mechanics. In our analyses, yield stress was introduced. Above this stress, we hypothesize that the fibers begin to fail and a failed fiber is replaced by a new fiber. This replacement is analogous to a micro-crack or an earthquake and its iteration is analogous to stick–slip motion. Below the yield stress, we assume that no fiber failure occurs, and the material behaves elastically. We show that deformation above yield stress under a constant strain rate for a sufficient amount of time can be modeled as an equation similar to that used for non-Newtonian viscous flow. We expand our rheological model to treat viscoelasticity and consider a stress relaxation problem. The solution can be used to understand aftershock temporal decay following an earthquake. Our results provide justification for the use of a non-Newtonian viscous flow to model the continuum deformation of brittle materials.


Fracture Brittle deformation Rheology Fiber-bundle Yield stress Viscoelasticity 



The author thanks the Editor K. Ravi-Chandar and two anonymous reviewers for constructive comments. A part of this study was conducted under the auspices of the MEXT Program for Promoting the Reform of National Universities.


  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover, New YorkGoogle Scholar
  2. Alava MJ, Nukala PKVV, Zapperi S (2006) Statistical models of fracture. Adv Phys 55(3–4):349–476. doi: 10.1080/00018730300741518 CrossRefGoogle Scholar
  3. Anderson EM (1951) The dynamics of faulting and dyke formation with applications to Britain. Oliver and Boyd, EdinburghGoogle Scholar
  4. Ben-Zion Y, Lyakhovsky V (2002) Accelerated seismic release and related aspects of seismicity patterns on earthquake faults. Pure Appl Geophys 159:2385–2412. doi: 10.1007/s00024-002-8740-9 CrossRefGoogle Scholar
  5. Ben-Zion Y, Lyakhovsky V (2006) Analysis of aftershocks in a lithospheric model with seismogenic zone governed by damage rheology. Geophys J Int 165:197–210. doi: 10.1111/j.1365-246X.2006.02878.x CrossRefGoogle Scholar
  6. Coleman BD (1956) Time dependence of mechanical breakdown phenomena. J Appl Phys 27:862–866. doi: 10.1063/1.1722504 CrossRefGoogle Scholar
  7. Coleman BD (1958) Statistics and time dependence of mechanical breakdown in fibers. J Appl Phys 29:968–983. doi: 10.1063/1.1723343 CrossRefGoogle Scholar
  8. Daniels HE (1945) The statistical theory of strength of bundles of threads. Proc Roy Soc A 183:405–435. doi: 10.1098/rspa.1945.0011 CrossRefGoogle Scholar
  9. Dieterich JH (1994) A constitutive law for rate of earthquake production and its application to earthquake clustering. J Geophys Res 99:2601–2618. doi: 10.1029/93JB02581
  10. Dorn JE (1954) Some fundamental experiments on high temperature creep. J Mech Phys Solids 3:85–116. doi: 10.1016/0022-5096(55)90054-5 CrossRefGoogle Scholar
  11. England P, Molnar P (1997) Active deformation of Asia: from kinetics to dynamics. Science 278:647–650. doi: 10.1126/science.278.5338.647 CrossRefGoogle Scholar
  12. Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, New YorkCrossRefGoogle Scholar
  13. Guarino A, Garcimartin A, Ciliberto S (1998) An experimental test of the critical behavior of fracture precursors. Eur Phys J B 6:13–24. doi: 10.1007/s100510050521 CrossRefGoogle Scholar
  14. Halász Z, Kun F (2009) Fiber bundle model with stick–slip dynamics. Phys Rev E 80:027102. doi: 10.1103/PhysRevE.80.027102 CrossRefGoogle Scholar
  15. Hansen A, Hemmer PC, Pradhan S (2015) The fiber bundle model: modeling failure in materials. Wiley-VCH, WeinheimCrossRefGoogle Scholar
  16. Hamphill T, Campos W, Pilehvari A (1993) Yield-power law model more accurately predicts mud rheology. Oil Gas J 91:45–50Google Scholar
  17. Hemmer PC, Hansen A (1992) The distribution of simultaneous fiber failures in fiber bundles. J Appl Mech 59(4):909–914. doi: 10.1115/1.2894060 CrossRefGoogle Scholar
  18. Houseman G, England P (1986) Finite stain calculations of continental deformation: 1. Method and general results for convergent zones. J Geophys Res 91:3651–3663. doi: 10.1029/JB091iB03p03651 CrossRefGoogle Scholar
  19. Houwen O, Geehan T (1986) Rheology of oil-base mud. In: Paper no SPE15416, proceedings of the 61st SPE annual technical conference and exhibition. New Orleans, Louisiana, October 5–8, 1986Google Scholar
  20. Jackson J (2002) Faulting, flow, and the strength of the continental lithosphere. Int Geol Rev 44(1):39–61. doi: 10.2747/0020-6814.44.1.39 CrossRefGoogle Scholar
  21. Kachanov LM (1986) Introduction to continuum damage mechanics. Springer, Netherlands. doi: 10.1007/978-94-017-1957-5 CrossRefGoogle Scholar
  22. Kaneko Y, Lapusta N (2008) Variability of earthquake nucleation in continuum models of rate-and-state faults and implications for aftershock rates. J Geophys Res 113:B12312. doi: 10.1029/2007JB005154 CrossRefGoogle Scholar
  23. Karato S, Wu P (1993) Rheology of the upper mantle: a synthesis. Science 260:771–778. doi: 10.1126/science.260.5109.771 CrossRefGoogle Scholar
  24. King G (1983) The accommodation of large strains in the upper lithosphere of the Earth and other solids by self-similar fault systems: the geometrical origin of b-value. Pure Appl Geophys 121(5–6):761–815. doi: 10.1007/BF02590182 CrossRefGoogle Scholar
  25. King G, Oppenheimer D, Amelung F (1994) Block versus continuum deformation in the western United States. Earth Planet Sci Lett 128(3–4):55–64. doi: 10.1016/0012-821X(94)90134-1 CrossRefGoogle Scholar
  26. Kovács K, Hidalgo RC, Pagonabarraga I, Kun F (2013) Brittle-to-ductile transition in a fiber bundle with strong heterogeneity. Phys Rev E 87:042816. doi: 10.1103/PhysRevE.87.042816 CrossRefGoogle Scholar
  27. Kovács K, Nagy S, Hidalgo RC, Kun F, Herrmann HJ, Pagonabarraga I (2008) Critical ruptures in a bundle of slowly relaxing fibers. Phys Rev E 77:036102. doi: 10.1103/PhysRevE.77.036102 CrossRefGoogle Scholar
  28. Krajcinovic D (1996) Damage mechanics. Elsevier, AmsterdamGoogle Scholar
  29. Kun F, Raischel F, Hidalgo RC, Herrmann HJ (2006) Extensions of fibre bundle models. In: Modelling critical and catastrophic phenomena in geoscience, vol 705 of the series lecture notes in physics. Springer, Berlin Heidelberg, pp 57–92. doi: 10.1007/3-540-35375-5_3
  30. Kun F, Zapperi S, Herrmann HJ (2000) Damage in fiber bundle models. Eur Phys J B17:269–279CrossRefGoogle Scholar
  31. Lawn BR, Wilshaw TR (1975) Fracture of brittle solids. Cambridge University Press, CambridgeGoogle Scholar
  32. Lyakhovsky V, Ben-Zion Y, Agnon A (1997) Distributed damage, faulting, and friction. J Geophys Res 102(B12):27635–27649. doi: 10.1029/97JB01896 CrossRefGoogle Scholar
  33. Ma F, Kuang ZB (1995) Continuum damage mechanics treatment of constraint in ductile fracture. Eng Fract Mech 51(4):615–628. doi: 10.1016/0013-7944(94)00290-X CrossRefGoogle Scholar
  34. Manaker DM, Turcotte DL, Kellogg LH (2006) Flexure with damage. Geophys J Int 166:1368–1383. doi: 10.1111/j.1365-246X.2006.03067.x CrossRefGoogle Scholar
  35. Moreno Y, Correig AM, Gomez JB, Pacheco AF (2001) A model for complex aftershock sequences. J Geophys Res 106:6609–6619. doi: 10.1029/2000JB900396 CrossRefGoogle Scholar
  36. Morishita M, Kobayashi M, Yamaguchi T, Doi M (2010) Observation of spatio-temporal structure in stick–slip motion of an adhesive gel sheet. J Phys Condens Matter 22:365104. doi: 10.1088/0953-8984/22/36/365104 CrossRefGoogle Scholar
  37. Mura T (1969) Method of continuously distributed dislocations. In: Mura T (eds) Mathematical theory of dislocations. The American Society of Mechanical Engineers, pp 25–48Google Scholar
  38. Nakatani M (2001) Conceptual and physical clarification of rate and state friction: frictional sliding as a thermally activated rheology. J Geophys Res 106:13347–13380. doi: 10.1029/2000JB900453 CrossRefGoogle Scholar
  39. Nanjo K, Nagahama H (2000) Observed correlations between aftershock spatial distribution and earthquake fault lengths. Terra Nova 12(6):312–316. doi: 10.1046/j.1365-3121.2000.00329.x CrossRefGoogle Scholar
  40. Nanjo KZ, Turcotte DL (2005) Damage and rheology in a fiber bundle model. Geophys J Int 162:859–866. doi: 10.1111/j.1365-246X.2005.02683.x CrossRefGoogle Scholar
  41. Nanjo KZ, Turcotte DL, Shcherbakov R (2005) A model of damage mechanics for the deformation of the continental crust. J Geophys Res 110:B07403. doi: 10.1029/2004JB003438 CrossRefGoogle Scholar
  42. Nanjo KZ, Enescu B, Shcherbakov R, Turcotte DL, Iwata T, Ogata Y (2007) Decay of aftershock activity for Japanese earthquakes. J Geophys Res 112:B08309. doi: 10.1029/2006JB004754 CrossRefGoogle Scholar
  43. Newman WI, Phoenix SL (2001) Time-dependent fiber bundles with local load sharing. Phys Rev E 63:021507. doi: 10.1103/PhysRevE.63.021507 CrossRefGoogle Scholar
  44. Newman WI, Turcotte DL, Gabrielov AM (1995) Log-periodic behavior of a hierarchical failure model with applications to precursory seismic activation. Phys Rev E 52:4827–4835. doi: 10.1103/PhysRevE.52.4827 CrossRefGoogle Scholar
  45. Nicolas A, Poirier JP (1976) Crystalline plasticity and solid state flow in metamorphic rocks. Wiley, HobokenGoogle Scholar
  46. Omori F (1894) On the after-shocks of earthquakes. J Coll Sci Imp Univ Jpn 7:111–200Google Scholar
  47. Orowan E (1934) Zur Kristallplastizitat. Z Phys 89:605–659CrossRefGoogle Scholar
  48. Polanyi M (1934) Uber eine Art Gitterstorung, die einen Kristall Plastiach Machen Konnte. Z Phys 89:660–664CrossRefGoogle Scholar
  49. Reasenberg PA, Jones LM (1989) Earthquake hazard after a mainshock in California. Science 243:1173–1176. doi: 10.1126/science.243.4895.1173 CrossRefGoogle Scholar
  50. Reed TD, Pilehvari AA (1993) A new model for laminar, transitional, and turbulent flow of drilling muds. In: Paper no SPE25456, SPE production operations symposium, 21–23 March, Oklahoma, City. doi: 10.2118/25456-MS
  51. Scholz CH (2002) The mechanics of earthquakes and faulting, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  52. Shcherbakov R, Turcotte DL (2003) Damage and self-similarity in fracture. Theor Appl Fract Mech 39:245–258. doi: 10.1016/S0167-8442(03)00005-3
  53. Shcherbakov R, Turcotte DL, Rundle JB (2005) Aftershock statistics. Pure Appl Geophys 161:1051–1076. doi: 10.1007/s00024-004-2661-8 CrossRefGoogle Scholar
  54. Skelland AH (1967) Non-Newtonian flow and heat transfer. Wiley, New YorkGoogle Scholar
  55. Skrzypek JJ, Ganczarski A (1999) Modeling of material damage and failure of structures: theory and applications. Springer, New YorkCrossRefGoogle Scholar
  56. Smith RL, Phoenix SL (1981) Asymptotic distributions for the failure of fibrous materials under series-parallel structure and equal load-sharing. J Appl Mech 48(1):75–82. doi: 10.1115/1.3157595 CrossRefGoogle Scholar
  57. Sornette D, Ouillon G (2005) Multifractal scaling of thermally activated rupture processes. Phys Rev Lett 94(3):038501. doi: 10.1103/PhysRevLett.94.038501 CrossRefGoogle Scholar
  58. Sornette D, Virieux J (1992) Linking short-timescale deformation to long-timescale tectonics. Nature 357:401–404. doi: 10.1038/357401a0 CrossRefGoogle Scholar
  59. Taylor GI (1934) The mechanism of plastic deformation of crystals. Proc Roy Soc (London) Ser A 145:362–415CrossRefGoogle Scholar
  60. Thatcher W (1995) Microplate versus continuum descriptions of active tectonic deformation. J Geophys Res 100(B3):3885–3894. doi: 10.1029/94JB03064 CrossRefGoogle Scholar
  61. Turcotte DL, Glasscoe MT (2004) A damage model for the continuum rheology of the upper continental crust. Tectonophysics 383:71–80. doi: 10.1016/j.tecto.2004.02.011 CrossRefGoogle Scholar
  62. Turcotte DL, Newman WI, Shcherbakov R (2003) Micro and macroscopic models of rock fracture. Geophys J Int 152:718–728. doi: 10.1046/j.1365-246X.2003.01884.x CrossRefGoogle Scholar
  63. Utsu T (1961) A statistical study on the occurrence of aftershocks. Geophys Mag 30:521Google Scholar
  64. Voyiadjis GZ, Kattan PI (1999) Advances in damage mechanics: metals and metal matrix composites. Elsevier, New YorkGoogle Scholar
  65. Yamaguchi T, Ohmata S, Doi M (2009) Regular to chaotic transition of stick–slip motion in sliding friction of an adhesive gel-sheet. J Phys Condens Matter 21:205105. doi: 10.1088/0953-8984/21/20/205105 CrossRefGoogle Scholar
  66. Zapperi S, Vespignani A, Stanley HE (1997) Plasticity and avalanche behaviour in microfracturing phenomena. Nature 388:658–660CrossRefGoogle Scholar
  67. Zamora M, Lord DL (1974) Practical analysis of drilling mud flow in pipes and annuli. In: Paper no SPE-4976, fall meeting of the society of petroleum engineers of AIME, 6–9 October, Houston, Texas. doi: 10.2118/4976-MS

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Global Center for Asian and Regional ResearchUniversity of ShizuokaShizuokaJapan
  2. 2.Institute of Advanced SciencesYokohama National UniversityYokohamaJapan

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