Skip to main content

A smeared seismicity constitutive model

Abstract

The classical application of rate and state dependent frictional constitutive laws has involved the instabilities developed between two sliding surfaces. In such a situation, the behaviour and evolution of asperities is the controlling mechanism of velocity weakening. However, most faults have a substantial thickness and it would appear that it is the bulk behaviour of the fault gouge, at whatever scale, that is important. The purpose of this paper is to explore how bulk frictional sliding behaviour may be described. We explore here the consequences of applying the rate and state framework initially developed to describe the frictional behaviour at the interface between two interacting sliding blocks, to frictional behaviour within a layer of gouge that has bulk elastic-plastic constitutive behaviour. The approach taken here is to replace the relative sliding velocity in the classical formulation with the maximum shear strain rate, D, and the characteristic length with a characteristic shear strain, γ c . This means that the frictional behaviour of the bulk material now evolves with shear strain rate, D, over a characteristic shear strain, γ c . This approach still does not address the problem of reproducing natural recurrence times between instabilities, but perhaps places the problem in a new framework.

References

  1. Aagaard, B. T., T. H. Heaton, and J. F. Hall, Dynamic earthquake ruptures in the presence of lithostatic normal stresses: Implications for friction models and heat production, Bulletin of the Seismological Society of America, 91, 1765–1796, 2001.

    Article  Google Scholar 

  2. Amontons, G., Histoire de l’Academie Royale des Sciences avec les Memoires de Mathematique et de Physique, 203–222, 1699.

    Google Scholar 

  3. Behn, M. D., J. Lin, and M. T. Zuber, A continuum mechanics model for normal faulting using a strain-rate softening rheology: Implications for thermal and rheological controls on continental and oceanic rifting, Earth and Planetary Science Letters, 202, 725–740, 2002.

    Article  Google Scholar 

  4. Ben-Zion, Y, Stress, slip, and earthquakes in models of complex single-fault systems incorporating brittle and creep deformations, Journal of Geophysical Research, 101, 5677–5706, 1996.

    Article  Google Scholar 

  5. Ben-Zion, Y., Dynamic ruptures in recent models of earthquake faults, Journal of the Mechanics and Physics of Solids, 49, 2209–2244, 2001.

    Article  Google Scholar 

  6. Ben-Zion, Y. and J. R. Rice, Slip patterns and earthquake populations along different classes of faults in elastic solids, Journal of Geophysical Research, 100, 12,959–12,983, 1995.

    Article  Google Scholar 

  7. Besuelle, P. and J. W. Rudnicki, Localization: Shear Bands and Compaction Bands. Mechanics of Fluid Saturated Rocks, edited by Y. Gueguen and M. Bouteca, International Geophysics Series, 89, Elsevier Academic Press, Amsterdam, 446 pp., 2004.

  8. Biggs, J. M., Introduction to Structural Dynamics, New York: McGraw-Hill, 1964.

    Google Scholar 

  9. Borja, R. I. and A. Aydin, Computational modeling of deformation bands in granular media. I. Geological and mathematical framework, Computer Methods in Applied Mechanics and Engineering, 193, 2667–2698, 2004.

    Article  Google Scholar 

  10. Bowden, F. P. and D. Tabor, The Friction and Lubrication of Solids, Part I, Clarendon Press, Oxford, 1950.

    Google Scholar 

  11. Chester, F. M. and N. G. Higgs, Multimechanism friction constitutive model for ultrafine quartz gouge at hypocentral conditions, J. Geophys. Res., 97, 1859–1870, 1992.

    Article  Google Scholar 

  12. Cochard, A. and R. Madariaga, Dynamic faulting under rate-dependent friction, Pure and Applied Geophysics, 142, 419–445, 1994.

    Article  Google Scholar 

  13. Cochard, A. and R. Madariaga, Complexity of seismicity due to highly rate-dependent friction, Journal of Geophysical Research, 101, 25,321–25,336, 1996.

    Article  Google Scholar 

  14. Cundall, P. A., Explicit finite difference methods in geomechanics, in Numerical Methods in Engineering, 1, 132–150, 1976.

    Google Scholar 

  15. Desai, C. S. and J. T. Christian, Numerical Methods in Geomechanics, McGraw-Hill, New York, 783 pp., 1977.

    Google Scholar 

  16. Dieterich, J. H., Time-dependent friction and the mechanics of stick slip, Pageoph, 116, 790–806, 1978.

    Article  Google Scholar 

  17. Dieterich, J. H., Modelling of rock friction. Experimental results and constitutive equations, J. Geophys. Res., 84, 2161–2168, 1979.

    Article  Google Scholar 

  18. Estrin, Y. and Y. Brechet, On a model of frictional sliding, Pure and Applied Geophysics, 147, 745–762, 1996.

    Article  Google Scholar 

  19. Gu, J.-C., J. R. Rice, A. L. Ruina, and S. T. Tse, Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction, J. Mech. Phys. Solids, 32, 167–196, 1984.

    Article  Google Scholar 

  20. Hirose, T. and T. Shimamoto, Growth of molten zone as a mechanism of slip weakening of simulated faults in gabbro during frictional melting, J. Geophys. Res., 2004 (accepted).

    Google Scholar 

  21. Hobbs, B. E., H. Tanaka, and Y. Iio, Acceleration of slip motion in deep extensions of seismogenic faults in and below the seismogenic region, Earth Planets Space, 54, 1195–1205, 2002.

    Article  Google Scholar 

  22. Hobbs, B. E., A. Ord, K. Regenauer-Lieb, and B. Drummond, Fluid reservoirs in the crust and mechanical coupling between the upper and lower crust, Earth Planets Space, 56, this issue, 1151–1161, 2004.

    Article  Google Scholar 

  23. Horowitz, F. G. and A. Ruina, Slip patterns in a spatially homogeneous fault model, J. Geophys. Res., 94, 10279–10298, 1989.

    Article  Google Scholar 

  24. Iio, Y., T. Sagiya, and Y. Kobayashi. What controls the occurrence of shallow intraplate earthquakes?, Earth Planets Space, 56, this issue, 1077–1086, 2004.

    Article  Google Scholar 

  25. ITASCA, FLAC, Fast Lagrangian Analysis of Continua, User’s Guide, Version 4.00. ITASCA, Minnesota, USA, 2002.

    Google Scholar 

  26. Ito, H., G. Beroza, K. Fujimoto, and Y. Ogawa, Preface, Earth Planets Space, 54, 999–100, 2002.

    Article  Google Scholar 

  27. Jaeger, J. C., Elasticity, Fracture and Flow, Chapman and Hall, London, 268 pp., 3rd edition, 1969.

    Google Scholar 

  28. Lavenda, B. H., Thermodynamics of Irreversible Processes, Macmillan Press Ltd., London, 182 pp., 1978.

    Google Scholar 

  29. Lorig, L. J. and B. E. Hobbs, Numerical modelling of slip instability using the distinct element method with state variable friction laws, Int. J. Rock Mech. Min. Sci. And Geomech. Abstr., 27, 525–534, 1990.

    Article  Google Scholar 

  30. Lyakhovsky, V., Scaling of fracture length and distributed damage, Geophysical Journal International, 144, 114–122, 2001.

    Article  Google Scholar 

  31. Lyakhovsky, V., Y. Ben-Zion, and A. Agnon, Earthquake cycles, fault zones, and seismicity patterns in a rheologically layered lithosphere, Journal of Geophysical Research, 106, 4103–4120, 2001.

    Article  Google Scholar 

  32. Lysmer, J. and R. L. Kuhlemeyer, Finite dynamic model for infinite media, J. Eng. Mech., 95, 859–877, 1969.

    Google Scholar 

  33. MacCurdy, E. Da Vinci, L. Notebooks. Translation into English, Jonathan Cape, London, 1938.

    Google Scholar 

  34. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Inc., New Jersey, 713 pp., 1969.

    Google Scholar 

  35. Marone, C., Laboratory-derived friction laws and their application to seismic faulting, Annu. Rev. Earth Planet. Sci., 26, 643–696, 1998.

    Article  Google Scholar 

  36. Neumann, G. and M. Zuber, A continuum approach to the develoment of normal faults, in Proc. 35th US Symposium on Rock Mechanics, Lake Tahoe, Nevada, edited by J. Daemen and R. Schultz, pp. 191–198, Balkema, 1995.

  37. Noda, H. and T. Shimamoto, Thermal pressurization and slip-weakening distance of a fault: An example of the Hanaore fault, Southwest Japan, Bull. Seism. Soc. Amer., 2004 (submitted).

    Google Scholar 

  38. Ord, A., Deformation of Rock: A pressure-sensitive, dilatant material, Pure and Applied Geophysics, 137, 337–366, 1991.

    Article  Google Scholar 

  39. Otter, J. R. H., A. C. Cassell, and R. E. Hobbs, Dynamic Relaxation, Paper No. 6986, Proc. Inst. Civil Eng., 35, 633–656, 1966.

    Google Scholar 

  40. Regenauer-Lieb, K. and D. Yuen, Positive feedback of interacting ductile faults from coupling of equation of state, rheology and thermal-mechanics, Physics of Earth and Planetary Interiors, 142, 113–135, 2004.

    Article  Google Scholar 

  41. Rice, J. R., Constitutive relations for fault slip and earthquake instabilities, Pure and Applied Geophysics, 121, 443–475, 1983.

    Article  Google Scholar 

  42. Rice, J. R., Spatio-temporal complexity of slip on a fault, Journal of Geophysical Research, 98, 9885–9907, 1993.

    Article  Google Scholar 

  43. Rice, J. R. and Y. Ben-Zion, Slip complexity in earthquake fault models, Proc. Natl. Acad. Sci. USA, 93, 3811–3818, 1996.

    Article  Google Scholar 

  44. Rice, J. R. and S. T. Tse, Dynamic motion of a single degree of freedom system following a rate and state dependent friction law, Journal of Geophysical Research, 91, 521–530, 1986.

    Article  Google Scholar 

  45. Rice, J. R., N. Lapusta, and K. Ranjith, Rate and state dependent friction and the stability of sliding between elastically deformable solids, Journal of the Mechanics and Physics of Solids, 49, 1865–1898, 2001.

    Article  Google Scholar 

  46. Ruina, A. L., Friction laws and instabilities: a quasi-static analysis of some dry friction behaviour. Ph.D. Thesis, Division of Engineering, Brown University, 1980.

    Google Scholar 

  47. Ruina, A. L., Slip instability and state variable friction laws, J. Geophys. Res., 88, 10259–10270, 1983.

    Article  Google Scholar 

  48. Sato, H., T. Imaizumi, T. Yoshida, H. Ito, and A. Hasegawa, Tectonic evolution and deep to shallow geometry of Nagamachi-Rifu active fault system, NE Japan, Earth Planets Space, 54, 1039–1043, 2002.

    Article  Google Scholar 

  49. Shimamoto, T., Transition between frictional slip and ductile flow for halite shear zones at room temperature, Science, 231, 711–714, 1986.

    Article  Google Scholar 

  50. Sibson, R. H., Interactions between temperature and pore-fluid pressure during earthquake faulting and a mechanism for partial or total stress relief, Nature Physical Science, 243, 66–68, 1973.

    Article  Google Scholar 

  51. Tse, S. T. and J. R. Rice, Crustal earthquake instability in relation to the depth variation of frictional slip properties, Journal of Geophysical Research, 91, 9452–9472, 1986.

    Article  Google Scholar 

  52. Vermeer, P. A. and R. De Borst. Non-associated plasticity for soils, concrete and rock, Heron, 29, 3–64, 1984.

    Google Scholar 

  53. Wilkins, M. L. Calculation of elastic-plastic flow, in Methods in Computational Physics, 3, Fundamental Methods in Hydrodynamics, pp. 211–263, edited by Alder, New York: Academic Press, 1964.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Ord.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ord, A., Hobbs, B.E. & Regenauer-Lieb, K. A smeared seismicity constitutive model. Earth Planet Sp 56, 1121–1133 (2004). https://doi.org/10.1186/BF03353331

Download citation

Key words

  • Seismicity
  • constitutive model
  • rate and state dependant friction
  • bulk frictional elastic-plastic behaviour