Skip to main content
Earth, Planets and Space
Download PDF

Volume 56 Supplement 12

Special Issue: Slip and Flow Processes in and below the Seismogenic Region (2)

  • Article
  • Published:

A smeared seismicity constitutive model

Earth, Planets and Space volume 56, pages 1121–1133 (2004)Cite this article

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

  • 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 

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

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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.

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

    Google Scholar 

  • 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 

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

    Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Book  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • 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.

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

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

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

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

    Google Scholar 

  • 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

Authors and Affiliations

  1. Predictive Mineral Discovery Cooperative Research Centre, CSIRO Exploration and Mining, PO Box 1130, Bentley, WA, 6102, Australia

    A. Ord, B. E. Hobbs & K. Regenauer-Lieb

  2. Geophysics and Geodynamics, Johannes Gutenberg-Universitat Mainz, 55099, Mainz, Germany

    K. Regenauer-Lieb

Authors
  1. A. Ord
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. E. Hobbs
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Regenauer-Lieb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ord.

Rights and permissions

Reprints and permissions

About this article

Check for updates. 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

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1186/BF03353331

Key words