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Simulation of Frictional Strength and Steady Relaxation Using the Rate and State Dependent Friction Model

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Abstract

In this paper, frictional strength of hard solids, such as rock–rock sliding surfaces, is studied as a function of waiting time and shearing velocity. A one dimensional spring–mass sliding system is numerically simulated under the quasistatic condition using the rate and state dependent friction model. It is established that frictional strength varies linearly with the logarithm of waiting time (also known as time of stationary contact or relaxation time, etc.) as well as logarithm of shearing velocity. Analytical expression developed for frictional strength is found to be valid only in the case of high stiffness of the connecting spring. In the steady relaxation simulation, a steadily sliding mass is suddenly brought to zero velocity and relaxation of the interfacial stress and corresponding velocity at the sliding interface is studied as a function of relaxation time in the velocity strengthening regime of friction. A mathematical relation is derived between state variable and waiting time using the concept of steady relaxation. The relaxation model is also compared with the experimental data from the literature. Finally, the present study enables one to unify the slide–hold–slide friction experiments.

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References

  • Baumberger, T., Heslot, F., and Perrin, B. (1994), Crossover from creep to inertial motion in friction dynamics, Nature 367, 544–546.

  • Baumberger, T., and Guther, L. (1996), Creep like relaxation at the interface between rough solids under shear, J. Phys. 6, 1021–1030.

    Google Scholar 

  • Berthoud, P., and Baumberger, T. (1998), Shear stiffness of a solid–solid multi contacts interface, Proc. R. Soc. Lond. A 454, 1615–1634.

  • Berthoud, P., Baumberger, T., Sell, C. G., and Hiver, J. M. (1999), Physical analysis of the state -and –rate dependent friction law: Static friction, Phys. Rev. B 59, 14313–14327.

    Google Scholar 

  • Baumberger, T., Caroli, C., and Ronsin, O. (2003), Selfhealing pulses and the friction of gelatin gels, Euro. Phys. J. E. 11, 85–93.

    Google Scholar 

  • Baumberger, T., and Caroli, C. (2006), Solid friction: From stick–slip down to pinning and aging. Adv. Phys. 55, 273–348.

    Google Scholar 

  • Dieterich, J. H. (1979), Modeling of rock friction-1. Experimental results and constitutive equations. J. Geophys. Res. 84, 2161–2168 a.

    Google Scholar 

  • Dieterich, J. H. (1972), Time-dependent friction in rocks, J. Geophys. Res. 77, 3690–3697.

    Google Scholar 

  • Dieterich, J. H. (1978), Rock friction and mechanics of stick–slip, Pure Appl. Geophys. 116, 790–805.

  • Heslot, F., Baumberger,T., Perrin B., Caroli B., and Caroli, C. (1994), Creep, stick–slip, and dry friction dynamics: experiments and a heuristic model, Phys. Rev. E 49, 4973–4988.

    Google Scholar 

  • Gu, J. C., Rice, J. R., Ruina, A. L., and Tse, S. T. (1984), 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.

    Google Scholar 

  • Karner, S. L., and Marone, C. (2001), Frictional strengthening in simulated fault gouge: Effect of shear load perturbations, J. Geophys. Res. 106, B9, 19319–19337.

    Google Scholar 

  • Kato, N., Yamamoto, K., Yamamoto, H., and Hirasawa, T. (1992), Strain rate effects on frictional strength and the slip nucleation process, Tectonophysics, 211, 269–282.

  • Kato, N., and Hirasawa, T. (1996), Effect of strain rate and strength non-uniformity on the slip nucleation process: A numerical experiment, Tectonophysics, 265, 299–311.

  • Johnson, T. (1981), Time dependent friction of granite: Implications for precursory slip on faults, J. Geophys. Res. 86, 6017–6028.

    Google Scholar 

  • Lapusta, N., Rice, J.R., Ben-Zion, Y., and Zheng, G. (2000), Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on faults with rate-and state dependent friction, J. Geophys. Res. 105, 23765–23789.

    Google Scholar 

  • Marone, C. (1998), Laboratory derived friction laws and their application to seismic faulting, Ann. Rev. Earth Plan. Sci. 26, 643–696.

    Google Scholar 

  • Marone, C. (1997), On the rate of frictional healing and the constitutive law for time-and slip-dependent friction. Int. J. Rock Mech. Min. Sci. 34:3–4, paper No. 187.

    Google Scholar 

  • Müser, M. H. (2008), How static is static friction? Proc. Nat. Am. Sci., 105, 13187–13188.

  • Persson, B. N. J. Sliding friction: Physical principles and applications (Springer, Heidelberg 2000).

  • Putelat,T., Dawes, J. H. P., and Willis, J.R. (2007), Sliding modes of two interacting frictional interfaces, J. Mech. Phys. Sol., 55, 2073–2105.

    Google Scholar 

  • Putelat, T., Dawes, J. H. P., and Willis, J.R. (2010), Regimes of frictional sliding of a spring-block system, J. Mech. Phys. Solids., 58, 27–53.

    Google Scholar 

  • Putelat, T., Dawes, J. H. P. and Willis, J. R. (2011), On the microphysical foundations of rate-and-state friction, J. Mech. Phys. Solids, 59, 1062–1075.

    Google Scholar 

  • Ronsin, O., and Coeyrehoucq, K. L. (2001), State, rate and temperature-dependent sliding friction of elastomers, Proc. R. Soc. Lond. A 457, 1277–1293.

    Google Scholar 

  • Rice, J.R., Lapusta, N., and Ranjith, K. (2001), Rate and state dependent friction and the stability of sliding between elastically deformable solids, J. Mech. Phys. Solids, 49, 1865–1898.

    Google Scholar 

  • Ranjith, K., Rice, and J. R. (1999), Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction, J. Mech. Phys. Solids, 47, 1207–1218.

    Google Scholar 

  • Rice, J.R., and Ben-Zion,Y. (1996), Slip complexity in earthquake fault models, Proc. Nat. Acad. Sci., 93, 3811–3818.

  • Rice, J. R., and Ruina, A. L. (1983), Stability of steady frictional slipping, J. App. Mech., 50, 343–349.

    Google Scholar 

  • Ruina, A. L. (1983), Slip instability and state variable friction law, J. Geophys. Res. 88, B12, 10359–10370.

    Google Scholar 

  • Ruina, A. L. Friction laws and instabilities: A quasistatic analysis of some dry frictional behavior (PhD thesis, Brown University, Providence, R.I. 1980)

  • Rabinowicz, E. (1951), The nature of the static and kinetic coefficient of friction, J. App. Phys. 22, 1373–1379.

  • Rabinowicz, E. (1958), The intrinsic variables affecting the stick–slip process, Proc. Phys. Soc. Lond. 71, 665–675.

  • Singh, A.K., and Juvekar,V.A. (2011), Steady dynamic friction at elastomers-hard solid interface: A model based on population balance of bonds, Soft Matter 7, 10601–10611.

    Google Scholar 

  • Scholz, C. H. The Mechanics of Earthquakes and Faulting, (Cambridge University Press, Cambridge 1990)

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Acknowledgments

First author is thankful to Dr. K. Ranjith for introducing the rate and state dependent friction model during his initial stage of doctoral program at IIT Bombay. We are grateful to Prof. Vinay A. Juvekar, Department of Chemical Engineering, IIT Bombay for his critical and fruitful suggestions concerning the present study. We also appreciate the efforts of Ms. Asfiya Q. C., IIT Bombay for improvement and timely correction of the manuscript.

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Correspondence to Arun K. Singh.

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Singh, A.K., Singh, T.N. Simulation of Frictional Strength and Steady Relaxation Using the Rate and State Dependent Friction Model. Pure Appl. Geophys. 170, 247–257 (2013). https://doi.org/10.1007/s00024-012-0493-5

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  • DOI: https://doi.org/10.1007/s00024-012-0493-5

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