The effect of inertia, viscous damping, temperature and normal stress on chaotic behaviour of the rate and state friction model

Article
  • 12 Downloads

Abstract

A fundamental understanding of frictional sliding at rock surfaces is of practical importance for nucleation and propagation of earthquakes and rock slope stability. We investigate numerically the effect of different physical parameters such as inertia, viscous damping, temperature and normal stress on the chaotic behaviour of the two state variables rate and state friction (2sRSF) model. In general, a slight variation in any of inertia, viscous damping, temperature and effective normal stress reduces the chaotic behaviour of the sliding system. However, the present study has shown the appearance of chaos for the specific values of normal stress before it disappears again as the normal stress varies further. It is also observed that magnitude of system stiffness at which chaotic motion occurs, is less than the corresponding value of critical stiffness determined by using the linear stability analysis. These results explain the practical observation why chaotic nucleation of an earthquake is a rare phenomenon as reported in literature.

Keywords

Rate and state friction inertia viscous damping temperature normal stress chaotic motion of rock sliding 

List of notations and abbreviations

\(\tau \)

Dimensional frictional stress (P\(_{\mathrm{a}}\))

\(\tau ^{*}\)

Reference frictional stress (P\(_{\mathrm{a}}\))

\(\sigma _n\)

Effective normal stress (P\(_{\mathrm{a}}\))

\(\mu _{*}\)

Reference coefficient of friction

\(\theta _1\), \( \theta _2\)

State variable related to asperity contact of the sliding interface

\(a,b_1\), \(b_2\)

Constants related to rate and state friction

\(Q_a ,Q_{b_1}\), \( Q_{b_2}\)

Activation energies corresponding to a, \(b_1\) and \(b_2\)

\(c_1\)

Nondimensional term relates to frictional heating

\({c}_2\)

Nondimensional term relates to heat conduction

c

\((1-\beta _1 q_1 ){Q_a }/{RT^{*}}+(1-\beta _2 q_2 ){Q_a }/{RT^{*}}\), \(\rho ={L_1 }/{L_2}\)

K

Stiffness of connecting spring (\(\hbox {P}_{\mathrm{a}}\,\,\hbox {m}^{-1}\))

k

\({KL_1 }/{\sigma _n }a\), non-dimensional spring stiffness

r

Ratio of inertial time to frictional characteristic time

r

\(\sqrt{{m}v_{*}^2}/{\sigma _n aL_1}\)

\(\hat{\gamma }\)

\({\gamma v_{*}}/{\sigma _n a}\)

\(\hat{\gamma }\)

Nondimensional viscous damping coefficient

m

Mass of the sliding block (kg)

\(\beta _1\)

\({b_1 }/a\)

\(\beta _2\)

\({b_2 }/a\)

\(q_1\)

\({Q_{b_1 } }/{Q_a }\)

\(q_2\)

\({Q_{b_2}}/{Q_a}\)

\(\psi \)

\(\tau /{\sigma a}\)

\(\phi \)

\(\hbox {ln}(v/{v_{*} })\)

\(d_c\)

Critical slip distance (m)

\(v_0\)

Pulling velocity (\(\hbox {ms}^{-1}\))

R

Universal gas constant (\(\hbox {J}\ \hbox {K}^{-1}\hbox {mol}^{-1}\))

\(T_{*}\)

Reference temperature (K)

T

\(tv_{*}/{L_1}\)

\(T_s\)

Temperature of the sliding interface (K)

\(\hat{{T}}_s\)

\(T_{s}/T_{*}\)

Notes

Acknowledgements

This work is supported by NRDMS-DST (order No. NRDMS//02/43/016(G)). The authors would like to thank Prof. Vinay A Juvekar, IIT Bombay for his useful discussion and suggestions for the improvement of the present manuscript.

References

  1. Becker T W 2000 Deterministic chaos in two state-variable friction slider and effect of elastic interaction; Geocomplex. Phys. Earthq. 120 5.CrossRefGoogle Scholar
  2. Burridge R and Knopoff L 1967 Model and theoretical seismicity; Bull. Seismol. Soc. Am. 57(3) 341–371.Google Scholar
  3. Brace W F and Byerlee J D 1966 Stick-slip as a mechanism of earthquake; Science 153 990–992.CrossRefGoogle Scholar
  4. Brittany A, Erickson B A, Birnir B and Lavallée D 2011 Periodicity, chaos and localization in a Burridge–Knopoff model of an earthquake with rate-and-state friction;Geophys. J. Int. 187(1) 178–198.CrossRefGoogle Scholar
  5. Carlson J M and Langer J S 1989 Properties of earthquake generated by fault dynamics; Phys. Rev. Lett. 62(22) 2632–2635.CrossRefGoogle Scholar
  6. Carlson J M and Langer J S 1989 Mechanical model of an earthquake fault; Phys. Rev. A. 40(11) 6470–6484.CrossRefGoogle Scholar
  7. Chau K T 1995 Landslides modelled as bifurcations of creeping slopes with nonlinear friction law; Int. J. Solid Struct. 32(23) 3451–3464.CrossRefGoogle Scholar
  8. Cochard A, Bureau L and Baumberger T 2003 Stabilization of frictional sliding by normal load modulation; J. Appl. Mech. 70(2) 220–226,  https://doi.org/10.1115/1.1546241.CrossRefGoogle Scholar
  9. Dieterich J D 1979 Modeling of rock friction: 1. Experimen results and constitutive equation; J. Geophys. Res. 84(B5) 2161–2168.CrossRefGoogle Scholar
  10. Dupont P E and Bapna D D 1994 Stability of the sliding frictional with varying normal force; J. Vib. Acoust. 116(2) 237–242.CrossRefGoogle Scholar
  11. Faillettaz J, Sornette D and Funk M 2010 Gravity-driven instabilities: Interplay between state- and velocity-dependent frictional sliding and stress corrosion damage cracking; J. Geophys. Res. 115 B03409,  https://doi.org/10.1029/2009JB006512.CrossRefGoogle Scholar
  12. Gu Y and Wong T-F 1994 Nonlinear dynamics of the transition from stable sliding to cyclic stick-slip in rock in nonlinear dynamics and predictability of geophysical phenomena; In: Geophysical monograph (ed.) Newman W, AGU, Washington DC 8 15–35.Google Scholar
  13. 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.CrossRefGoogle Scholar
  14. Huang J and Turcotte D L 1990 Are earthquakes an example of deterministic chaos?; Geophys. Res. Lett. 17 223.CrossRefGoogle Scholar
  15. Helmstetter A, Sornette D, Grassol J-R, Andersen J V, Gluzman S and Pisarenko V 2004 Slider-block friction model for landslides: Application to Vaiont and La Clapi‘ere landslides; J. Geophys. Res. 109 B202409.CrossRefGoogle Scholar
  16. Kawamura H, Ueda Y, Kakui S, Morimoto S and Yamamoto T 2017 Statistical properties of the one-dimensional Burridge–Knopoff model of earthquakes obeying the rate- and state-dependent friction law; Phys. Rev. E 95 042122.Google Scholar
  17. Linker M F and Dieterich J H 1992 Effect of variable normal stress on rock friction: Observation and constitutive equation; J. Geophys. Res. 97 4923–4940.CrossRefGoogle Scholar
  18. Lakshmanan M and Rajasekar S 2003 Nonlinear Dynamics (Integrability, Chaos and Patterns); Springer, Heidelberg.Google Scholar
  19. Liu Y 2007 Physical basis of aseismic deformation transients in subduction zones; PhD thesis, Harvard University.Google Scholar
  20. Marone C 1998 Laboratory-derive friction laws and their application to seismic faulting; Ann. Rev. Earth Planet. Sci. 26 643–696.CrossRefGoogle Scholar
  21. Niu Z-B and Chen D M 1994 Period-doubling bifurcation and chaotic phenomena in a single degree of freedom elastic system with a two-state variable friction law; In: Nonlinear Dynamics and Predictability of Geophysical Phenomena (eds) Newman W I, Gabrielov A and Turcotte D L, AGU, Washington DC, https://doi.org/10.1029/GM083p0075.
  22. Niu Z B and Chen D M 1995 Lyapunov exponent and dimension of the strange attractor of elastic frictional system; Acta Seimol. Sinica 8 575–584.Google Scholar
  23. Ranjith K and Rice 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.CrossRefGoogle Scholar
  24. Rice J R 1996 Slip complexity in dynamic model of earthquake faults; Proc. Nat. Acad. Sci. USA 93 3825–3829.CrossRefGoogle Scholar
  25. Rice J R and Ruina A L 1983 Stability of steady frictional slipping; J. Appl. Mech. 50 343–349.CrossRefGoogle Scholar
  26. Rice J R 1993 Spatio-temporal complexity of slip on a fault; J. Geophys. Res. 98 9885–9907.CrossRefGoogle Scholar
  27. Ruina A L 1983 Slip instability and state variable friction laws; J. Geophys. Res. 88 10359–10370.CrossRefGoogle Scholar
  28. Rice J R and Tse S T 1986 Dynamic motion of a single degree of freedom system following a rate and state dependent friction law; J. Geophys. Res. 91 521–530.CrossRefGoogle Scholar
  29. Segall P and Rice J R 2006 Does shear heating of pore fluid contribute to earthquake nucleation?; J. Geophys. Res. 111 B09316.CrossRefGoogle Scholar
  30. Shelly D R 2010 Periodic, chaotic, and doubled earthquake recurrence intervals on the deep san andreas fault; Science 238(11) 1385–1388.CrossRefGoogle Scholar
  31. Singh A K, Kanthola A and Singh T N 2012 Prediction of slope stability with an advanced friction model; Int. J. Rock. Mech. Min. Sci. 55 164–167.Google Scholar
  32. Singh A K and Singh T N 2013 Friction strength and steady relaxation using the rate and state dependent friction model; Pure Appl. Geophys. 170(3) 247–257.CrossRefGoogle Scholar
  33. Singh A K and Singh T N 2016 Stability of the rate, state, and temperature dependent friction model and its application; Geophys. J. Int. 205 636–647.CrossRefGoogle Scholar
  34. Sinha N and Singh A K 2016 Numerical study on chaotic behavior of slip and ageing laws of the rate and state friction; In: Dynamics, Vibration and Control, Narosa Publishing House, pp. 162–165.Google Scholar
  35. Sinha N and Singh A K 2016 Linear and nonlinear stability analysis of rate state friction model with three state variables; Nonlin. Process Geophys. Discuss.,  https://doi.org/10.5194/npg-2016-11.
  36. Strogatz S H 1994 Nonlinear dynamics and chaos: With application to physics, biology, chemistry and Engineering; Hachette, UK.Google Scholar
  37. Tolstoi D M 1967 Significance of the normal degree of freedom and natural normal vibrations in contact friction; Wear 10 199–213.CrossRefGoogle Scholar
  38. Xuejun G 2013 Bifurcation behaviors of the two-state variable frictional law of a rock mass system; Int. J. Bifurcation Chaos. 23 11,  https://doi.org/10.1142/S0218127413501848.CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringVisvesvaraya National Institute of TechnologyNagpurIndia
  2. 2.Department of Earth SciencesIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations