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

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## 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.

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