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

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.

Appendices

Appendix

A1. Linear stability of 2sRSTF model

In this section, we have carried out linear stability of the 2sRSTF model for predicting the critical stiffness at which sliding behaviour changes from stable to unstable. The system of differential equations in equation (8) is linearized about the equilibrium point or fixed point. This is obtained by equating the time derivatives in equation (8) equal to zero and solving the system of algebraic equations numerically. Moreover, the corresponding Jacobian matrix of equation (8) is obtained as:

The characteristic polynomial is obtained by expanding the above Jacobian matrix and this is given by $s_0 \lambda ^{4}+s_1 \lambda ^{3}+s_2 \lambda ^{2}+s_3 \lambda +s_4 =0$, where the co-efficient of the polynomial are expressed as:

$$\begin{aligned} s_0= & {} 16, \\ s_1= & {} 8\left( c_2 +e^{\phi }-\beta _1 e^{\phi }+ke^{\phi }+\rho e^{\phi }-\rho \beta _2 e^{\phi }\right. \\&\left. -{cc_1 e^{\phi }\psi }/{\hat{T}_s^2 } \right) ,\\ s_2= & {} 4\left( c_2 e^{\phi }-\beta _1 c_2 e^{\phi }+c_2 ke^{\phi }+ke^{2\phi }+c_2 \rho e^{\phi }\right. \\&-\beta _2 c_2 \rho e^{\phi }+\rho e^{2\phi }-\rho \beta _1 e^{\phi }-\rho \beta _2 e^{2\phi }+\rho ke^{2\phi }\\&\left. +{cc_1 ke^{2\phi }}/{\hat{T}_s^2 }...-{cc_1 e^{2\phi }\psi }/{\hat{T}_s^2 }-{cc_1 \rho e^{2\phi }\psi }/{\hat{T}_s^2 }\right) ,\\ s_3= & {} 2\left( c_2 ke^{2\phi }+c_2 \rho e^{2\phi }-\beta _1 c_2 \rho e^{2\phi }-\beta _2 c_2 \rho e^{2\phi }\right. \\&+c_2 \rho ke^{2\phi } +\rho ke^{3\phi }+{cc_1 ke^{3\phi }}/{\hat{T}_s^2 }\\&\left. +{cc_1 k\rho e^{3\phi }}/{\hat{T}_s^2 }...-{cc_1 \rho e^{3\phi } \psi }/{\hat{T}_s^2 }\right) \\ s_4= & {} \left( {c_2 k\rho e^{3\phi }+{cc_1 k\rho e^{4\phi }}/{\hat{T}_s^2 }} \right) , \end{aligned}$$

The Routh–Hurwitz criterion is applied on above characteristic polynomial to obtain the governing algebraic equations for stability, namely, $s_1 s_2 -s_0 s_3 =0$ and $s_1 s_2 s_3 -s_0 s_3^2 -s_4 s_1^2=0$. These two coupled non-linear algebraic equations are, in turn, solved numerically for critical stiffness.

A2. Linear stability of 2sRSNF model

In this section, we have also carried out linear stability of the 2sRSNF model for predicting the critical stiffness of the system. We followed the same procedure for evaluating critical stiffness of the 2sRSTF friction model as is done in the case of 2sRSTF. The system of equilibrium point of equation (9) is obtained about the steady sliding as $\phi _{ss} =e^{v_0 }$, $\hat{\theta }_{2ss} =-\phi _{ss}$, $\hat{\sigma }_{nss} =1$ and $\psi _{ss} =\hat{\sigma }_{nss} [\hat{\mu }_{*} +({1-\beta _1 -\beta _2 } )\phi _{ss} ]$. This is now used to obtain the Jacobian matrix as following:

$$\begin{aligned} J_0 =\left[ {\begin{array}{llll} J_{11}&{} ,J_{12}&{} ,J_{13}&{} ,J_{14} \\ J_{21}&{} ,J_{22}&{} ,J_{23}&{} ,J_{24} \\ J_{31}&{} ,J_{32}&{} ,J_{33}&{} ,J_{34} \\ J_{41}&{} ,J_{42}&{} ,J_{43}&{} ,J_{44} \\ \end{array}} \right] _{(\phi _{{ ss}} ,\psi _{{ ss}} ,\hat{\theta }_{2ss} ,\hat{\sigma }_{nss})} \end{aligned}$$

where abbreviations are defined as:

$$\begin{aligned} J_{11}= & {} {-ke^{\phi _{ss} }}/{\hat{\sigma }_{nss} }-{ke^{\phi _{ss} }\psi _{ss} \tan \alpha }/{\hat{\sigma }_{nss}^2 }\\&+\rho \beta _2 e^{\phi _{ss} }+(\beta _1 -1)e^{\phi _{ss} }, \\ J_{12}= & {} {e^{\phi _{ss} }}/{\hat{\sigma }_{nss} },~~J_{13} =(\rho -1)\beta _2 e^{\phi _{ss} },\\ J_{14}= & {} {-e^{\phi _{ss} }\psi _{ss} }/{\hat{\sigma }_{nss}^2 },\\ J_{21}= & {} -ke^{\phi _{ss} },~~J_{22} =0,\\ J_{23}= & {} 0,~~J_{24} =0 \\ J_{31}= & {} -\rho e^{\phi _{ss} },~~ J_{32} =0,\\ J_{33}= & {} -\rho e^{\phi _{ss} },~~ J_{34} =0\\ J_{41}= & {} ke^{\phi _{ss} }\tan \alpha ,~~ J_{42} =0,\\ J_{43}= & {} 0,~~J_{44}=0. \end{aligned}$$

The characteristic equation is given $|{J_0 -\lambda I} |=0$, where $\lambda$ and I are eigen value and identity matrix, respectively. The characteristic equation is expanded for getting the characteristic polynomial as $a_0 \lambda ^{4}+a_1 \lambda ^{3}+a_2 \lambda ^{2}+a_3 \lambda +a_4 =0$. The co-efficient of the polynomial is defined as the following

$$\begin{aligned} a_0= & {} 16,\\ a_1= & {} 8\left( e^{\phi }-\beta _1 e^{\phi }+{ke^{\phi }}/{\hat{\sigma }_n }+\rho e^{\phi }\right. \\&\left. -\beta _2 \rho e^{\phi }+{\psi e^{\phi }k\tan \alpha }/{\hat{\sigma }_n^2 } \right) ,\\ a_2= & {} 4\left( k{e^{2\phi }}/\hat{\sigma }_n +\rho e^{2\phi }-\beta _1 \rho e^{2\phi }\right. \\&\left. -\beta _2 \rho e^{2\phi }+{k\rho e^{2\phi }}/\hat{\sigma }_n +{k\psi e^{2\phi }\tan \alpha }/\hat{\sigma }_n^2\right. \\&\left. +{k\psi \rho e^{2\phi }\tan \alpha }/{\hat{\sigma }_n^2 } \right) ,\\ a_3= & {} 2\left( {k\rho e^{3\phi }}/\hat{\sigma }_n +{k\psi \rho e^{3\phi }\tan \alpha }/{\hat{\sigma }_n^2}\right) ,\\ a_4= & {} 0. \end{aligned}$$

Finally, the Routh–Hurwitz criterion is used to obtain critical stiffness $k_{cr}$ at which sliding behaviour of the system changes from stable to unstable or vice versa. The Routh and Hurwitz criterion leads to the following system of coupled algebraic equations as $a_1 a_2 -a_0 a_3 =0$ and $a_1 a_2 a_3 -a_0 a_3^2 -a_4a_1^2 =0$. These equations are, in turn, solved numerically for critical stiffness $k_{cr}$ of the sliding system.