Dynamic Stability of the Rate, State, Temperature, and Pore Pressure Friction Model at a Rock Interface

Abstract

In this article, we study numerically the dynamic stability of the rate, state, temperature, and pore pressure friction (RSTPF) model at a rock interface using standard spring-mass sliding system. This particular friction model is a basically modified form of the previously studied friction model namely the rate, state, and temperature friction (RSTF). The RSTPF takes into account the role of thermal pressurization including dilatancy and permeability of the pore fluid due to shear heating at the slip interface. The linear stability analysis shows that the critical stiffness, at which the sliding becomes stable to unstable or vice versa, increases with the coefficient of thermal pressurization. Critical stiffness, on the other hand, remains constant for small values of either dilatancy factor or hydraulic diffusivity, but the same decreases as their values are increased further from dilatancy factor $(\sim \;10^{ - 4} )$ and hydraulic diffusivity $(\sim \;10^{ - 9} \;{\text{m}}^{2} \;{\text{s}}^{ - 1} )$. Moreover, steady-state friction is independent of the coefficient of thermal pressurization, hydraulic diffusivity, and dilatancy factor. The proposed model is also used for predicting time of failure of a creeping interface of a rock slope under the constant gravitational force. It is observed that time of failure decreases with increase in coefficient of thermal pressurization and hydraulic diffusivity, but the dilatancy factor delays the failure of the rock fault under the condition of heat accumulation at the creeping interface. Moreover, stiffness of the rock-mass also stabilizes the failure process of the interface as the strain energy due to the gravitational force accumulates in the rock-mass before it transfers to the sliding interface. Practical implications of the present study are also discussed.

Appendix

In this section, we discuss the linear stability analysis of the RSTPF model in Eq. (7). We carry out linear stability about the equilibrium point of Eq. (7). The equilibrium point $\left( {\phi_{\text{ss}} , \, \hat{p}_{\text{ss}} , { }\psi_{\text{ss}} ,\hat{T}_{\text{ss}} } \right)$ is corresponding to steady sliding given by the following:

$$\phi_{\text{ss}} = e^{{\upsilon_{0} }} , \, \hat{p}_{\text{ss}} = \hat{p}_{\infty } , \, \hat{T}_{\text{ss}} = \frac{{c_{1} }}{{c_{2} }}e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} + \hat{T}_{\infty } ,{\text{ and }}\psi_{\text{ss}} = \left( {\hat{\sigma }_{n} - \hat{p}_{\text{ss}} } \right)\left[ {\hat{\mu }_{ * } + c(\hat{T}_{\text{ss}}^{ - 1} - 1) - (\beta - 1)\phi_{\text{ss}} } \right],$$

and Jacobian matrix about the equilibrium point is expressed as follows:

$$J_{0} = \left[ \begin{aligned} J_{11} , \, J_{12} , \, J_{13} , \, J_{14} \hfill \\ J_{21} , \, J_{22} , { }J_{23} , { }J_{24} \, \hfill \\ J_{31} , \, J_{32} , { }J_{33} , { }J_{34} \hfill \\ J_{41} , \, J_{42} , { }J_{43} , { }J_{44} \hfill \\ \end{aligned} \right]_{{ (\phi_{\text{ss}} ,\psi_{\text{ss}} ,\hat{T}_{\text{ss}} ,\hat{p}_{\text{ss}} )}} { ,}$$

where the terms are defined as follows:

$$\begin{aligned} J_{11} & = - ke^{{\phi_{\text{ss}} }} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} + \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \left[ {\varLambda^{\prime}c_{1} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\beta - 1)} \right] + (b - 1)e^{{\phi_{\text{ss}} }} + cc_{1} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} \hat{T}_{\text{ss}}^{ - 2} , \\ J_{12} & = \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \left[ {\varLambda^{\prime}c_{1} e^{{\phi_{\text{ss}} }} - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} } \right] + (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} e^{{\phi_{\text{ss}} }} + cc_{1} e^{{\phi_{\text{ss}} }} \hat{T}_{\text{ss}}^{ - 2} , \\ J_{12} & = \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \left[ {\varLambda^{\prime}c_{1} e^{{\phi_{\text{ss}} }} - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} } \right] + (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} e^{{\phi_{\text{ss}} }} + cc_{1} e^{{\phi_{\text{ss}} }} \hat{T}_{\text{ss}}^{ - 2} , \\ J_{12} & = \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \left[ {\varLambda^{\prime}c_{1} e^{{\phi_{\text{ss}} }} - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} } \right] + (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} e^{{\phi_{\text{ss}} }} + cc_{1} e^{{\phi_{\text{ss}} }} \hat{T}_{\text{ss}}^{ - 2} , \\ J_{13} & = - \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} (\varLambda^{\prime}c_{2} + \hat{\varepsilon }ce^{{\phi_{\text{ss}} }} \beta^{ - 1} \hat{T}_{\text{ss}}^{ - 2} ) + c\hat{T}_{\text{ss}}^{ - 2} e^{{\phi_{ss} }} - cc_{2} \hat{T}_{\text{ss}}^{ - 2} , \\ J_{14} & = \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \left[ { - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} - c_{h} } \right] + e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} ,J_{31} = c_{1} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} ,J_{32} = c_{1} e^{{\phi_{\text{ss}} }} , \\ J_{21} & = - ke^{{\phi_{\text{ss}} }} ,J_{22} = 0,J_{23} = 0,J_{24} = 0,J_{33} = - c_{2} ,J_{34} = 0,J_{41} = \left[ {\varLambda^{\prime}c_{1} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\beta - 1)} \right], \\ J_{42} & = \varLambda^{\prime}c_{1} e^{{\phi_{\text{ss}} }} - \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} ,J_{43} = - \varLambda^{\prime}c_{2} - \hat{\varepsilon }ce^{{\phi_{\text{ss}} }} \beta^{ - 1} \hat{T}_{\text{ss}}^{ - 2} ,J_{43} = - \hat{\varepsilon }\psi_{\text{ss}} e^{{\phi_{\text{ss}} }} \beta^{ - 1} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} - c_{\text{h}} . \\ \end{aligned}$$

The characteristic polynomial equation is obtained by expanding the Jacobian matrix $J_{0}$ for eigenvalues $\lambda_{s}$ as $a_{0} \lambda^{4} + a_{1} \lambda^{3} + a_{2} \lambda^{2} + a_{3} \lambda + a_{4} = 0$, where coefficients are defined as follows:

$$a_{0} = 16,$$

$$a_{1} = 8\left[ {c_{2} + c_{\text{h}} + e^{{\phi_{\text{ss}} }} - \beta e^{{\phi_{\text{ss}} }} + e^{{\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} - cc_{1} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} \hat{T}_{\text{ss}}^{ - 2} + \hat{\varepsilon }e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} - c_{1} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}}^{2} \varLambda^{\prime}(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} } \right],$$

$$a_{2} = 4\left[ \begin{aligned} & c_{2} c_{\text{h}} + c_{2} e^{{\phi_{\text{ss}} }} - \beta c_{2} e^{{\phi_{\text{ss}} }} + c_{\text{h}} e^{{\phi_{\text{ss}} }} - \beta c_{\text{h}} e^{{\phi_{\text{ss}} }} + cc_{1} ke^{{2\phi_{\text{ss}} }} \hat{T}_{\text{ss}}^{ - 2} + c_{2} e^{{\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} + c_{\text{h}} e^{{\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} .. \\ & + \;e^{{2\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} - cc_{1} c_{\text{h}} e^{{\phi_{\text{ss}} }} \psi_{\text{ss}} \hat{T}_{\text{ss}}^{ - 2} - cc_{1} e^{{2\phi_{\text{ss}} }} \psi_{\text{ss}} \hat{T}_{\text{ss}}^{ - 2} + c_{2} e^{{\phi_{\text{ss}} }} \hat{\varepsilon }\psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} + c_{1} e^{{2\phi_{\text{ss}} }} k\varLambda^{\prime}\psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} .. \\ & - \;c_{1} e^{{2\phi_{\text{ss}} }} \varLambda^{\prime}\psi_{\text{ss}}^{2} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \\ \end{aligned} \right],$$

$$a_{3} = 2\left[ \begin{aligned} & c_{2} c_{\text{h}} e^{{\phi_{\text{ss}} }} - \beta c_{2} c_{\text{h}} e^{{\phi_{\text{ss}} }} + cc_{1} c_{\text{h}} ke^{{2\phi_{\text{ss}} }} \hat{T}_{\text{ss}}^{ - 2} + cc_{1} e^{{3\phi_{\text{ss}} }} k\hat{T}_{\text{ss}}^{ - 2} + c_{2} c_{\text{h}} e^{{\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} + c_{2} e^{{2\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} .. \\ & + \;c_{\text{h}} e^{{2\phi_{\text{ss}} }} k(\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} - cc_{1} c_{\text{h}} e^{2\phi } \psi T^{ - 2} + c_{1} e^{3\phi } k\varLambda^{\prime}\psi_{\text{ss}} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 2} \\ \end{aligned} \right],$$

$$a_{4} = \left[ {cc_{1} c_{\text{h}} ke^{{3\phi_{\text{ss}} }} \hat{T}_{\text{ss}}^{ - 2} + c_{2} c_{\text{h}} ke^{{2\phi_{\text{ss}} }} (\hat{\sigma }_{\text{n}} - \hat{p}_{\text{ss}} )^{ - 1} } \right].$$

Finally, using the Routh–Hurwitz criterion for dynamic stability, the above coefficients as $a_{1} a_{2} - a_{0} a_{3} \;{\text{and}}\;a_{1} a_{2} a_{3} - a_{0} a_{3}^{2} - a_{4} a_{1}^{2} = 0$. These nonlinear equations are, in turned, solved numerically for critical stiffness $k_{\text{cr}}$ of the sliding system.