Appendix 1: Exponential Time Differencing Method for the SLIP Law
Governing Equations
A spring-slider system with one-degree-of-freedom consists of Eqs. (4)–(7) in the main text:
$$m\dot{v} = F_{\text{s}} - F_{\text{t}}$$
(14)
$$\dot{F}_{\text{s}} = k(v_{\text{L}} - v)$$
(15)
$$F_{\text{t}} = F_{0} + A\ln \left( {\frac{v}{{v_{0} }}} \right) + \theta$$
(16)
$$\dot{\theta } = \frac{v}{{L_{\text{c}} }}\left( { - B\ln \left( {\frac{v}{{v_{0} }}} \right) - \theta } \right),$$
(17)
where dots on the top represent derivatives with respect to time t, \(A = aF_{\text{n}}\) and \(B = bF_{\text{n}}\) are parameters representing the direct and evolution effects, respectively, and the definitions of the other characters are the same as those in the main text. The time-derivative of Eq. (16) is:
$$\dot{F}_{\text{t}} = A\frac{{\dot{v}}}{v} + \dot{\theta }.$$
(18)
Now we normalize these equations by introducing nondimensional parameters. The slip rate shall be normalized by the steady-state velocity:
$$w = {v \mathord{\left/ {\vphantom {v {v_{0} }}} \right. \kern-0pt} {v_{0} }}.$$
(19)
The nondimensional time shall be defined as:
$$\tau = {{v_{0} t} \mathord{\left/ {\vphantom {{v_{0} t} {L_{\text{c}} }}} \right. \kern-0pt} {L_{\text{c}} }}.$$
(20)
The nondimensional frictional stress difference from the steady-state is:
$$g = \frac{{F_{\text{t}} - F_{0} }}{A},$$
(21)
and the nondimensional spring force difference from the steady-state is:
$$h = \frac{{F_{\text{s}} - F_{0} }}{{kL_{\text{c}} }}.$$
(22)
The nondimensional state variable is:
$$\varPsi = {\theta \mathord{\left/ {\vphantom {\theta B}} \right. \kern-0pt} B}.$$
(23)
The nondimensional equations then take the form of:
$$w^{\prime } = Ch - Dg$$
(24)
$$h^{\prime } = E - w$$
(25)
$$g^{\prime } = C\frac{h}{w} - D\frac{g}{w} - \beta w\left( {\ln (w) + \varPsi } \right)$$
(26)
$$\varPsi^{\prime } = w\left( { - \ln (w) - \varPsi } \right),$$
(27)
where primes represent derivatives with respect to the nondimensional time. The parameters in the nondimensional equations are:
$$\beta = {B \mathord{\left/ {\vphantom {B A}} \right. \kern-0pt} A}$$
(28)
$$D = \frac{{AL_{\text{c}} }}{{mv_{0}^{2} }}$$
(29)
$$C = \frac{{kL_{\text{c}}^{ 2} }}{{mv_{0}^{2} }} = \frac{{kL_{\text{c}} }}{A}\frac{{AL_{\text{c}} }}{{mv_{0}^{2} }} = \kappa_{1} D$$
(30)
$$E = {{v_{\text{L}} } \mathord{\left/ {\vphantom {{v_{\text{L}} } {v_{0} }}} \right. \kern-0pt} {v_{0} }}.$$
(31)
D represents the significance of the direct effect with respect to the inertia, \(\kappa_{1}\) is the nondimensional spring constant, and C represents the significance of spring stiffness with respect to inertia. Note that \(C^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\) is proportional to the natural angular frequency of the harmonic oscillator.
There are four Eqs. (24)–(27), but we have one constraint from the friction law:
$$w = \exp \left( {g - \beta \varPsi } \right).$$
(32)
Therefore, the system is a three-dimensional ordinary differential equation. Since Eq. (32) describes the relation between w, g, and \(\varPsi\), we have only to integrate Eq. (25) and additional two equations among Eqs. (24), (26), and (27).
Exponential Time Differencing Method
By integrating Eqs. (25)–(27) with (32), we obtained the following second-order accurate (in terms of \(\tau_{1}\)) expressions at \(\tau_{1}\), supposing we have \(h_{0}\), \(g_{0}\), and \(\varPsi_{0}\) at \(\tau = 0\):
$$h_{1}^{*} = h_{0} + (E - w_{0} )\tau_{1}$$
(33)
$$g_{1}^{*} = g_{0} \exp ( - \tau_{1} /\tau_{\rm{g0}} ) + g_{\rm{ss0}} (1 - \exp ( - \tau_{1} /\tau_{\rm{g0}} ))$$
(34)
$$\varPsi_{1}^{*} = \varPsi_{0} \exp ( - \tau_{1} /\tau_{\Psi 0} ) + \varPsi_{\text{ss0}} (1 - \exp ( - \tau_{1} /\tau_{\Psi 0} )),$$
(35)
where
$$w_{0} = \exp \left( {g_{0} - \beta \varPsi_{0} } \right)$$
(36)
$$g_{\text{ss0}} = \frac{C}{D}h_{0} - \frac{\beta }{D}w_{0}^{2} \left( {\ln (w_{0} ) + \varPsi_{0} } \right)$$
(37)
$$\tau_{{{\text{g}}0}} = w_{0} /D$$
(38)
$$\varPsi_{{{\text{ss}}0}} = - \ln (w_{0} )$$
(39)
$$\tau_{\Psi 0} = 1/w_{0} .$$
(40)
Note that \(g_{{{\text{ss}}0}}\) and \(\varPsi_{{{\text{ss}}0}}\) are (pseudo-)steady-state values which would be achieved if \(g\) and \(\varPsi\) were only variables in Eqs. (26) and (27), respectively. \(\tau_{\text{g}}\) and \(\tau_{\Psi }\) are decay time constants. Then we can estimate \(w\) at \(\tau_{1}\) as:
$$w_{1}^{*} = \exp \left( {g_{1}^{*} - \beta \varPsi_{1}^{*} } \right).$$
(41)
Adopting those starred values leads to a first-order integration scheme, but we can iterate this scheme to increase the order of accuracy (see Noda and Lapusta 2010). In the second-order scheme, we integrate the first half time-step using values at \(\tau = 0\):
$$h_{1/2} = h_{0} + (E - w_{0} )\tau_{1} /2$$
(42)
$$g_{1/2} = g_{0} \exp ( - \tau_{1} /2\tau_{\rm{g0}} ) + g_{{{\rm{ss}}0}} (1 - \exp ( - \tau_{1} /2\tau_{\rm{g0}} ))$$
(43)
$$\begin{aligned} \varPsi_{1/2} &= \varPsi_{0} \exp ( - \tau_{1} /2\tau_{\Psi 0} )\nonumber\\&\quad+ \varPsi_{{{\text{ss}}0}} (1 - \exp ( - \tau_{1} /2\tau_{\Psi 0} ))\end{aligned}$$
(44)
and then the latter half using the starred values estimated above
$$h_{1}^{**} = h_{1/2} + (E - w_{1}^{*} )\tau_{1} /2$$
(45)
$$g_{1}^{**} = g_{1/2} \exp ( - \tau_{1} /2\tau_{\rm{g1}}^{*} ) + g_{{{\rm{ss}}1}}^{*} (1 - \exp ( - \tau_{1} /2\tau_{\rm{g1}}^{*} ))$$
(46)
$$\begin{aligned} \varPsi_{1}^{**} &= \varPsi_{1/2} \exp ( - \tau_{1} /2\tau_{\Psi 1}^{*} ) \nonumber\\&\quad+ \varPsi_{\text{ss1}}^{*} (1 - \exp ( - \tau_{1} /2\tau_{\Psi 1}^{*} )) \end{aligned}$$
(47)
where
$$g_{\text{ss1}}^{*} = \frac{C}{D}h_{1}^{*} - \frac{\beta }{D}w_{1}^{*2} \left( {\ln (w_{1}^{*} ) + \varPsi_{1}^{*} } \right)$$
(48)
$$\tau_{{{\text{g}}1}}^{*} = w_{1}^{*} /D$$
(49)
$$\varPsi_{{{\text{ss}}1}}^{*} = - \ln (w_{1}^{*} )$$
(50)
$$\tau_{\Psi 1}^{*} = 1/w_{1}^{*} .$$
(51)
We then estimate w at \(\tau_{1}\) as:
$$w_{1}^{**} = \exp \left( {g_{1}^{**} - \beta \varPsi_{1}^{**} } \right).$$
(52)
Variable Time Step for the Exponential Time Differencing Method
To decrease the calculation time, we use variable time steps determined by the following equation:
$$\Delta \tau = \hbox{min} \left( {\Delta \tau_{\hbox{max} } , { }\Delta \tau_{\text{cnst}} {{v_{\hbox{max} } } \mathord{\left/ {\vphantom {{v_{\hbox{max} } } v}} \right. \kern-0pt} v}} \right).$$
(53)
The time interval is \(\Delta \tau_{\hbox{max} }\) when the slip rate of the block v is very small. When v is nearly \(v_{\hbox{max} }\), the time interval is smaller than the time interval \(\Delta \tau_{\text{cnst}}\) for appropriate calculations with the constant time interval. We set \(\Delta \tau_{\hbox{max} } = 0.01\), \(\Delta \tau_{\text{cnst}} = 10^{ - 5}\), and \(v_{\hbox{max} } = 0.05{\text{ m/s}}\), which is consistent with the maximum slip rate in the large-scale experiments. Note that the slip increment for a time step is a fixed fraction of \(L_{\text{c}}\) because \(\tau\) is equal to \({{v_{0} t} \mathord{\left/ {\vphantom {{v_{0} t} {L_{\text{c}} }}} \right. \kern-0pt} {L_{\text{c}} }}\) and \(\Delta \tau\) is inversely proportional to v.
Comparison with the Runge–Kutta Method
The parameters of an example problem are \(v_{\text{L}} = 0.1{\text{ mm/s}}\), \(b = 0.0092\), and \(L_{\text{c}} = 0.5 \, \upmu {\text{m}}\). Simulations were carried out until \(t = 20{\text{ s}}\). Figure 18 shows the stick–slip behaviors obtained by the Runge–Kutta method and our exponential time differencing method. They are almost identical; our numerical method works properly. The calculation time necessary for our method was approx. 1/10,000 of that needed to use the Runge–Kutta method.
Appendix 2: Exponential Time Differencing Method for the AGING Law
We developed the same method for the Aging law as for the Slip law. The definitions of the characters in the following equations are the same as those in “Appendix 1”. We also used the same variable time step stated in “Appendix 1.3”.
Governing Equations
A spring-slider system with one-degree-of-freedom consists of Eqs. (4), (5), (8), and (9) in the main text:
$$m\dot{v} = F_{\text{s}} - F_{\text{t}}$$
(54)
$$\dot{F}_{\text{s}} = k(v_{\text{L}} - v)$$
(55)
$$F_{\rm{t}} = F_{0} + A\ln \left( {{v \mathord{\left/ {\vphantom {v {v_{0} }}} \right. \kern-0pt} {v_{0} }}} \right) + B\ln \left( {{{v_{0} \theta } \mathord{\left/ {\vphantom {{v_{0} \theta } {L_{\text{c}} }}} \right. \kern-0pt} {L_{\text{c}} }}} \right)$$
(56)
$$\dot{\theta } = 1 - \frac{v\theta }{{L_{\text{c}} }},$$
(57)
where the definitions of the other characters are the same as those in “Appendix 1” and the main text. The time-derivative of Eq. (56) is:
$$\dot{F}_{\text{t}} = A\frac{{\dot{v}}}{v} + B\frac{{\dot{\theta }}}{\theta }.$$
(58)
Now we normalize these equations by introducing nondimensional parameters. Nondimensional slip rate, time, frictional stress difference from the steady-state, and spring force difference from the steady-state shall be defined as Eqs. (19)–(22). The nondimensional state variable is:
$$\varPsi = {{v_{0} \theta } \mathord{\left/ {\vphantom {{v_{0} \theta } {L_{\text{c}} }}} \right. \kern-0pt} {L_{\text{c}} }}.$$
(59)
The nondimensional equations then take the form of Eqs. (24), (25) and
$$g^{\prime } = C\frac{h}{w} - D\frac{g}{w} - \beta w + \beta \frac{1}{\varPsi }$$
(60)
$$\varPsi^{\prime } = 1 - w\varPsi .$$
(61)
The parameters in the nondimensional equations, \(\beta\), D, C, and E, are Eqs. (28)–(31).
There are four Eqs. (24), (25), (60) and (61), but we have one constraint from the friction law:
$$w = \exp \left( {g - \beta \ln (\varPsi )} \right).$$
(62)
Therefore, the system is a three-dimensional ordinary differential equation. Since Eq. (62) describes the relation between w, g, and \(\varPsi\), we have only to integrate Eq. (25) and additional two equations among Eqs. (24), (60), and (61).
Exponential Time Differencing Method
By integrating Eqs. (25), (60), (61) with (62), we obtained the second-order accurate (in terms of \(\tau_{1}\)) Eqs. (33)–(35), where \(\tau_{\text{g0}}\) and \(\tau_{\Psi 0}\) are Eqs. (38) and (40), respectively, and
$$w_{0} = \exp \left( {g_{0} - \beta \ln (\varPsi_{0} )} \right)$$
(63)
$$g_{\text{ss0}} = \frac{C}{D}h_{0} + \frac{\beta }{D}\frac{{w_{0} }}{{\varPsi_{0} }} - \frac{\beta }{D}w_{0}^{2}$$
(64)
$$\varPsi_{{{\text{ss}}0}} = {1 \mathord{\left/ {\vphantom {1 {w_{0} }}} \right. \kern-0pt} {w_{0} }}.$$
(65)
Note that \(g_{\text{ss0}}\) and \(\varPsi_{\text{ss0}}\) are steady-state values which would be achieved if only \(g\) and \(\varPsi\) were variables in Eqs. (60) and (61), respectively. \(\tau_{\text{g}}\) and \(\tau_{\Psi }\) are decay time constants. Then we can estimate \(w\) at \(\tau_{1}\) as:
$$w_{1}^{*} = \exp \left( {g_{1}^{*} - \beta \ln \left( {\varPsi_{1}^{*} } \right)} \right).$$
(66)
Adopting those starred values leads to a first-order integration scheme, but we can iterate this scheme to increase the order of accuracy (see Noda and Lapusta 2010). In the second-order scheme, we integrate the first half time-step using values at \(\tau = 0\), Eqs. (42)–(44), and then the latter half using the starred values estimated above, Eqs. (45)–(47), where \(\tau_{\rm{g1}}^{*}\) and \(\tau_{\Psi 1}^{*}\) are Eqs. (48) and (50), respectively, and
$$g_{\text{ss1}}^{*} = \frac{C}{D}h_{1}^{*} + \frac{\beta }{D}\frac{{w_{1}^{*} }}{{\varPsi_{1}^{*} }} - \frac{\beta }{D}w_{1}^{*2}$$
(67)
$$\varPsi_{\text{ss1}}^{*} = 1/w_{1}^{*} .$$
(68)
We then estimate w at \(\tau_{1}\) as:
$$w_{1}^{**} = \exp \left( {g_{1}^{**} - \beta \ln \left( {\varPsi_{1}^{**} } \right)} \right).$$
(69)
Appendix 3: Estimation of Gouge Production Rate
To evaluate effect of the increasing gouge layer thickness, we estimated the gouge production rate from the volume of collected gouge material and amount of mechanical works done during the experiments. Mass of the gouge materials collected after the experiment LB01-111 was 12.0009 g. This gouge was produced by four frictional experiments, LB01-104, -106, -108, and -111. See Fukuyama et al. (2014) for the details of the experimental conditions. From the measured mass, we estimated the volume of the produced gouge material to be 6.330 × 10−6 m3 under the assumption that effective density of the gouge material of metagabbro is equal to 1896 kg/m3 following Yamashita et al. (2015). Total amount of mechanical works was calculated as 1.086 × 105 J from \(F_{\text{s1}}\) integrated over the entire slip distances in the four experiments. As the result, we estimated the gouge production rate to be 5.828 × 10−11 m3/J.
From this estimated rate, we can calculate the averaged thicknesses of the gouge layer for W1 and W2 in LB01-127 as 5.013 × 10−7 and 1.303 × 10−6 m, respectively. Note that we estimated these thicknesses assuming the produced gouge materials are uniformly distributed over the fault surface. Actually, the gouge material was locally produced in and around the generated grooves as revealed by Yamashita et al. (2015). Therefore, these thicknesses could be minimum estimates.