We use the same model as that by Kato (2004), and Yoshida (2018), who describe the simulation process in detail. A 2D planar fault is loaded at a constant rate *V*_{pl} in the *x*-direction, and the fault plane is divided into 256 × 256 square cells with areas of 0.5 km × 0.5 km. The shear stress *τ* on cell (*i, j*) is given by:

$$\tau \left( {i, j} \right) = \mathop \sum \limits_{k,l} K\left( {i - k,j - l} \right)\left[ {u\left( {k,l} \right) - V_{pl} t} \right] - \frac{G}{2\beta }V\left( {i,j} \right),$$

(15)

where *u*(*k, l*) is the slip at cell (*k, l*) in the x-direction, *K*(*i*-*k, j*-*l*) is the static shear stress at the center of cell (*i, j*) due to the uniform unit slip in the *x*-direction over cell (*k, l*), *V *= d*u*/d*t* is the slip velocity, *G* is the rigidity, and β is the S wave speed. In all of our simulations, we assumed that *V*_{pl} = 0.085 m/year, *G *= 30 GPa, and β = 3.5 km/s. The second term on the right-hand side of Eq. (15) represents the approximate reduction in shear stress due to wave radiation, as introduced by Rice (1993).

We first simulate the unperturbed earthquake cycle using constitutive law (1), normal stress-dependent Nagata law (5), and Eq. (15). Assuming *a* = 0.03, *a*–*b* = − 0.012 and 0.0014 (inside and outside the asperity of radius *R* = 3 km, respectively), *σ* = 50 MPa, *L* = 0.003 m, *α* = 4.2, and *c* = 20, we obtained the earthquake cycle shown in Fig. 2. Earthquakes occur repeatedly with a recurrence interval *T*_{r} = 20 years. Bhattacharya et al. (2015) estimated the value of *c* at 10 to 100 based on laboratory experiments using a rock interface with gouge, which is larger than the value of 2 estimated by Nagata et al. (2012) from bare rock experiments. We obtain *α* = 4.2 using the scaling relationship with *α* = 0.2 for the aging law and *c* = 20. In this section, we apply Coulomb stress perturbations of various amplitudes assuming the same friction parameters. Figure 2 shows the evolution of the shear stresses and slip velocities on a log scale, and that of the displacements at the three points indicated in Fig. 2d.

Next, we apply dynamic shear and/or normal stress perturbation:

$$\begin{aligned} & \tau_{d} \left( t \right) = S_{d} \sin \left( {\frac{2\pi }{{T_{p} }}t} \right), \\ & \sigma_{d} \left( t \right) = N_{d} \sin \left( {\frac{2\pi }{{T_{p} }}t} \right)\quad 0 \, \le t \le T_{p} . \\ \end{aligned}$$

(16)

The corresponding dynamic Coulomb stress perturbation CFF_{d}(*t*) is

$${\text{CFF}}_{d} \left( t \right) = \Delta {\text{CFF}}_{d} \sin \left( {\frac{2\pi }{{T_{p} }}t} \right),\quad 0 \, \le t \le T_{p} ,$$

(17)

with

$$\Delta {\text{CFF}}_{d} = S_{d} - \left( {\mu - \frac{\alpha }{1 + c}} \right)N_{d} ,$$

(18)

where *S*_{d} and *N*_{d} are the amplitude of the shear and normal stress perturbation, respectively, *T*_{p} is the period and *t *= 0 is taken to be the time when the perturbation is applied.

Figure 3a shows the Coulomb stress perturbation applied at 7/8 of the recurrence interval *T*_{r} ~ 20 years, as indicated with the arrow in Fig. 2a. The assumed parameters are *S*_{d} = 0 MPa, *N*_{d} = − 1.48 MPa, *μ* = 0.7, *α* = 4.2, *c* = 20, and *T*_{p} = 100 s. Figure 3b, c shows the slip velocity and displacement, respectively, at the three points in the asperity shown in Fig. 2d.

Dieterich (1994) represented the time to instability as a function of slip velocity using a single spring-block model and pointed out that an increase in the slip velocity due to a static stress change could trigger an earthquake. This model can be extended to incorporate dynamic stress perturbation, which reduces the frictional strength, causing an increase in slip velocity. Figure 3b shows that the slip velocity after the perturbation ceased is higher than that before the perturbation starts.

Figure 4 shows that the slip accelerates after the perturbation starts at *t* = 0, with the triggered event occurring approximately 17.4 days later. The increase in slip velocity promotes the rupture nucleation growth, resulting in the reduction of the time to instability from 2.5 year to 17.4 days.

Figure 5a shows the relationship between the amplitudes of the stress perturbation and the time to instability. The values of the friction parameters are the same as those in Fig. 2. The red circles show the effects of the shear stress change without normal stress perturbation (*N*_{d} = 0), and the other circles represent the effects of the normal stress changes with *S*_{d} = 0. The assumed values of α are 0 (orange), 2.1 (green), and 4.2 (black), respectively.

The time to instability decreases, i.e., the triggering potential is high, as the stress amplitude increases. As small values of α results in large values of ΔCFF_{d} for negative *N*_{d} as denoted by Eq. (18), the values of *N*_{d} required to cause a certain time to instability decrease when α is small. The relationships of ΔCFF_{d} calculated using Eq. (18) and the time to instability are plotted in Fig. 5b, which shows that the curves for the four cases closely overlap. This means that the triggering potential is determined mainly by a value of ΔCFF_{d}.

Triggering by static stress change is also simulated. At *t *= 0, static stress jump is applied:

$$\begin{aligned} \tau_{s} \left( t \right) & = S_{s} t/T_{p} \quad 0 \le t \le T_{p} \\ & = S_{s} \quad t > T_{p} \\ \sigma_{s} \left( t \right) & = N_{s} t/T_{p} \quad 0 \le t \le T_{p} \\ & = N_{s} \quad t > T_{p} \\ \end{aligned}$$

(19)

The corresponding Coulomb stress jump is written as

$$\begin{aligned} {\text{CFF}}_{s} \left( t \right) & = \Delta {\text{CFF}}_{s} t/T_{p} \quad 0 \le t \le T_{p} \\ & = \Delta {\text{CFF}}_{s} \quad t > T_{p} , \\ \end{aligned}$$

(20)

with

$$\Delta {\text{CFF}}_{s} = S_{s} - \left( {\mu - \frac{\alpha }{1 + c}} \right)N_{s} .$$

(21)

Figure 6a shows the effect of the static stress changes *S*_{s} and *N*_{s} on the time to instability by assuming μ = 0.7, α = 4.2, *c* = 20, and the same friction parameters as in Fig. 2a. Figure 6b shows the relationship between ΔCFF_{s} and the time to instability. The two curves for the shear and normal stress jumps completely overlap. In the static case the triggering potential is a function of ΔCFF_{s}.

Dieterich et al. (2000) pointed out that a modified Coulomb stress change based on conventional RSF for static triggering is given by ΔCFF_{m} = *S*_{s} − (μ − α) *N*_{s}. If scaling relationship (13) is satisfied, ΔCFF_{m} takes the same value as ΔCFF_{S} evaluated using Eq. (21) for given *S*_{s} and *N*_{s}.

The change Δ_{s}ln(*V*/*V**) due to the static jump of CFF can be approximately obtained using Eq. (7),

$$\Delta_{S} \ln (V/V^{*} ) = \frac{{\Delta {\text{CFF}}_{s} }}{{a\sigma /\left( {1 + c} \right)}}.$$

(22)

Note that in the previous section, we derived some equations under the condition of *V* = const. to reproduce the experimental results obtained by Linker and Dieterich (1992). As *V* is not controlled in the earthquake cycle simulations, ln*V* increases in proportional to ΔCFF_{S}.

Comparing Figs. 5b and 6b, we find that values of ΔCFF_{s} for static change smaller than ΔCFF_{d} for dynamic change can trigger instability. This means that a value of ΔCFF itself cannot be used for evaluating the triggering potential to compare the static and dynamic perturbations. This is consistent with previous observations. For instance, Kilb et al. (2000) reported that the aftershocks of the 1992 Landers earthquake (*M*7.3) were possibly triggered not only by static stress changes, but also by dynamic stress changes with much larger amplitude than that of the static changes. In the case of dynamic triggering, the applied Coulomb stress becomes zero after the perturbation ceases, as implied by Eq. (19), and ΔCFF_{d} makes no direct contribution to the first term of the right-hand side of Eq. (7) for changing Δln(*V*/*V**).

Next, we consider the increases of *V* when the following dynamic perturbation CFF_{d}(*t*):

$$\begin{aligned} {\text{CFF}}_{d} \left( t \right) & = \tau_{d} \left( t \right) - \left( {\mu - \frac{\alpha }{1 + c}} \right)\sigma_{d} \left( t \right)\quad 0 < t < T_{n} \\ & = 0 \quad t > T_{n} . \\ \end{aligned}$$

(23)

The velocity when the perturbation is applied, based on Eq. (7), is approximately

$$\ln \left( {V_{d} \left( t \right)/V_{i} } \right) = \frac{{{\text{CFF}}_{d} \left( t \right)}}{{a\sigma /\left( {1 + c} \right)}},$$

(24)

or

$$V_{d} \left( t \right) = V_{i} { \exp }\left[ {\frac{{{\text{CFF}}_{d} \left( t \right)}}{{a\sigma /\left( {1 + c} \right)}}} \right],$$

(24b)

where *V*_{i} is the velocity just before applying the perturbation. The displacement during the perturbation is given by

$$\Delta u = \mathop \smallint \nolimits_{0}^{{T_{n} }} V_{d} {\text{d}}t.$$

(25)

As CFF_{d}(*t*) = 0 after the perturbation ceases, Eq. (8) approximately leads to

$$\Delta \ln (V/V^{*} ) = \frac{b}{aL}\Delta u = \frac{b}{aL}\mathop \smallint \nolimits_{0}^{{T_{n} }} V_{d} \left( t \right){\text{d}}t.$$

(26)

Here, we ignored the integral of 1/θ in Eq. (8) because the values were less than 5% of values estimated using the right-hand side of Eq. (26) in the numerical simulations in Fig. 5. Moreover, Δ*τ* has a small value after the perturbation because of the displacement. Heterogeneous displacement inside the asperity can cause both stress release and stress concentration depending on location and heterogeneity.

We evaluate the equivalent ΔCFF_{eq} for the dynamic perturbation, which results in the same velocity increase as that obtained in the case of static stress perturbation when ΔCFF_{s} = ΔCFF_{eq}. Comparing Eqs. (22) and (26), ΔCFF_{eq} is represented by

$$\begin{aligned} \Delta {\text{CFF}}_{\text{eq}} & = \frac{b\sigma }{{L\left( {1 + c} \right)}}\mathop \smallint \nolimits_{0}^{{T_{n} }} V_{d} \left( t \right){\text{d}}t, \\ & = \frac{{b\sigma V_{i} }}{{L\left( {1 + c} \right)}}\mathop \smallint \nolimits_{0}^{{T_{n} }} { \exp }\left[ {\frac{{{\text{CFF}}_{d} \left( t \right)}}{{a\sigma /\left( {1 + c} \right)}}} \right]{\text{d}}t. \\ \end{aligned}$$

(27)

ΔCFF_{eq} is independent of a value of *c* including *c* = 0 (aging law) when the scaling relations (13) and (14) are satisfied. Figure 7 shows ΔCFF_{eq} using the average velocity over the asperity before the perturbation starts as *V*_{i}. It is found that the triggering potentials for the dynamic and static perturbations are similar when ΔCFF_{s} = ΔCFF_{eq}.

Effects of stress perturbation timing were briefly discussed by Yoshida (2018) assuming constant normal stress. As the effects of timing may not be changed essentially by including the normal stress dependence, we do not discuss this subject here.