# Nucleation Mechanism and Rupture Dynamics of Laboratory Earthquakes at Different Loading Rates
## 3 Results
### Effect of Loading Rate on Earthquake Nucleation Process
#### 3.3.2 Critical Nucleation Length  
The critical nucleation length of the laboratory earthquakes can be determined from the rupture front in Figure 11. Theoretically, critical nucleation length is related to the normal stress and frictional parameters [19]. As shown in Figure 12, the critical nucleation length decreases with the increase in the loading rate when the loading rate is lower than 3.0 \(\upmu\)m/s, which is qualitatively consistent with [PERSON]'s results [35]. This result suggests that it may be more difficult to identify the nucleation phase of earthquakes on faults with high loading rates than those with slow loading rates.  
Previous studies explored the underlying mechanism of the loading rate influence on the nucleation size to some extent [33; 35]. However, these interpretations were mainly based on qualitative analysis. In this study, we attempted to explain why the nucleation size depends on the loading rate in a quantitative manner. Assuming that the fault friction is governed by the slip-weakening law [44], the critical rupture length can be expressed as follows:  
\[\mathrm{L_{c}}=(1+\
u)\cdot(\tau_{\mathrm{s}}-\tau_{\mathrm{d}})\cdot\mathrm{ G}\cdot\mathrm{dc}/\pi\cdot(\tau_{0}-\tau_{\mathrm{d}})^{2} \tag{6}\]  
where \(\
u\) is the [PERSON]'s ratio, \(G\) is the shear modulus, and \(d_{c}\) is the critical slip distance; \(\tau_{\mathrm{s}}=\mu_{\mathrm{s}}\sigma_{\mathrm{n}}\) and \(\tau_{\mathrm{d}}=\mu_{\mathrm{d}}\sigma_{\mathrm{n}}\) are the static and dynamic friction strengths, respectively; and \(\tau_{0}=\mu^{0}\sigma_{\mathrm{n}}\) is the initial friction strength.  
From Figure 7b, we can see that the loading rate has an effect on the fault strength. It was found that the loading rate influences the sliding mode in a specific range of the normal stress [21]. It was also found that the shear stress drop decreases with the loading rate, and the sliding mode transforms from stick-slip to stable sliding when the loading rate reaches a critical value. This is partly due to the fact that a higher loading rate may prevent the action of adhesion forces on the interlocking asperities. According to the rate  
Figure 11: Diagram illustrating the spatiotemporal distribution of rupture fronts, denoted by asterisk markers, at a loading rate of 0.5 \(\upmu\)m/s. The red line provides a linear fit of these rupture points, with its slope signifying the reciprocal of the rupture propagation velocity. The chart distinguishes between the dynamic propagation at 1908 m/s and the more gradual quasi-static propagation with an approximate velocity of 11 m/s, notably below the S-wave velocity.  
and state friction constitutive law, the change in the sliding velocity leads to the change in the coefficient of friction. Thus, we can assume that the loading rate affects the static friction coefficient (\(\mu_{\text{s}}\)), and the relationship between \(\mu_{\text{s}}\) and the loading rate has the same form as that in the rate-and-state friction constitutive law:  
\[\mu_{\text{s}}=\text{a}\mathbf{ln}(\text{b}\text{V}_{\text{L}}) \tag{7}\]  
where \(\mathbf{a}\) and \(\mathbf{b}\) are the constants. Equation (5) can be transformed as follows:  
\[\begin{split}\text{L}=\text{L}_{\text{c}}\sigma_{\text{n}}& =(1+\
u)\cdot(\mu_{\text{s}}-\mu_{\text{d}})\cdot\text{G}\cdot \text{dc}/\pi\cdot(\mu^{\text{P}}-\mu_{\text{d}})^{2}\\ &=(1+\
u)\cdot(\text{a}\mathbf{ln}(\text{b}\text{V}_{\text{L}})-\mu_{ \text{d}})\cdot\text{G}\cdot\text{dc}/\pi\cdot(0.048\,\text{V}_{\text{L}}{}^{-0.455}+0.414-\mu_{\text{d}})^{2}\end{split} \tag{8}\]  
Fitting the data in Figure 12 based on Equation (4) yields that \(\alpha=-0.243\), \(b=0.156\), \(\mu_{d}=0.16\), \(d_{c}=19.6\)\(\mu\)m, \(\mu^{\text{P}}=0.048\)\(\text{V}_{\text{L}}{}^{-0.455}+0.414\), and \(\Delta\mu=0.101\)\(\text{V}_{\text{L}}{}^{-0.281}\).