## 4 Discussion  
### Granite Instability Locked Bodies Fracture Nucleation Mechanism  
As a kind of original heterogeneous material, the rock contains many structures with different strengths and scales due to the formation conditions, different mineral composition, burial conditions, geological tectonic movement and other factors, among them the structure with high strength controlling the stability of the sample which called locked bodies [17]. The process of macroscopic cracking and expansion of granite samples are the process of continuous fracture of locked bodies inside the rock mass. Combined with the AE localization map (Figure 4), it can be seen that the fracture process of the first locked solid of granite also undergoes a series of microscopic plastic yielding processes such as micro-fracture generation, expansion, and nucleation. At the microscopic scale, the micro-fracture rate inside the rock mass under load can be obtained by the Arrhenius Equation:  
\[v_{c}(t)=A\exp(\frac{A_{1}E\int_{0}^{t}\varepsilon(t)dt-U_{0}}{kT}), \tag{1}\]  
where in: \(v_{c}\) is the micro-fracture rate; \(A\), \(A_{1}\) are constant; \(T\) is the absolute temperature of the rock mass; \(k\) is the Boltzmann constant; \(U_{0}\) is the fracture activation energy; \(\dot{\varepsilon}(t)\) is the strain rate; and \(E\) is the elastic modulus. In order to reduce the influence of unnecessary factors, it can be considered that process is isothermal (\(T\) is a constant), when the rock mass is subjected to load, and the strain loading rate is constant (\(\dot{\varepsilon}(t)=\dot{\varepsilon}\)). Then, when the sample is loaded to time \(t\), the cumulative micro-fracture count can be shown as:  
\[V_{c}(t)=\int_{0}^{t}v_{c}dt=\frac{kTA}{A_{1}E\dot{\varepsilon}}\exp(-\frac{U _{0}}{kT})[\exp(\frac{A_{1}E\dot{\varepsilon}t}{kT})-1]. \tag{2}\]  
As the loading progresses, \(V_{c}\) increases exponentially. According to the fracture density criterion proposed by [PERSON] et al. [18], it is considered that, when the number of micro-fractures in the area around the locked bodies reaches the fracture limit, the locked bodies will break:  
\[V_{c}(t)\big{|}_{r\leq A_{2}r_{i}}\geq\frac{N_{si}}{\frac{4}{3}\pi r_{i}^{3}}, \tag{3}\]  
where in: \(r_{i}\) is the radius of the \(i\)th locked solid; \(V_{c}(t)\big{|}_{r\leq A_{2}r_{i}}\) is the same sphere as the \(i\)th locked bodies, the radius is the total number of micro-fractures in the \(A_{2}r_{i}\) sphere, wherein \(A_{2}\) is a constant, \(A_{2}>1\); \(N_{si}\) is the critical point of the \(i\)th locked bodies fracture threshold. Joint Equation (3), the critical stress of the \(i\)th locked bodies fracture is:  
\[\sigma_{ci}=\frac{kT}{A_{1}}\ln\Bigg{[}\frac{3N_{si}A_{1}E\dot{\varepsilon}}{ 4\pi r_{i}^{3}kTA}\exp(\frac{U_{0}}{kT})+1\Bigg{]}. \tag{4}\]  
Since the sample is a regular cube, the geometric center of the sample is most prone to stress concentration and the crack in this area first develops. When the surrounding defect density exceeds the critical threshold of the locked bodies fracture (as known from Equation (3)), the locked bodies in geometric center of granite sample firstly yield to fracture, and the AE signal at the initial stage of loading is generated. As the stress level increases, a large number of micro-fractures accumulate around the solid, and the number of cracks increases exponentially with the increase of the stress, and the micro-stress increment produces more defects. It is believed that each locked body is in the yield critical state, and any slight increase in stress will cause the interlocking development of the fracture. When the micro-fracture exceeds the locked bodies stability threshold, the locked bodies yields and is constrained to the next locked bodies. When the last locked bodies of the rock mass breaks, due to the fracture is no longer restrained, the stress is instantaneously released, the crack rapidly expands, and the sample shows macroscopic cracking and transient instability, the stress at this time is the peak stress of the rock mass.