# Characteristics of Acoustic Emission Caused by Intermittent Fatigue of Rock Salt
## 4 Results and Discussion
### Intermittent Fatigue Life Model for Rock Salt Based on Damage  
Damage variable D represents the status of material degradation, defined by [PERSON] [33] as the ratio of A\({}_{d}\) (area of micro-defects on an instantaneous load-bearing surface) and A (sectional area of the initial undamaged condition):  
\[\text{D}=\frac{\text{A}_{d}}{\text{A}} \tag{3}\]  
We assume that the samples of rock salt have no initial damage. The sum of the accumulated AE counts is then N\({}_{total}\) when the entire section A fails. The AE rate for infinitesimal failures per unit area is then given by:  
\[v_{b}=\frac{\text{N}_{total}}{\text{A}} \tag{4}\]  
If we ignore the magnitude of each AE count, then when the failed (damaged) area of the section reaches A\({}_{d}\), the number of accumulated AE counts is given by  
\[\text{N}=v_{b}\text{A}_{d}=\text{N}_{total}\frac{\text{A}_{d}}{\text{A}} \tag{5}\]  
Equations (3) and (5) are simultaneous, so we have a relationship between the number of AE counts and damage variable D:  
\[\text{D}=\frac{\text{N}}{\text{N}_{total}} \tag{6}\]  
Equation (6) indicates that there is a theoretical consistency between the number of AE counts and material damage. According to the relevant descriptions of the [PERSON] effect [34; 35; 36] and considering the opportunities of the AE to re-emerge, the number of AE counts N\({}_{i}\) for the \(i\)th re-loading is  
\[\frac{\text{N}}{\text{N}_{total}}=\langle\text{D}-\text{D}_{i-1}\rangle \tag{7}\]  
In Equation (7), D\({}_{i-1}\) represents the damage variable generated by the last loading, and \(\langle\bullet\rangle\) is defined as  
\[\langle x\rangle=\begin{cases}x,\ x\geq 0\\ x,\ x<0\end{cases} \tag{8}\]  
We then introduce the damage law expressed by strain \(\epsilon\)[37]:  
\[\text{dD}=\begin{cases}f(\epsilon)\text{d}\epsilon,\ \epsilon=\zeta\ and\ \text{d} \epsilon>0\\ 0,\ \text{d}\epsilon<0\end{cases} \tag{9}\]  
In Equation (9), \(f(\epsilon)\) is a continuous positive definite function of \(\epsilon\) and represents the damage evolution function under the ideal state, and \(\zeta\) is the threshold value of variable damage strain. d\(\epsilon>0\) is always correct under monotonic once-off loading. We assume that there is no initial damage, which means that \(D=\varepsilon=\zeta=0\). Equations (6) and (9) are simultaneously based on this boundary condition, so we then have  
\[\frac{N}{N_{total}}=D=F(\varepsilon)=\int_{0}^{\varepsilon}f(x)d x \tag{10}\]  
Using the constitutive modeling method established by [PERSON] [38; 39], [PERSON] [40], [PERSON] [41], and [PERSON] [42], we obtain the accumulative damage function as  
\[D_{i}=\sum_{i=1}^{n}\frac{N_{i}}{N_{total}}=1-\left(1-i^{\alpha}\right)^{b} \tag{11}\]  
In Equation (11), \(D_{i}\) is the accumulative damage of the \(i\)th circle, and a and b are constants. Figure 13 shows the accumulated damage curves of tests carried out at the various fatigue intervals calculated from Equation (11).  
Figure 13 shows that the damage experience comprises three sequential phases: deceleration-stabilization-acceleration. Most important is the stable damage phase because of its guiding role during the entire damage process. The black lines in Figure 13 represent the stable damage phase, the endings of which direct the trends of the accelerated damage phases. The stable damage phases are fitted by a linear function of the form \(y=k_{d}x+c\), and the relationship between \(k_{d}\) and the fatigue time intervals is presented in Figure 14.  
The development of \(k_{d}\) can be divided into two periods. In the first period, \(k_{d}\) increases as the duration of the intervals extends to 900 s and the accelerating effect of the time interval on fatigue damage takes place: the existence of the interval obviously influences the progression of the fatigue.  
Figure 13: Accumulated damage curves-\(D_{i}\).