## 4 Uniaxial Loading of Critically Damaged Coal Samples
### Acoustic Emission Signal Analysis  
The uniaxial compression of a coal sample is accompanied by the emission of an acoustic signal. The relevant data measured using AE acoustic emission is used to more accurately describe the internal crack expansion during the damage of the coal sample. In this paper, the acoustic emission impact number and \(b\)-value are used to analyze the internal damage characteristics of coal samples during the loading process. The number of acoustic emission hits refers to any acoustic emission signal that passes the threshold and results in accumulated data for one system channel. Variations in \(b\)-values determine the different scale variations of micro-ruptures within the coal rock body. A decrease in \(b\)-values implies an increase in the proportion of large events within the coal rock body and an increase in large-scale micro-ruptures. Acoustic emission \(b\)-values (hereafter referred to as \(b\)-values) have previously been widely used in seismology and are commonly used to study the relationship between magnitude and frequency [26; 27]. Commonly used ways to calculate the acoustic emission \(b\)-value are linear least squares, exponential least squares, and maximum likelihood estimation. One of the least squares G-R expressions is as follows [19; 28]:  
\[\log_{10}N=a-bM \tag{1}\]  
where \(N\) is the total number of earthquakes of magnitude higher than \(M\). \(M\) is the amplitude of the earthquake on the Richter scale (which is a logarithmic scale) and \(b\) is the \(b\)-value. When calculating the \(b\)-value during rock damage, since there is no concept of seismic magnitude, the value obtained by dividing the acoustic emission amplitude by 20 is usually equivalent to the seismic magnitude, making the distribution of the acoustic emission amplitude catalog and the seismic magnitude catalog roughly similar. The equation for calculating the acoustic emission \(b\)-value of rock damage using the least squares method is modified as follows [29; 30]:  
\[\log_{10}[N(A/20)]=a-b\times(A/20) \tag{2}\]  
where \(A\) is the acoustic emission amplitude in db; \(N(A/20)\) is the cumulative number of acoustic emission events of amplitude; \(a\) is the empirical constant; \(b\) is the acoustic emission \(b\) value. According to Equation (3), the equation that can be used to calculate the rock damage acoustic emission \(b\)-value using the minimum likelihood method is modified as:  
\[b=\log_{10}e/(\overline{M}-M_{\rm min}) \tag{3}\]  
where \(\overline{M}\) is the mean magnitude; \(M_{\rm min}\) is the minimum magnitude. According to the statistics of the number of events of acoustic emission, the total number of acoustic emission events detected in the uniaxial reloading process of coal samples with different degrees of damage varied greatly, so when selecting the number of events of the samples, the samples were selected according to the total number of events of the specimen itself, and the sample window was selected between \(1/10\) and \(1/14\) of the total number of samples, and the window of the no-damage, 5 Mpa-60%, 10 Mpa-60%, and 15 Mpa-60% coal samples was 70, 150, 100, and 70, respectively. In addition, based on Equations (5) and (6), the \(b\)-value calculation formula was obtained as:  
\[b=(20\log_{10}e)/(\overline{A}-A_{\rm min}) \tag{4}\]  
where \(\overline{A}\) is the average amplitude; \(A_{\rm min}\) is the minimum amplitude. The \(b\)-value, number of impacts, and cumulative number of impacts of the coal sample were calculated from the acoustic emission signals corresponding to the characteristic curves and the above equations, and the curves of stress, \(b\)-value, number of impacts, and cumulative number of impacts versus time for different paths were plotted; see Figure 8.  
The figure shows that the acoustic emission signal of the critically damaged coal sample has a strong regularity, which is divided into four stages: I for the stability zone, II for the growth zone, III for the secondary stability zone, and IV for the steep growth zone.