# Mechanical and Acoustic Emission (AE) Characteristics of Rocks under Biaxial Confinements
## 3 Results and Discussion
### AE Characteristics  
#### 3.4.1 AE Hit Rate, AE Count Rate, and Dynamic \(b\)-Value  
AE hit rate and AE count rate refers to the cumulative AE hit and AE count of the specimen in one second during the process of rock failure, and they reflect the damage degree of the specimen under loading. The dynamic \(b\)-value refers to the ratio of AE hit with low amplitude signal to AE hit with high amplitude signal during loading, and the higher dynamic \(b\)-value reflects the greater proportion of AE hit with low amplitude signal [28; 46; 47; 48]. The AE data of the two sensors have a good consistency for most specimens in the biaxial compressive tests. Therefore, the data from \"S1\" was presented in this paper. According to Gutenberg-Richter (G-R) distribution [49; 50], the relation between the number of earthquakes above a given magnitude, \(N(M)\), and the magnitude, \(M\), can be drawn as:  
\[\lg N(M)=a-b(M-M_{1});M\geq M_{1}, \tag{2}\]  
where \(M_{1}\) is a lower threshold, below which magnitudes are not considered; \(a\) is a constant and \(b\) is referred to as the \(b\)-value. When calculating the AE \(b\)-value of rock damage, AE amplitude is generally used to reflect AE signal, and the equivalent earthquake magnitude is obtained by dividing AE amplitude by 20. So, Equation (2) can be modified into Equation (3):  
\[\lg[N(A/20)]=a-b(A/20), \tag{3}\]  
where \(A\) is the amplitude of the AE hit; and \(N\) is the cumulative number of AE hits with an amplitude of \(A\). The AE amplitude data set of each rock specimen was evenly divided into 13 parts (A1, A2, A3,..., A13) according to the overall loading time, and the dynamic \(b\)-value of two adjacent parts was calculated successively from A1 to A13 at a pace of 1. The changes of the AE hit rate, AE count rate, and dynamic \(b\)-value with time are shown in Figure 9, and the stress-strain curves of rock specimens are also drawn. In the initial stage of loading, the natural cracks in the rock are mainly closed without significant damage, so there are few AE signals at this stage. With the continuous increase of \(\sigma_{1}\), the internal cracks in the rocks begin to develop and propagate, and AE signals at this stage gradually increase. Based on the analysis of data of AE hit rate, AE count rate, and dynamic \(b\)-value, the main conclusions are summarized as follows:  
Figure 8: Strength characteristics including (**a**) biaxial compressive strengths and the associated fitting curves, and (**b**) the increasing rate of \(\sigma_{\rm bcs}\) for granite, marble, and sandstone. (1) In the same rock, the AE hit rate and AE count rate have roughly the same change law during loading, while the value of AE count rate is 10 times as much as AE hit rate. Under uniaxial compression, many peaks appear in the AE hit rate and AE count rate curves (Figure 9a,d,g), which indicates that the internal cracks are well developed, and more fracture surfaces are produced, which makes the rock failure more thorough. (2) The AE hit rate and AE count rate increase rapidly when \(\sigma_{1}\) approaches \(\sigma_{\text{bcs}}\) until the failure occurring, which indicates that the failure of the rock specimens is caused by the increase of its internal cumulative damage to a specific threshold value. With the increase of \(\sigma_{2}\), the peak values of AE hit rate and AE count rate appear later, indicating that \(\sigma_{2}\) can slow down the growth rate of the damage degree of rock under the increasing load (Figure 9b,c,e,f,h,i). (3) The dynamic \(b\)-value of the three types of rocks are all distributed in the range of 1-3 (mainly in 1-2), which is related to the AE hit rate and AE count rate. The dramatic increase in the AE hit rates is always accompanied by a significant decrease in the dynamic \(b\)-value, which is a phenomenon that also occurs before the rock failure where the peaks of AE hit rate and AE count rate appear. This behavior has been observed in brittle materials [51; 52; 53].