# Mechanical and Acoustic Emission (AE) Characteristics of Rocks under Biaxial Confinements
## 3 Results and Discussion
### Biaxial Compressive Strength  
The biaxial compressive strength \(\sigma_{\rm bcs}\) of granite, marble, and sandstone specimens are listed in Table 3 and shown in Figure 8a. The \(\sigma_{\rm bcs}\) of the three rocks increase initially and then decrease with the increase of \(\sigma_{2}\). There exists a maximum value of the \(\sigma_{\rm bcs}\), named as \(\sigma_{\rm bcs,peak}\), which is the peak value of \(\sigma_{\rm bcs}\) in the variation range of \(\sigma_{2}\). The ratios of the \(\sigma_{\rm bcs,peak}\) to the uniaxial compressive strengths of granite, marble, and sandstone are approximately the same as 1.5-1.6. The peak points of \(\sigma_{\rm bcs}\) corresponding to \(\sigma_{2}\) for granite, marble, and sandstone are about 20, 20, and 10 MPa, respectively, which is much smaller than that in the literature [18; 19; 20; 21; 22]. The first derivative curves of the fitting function of \(\sigma_{\rm bcs}\) of three rocks are drawn as Figure 8b, which show clearly that the increasing rate of \(\sigma_{\rm bcs}\) decreases with the increase of \(\sigma_{2}\). Furthermore, the increasing rate of \(\sigma_{\rm bcs}\) for granite is always greater than that of marble and sandstone, and the increasing rate for sandstone is the smallest. The maximum increasing rate \(\sigma_{\rm bcs}\) for granite, marble, and sandstone are 4, 2.5, and 1.5, respectively, which indicates that the \(\sigma_{2}\) has a lower effect on the \(\sigma_{\rm bcs}\) of rocks with stronger plasticity.  
For all rocks, the increasing rate of \(\sigma_{\rm bcs}\) is 0 at the \(\sigma_{2}\) corresponding to the turning point of \(\sigma_{\rm bcs}\). There is an optimal solution of \(\sigma_{2,\rm peak}\) corresponding to the maximum of \(\sigma_{\rm bcs}\), and the \(\sigma_{2,\rm peak}\) of granite, marble, and sandstone are 25.2, 24.6, and 12.3 MPa, respectively. The \(\sigma_{2,\rm peak}\) is of great significance for disaster prevention and control in the stress adjustment area after underground engineering excavation. At the stage of the increasing rate of \(\sigma_{\rm bcs}\) is negative, the curve of each of the rock shows a similar trend with decreasing first and then increasing slowly, and asymptotically approaches to zero, which demonstrates that the \(\sigma_{\rm bcs}\)  
Figure 7: Lateral strain curves: (**a**–**c**) granite; (**d**–**f**) marble; (**g**–**i**) sandstone.  
decrease slowly when it increases to a certain extent. The minimum increasing rate \(\sigma_{\rm bcs}\) for granite, marble, and sandstone are \(-1\), \(-0.5\), and \(-0.5\), respectively, which indicates that the \(\sigma_{2}\) has less influence on \(\sigma_{\rm bcs}\) for the rock with lower strength under higher \(\sigma_{2}\).  
Based on the results of the biaxial compressive strengths \(\sigma_{\rm bcs}\) of the three rocks, a relationship between the \(\sigma_{\rm bcs}\) and \(\sigma_{2}\), i.e., an exponent strength criterion, was proposed as shown in Equation (1), which has a high degree of adaptability to the strength data of the three rocks obtained from the tests, and has a good reference value when evaluating the \(\sigma_{\rm bcs}\) of the rocks:  
\[\sigma_{\rm bcs}=\sigma_{\rm ucs}+\mathrm{A}\sigma_{\rm ucs}\sigma_{2}\exp(1- \mathrm{B}\sigma_{2}), \tag{1}\]  
where both A and B are material constants.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rock Type** & **Specimen NO.** & \(\sigma_{3}\) **(MPa)** & \(\sigma_{2}\) **(MPa)** & \(\sigma_{1}\) **(MPa)** & \(\sigma_{\rm bcs}\) **(MPa)** \\ \hline \multirow{9}{*}{Granite} & g-0-1 & 0 & 0 & 86.12 & 86.12 \\  & g-0-2 & 0 & 0 & 93.68 & 93.68 \\  & g-5-1 & 0 & 5 & 103.00 & 103.00 \\  & g-5-2 & 0 & 5 & 9