# An Advanced Shear Strength Criterion for Rock Discontinuities Considering Size and Low Shear Rate
## 3 Results
### Comparison Experimental Data with Predicted Results  
#### 3.2.1 Rate-Dependent Friction Angle \(\varphi_{\mathrm{v}}\)  
In this section, we verified the feasibility of the new shear strength criterion. As mentioned in Section 2.3, the values of arctan (\(\tau/\sigma\)) and \(JRC_{\mathrm{n}}\cdot\log_{10}\) (\(JCS_{\mathrm{n}}\)/\(\sigma\)) for the granite discontinuity samples were calculated based on Equation (6) and Table 3, as listed in Table 1. On the basis of Equations (6) and (7), values of the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) for granite discontinuity samples were calculated and the average values were listed in Table 2. According to Equation (7), the mathematical fitting between the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) for granite discontinuity samples and the shear rate \(V\) was conducted. The relationship between \(\log_{10}\) (\(1/V\)) and the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) was shown in Figure 5.  
The linear regression analysis was performed on the test data in Figure 5:  
\[\varphi_{\mathrm{v}}=-6.2538\log_{10}(1/V)+24.188, \tag{8}\]  
where the squared value of the correlation coefficient \(R^{2}\) = 0.834. According to Equation (8) and Figure 5, a strong linear relationship between the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) and \(\log_{10}\) (\(1/V\)) is presented, with a high value of \(R^{2}\) (0.834). There is an absolute difference between theoretically predicted results and experimental data, wherein exists a ratio within 11.7% in comparison with the theoretically predicted results. Moreover, the intercept value of the fitting Equation (8) is 24.188\({}^{\circ}\). It is very close to the average value of the basic friction angle (25\({}^{\circ}\)) for granite discontinuity samples. Therefore, the relationship between the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) of granite discontinuity samples and \(\log_{10}\) (\(1/V\)) can be fitted by Equation (7) in a good manner. To check the generality of the new equation, we introduced the indoor direct shear test data of cement discontinuities under the different shear rates and constant normal load condition [51]. The geometric and strength parameters, such as the sample morphology, length, maximal undulated amplitude, uniaxial compressive strength and basic friction angle of cement discontinuities with regular sawtooth shapes were listed in Table 4. By utilizing Equations (2) to (4), the values of \(JRC_{\mathrm{n}}\) and \(JCS_{\mathrm{n}}\) for cement discontinuity samples were calculated respectively, as listed in Table 4.  
Based on Equation (6) and Table 4, the values of arctan (\(\tau/\sigma\)) and \(JRC_{\mathrm{n}}\cdot\log_{10}\) (\(JCS_{\mathrm{n}}\)/\(\sigma\)) for cement discontinuity samples were calculated, as listed in Table 3 (after [51]). On the basis of Equations (6) and (7), values of the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) for cement discontinuity samples were calculated and the average values were listed in Table 2. According to Equation (7), the mathematical fitting between the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) for cement discontinuity samples and the shear rate \(V\) was performed. Figure 6 shows the relationship between \(\log_{10}\) (\(1/V\)) and rate-dependent friction angle \(\varphi_{\mathrm{v}}\). Figure 5: Relationship between average value of the rate-dependent friction angle \(\varphi_{\mathrm{v}}\) and \(\log_{10}\) (\(1/V\)) of granite discontinuity samples. According to Figure 6, there is a solid linear relationship between the rate-dependent friction angle \(\varphi_{\rm v}\) and \(\log_{10}\) (\(1/V\)):  
\[\varphi_{\rm v}=1.4288\log_{10}(1/V)+36.259, \tag{9}\]  
where the squared value of the correlation coefficient \(R^{2}=0.9976\). The intercept value of Equation (9) is \(36.259^{\circ}\), which is very close to the average value of the basic friction angle (\(36.22^{\circ}\)) for cement discontinuity samples [51]. Furthermore, the value of \(R^{2}\) can reach \(0.9976\).