# An Advanced Shear Strength Criterion for Rock Discontinuities Considering Size and Low Shear Rate
## 2 Method  
### Establishment of the New Shear Strength Equation  
As mentioned in Section 1, compared with Equations (a), (b), (c) and (e) [48; 51; 53; 55], Equation (d) proposed in [54] has an advantage of dimension unity, which can be used as the basis for establishing the new shear strength criterion of peak shear strength for the rock discontinuities considering both the size and shear rate effects. Both the discontinuity roughness coefficient (\(JRC\)) and the discontinuity wall compressive strength (\(JCS\)) are the fatal parameters in predicting the shear strength of the rock discontinuities [21; 37]. Therefore, it is crucial to acquire the values of \(JRC\) and \(JCS\) accurately. In order to establish the new shear strength criterion taking into account the size and shear rate effects, first of all, it is indispensable to study the effects of the discontinuity length and the undulated amplitude on the values of \(JRC\) and \(JCS\). In previous studies, [PERSON] and [PERSON] (1982) proposed the empirical equation of \(JRC_{\text{n}}\) considering the effect of the discontinuity length on the shear strength [40]:  
\[JRC_{\text{n}}=JRC_{0}\left(\frac{L_{\text{n}}}{L_{0}}\right)^{-0.02 JRC_{0}}, \tag{1}\]  
where \(JRC_{\text{n}}\) is the roughness coefficient of the shear contact surfaces for the discontinuity with the length of \(L_{\text{n}}\); \(JRC_{0}\) is the roughness coefficient of the shear contact surfaces for the discontinuity of which length is \(L_{0}\). In general, \(L_{0}\) equates 100 mm. In Equation (1), the \(JRC_{0}\) is calculated by the mathematical relation between \(JRC_{0}\) and average value of entire secant angles, i.e., \(\beta_{100\%}\) for the shear contact surfaces of the irregular undulated discontinuity studied by [35], as shown in Equation (2):  
\[JRC_{0}=2.9206\beta_{100\%}^{0.7592}-3.1. \tag{2}\]  
For discontinuity samples with regular sawtooth shapes, the \(\beta_{100\%}\) in Equation (2) can be degraded into the undulated angle of the shear contact surfaces for the discontinuity. As mentioned in Section 1, the \(JRC\) value increased with increasing the undulated amplitude under the same discontinuity length condition [43; 45; 46]. However, the effect of the undulated amplitude was not taken into account in Equation (1), which needs to be improved. Hence, the maximal undulated amplitude was introduced as a correction parameter into Equation (1). The modified equation is assumed as follows:  
\[JRC_{\text{n}}=JRC_{0}\left(\frac{L_{\text{n}}/h_{\text{n}}}{L_{0}/h_{0}} \right)^{-0.02 JRC_{0}}, \tag{3}\]  
where \(h_{\text{n}}\) is the maximal undulated amplitude of the shear contact surfaces for the discontinuity when it has the length of \(L_{\text{n}}\); \(h_{0}\) is the maximal undulated amplitude of the shear contact surfaces for the discontinuity with the length of \(L_{0}\); In general, \(L_{0}\) is equal to 100 mm. Besides Equation (1), [PERSON] and [PERSON] (1982) also put forward the empirical equation of \(JCS_{\rm n}\) considering the discontinuity length [40]:  
\[JCS_{\rm n}=JCS_{0}\bigg{(}\frac{L_{\rm n}}{L_{0}}\bigg{)}^{-0.03 JRC_{0}}, \tag{4}\]  
where \(JCS_{\rm n}\) is the wall compressive strength for the discontinuity as its length is \(L_{\rm n}\); \(JCS_{0}\) is the wall compressive strength for the discontinuity with the length of \(L_{0}\); In general, \(L_{0}\) equates 100 mm.