# An Advanced Shear Strength Criterion for Rock Discontinuities Considering Size and Low Shear Rate
## 1 Introduction
First of all, the dimension unity could not be fulfilled in Equations (a), (b), (c) and (e) proposed by [48; 51; 53; 55]. Thus, the physical meaning behind these equations was not clear \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample Type** & **Shear Rate (mm/s)** & **Empirical Equations** & **References** \\ \hline Granite, Diamond-rock & 0–0.1 & \(\mu=\mu_{0}+A\log(\frac{R_{d}}{\
u}+1)\dots\) (a) & [48] \\ Cement & 0.02–0.8 & \(\tau=0.883\sigma\tan\bigl{(}\varphi_{0}+7.52\varepsilon\varepsilon_{0}^{0.37} \bigr{)}V^{-0.032}\dots\) (b) & [51] \\ Cement mortar & 0.01–0.4 & \(\tau=0.982\sigma\tan\bigl{|}\varphi_{0}+4.970(JRC_{0})^{4\sigma}\bigl{|}g( \frac{ICS}{\sigma})\bigr{|}V^{-0.06}\dots\) (c) & [53] \\ Syenite, Sandstone, & & & \\ Cement, Gypsum, & 0.005–0.8 & \(\tau=\sigma\tan\Bigl{(}\log_{10}\Bigl{(}\big{(}\frac{ICS}{\sigma}\big{)}^{JRC} \bigl{(}\frac{\
u}{\
u}\bigr{)}^{M}\Bigr{)}+\varphi_{0}\Bigr{)}\dots\) (d) & [54] \\ Concrete & & & \\ Hot metamorphic & & & \\ limestone, Chemically & 0.0017–0.83 & \(\mu=\mu_{0}^{\prime}+a\ln(\sigma_{0})-b\ln(\sigma_{0})\dots\) (e) & [55] \\ deposited limestone & & & \\ \hline Symbols in Table 1: \(\mu\) is the coefficient of sliding friction; \(\mu_{0}\) is the constant coefficient of sliding friction; \(\mu_{0}^{\prime}\) is the friction coefficient when the normal stress is 1 MPa and the shear rate is 1 m/s; \(A\), \(B\) are constants related to the static friction strength; \(a,b\) are material constants; \(M\) is related to physical properties and the microgeometric morphology of the discontinuity; \(d_{c}\) is the critical displacement, which is positively correlated with the discontinuity roughness; \(V\) is the shear rate; \(V_{0}\) is the constant shear rate; \(\tau\) is the peak shear strength; \(\sigma\) is the normal stress; \(\varphi_{0}\) is the basic friction angle of the discontinuity; \(\varphi_{0}\) is the residual friction angle; \(a_{0}\) is the undulated angle; \(JRC\) is the discontinuity roughness coefficient; \(JRC_{0}\) is the initial roughness coefficient of the discontinuity; \(JCS\) is the discontinuity wall compressive strength. Note: In Equation (a), \(\mu_{0}\) approximately equates 0.71, \(d_{c}\) is equal to 0.005 mm, \(A\) equates 0.02, \(B\) is equal to 1 [48]. In Equation (d), the units of \(V_{0}\) and \(V\) are mm/s, \(V_{0}\) equates 1 mm/s [54]. & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Empirical equations of shear strength considering the shear rate. In addition, the size effects of both the discontinuity length and the undulated amplitude were not taken into account in Equations (a), (b), (c) and (e). Although the dimension unity and the size effect induced by the discontinuity length were considered in Equation (d) proposed by [54], the influence of the discontinuity undulated amplitude was ignored. Hence, these empirical equations have some limitations in the practical applications. The purpose of this study is to establish a more feasible and reasonable criterion on peak shear strength of the rock discontinuities taking both the size and shear rate effects into account simultaneously. This paper is structured as follows. Section 1 reviews previous studies on the shear strength of the rock discontinuities and analyzes the limitations of these studies. Section 2 introduces the methods to establish the shear strength criterion and to design the verification laboratory tests. Section 3 shows the results in detail. Discussion and conclusions are given in Sections 4 and 5.