# Experimental and Numerical Investigations on the Slate Shearing Mechanical Behavior
## 5 Discussion  
In order to assess the shear strength of slate using the anisotropic shear failure criterion initially proposed for shale [32], it is imperative to ascertain the anisotropic state parameter \(A\) to capture both the inherent bedding-induced anisotropy and the stress-induced anisotropy observed in the tests. When \(\tau=0\) MPa, \(A\) can be further simplified as expressed in Equation (2). \[A=F_{ij}^{\pi}d_{ij}=-\frac{\Delta}{3+\Delta}\frac{3}{\sqrt{5}}\bigg{(}\cos 2 \theta+\frac{1}{9}\bigg{)} \tag{2}\]  
\[\theta=\beta-\frac{1}{2}\text{arctan}\bigg{(}\frac{2\tau}{\sigma_{n}}\bigg{)} \tag{3}\]  
\[f_{\rm c}=f_{\rm 0}\bigg{[}1+\bigg{(}-\frac{\Delta}{3+\Delta}\frac{3}{\sqrt{5}} \bigg{(}\cos 2\theta+\frac{1}{9}\bigg{)}\bigg{)}+f_{\rm c1}\bigg{(}-\frac{ \Delta}{3+\Delta}\frac{3}{\sqrt{5}}\bigg{(}\cos 2\theta+\frac{1}{9}\bigg{)} \bigg{)}^{2}\bigg{]} \tag{4}\]  
Figure 16: Local magnification of force chain at peak stress; (**a**) \(\beta=45^{\circ}\) and (**b**) \(\beta=60^{\circ}\). where \(f_{\rm c}\) represents the uniaxial compressive strength, and \(f_{\rm c0}\),\(f_{\rm c1}\), and \(f_{\rm c2}\) are unknown material parameters. The Mohr-Coulomb criterion, widely used to describe the shear strength of geotechnical materials, is expressed as Equation (5). \[\tau=c+f\sigma_{n} \tag{5}\]  
For transversely isotropic geomaterials, the internal friction angle and cohesion are related to the bedding orientation or, more directly, the anisotropic state. From Equation (4), it is reasonably assumed that the M-C strength parameters of a given anisotropic geomaterial follow the same variation trend as \(A\). Therefore, the internal friction coefficient and cohesion are considered as given in Equations (6) and (7). \[f(A)=f_{0}\Big{[}1+f_{1}\cdot A+f_{2}\cdot(A)^{2}\Big{]} \tag{6}\]  
\[c(A)=c_{0}\Big{[}1+c_{1}\cdot A+c_{2}\cdot(A)^{2}\Big{]} \tag{7}\]  
where \(f_{0}\), \(f_{1}\), \(f_{2}\), \(c_{0}\), \(c_{1}\), and \(c_{2}\) are material parameters that can be determined through a numerical approximation method using direct shear tests on slate samples with different orientations. Substituting Equations (6) and (7) into the M-C criterion in Equation (5) yields a new anisotropic shear failure criterion. The uniaxial compressive strength of slate samples in different bedding directions can be used to calibrate \(\delta\). The compressive strength values of slate with bedding directions of \(0^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\) are 70 MPa, 30 MPa, and 90 MPa, respectively. Then, the unknown parameters are determined using the least square method as \(\Delta=2.952\), \(f_{\rm c0}=31.71\), \(f_{\rm c1}=3.56\), and \(A\) can be calculated. Finally, six material parameters can be obtained, i.e., \(c_{0}=10.798\), \(c_{1}=1.118\), \(c_{2}=0.508\), \(f_{0}=0.808\), \(f_{1}=-0.693\), and \(f_{2}=0.256\). The theoretical estimates of the shear strength for specimens with different bedding orientations are shown in Figure 17. This criterion can capture the slate shear strength sensitivity to \(\beta\) and \(\sigma_{\rm n}\). However, the estimated values exhibit significant disparities from experimental and simulation results. A macroscopic analysis suggests that the literature uses shale, a sedimentary rock, whereas it is a metamorphic rock derived from an original sedimentary rock composed of clay or volcanic ash. Sedimentary rocks are usually formed by the deposition of sediments such as clasts, minerals, and organic material in natural processes. Organic matter such as asphalt and carbon are also sometimes present. The mineralogical characteristics of metamorphic rocks are related to the composition of the original rocks, but some minerals may change during metamorphism. For example, under high temperatures and high pressure, feldspar may become amphibole, and quartz may change to form new minerals such as mica. In addition, the bedding distribution of shale samples is more uneven than that of slate, affecting the accuracy of the estimation results [38; 39; 40; 41; 42].