## 3 Basic Theory of Slope Characterization Models
### Basic Theory of Micrometric Weibull Distribution Modeling of Nonhomogeneous Slopes
#### 3.2.1 Petrographic Relationships of Rocks  
The failure process of rock materials can be viewed as the progressive damage and evolution of internal defects within microscopic structural units under external forces. In uniaxial compression tests of rock specimens, the strength of each microelement follows statistical laws. Based on the expression of damage parameters D, the constitutive relationship of rock fracture process can be derived as follows:  
\[\sigma=E_{0}\varepsilon(1-\text{D})=E_{0}\varepsilon\bigg{\{}1-\frac{1}{\text {s}\sqrt{2\pi}}\!\int_{0}^{\varepsilon}\exp\left[-\frac{1}{2}\bigg{(}\frac{x- \varepsilon_{0}}{\text{s}}\bigg{)}\right]dx\bigg{\}} \tag{5}\]  
where \(\sigma\) is the load on the rock. \(\varepsilon\) is the element strain. \(E_{0}\) denotes the initial elastic modulus. \(\varepsilon_{0}\) is the mean strain of element. s is the standard deviation parameter that characterizes the distribution of damage.  
If we assume that the strength distribution function of the microelements follows a Weibull distribution, we can derive the constitutive equation describing the stress-strain relationship of the rock:  
\[\sigma=E_{0}\varepsilon(1-\text{D})=E_{0}\varepsilon\cdot\exp\left[-\left( \frac{\varepsilon}{\varepsilon_{0}}\right)^{\beta}\right] \tag{6}\]  
Figure 4: Mohr-Coulomb yield criterion for joint surfaces.  
where \(\beta\) is the shape parameter, indicating the degree of non-uniformity of the distribution, and the larger its value indicates the more uniform distribution. Equation (6) takes derivatives of \(\varepsilon\) on both sides, it can be obtained that the derivative at the peak is zero:  
\[\frac{\varepsilon_{\max}}{\varepsilon_{0}}=\left(\frac{1}{\beta}\right)^{\frac {1}{\beta}} \tag{7}\]  
where \(\varepsilon_{\max}\) is the strain at the peak of the strength of the element.  
Similarly, we can derive the following expression from Equation (6):  
\[\beta=\frac{1}{\ln\frac{E_{0}\varepsilon_{\max}}{\sigma_{\max}}}=\frac{1}{\ln E _{0}-\ln E_{0\max}} \tag{8}\]  
where \(E_{0\max}=\frac{\sigma_{\max}}{\varepsilon_{\max}}\), is the reciprocal of the ratio of the strain at the peak strength of the element to the corresponding stress.  
Furthermore, an expression in terms of extrema can be derived for the constitutive equation:  
\[\sigma_{\max}=E_{0}\varepsilon_{\max}(1-D)=E_{0}\varepsilon_{\max}\cdot\exp \left(-\frac{1}{\beta}\right) \tag{9}\]  
From Equations (7) and (9), it is evident that the strength distribution model based on the Weibull function relates the damage parameter and stress state at a given moment to the material's elastic modulus, peak strength, and strain. Equation (8) indicates that \(\beta\) is a function of both the elastic modulus and the secant modulus at the peak point. Moreover, when the elastic modulus is constant, a larger \(\beta\) leads to a larger \(\varepsilon_{\max}\), suggesting greater brittleness of the rock material. Conversely, a smaller \(\beta\) results in a smaller \(\varepsilon_{\max}\), indicating greater plasticity of the rock material. The shape parameter significantly influences the evolution of material damage.  
For the parameters \(\beta\) and \(\varepsilon_{0}\) in the damage eigen structure equation can be determined from the peak points \(\sigma_{\text{c}}\) and \(\varepsilon_{0}\) of the full stress-strain curve, which can be obtained as:  
\[\beta=\left(\ln\frac{E\varepsilon_{\text{c}}}{\sigma_{\text{c}}}\right)^{-1} \tag{10}\]  
where \(\sigma_{\text{c}}\), \(\varepsilon_{\text{c}}\) are the stress and strain corresponding to the peak point of the whole stress-strain curve under uniaxial loading condition. \(E\) is the modulus of elasticity.