## 3 Basic Theory of Slope Characterization Models
### Basic Guidelines for Ubiquitous-Joint Model of Layered Slopes
#### 3.1.1 Elasticity Criterion  
In the Cartesian coordinate system, \(x\), \(y\), \(z\) global coordinates are used to represent the orientation of joints (as depicted in Figure 3). In the local coordinate system, \(x^{\prime}\) denotes the dip direction of the weak plane, \(y^{\prime}\) represents the strike direction of the weak plane, and \(z^{\prime}\) signifies the normal direction of the weak plane [30].  
The relationship between stress in generalized coordinates and stress in local coordinates can be expressed as follows:  
\[\sigma^{\prime}=C^{T}\sigma C \tag{1}\]  
where \(\sigma\) is the stress tensor in the generalized coordinate system (global coordinate system), \(\sigma^{\prime}\) is the stress tensor in the weak plane or local coordinate system, and \(C\) is the direction cosine matrix, describing the directional relationship between generalized coordinates and local coordinates. Each element in \(C\) represents the cosine value between the local coordinate system direction (\(x^{\prime}\),\(y^{\prime}\),\(z^{\prime}\)) and the generalized coordinate system direction (\(x\),\(y\),\(z\)), which can be expressed by Equation (2). \(C^{T}\) is the transpose matrix of \(C\).  
\[C=\begin{bmatrix}\cos(x^{\prime},x)&\cos(x^{\prime},y)&\cos(x^{\prime},z)\\ \cos(y^{\prime},x)&\cos(y^{\prime},y)&\cos(y^{\prime},z)\\ \cos(z^{\prime},x)&\cos(z^{\prime},y)&\cos(z^{\prime},z)\\ \end{bmatrix} \tag{2}\]  
Figure 2: Blasting vibration signal processing. (**a**) Frequency spectrum analysis of particle vibration velocity; (**b**) variation of particle vibration velocity in \(x\), \(y\), and \(z\) directions with time; (**c**) three-directional particle vibration velocity vector sum.  
The generalized stress vector describing the failure of a weak surface has four components, namely \(\sigma_{1^{\prime}1^{\prime}}\), \(\sigma_{2^{\prime}2^{\prime}}\), \(\sigma_{3^{\prime}3^{\prime}}\), and \(\tau\). Among them, \(\sigma_{1^{\prime}1^{\prime}}\), \(\sigma_{2^{\prime}2^{\prime}}\), and \(\sigma_{3^{\prime}3^{\prime}}\) are the normal stresses in the normal direction (\(x^{\prime}\),\(y^{\prime}\),\(z^{\prime}\)) of the weak plane local coordinate system. \(\tau\) is the shear stress in the weak plane local coordinate system. The generalized strain vector describing the failure of weak surfaces has four components, namely \(\varepsilon_{1^{\prime}1^{\prime}}\), \(\varepsilon_{2^{\prime}2^{\prime}}\), \(\varepsilon_{3^{\prime}3^{\prime}}\), and \(\gamma\), Among them, \(\varepsilon_{1^{\prime}1^{\prime}}\), \(\varepsilon_{2^{\prime}2^{\prime}}\), and \(\varepsilon_{3^{\prime}3^{\prime}}\) are the normal strain in the normal direction in the weak plane local coordinate system, and \(\gamma\) is the shear strain in the weak plane local coordinate system. The incremental expression of [PERSON]'s law for the weak surfaces is as follows:  
\[\left\{\begin{array}{l}\Delta\sigma_{1^{\prime}1^{\prime}}=\alpha_{1}\Delta e ^{\mathrm{e}}_{1^{\prime}1^{\prime}}+\alpha_{2}\big{(}\Delta e^{\mathrm{e}}_{2 ^{\prime}2^{\prime}}+\Delta e^{\mathrm{e}}_{3^{\prime}3^{\prime}}\big{