## 2 Physically Based Dynamic Models for Modeling Complex Landslides
### A Multi-Block Model for Slowly Sliding Landslide
#### 2.1.2 Motion Equations of Gravity-Driven Sliding Blocks  
The main factors affecting block motion include gravity, \(\mathbf{W}\); anti-sliding force, \(\mathbf{R}\); and the interaction forces between blocks, \(\mathbf{P}\). The effect of vertical shear stress between blocks was ignored due to their small values. Self-weight is the main driving force that causes the mass to slide. The anti-sliding factor is mainly due to the friction at the basal terrain. It is assumed that the resultant horizontal stress, \(\mathbf{P}\), of a block results from the difference in  
Figure 2: A virtual terrain in the global coordinate system (GCS). lateral earth pressure acting on both sides of the block (Figure 1d). For granular materials, the resultant horizontal stress vector can be expressed as  
\[\mathbf{P}=\mathbf{k}\
abla(\rho gh), \tag{6}\]  
where \(\
abla=(\partial/\partial x,\partial/\partial y)\) is the gradient operator, and \(\mathbf{k}=(k_{x},k_{y})\) is the vector of the lateral earth pressure coefficients [26]. It is assumed that the Coulomb friction law should be obeyed when the block slides along the basal terrain. Then, the net force for a block can be expressed as [42]  
\[\mathbf{F}=(\mathbf{g}\cdot\mathbf{n})\mathbf{n}-\frac{\mathbf{u}}{\mid \mathbf{u}\mid}(\mathbf{g}\cdot\mathbf{n})\text{tan}\delta-\mathbf{k}\
abla( \rho gh), \tag{7}\]  
where \(\delta\) is the basal friction angle between the landslide and basal terrain. From Equation (7), the acceleration components of a block with unit weight in \(x\) and \(y\) directions can then be derived:  
\[a_{x}=g\text{sin}\alpha_{x}-\frac{u_{x}}{\mid\mathbf{u}\mid}(g\text{cos} \theta\text{tan}\delta)-k_{x}\frac{\partial h}{\partial x}, \tag{8}\]  
\[a_{y}=g\text{sin}\alpha_{y}-\frac{u_{y}}{\mid\mathbf{u}\mid}(g\text{cos} \theta\text{tan}\delta)-k_{y}\frac{\partial h}{\partial y}, \tag{9}\]  
where \(a_{x}\) and \(a_{y}\) represent the acceleration components in the \(x\) and \(y\) directions, respectively. Due to the assumption of unit weight, the product term \(\rho g\) disappears from the third term on the right-hand side of Equations (8) and (9). At the initial stage of sliding, the landslide maintains high integrity, and its internal relative displacement is small. A large portion of the slope has almost the same displacement. For landslides, the grid size of a digital topographic map (DEM) may be at the meter level, while the characteristic size of a block with approximate displacement may be on the scale of tens of meters. This scale range may include multiple cells in the digital terrain model. Therefore, the initial landslide can be divided into several groups, and all the cells in each group have the same velocity characteristics, which is closer to the displacement characteristics of the real landslide. For example, in Figure 1b, one group consists of nine blocks, and the actual number of blocks may be less than this in the irregular margin of the landslide. Since all blocks within a group should have consistent physical quantities, the overall acceleration component of the group can be viewed as the averaged acceleration components of each block within the group. For example, when a group consists of nine blocks, the averaged angle between the tangent plane of the basal bed and the \(x\) and \(y\) axes can be expressed as  
\[\alpha_{x}=\arctan\left(\sum_{i=1}^{9}\left(h_{i}\text{tan}a_{x}^{i}\right)/ \sum_{i=1}^{9}h_{i}\right), \tag{10}\]  
\[\alpha_{y}=\arctan\left(\sum_{j=1}^{9}\left(h_{j}\text{tan}a_{y}^{j}\right)/ \sum_{j=1}^{9}h_{j}\right), \tag{11}\]  
where the tangent values of the basal terrain of individual blocks can be obtained from Equations (4) and (5).