# Mechanics and Stability of Force Chain Arch in Excavated Granular Material
## 3 Evolution of Force Chain Networks during the Solid-Fluid Transition Process
### Force Analysis of the Force Chain Arch Formed by Lateral Particle Flow  
The force chain arch effect, resulting from the solid-fluid transition, can extend horizontally. Particles were released laterally [36] from the right edge of the model (indicated by the red arrow in Figure 10a, with green dots representing the release points) using the biaxial compression bidirectional flow experimental apparatus for granular materials. This visually demonstrated the evolution from localized slippage to landslides and other geological disasters in slope bodies. The trajectory of the force chain arch is determined using the spatial autocorrelation coefficient method, and then, it is fitted, as shown in Figure 10b,c. Similarly, force analysis is conducted on the force chain arch formed by the lateral release of particles. The force chain arch ACB represents the form of a semi-arch and can be fitted with the equation of a parabola opening to the right, with the arch span denoted as h and the arch height as \(l\). The maximum value for \(l\) is \(L_{0}\) = 760 mm, which is the width of the model. The mechanical analysis is depicted in Figure 11a, where the horizontal reaction force at the arch foot near the release point is \(F_{H}\); the vertical reaction force is \(F_{V}\); and, due to the vertical load at the top, the stress is transmitted vertically downward as \(F_{R}\). Based on the static equilibrium condition at the arch foot near the right-side release point, it follows that  
\[\left\{ \begin{array}{l}\sum F_{x}=q_{x}h-F_{H}=0\\ \sum F_{y}=F_{V}-q_{y}(L_{0}-l)-F_{R}=0\\ \sum M=F_{H}h-q_{y}\frac{(L_{0}\ ^{2}-l^{2})}{2}-F_{V}l-q_{x}h\frac{h}{2}=0 \end{array} \right. \tag{15}\]  
The solution of Equation (15) yields  
\[\left\{ \begin{array}{l}F_{H}=q_{x}h\\ F_{V}=\frac{h^{2}}{2l}q_{x}-\frac{(L_{0}\ ^{2}-l^{2})}{2l_{x}}q_{y}\\ F_{R}=\frac{h^{2}}{2l}q_{x}-\frac{(L_{0}-l)^{2}}{2l}q_{y}\end{array} \right. \tag{16}\]  
As shown in Figure 11b, at any point C (\(x\),\(y\)) on the arch, the normal force is \(F_{N}\); the tangential force is \(F_{T}\); and the angle between the line connecting C to the upper right corner and the horizontal direction is \(\theta\). Figure 10: (**a**) Network of force chains during lateral flow of granular material; (**b**) Fitting of the force chain arch (lateral release of particles); (**c**) The fitting function. The forces at any point C (\(x\),\(y\)) on the arch fulfill the equilibrium condition:  
\[ \begin{array}{c}\sum F_{x}=F_{T}\textit{sin}\theta(y)-F_{N}\textit{cos}\theta(y )-F_{H}+q_{x}h(y)=0\\ \sum F_{y}=F_{V}-F_{N}\textit{sin}\theta(y)-F_{T}\textit{cos}\theta(y)-q_{y}l (y)=0\end{array}  \tag{17}\]  
which leads to the following equation  
\[ \begin{bmatrix}\textit{cos}\theta(y)&-\textit{sin}\theta(y)\\ \textit{sin}\theta(y)&\textit{cos}\theta(y)\end{bmatrix}\begin{bmatrix}F_{N} \\ F_{T}\end{bmatrix} = \begin{bmatrix}q_{x}h(y)-F_{H}\\ -q_{y}l(y)+F_{V}\end{bmatrix}  \tag{18}\]  
and finally to Equation (19)  
\[ \begin{array}{c}\begin{bmatrix}F_{N}\\ F_{T}\end{bmatrix}=\begin{bmatrix}\textit{cos}\theta(y)&-\textit{sin}\theta(y) \\ \textit{sin}\theta(y)&\textit{cos}\theta(y)\end{bmatrix}^{-1}\begin{bmatrix}q_{x }h(y)-F_{H}\\ -q_{y}l(y)+F_{V}\end{bmatrix}\\ =\begin{bmatrix}\begin{pmatrix}(q_{y}l(y)+F_{V})\textit{sin}\theta(y)+(q_{x}h(y) -F_{H})\textit{cos}\theta(y)\\ (-q_{y}l(y)+F_{V})\textit{cos}\theta(y)-(q_{x}h(y)-F_{H})\textit{sin}\theta(y) \end{bmatrix}\end{array} \end{array} \tag{19}\]  
Substituting Equation (16) provides the expressions for the normal and tangential forces at any point on the arch:  
\[ \begin{cases}F_{N}=\begin{bmatrix}\frac{y^{2}}{2}q_{x}-\frac{(L_{0}\ ^{2}-l^{2})}{2l}q_{y}+q_{y}l(y) \textit{sin}\theta(y)+q_{x}[(h(y)-h)\textit{cos}\theta(y)\\ F_{T}=\begin{bmatrix}\frac{y^{2}}{2}q_{x}-\frac{(L_{0}\ ^{2}-l^{2})}{2l}q_{y}-q_{y}l(y) \textit{cos}\theta(y)-q_{x}[(h(y)+h)\textit{sin}\theta(y)\end{array}\end{cases}