# Research on Grouting Pressure of Earth-Rock Dams Reinforced by Polymer Based on Discrete Elements
## 3 Principle of Fluid-Solid Coupling Model
### Fluid-Solid Coupled Model  
The DEM is a simulation method specially used to solve the problem of discontinuous media. The discrete element model is composed of a series of particle units that follow [PERSON]'s second law of motion (Figure 2a). The model particles of the most basic linear elastic model are connected by breakable springs, and the force only appears at the contact points between adjacent particles (Figure 2b,c). Figure 2: Schematic diagram of the discrete element particle model: (**a**) particle packing model; (**b**) normal force between particles; (**c**) tangential force between particles. The normal force \(F_{\rm n}\) and the tangential force \(F_{\rm s}\) between the unit particles are realized by the interaction of the spring force, where the normal force \(F_{\rm n}\) is obtained from Equation (4) [17], and the tangential force \(F_{\rm s}\) is obtained from Equation (5) [18]. \[F_{\rm n}=\left\{ \begin{array}{ll}K_{\rm n}(r-R),r<r_{0}\ (\rm normal\ connection,\ a)\\ K_{\rm n}(r-R),r<R\ (\rm connecting\ fracture,\ b)\\ 0,\qquad\quad\quad r\geq R\ (\rm connecting\ fracture,\ c)\end{array} \right. \tag{4}\]  
In the formula, \(K_{\rm n}\) is the normal spring stiffness, \(R\) is the equilibrium distance between particles (the sum of the radius of the two particles), \(r_{0}\) is the fracture threshold, and \(r\) is the current distance between particles. When the distance between particles \(r\) is less than \(r_{0}\) (Equation (4a)), the particles are normally connected and subjected to tension or pressure; when the distance between particles \(r\) is greater than \(r_{0}\) but less than \(R\), the connection is broken, the tension disappears, and there is only a pressure effect (Equation (4b)). When the particle distance \(r\) is greater than \(R\), the spring breaks, but when the two particles return to the compressed contact state, the pressure still exists (Equation (4c)). \[F_{\rm s}=K_{\rm s}X_{\rm s} \tag{5}\]  
In the formula, \(K_{\rm s}\) is the shear spring stiffness, and \(X_{\rm s}\) is the tangential relative displacement. The tangential force and shear deformation between particles are simulated by a tangential spring perpendicular to the normal spring. Between normally connected particles, the maximum tangential force \(F_{\rm smax}\) determined by the Mohr-Coulomb criterion is  
\[F_{\rm smax}=F_{\rm s_{0}}-\mu_{\rm p}F_{\rm n} \tag{6}\]  
In the formula, \(F_{\rm s0}\) is the shear resistance inside the particle, and \(\mu_{\rm p}\) is the friction coefficient. When the tangential external force on the particles is greater than the maximum tangential force \(F_{\rm smax}\), the connection between the particles will break. The tangential force \(F_{\rm s}\) is limited to less than or equal to the maximum tangential force \(F_{\rm smax}\). Therefore, when the particle connection is broken, \(F_{\rm s}\) should be less than or equal to the limit friction:  
\[F_{\rm smax}^{\prime}=\ -\mu_{\rm p}F_{\rm n} \tag{7}\]  
When the tangential external force is greater than the ultimate friction force \(F_{smax}^{\prime}\), the two particles begin to slide relatively, and the sliding friction force is equal to \(F_{smax}^{\prime}\). If the two particles are separated by fracture, all the forces between them will become 0. In MatDEM, particles are usually given five macromechanical properties of materials: Young's modulus \(E\), Poisson's ratio \(v\), compressive strength \(C_{\rm u}\), tensile strength \(T_{\rm u}\), and internal friction coefficient \(\mu\). According to the macro-micromechanical transformation formula [19] in MatDEM, the micromechanical parameters of the material can be determined, such as the positive stiffness \(K_{\rm n}\), the tangential stiffness \(K_{\rm s}\), the fracture displacement \(X_{\rm b}\), the initial shear force \(F_{\rm s0}\), and the friction coefficient \(\mu_{\rm p}\), see Equations (8)-(13).