# Prediction of the Dangerous Range of Dumps Based on a Dynamic Process
## 3 Introduction of the Calculation Method
### Introduction of the SPH Method  
The smoothed particle hydrodynamics (SPH) method solves partial differential equations or integral equations through a series of interaction points carrying material information. It is a meshless, stable, and adaptive Lagrangian dynamics solution method and obtains accurate and stable numerical solutions. It has been widely used in the field of simulation calculations of disasters such as landslides and collapses.  
Based on the Lagrangian description, the basic rules of mass conservation, momentum conservation, and energy conservation can be written as Navier-Stokes partial differential equations, as shown below:  
\[\left\{\begin{array}{l}\frac{D\rho}{Dt}=-\rho\frac{\partial\mathbf{v}^{\beta}}{ \partial\mathbf{x}^{\beta}}\\ \frac{D\sigma^{\alpha}}{Dt}=\frac{1}{\rho}\frac{\partial\sigma^{\alpha\beta}} {\partial\mathbf{x}^{\beta}}+\mathbf{F}^{\alpha}\end{array}\right. \tag{14}\]  
where the superscript represents the coordinate direction, \(\rho\) is the density, \(\mathbf{v}\) is the velocity, \(\sigma\) is the stress tensor, \(\mathbf{F}\) is the volume stress component, and \(t\) is the time. The SPH form of the equation can be written as  
\[\frac{D\rho_{i}}{Dt}=\rho_{i}\sum_{j=1}^{N}\frac{m_{j}}{\rho_{j}}v_{ij}^{ \beta}\frac{\partial W_{ij}}{\partial\mathbf{x}_{i}^{\beta}} \tag{15}\]  
\[\frac{D\mathbf{v}_{i}^{\alpha}}{Dt}=\sum_{j=1}^{N}m_{j}(\frac{\sigma_{i}^{\alpha \beta}}{\rho_{i}^{2}}+\frac{\sigma_{j}^{\alpha\beta}}{\rho_{j}^{2}})\frac{ \partial W_{ij}}{\partial\mathbf{x}_{i}^{\beta}}+\mathbf{F}_{i}^{\alpha} \tag{16}\]  
In addition, the momentum equation in the form of the SPH method can be written as  
\[\mathbf{\sigma}^{\alpha\beta}=-p\delta^{\alpha\beta}+2\Bigg{(}\eta+\frac{\tau_{ min}}{\dot{\gamma}^{\alpha\beta}}\Bigg{)} \tag{17}\]  
where \(p\) is the pressure of rock and soil, \(\sigma\) is the stress tensor, \(\delta^{\alpha\beta}\) is the Kronecker symbol, \(\eta\) is the viscosity coefficient of large deformation of rock and soil, \(\tau_{min}\) is the yield stress, \(\dot{\gamma}\) is the pure shear strain rate, and \(\dot{\epsilon}^{\alpha\beta}\) is the deviatoric strain rate tensor, The Greek letter superscripts \(\alpha\) and \(\beta\) represent the coordinate direction. The pure shear strain rate \(\dot{\gamma}\) and the deviatoric strain rate tensor \(\dot{e}^{\alpha\beta}\) are defined as  
\[\dot{\gamma}=\sqrt{2\dot{e}^{\alpha\beta}\dot{e}^{\alpha\beta}} \tag{18}\]  
\[\dot{e}^{\alpha\beta}=\dot{e}^{\alpha\beta}-\frac{1}{3}\dot{e}^{\alpha\beta} \delta^{\alpha\beta} \tag{19}\]  
where \(\dot{e}^{\alpha\kappa}\) is the volumetric strain rate and \(\dot{e}^{\alpha\beta}\) is the strain rate tensor, which is defined as  
\[\dot{\epsilon}^{\alpha\beta}=\frac{1}{2}\left(\frac{\partial v^{\alpha}}{ \partial x^{\beta}}+\frac{\partial v^{\beta}}{\partial x^{\alpha}}\right) \tag{20}\]  
where \(v^{\alpha}\), \(v^{\beta}\) are the fluid velocities, and \(x^{\alpha}\) and \(x^{\beta}\) are fluid coordinates.  
The yield strength based on the Drucker-Prager yield criterion can be defined as  
\[\tau_{min}=-\mu p+c \tag{21}\]  
where \(\mu\) is the friction coefficient of rock and soil, and \(c\) is the cohesion.