# DEM Analysis of Track Ballast for Track Ballast-Wheel Interaction Simulation
## 2 Theoretical Background for Ballast Modeling and Analysis
### DEM Analysis  
In a DEM analysis, the individual objects are modeled as rigid bodies. The simplest way to model gravel is to use spherical rigid bodies. For modeling gravel pieces as simple rigid bodies, an appropriate contact model should be used to consider the elasticity of the actual object. Hence, after modeling the rigid bodies using the contact model between them, the analysis can be conducted. The analysis is performed based on [PERSON]'s second law. Hence, the increments in the velocity and displacement of the individual bodies due to the applied loads are determined by calculating their acceleration. Figure 3 shows the general procedure of the DEM analysis based on an explicit structural dynamic analysis. At the preparation stage, the initial positions and velocities of the rigid bodies are defined. Because all the objects are considered as individual rigid bodies, initial overlapping between contacted objects is not allowed. Therefore, if the initial position of each object is defined, the initial analysis should be conducted to determine the appropriate position without any overlapping. To avoid the initial analysis and define the appropriate initial positions of the objects, a numerical method such as particle packing can be used. The details of particle packing are described in the next section. Once the initial conditions are defined, including the initial positions, velocities, and boundary conditions of the domain, the dynamic analysis can be conducted to obtain the dynamic structural responses of all the objects, including the rigid bodies and deformable structural members, in the model. Again, the DEM analysis is suitable for dynamic problems because the displacement increments are calculated by integrating the acceleration increments. As shown in Figure 2, the collision between individual bodies is verified to determine whether the contact force and torque should be calculated or not. If the force and torque of the object are not calculated, the acceleration increment of the object is calculated using the force and moment determined at the previous time step. If the force and torque have to be calculated, the forces are calculated based on the pre-defined contact model. Various contact models exist for DEM analysis. Among these, the Hertz-based model was used in this study because the model can be effectively applied for considering contact behaviors of elastic bodies modelled as rigid spherical bodies. The applied contact model is shown in Figure 4.  
Figure 4: Applied contact model for considering contact between the rigid particles. Figure 3: General procedure of discrete-element (DEM) analysis based on explicit structural dynamic analysis. Using the contact model, the elastic behavior of the object can be taken into account, even though the objects are modeled as rigid bodies for numerical simplicity. Thus, the stiffness for normal contact is estimated considering the elastic modulus and the size of the objects in contact. In addition, the friction between the objects in contact is considered as a tangential contact force. Once the friction coefficient for the objects in contact is defined, the friction force can be calculated because the normal force has been calculated. In this simulation, a simple linear friction model was applied. Consequently, all the force components required for defining the motion equation of each object, including inertia and damping forces, were calculated and considered. The normal contact force \(F_{n}\) between two spherical particles in contact can be calculated using the Hertz solution, as shown in Equation (1):  
\[F_{n}=\frac{4}{3}E^{*}\sqrt{R}\sqrt{\delta^{3}}, \tag{1}\]  
where \(R=\frac{R_{1}R_{2}}{R_{1}+R_{2}}\); \(\frac{1}{E^{*}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}\); \(E_{i}\) is the [PERSON]'s modulus of particle \(i\); \(v_{i}\) is the [PERSON]'s ratio of particle \(i\); and \(\delta\) is the normal overlap between two spherical particles in contact.