## 4 Numerical Methodology  
### Coupling FE Model  
The high-speed railway train-unballasted track-subgrade coupling FE model was established based on ABAQUS. Because CRTS III [25] plate is used in ballastless track structure in China, Herts non-linear elastic contact is used in normal direction, and Coulomb friction model is used in tangential direction in numerical simulation. The dynamic loading in this study is mainly divided into two parts: normal stress and tangential stress. Figure 5: Typical failure modes of specimens. (**a**) The density is 0.235 g/cm; (**b**) The density was 0.286 g/cm\({}^{3}\); (**c**) The density is 0.323 g/cm\({}^{3}\). Figure 6: Variation of elastic modulus and polymer density. 1. Normal stress. Wheel-rail contact is simplified to Hertz contact. The normal contact force between wheel and rail is described by nonlinear elastic of Hertz contact theory. \[p(t)=\left[\frac{1}{G}\Delta Z(t)\right]^{\frac{3}{2}} \tag{1}\]  
where, G is the contact constant of wheel-rail (m/N\({}^{2/3}\)), and its value is based on Table 2. \(\Delta Z(t)\) is the elastic compression between wheels and rails (m). 2. Tangential stress. The tangential action of wheel-rail contact surface is realized by establishing friction model of wheel-rail. According to [PERSON]'s ethics, the friction between wheel and rail is equal to the product of the normal force of wheel and rail and the friction coefficient,  
\[F=\mu p(t) \tag{2}\]  
where, \(\mu\) is the friction coefficient between wheel and rail,  
\[\mu=\mu_{k}+(\mu_{s}-\mu_{k})e^{-\beta\
u} \tag{3}\]  
where, \(\mu_{s}\) is the static friction factor, \(\mu_{k}\) is the dynamic friction factor, \(\beta\) is an exponential attenuation coefficient, and \(\
u\) is the relative sliding velocity between slave planes. The high-speed railway train is simulated with multi-rigid body structure (as shown in Figure 7). The vehicle model system consists of car body, bogie, wheelset, and two-system suspension system. The car body, bogie, and wheelset are regarded as rigid bodies. In the simulation, in order to facilitate quantitative research, the elastic deformation of the car body, bogie, and wheelset during the vibration process is neglected. In addition, the primary suspension system and the secondary suspension system are simulated by connecting elements. Finally, through the connection of suspension system, the car body, bogie, and wheelset are combined to form a multi-rigid body vehicle model [26]. Two types of elements are first-order linear elements. The CRTSIII unballasted track structure consisted of rails, fasteners, slab track, self-compacti \begin{table}
\begin{tabular}{c c c} \hline \hline
**Tread Type** & **Wheel of Conical Tread** & **Wheel of Abrasive Tread** \\ \hline Contact constant of wheel-rail & \(\text{G}=4.57\text{R}^{-0.149}\times 10^{-8}(\text{m}/\text{N}^{2/3})\) & \(\text{G}=3.86\text{R}^{-0.115}\times 10^{-8}(\text{m}/\text{N}^{2/3})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Contact constant value of the wheel-rail. Figure 7: The vehicle model. Eight-node reduced integration solid elements were used to simulate the rail, and connection element was used to simulate the stiffness and damping characteristics of the fastener; the rotational freedom of the end of the connection element was restrained to simulate the restraint effect of the fastener on the rail. Solid elements of eight-node hexahedron were used to simulate the slab track. The slabs on the subgrade were connected longitudes, and the adjacent slabs were connected with steel bars. The steel bars were simulated with beam elements, and the Embedded region was used as the constraint between the steel bars and the slab track, as shown in Figure 8a. Each adjacent two track board is filled with resin mortar with a gap of 2 cm. Two convex plates were set under the corresponding self-compaction concrete layer under each slab track, as shown in Figure 8b. The geometric size of the convex plate was 1000 mm (transverse) \(\times\) 700 mm (longitudinal) \(\times\) 100 mm (depth).