# Field Investigation and Rapid Deterioration Analysis of Heavy Haul Corrugation
## 4 Multi-Crack Expansion Analysis Considering Grinding Marks
### Establishment of a Multi-Crack Model  
For the process of crack simulation based on the classical theory of fracture mechanics, knowing how to divide crack elements and how to solve the singularity of the crack's tip are key challenges. In order to solve the singularity of the crack tip, a 1/4 singular element was used to describe the tip of the cracks. The elements with the types C3D15 and C3D20 supported by commercial finite element software ABAQUS are used to mesh the template part of the crack. The total element number is around 1.5 million and the minimum distance at the crack tip is 0.01 mm.  
As shown in Figure 7, the depth of the crack is set to 0.1 mm when the crack length is set as 10 mm from the field investigation. The crack angle was 20\({}^{\circ}\) from the longitudinal direction of the rail, and five parallel cracks were considered for the railroad finite element model. For the model with the grinding marks, the width and depth of the grinding marks were set to 0.2 mm, according to the measurement results by the portable optical microscope in the field shown in Figure 8. The grinding marks were set to 15 mm and perpendicular to the train's movement direction, respectively, according to the on-site studies through photographs and the setting location of the grinding wheel.  
\begin{table}
\begin{tabular}{c c} \hline
**Types** & **Parameters** \\ \hline Vehicle type & C80 \\ Axle load & 25 t \\ Distance between bogie centers & 8300 mm \\ Wheelbase & 1830 mm \\ Rail type & 75 kg/m \\ Elastic modulus of rail & 205.9 GPa \\ Wheel tread type & LM \\ \hline \end{tabular}
\end{table}
Table 3: Basic parameters of the model.  
For the load applied to the wheel and rail parts, the models of heavy haul railway C80 vehicles with an axle load of 25 t, based on multi-body dynamics, were established, and their basic parameters are shown in Table 1.  
The semi-Hertzian wheel-rail contact algorithm [31; 32] was used to calculate the wheel-rail contact state. The lateral creep force, longitudinal creep force, and contact surface normal force were applied to the railroad sub-model. Among these, the normal contact forces and their adhesion states to the straight line, transition curve, and circular curve sections were calculated using the rail profile measured by MiniProf, as shown in Figure 9. From the figure, we can see that the railhead of the inner rail is flatter than it is in the outer rail, and both of the profiles show a significant difference from the standard profile after grinding. The main reason for this is that the deformation of the railhead is severe, which provides an extra challenge to the grinding works when attempting to obtain the ideal profiles of rails after grinding.  
Figure 8: Measurement of the grinding marks by portable optical microscope: (**a**) portable optical microscope; (**b**) the picture of the grinding marks and the results after calibration.  
Figure 7: Schematic diagram of crack model with grinding marks.  
Figure 9: Measured profile after grinding by MiniProf.  
According to Figure 10, it is clear that, due to the difference between the actual rail grinding profile and the standard 75 kg/m profile, W/R contact occurred at the center of the railhead, which is different to the general W/R contact calculation between the new profile of the wheel and the rail.