# Inductive Thermography as Non-Destructive Testing for Railway Rails
## 5 Simulations of Realistic Head Checks  
In the previous sections it was shown that the experimental and the finite element simulation results correspond very well for the artificial cracks and how the simulation results help to interpret the experimentally achieved phase images. However, these EDM cuts are perpendicular to the rail gauge corner, whereas head checks on rails in service occur at a specific angle. This angle is affected by the local load situation (track curvature and corresponding wheel-rail interaction) and the rail material. Therefore, further simulations were carried out, where the head check shows an angle to gauge line (\(\alpha\)) and it has another penetration angle (\(\beta\)) into the material. Figure 6 shows a principle sketch with the definition of the different parameters characterizing the head checks pattern and which were varied in the simulations to investigate their effect on the phase images. In Figure 7 simulation results are presented for a crack with \(\alpha=45^{\circ}\) and \(\beta=30^{\circ}\). Figure 7a shows the simulated temperature distribution after 0.1 s inductive heating pulse and in Figure 7b half of the model is presented, whereby the inclination angle and the temperature distribution below the surface are also visible. A phase image is calculated from the simulated temporal change of the temperature, in the same way as it is done for the measurements. This image (Figure 7c) shows a good agreement with the experimental results of head checks. Figure 5: (**a**) CAD drawing of the EDM-cuts; (**b**) photo of EDM-cuts with different penetration angles; (**c**) phase image of EDM-cuts after 0.1 s inductive heating pulse; (**d**) simulation results of the slanted artificial cracks. Figure 6: The definition of the characteristic parameters at the surface (**a**) and in cross-section (**b**): \(\alpha\): inclination angle at the surface to the gauge corner; \(l\): length of the head check at the surface; \(c\): distance of the head checks; \(\beta\): inclination angle below the surface; \(d\): depth; \(o\): width of overlapped region. Further simulations were carried out to investigate how the inclination angle below the surface affects the phase images. In Figure 8 the temperature and phase images are compared for cracks with 1 mm depth but with different angles. The images on the top row show the results for a vertical crack (\(\beta=90^{\circ}\)). In this case the temperature and phase distribution are symmetrical around the crack position at \(x=0\) mm. As the inclination angle decreases, the length of the crack increases to \(d/sin(\beta)\) and the overlapped length \(o=d/tan(\beta)\). In the phase image this overlapping can be easily recognized: due to the small inclination angle the heat is captured for longer in this region, causing a low phase value. It is easy to see that the width of the low phase value region corresponds well to the width of the overlapped region. This gives the possibility of recognizing at the surface of the phase image in which direction the crack is inclined and how wide the overlapped region is. Figure 9 demonstrates how the angle below the surface (\(\beta\)), the depth and the width of the overlapped region influence the phase image. In the image at the top, the crack is modelled as a straight line, starting with an angle of \(30^{\circ}\). In many cases head checks change  
Figure 8: The simulation results for cracks with \(d=1\) mm depth. The inclination angles are varied from \(\beta=90^{\circ}\) to \(15^{\circ}\). The left column of images shows the temperature after 0.1 s inductive heating, and the right one the corresponding phase images. Figure 7: The simulated temperature results for one crack at the gauge corner after 0.1 s inductive heating pulse; (**a**) the whole 3D model; (**b**) half of the model; (**c**) calculated phase image from the surface temperature. their direction and, after a small starting angle, penetrate under a larger angle into the rail. The image in the middle of Figure 9 shows such a situation.