# Inductive Thermography as Non-Destructive Testing for Railway Rails
## 5 Simulations of Realistic Head Checks
The starting angle and the depth are the same, as in the case at the top image, but the overlapped region is shorter. This can be also noticed in the phase image, as the region with low phase value is narrower. In the case at the bottom of Figure 9, the crack is again modelled as a straight line, having the same depth and overlapped region as in the middle case, but its starting angle is larger with \(\beta=41^{\circ}\). As the overlapped region is the same is in the middle case, the width of the low phase region is also the same. Due to the slightly larger inclination angle to the surface, the phase value is slightly less at the crack position than in the case in the middle. In typical head check pattern the cracks are close to each other, therefore the question is how the phase distribution around a crack is influenced by the neighboring cracks. In Figure 10 two situations are compared: in one situation the cracks do not overlap each other in the depth and once in the other situation do. All the cracks have a depth of 1 mm and \(\beta=30^{\circ}\). If the distance between cracks is larger than the overlapped region by a crack, then all the phase values at the crack positions are the same and they are also equal to the phase profile around one single crack (see Figure 10a). If the cracks are closer to each other, then the heat is trapped longer between the cracks than would be the case for one single crack, and the phase value around the crack is less (see Figure 10b). In this way it is possible to tell from the phase image of more parallel cracks, whether they are overlapping each other below the surface or not. The simulations shown in Figures 8-10 were calculated in a 2D model, so the results are valid for long cracks. The head checks have usually a length about 6-10 mm at the surface. In the mid of the crack the phase values are the same as for a long crack, but at the crack tips additional hot spots occur [24]. In Figure 11 the results are presented simulated with a 3D model, where the crack is placed in the middle of a plane surface. For the 6 mm long and 1 mm deep crack the two inclinations angles are varied. If \(\alpha\) is \(90^{\circ}\), then the line of the crack is perpendicular to the line of the modelled coil (Figure 11a,c), which is lying in the x direction. The angle \(\beta\) is the inclination angle below the surface; if \(\beta=90^{\circ}\) the crack is vertical, which means  
inclination angle below the surface causes an asymmetric phase pattern (Figure 11c,d). The phase pattern of Figure 11d demonstrates the case, which is the most similar one to head checks, as it is presented in the next section. Figure 11: Simulated phase images for cracks with \(d=1\) mm depth and \(l=6\) mm length at the surface with different inclination angles at and below the surface. (**a**): the crack is perpendicular to the gauge corner (\(\alpha=90^{\circ}\)) and it is perpendicular to the surface (\(\beta=90^{\circ}\)); (**b**): the crack has an angle of \(45^{\circ}\) (\(\alpha\)) to the gauge corner and it is perpendicular to the surface (\(\beta=90^{\circ}\)); (**c**): the crack is perpendicular to the gauge corner (\(\alpha=90^{\circ}\)) and it has an angle of \(30^{\circ}\) to the surface (\(\beta\)); (**d**): the crack has an angle of \(45^{\circ}\) (\(\alpha\)) to the gauge corner and it has an angle of \(30^{\circ}\) to the surface (\(\beta\)). Figure 10: Simulation results for three cracks with \(d=1\) mm depth and inclination angle of \(\beta=30^{\circ}\). In figures (**a**) and (**b**) at the top: temperature distribution in the cross-section; mid: phase distribution at the surface; bottom: phase profile through the cracks at the surface along the x axis. (**a**): distance between the cracks is 2 mm, therefore they do not overlap each other below the surface; (**b**): distance of the cracks is 1 mm and overlapping each other in the depth.