# Inductive Thermography as Non-Destructive Testing for Railway Rails
## 3 Finite Element Simulations  
The ANSYS multiphysics simulation package [23] was used to model the inductive thermography measurements of the head checks in the rail. This package has the advantage that different physical phenomena can be simulated in a coupled way, as in this case electromagnetic induction is coupled with thermal calculation. In the first step inductive heating and the distribution of the induced eddy currents are modelled. Based on these results the Joule heating is calculated. In a further step the heat flow due to the generated Joule heating is modelled. The left image in Figure 3 shows the geometrical model of the gauge corner, where typically the head checks occur. The images in the middle and on the right side of Figure 3 show the temperature distribution after 0.1 s inductive heating pulse around a crack, which has at the deepest position 1.5 mm depth. The excitation frequency in the simulation model was 200 kHz and typical pearlitic steel material parameters were used for the rail piece. With these parameters the penetration depth of the eddy current is about 0.03 mm. As this depth is much less than the defect depth, the induced eddy currents flow around the defect, causing a higher current density and higher Joule heating in the vicinity of the crack. Additionally, the heat flow is also hampered by the crack and due to the heat accumulation the region around the crack has a higher temperature than the surrounding surface.  
The simulation was set up using 3D hexahedral grids with 20-node elements. The size of the grid was about 0.01 mm in the region close to the surface and around the crack, and the grid size continuously increased towards the regions far away from the crack with a maximum element size of 0.6 mm. The electromagnetic part of the model consists of 296,000 elements, of which 134,000 are modeling the rail piece, 12,000 the induction coil and 150,000 the surrounding air. The thermal modelling was only performed for the work-piece part; the elements of the coil and the air were not included in the thermal calculation.  
Figure 2: a: Top image: thermogram after an inductive heating pulse of 0.1 s; two points were marked: (**a**) point on a crack position, (**b**) point on sound surface; bottom image: phase image for the same measurement; b: temperature vs. time diagram of the heating and cool-down for the two marked points in (**a**).  
Several different numerical models were set up by varying the defect depth, its inclination angle below the surface and it's angle at the surface regarding the gauge corner. Further simulation results are presented in Section 5. The temporal change of the simulated temperature was also evaluated by Fourier transformation and this phase image was then compared to the experimental results. The goal of the finite element simulations was to investigate the influence of the crack parameters on the phase distribution. For artificial cracks with known characteristics the simulated and the experimental results can be easily compared, which was used to validate the simulation models. In a further step, this supports the usage of the simulation results also for real cracks where the depth and inclination angle below the surface are not known so that they can be estimated from the simulations. Additional models were also set up, varying the heating pulse duration and the excitation frequency [15]. These models were used to optimize the experimental setup, resulting in the usage of 200 kHz as the excitation frequency and a heating pulse duration between 0.1 and 0.2 s.