## 5 Inclined Cracks with Small Eddy Current Penetration Depth  
In many cases, the crack does not lie perpendicular to the surface, but inclined with an angle. Typical defects are rolling or forging laps, as shown in Figure 11a. Such cracks can be simulated with the simu.1 and simu.2 models [14], as listed in Table 2 as \"Case 3.\" The results of both models were compared for several cases and showed good agreement. Figure 11b shows the simu.2 calculated Joule heating for ferro-magnetic steel around a crack depth of \(d=1\) mm and a \(\theta=30^{\circ}\) inclination angle. As the eddy current penetration depth is very small (\(\delta=0.034\) mm), the current and the Joule heating follow the line of the crack. The heat becomes trapped in the corner with the smaller angle, as shown in Figure 12. From the other corner, which has an angle larger than \(90^{\circ}\) to the surface, the heat flows away much more quickly, resulting in an asymmetrical temperature distribution. As the phase value is a good measure for the heat flow rate, it is lower in the region with heat accumulation than that for a vertical crack. On the other side of the crack, which has a larger heat flow, the phase is higher than that for a vertical crack, so the phase distribution is asymmetrical (Figure 13). With a longer heating pulse, the heat flows away faster from the crack position, so the asymmetrical distribution decreases, and the difference between both sides becomes negligible. As shown earlier, \(d_{th}/2\) is approximately the distance where the phase value is at its minimum. In Figure 12, for both cases, the distance \(d_{th}/2\) is marked. In Figure 12a, this position (\(d_{th}/2=1.08\) mm) is close to the crack, so it has a strong influence. However, after a 1 s heat pulse (\(d_{th}/2=3.4\) mm, see Figure 12b), it is far away from the crack, and the temperature distribution and heat flow on both sides of the crack become very similar. Figure 13a compares the phase distribution after different heating pulse durations for the same crack. For the short heating pulse, the phase distribution is strongly asymmetrical, but with longer  
Figure 11: (a) Microscopic cross section of a rolling lap defect; (b) simulated Joule heating around a crack in a ferro-magnetic material, \(f=200\) kHz. Figure 12: Temperature distribution after 0.1 s (**a**) and after 1 s (**b**) heating pulses, simulated for the same crack as in Figure 11b. In the figures, \(d_{th}/2\) is also marked, which is approximately the distance where the phase is at its minimum. pulses, and the difference between both sides diminishes. For comparison in Figure 13b for the same pulse durations, the phase for the vertical crack is shown. With longer heating, the phase around the inclined crack becomes almost symmetrical, but the phase value at the crack position is higher than that for a vertical crack with the same depth. The larger the inclination angle \(\theta\), the longer the heat is trapped in the corner of the small angle, and the temperature and the phase distribution become strongly asymmetrical. Figure 14 demonstrates this behavior for different inclination angles. The difference between the phase minima of both sides \(\Delta\Phi\) is a measure for the asymmetry in the phase distribution. Figure 14a shows the phase for the short heating pulse of 0.1 s for three different angles. In Figure 15a, the dependency of \(\Delta\Phi\) on the inclination angle \(\theta\) is plotted. The phase contrast on the left side (\(\Phi_{1}\)) is smaller for large inclination angles, as the heat can flow away without almost any obstacles (Figure 15b). On the other side of the crack, \(\Phi_{2}\) increases with the \(\theta\) values. For a short heating pulse of 0.1 s, the mean value of these two is close to the phase contrast of the vertical crack [23; 26] (Figure 15b). However, this is only valid for short heating pulses. As shown in Figures 12 and 13a, the asymmetry in the phase decreases as pulse duration increases.