## 7 Vertical Cracks with Large Eddy Current Penetration Depth  
In the previous sections, an eddy current penetration depth was assumed, which is small compared to the crack depth. Since, for non-magnetic materials, the penetration depth is significantly larger, this assumption is no longer valid. For the simulation of this scenario, the simu.2 model can be used as it also considers the distribution of the eddy current and the Joule heating (see Table 2). If the crack is vertical, which is listed as \"Case 2\" in Table 2, then the eddy current is deviated by the crack and will not flow close to the crack corner (Figure 18a) [27; 28]. Therefore, the Joule heating in this region is less than that at the sound surface (Figure 18b). A hot spot occurs at the tip of the crack, but this is inside the sample and not visible at the surface. Figure 17: Results for a steel sample with five artificial defects, each of them with a 1 mm depth and inclination angles from left to right of 75\({}^{\circ}\), 60\({}^{\circ}\), 45\({}^{\circ}\), 30\({}^{\circ}\), and 0\({}^{\circ}\), respectively. (**a**) \(t_{pulse}\) = 0.1s; (**b**) \(t_{pulse}\) = 0.6 s; (**c**) profiles for different heating pulses. Figure 18: Simulation model (simu.2) results shown in close-up to a vertical surface crack with a depth of 1 mm. Material: non-magnetic steel, properties are listed in Table 1. The excitation frequency is 200 kHz. (**a**) Eddy current streamlines; (**b**) Joule-heating. In Figure 19, the Joule heating at the surface, normalized between 0 and 1, is plotted and compared for different crack and penetration depths. The deeper the crack, the broader the region around the crack, where the Joule heating is lower than it is at the sound surface. This is also valid for the penetration depth: the larger \(\delta\) is, the broader the region where the eddy current is pushed inside from the surface into the inner part of the material. Normalizing the distance to \(\sqrt{d\delta}\), the curves become almost identical (Figure 19b). Introducing \(\beta\) for the ratio crack depth to penetration depth  
\[\beta=\frac{d}{\delta} \tag{8}\]  
the full-width-half-maximum (FWHM) of the curves is approximately  
\[x_{FWHM}=\sqrt{d\delta}=\frac{d}{\sqrt{\beta}}. \tag{9}\]  
As the Joule heating at the surface in the corner of the crack is low, the temperature in this region immediately after the heat is switched on is less than that at the sound surface (Figure 20a). With increasing time, the heat diffusion equalizes this difference, the heat flows into the corners, and the temperature becomes more homogeneous (Figure 20b). Figure 21 compares the temperature curves at the crack position and at the sound surface. If the heating pulse is short (Figure 21a), then the crack is significantly cooler than the sound surface, as there is less Joule heating in this region. It is also observed that the temperature at the crack reaches its maximum later than the heating was switched off. Even if there is no longer any heating, due to the  
Figure 19: Joule heating at the surface for \(d=0.5\) and \(1\) mm crack depths and for two different penetration depths depending on the distance (**a**) and abscissa normalized to \(\sqrt{d\delta}\) (**b**). Figure 20: (**a**) Temperature distribution after 0.1 s and (**b**) 1 s heating pulses for the same scenario as shown in Figure 18.  
heat diffusion, the heat flows further into this cooler region and increases the temperature. If the heating pulse is long (Figure 21b), then, with increasing time, the difference between the temperature at the crack position and at the sound surface becomes negligibly small. Calculating the phase from the temperature functions, the area around the crack has lower values than that at the sound surface. In Figure 22a, the phase distribution for different pulse heating durations is plotted. The shorter the pulse, the larger the phase difference due to the crack.