## 2 The Models
Figure 1 shows the Joule heating and the temperature distribution calculated by this simulation model. It can be well observed that the Joule heating follows the line of the crack, and due to the heat accumulation in the crack corners, the temperature becomes higher in this region. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Electrical** & **Relative** & **Thermal** & **Thermal** & **Penetration** & **Th. Diffusion** \\  & **Conductivity** & **Permeability** & **Conductivity** & **Diffusivity** & **Depth \(\delta\),** & **Length \(d_{th}\),** \\  & \(\sigma\) & \(\mu_{r}\) & \(\lambda\) & \(\kappa\) & **If \(f=200\) kHz** & **If \(f=0.1\) s** \\  & \(\mathbf{(\Omega-^{-1}m^{-1})}\) & **(-)** & **(W \(\mathbf{m^{-1}K^{-1})}\)** & **(m\({}^{2}\) s\({}^{-1}\))** & **(mm)** & **(mm)** \\ \hline Steel ferro-magn. & \(1.85\times 10^{6}\) & 600 & 40 & \(1.16\times 10^{-5}\) & 0.034 & 2.16 \\ \hline Steel AISI304 non-magn. & \(7\times 10^{5}\) & 1 & 16 & \(4\times 10^{-6}\) & 1.35 & 1.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters. In simu.1 and simu.2, the crack is assumed to be a long crack, thereby the models have been set up in 2D in a cross-section to the defect. The size of the models was 12 \(\times\) 40 mm\({}^{2}\). In both model types, several parameters such as crack depth, heating pulse duration, and the inclination angle of the crack to the surface are varied. Table 2 summarizes the results of the models and for the cases to which they can be applied. In this study, the results for these four listed cases are presented and compared. Additionally, for these cases, measurement results are also shown and compared with the simulation models. To prove the validity of the simu.2 models, the results were compared for small penetration depths with the results of simu.1. Furthermore, these were compared with the analytical model for the situation where the crack is perpendicular to the surface. On one hand, the simu.2 models can be used to calculate general cases, but on the other hand, these models are complex and require very fine gridding, so high computer resources and long calculation times are needed. Therefore, in praxis, the simplest model should always be used, which is still appropriate to handle a specific case. It is worth noting that, in these models, only the heating due to the ohmic loss was considered. In ferro-magnetic materials, a loss due to hysteresis occurs. We also set up numerical models including this phenomenon, but our experience was that its effect on crack detection is negligibly small, so this was not included in the currently presented results. In the presented simulations, one constant value of magnetic permeability is considered. This is a simplified model, as the relative permeability value is a complex number and depends on the excitation frequency [21; 22]. Since only whether a penetration depth smaller than the crack depth or a penetration depth comparable to crack depth is decisive, the exact value of the penetration depth has no direct impact on the obtained results of our evaluations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Crack Types** & **Cases** & **Analytical Model** & \
\begin{tabular}{c} **FEM simu.1** \\ **(Surface Heat Flux)** \\ \end{tabular} &
\begin{tabular}{c} **FEM simu.2** \\ **(Inductive Heating)** \\ \end{tabular} \\ \hline \multirow{3}{*}{Vertical crack} & Case 1 & \multirow{3}{*}{\(\delta\) = 0} & \multirow{3}{*}{\(\delta\) = 0} & \multirow{3}{*}{\(\delta\) \(<\) \(d\)} \\  & \(\delta\) \(\sim\)0, \(\delta\) \(<<\) \(d\) & & & \\ \cline{2-5}  & Case 2 & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{\(\delta\) \(\sim\)\(d\)} \\  & \(\delta\) \(\sim\)\(d\) & & & \\ \hline \multirow{3}{*}{Inclined crack} & Case 3 & \multirow{3}{*}{-} & \multirow{3}{*}{\(\delta\) = 0} & \multirow{3}{*}{\(\delta\) \(<<\) \(d\)} \\  & \(\delta\) \(\sim\)\(d\) & & & \\ \hline \hline \end{tabular}
\end{table}