## 2 The Models  
In induction heating, eddy currents are induced in an electrically conductive sample. This current flows directly below the surface of the material in a shallow depth. Due to the ohmic resistance of the sample heat, the so-called Joule heat is generated below the surface inside the material. The penetration depth of the induced eddy current can be determined by  
\[\delta=\sqrt{\frac{1}{\pi\mu_{0}\mu_{r}\sigma f}} \tag{1}\]  
where \(\mu_{0}\) denotes the permeability of vacuum with the value of \(4\pi 10^{-7}\) Vs/Am; \(\mu_{r}\) is the relative magnetic permeability of the material; \(\sigma\) is its electrical conductivity; \(f\) is the excitation frequency. This penetration depth is strongly influenced, on one hand, by the material properties, and on the other hand, by the excitation frequency. Table 1 summarizes these values for two materials. This formula is only valid for the ideal case of a semi-infinite plane surface; otherwise, the eddy current distribution and its penetration depth is affected by the sample geometry. A non-uniform heat distribution initiates a diffusion process to equalize the temperature differences. Generally, a diffusion process, i.e., heat diffusion, can be characterized by the diffusion length, describing approximately how far the heat flows in a given time \(t\). \[d_{th}=2\sqrt{\kappa t} \tag{2}\]  
where \(\kappa\) denotes the thermal diffusivity of the material. In the case of ferro-magnetic material at a typical excitation frequency of 200 kHz, the penetration depth is negligibly small and can be well estimated as a surface heat flux. In a simplified model, assuming that the eddy current cannot flow through the crack and flows around it, the heating is applied not only to the surface of the sample but also to the sides of the crack. An analytical model was developed [6; 18; 19] for cases where a vertical crack is heated with a surface heat flux. This model describes very well the temperature distribution around a crack due to this additional heating. The sound surface can usually be modeled as a semi-infinite body, thereby the temperature increases as  
\[T_{sound}(t)=\frac{2Q}{\lambda}\sqrt{\frac{\kappa t}{\pi}} \tag{3}\]  
where \(Q\) denotes the applied surface heat flux. At the position of the crack with depth \(d\), the temperature is calculated according to [18]. \[T_{crack}(t)=T_{sound}(t)\left(1+\frac{1}{\sqrt{\pi}}\frac{d}{d_{th}}\ Ei{\left( \frac{d^{2}}{d_{th}}^{2}\right)}+\ erf{\left(\frac{d}{d_{th}}\right)}\right) \tag{4}\]  
where \(Ei{\left(\right)}\) is the so-called exponential integral function. The relative temperature contrast can be calculated for this case as  
\[C_{R}(t)=\frac{T_{crack}(t)-T_{sound}(t)}{T_{sound}(t)}=\frac{\alpha}{\sqrt{\pi }}\ Ei{\left(\alpha^{2}\right)}+\ erf{\left(\alpha\right)} \tag{5}\]  
where \(\alpha\) denotes the ratio  
\[\alpha=\frac{d}{d_{th}}. \tag{6}\]  
As can be seen, the contrast depends only on the ratio of the crack depth \(d\) to the thermal diffusion length \(d_{th}\), whereby this second one contains the dependency on the heating duration time \(t\), and the material dependent thermal diffusivity \(\kappa\). If the ratio \(\alpha=d/d_{th}\) is large, then \(T_{crack}(t)\) \(\sim\)2 \(T_{sound}(t)\) and \(C_{R}(t)\) becomes close to 1. To model more complex problems, such as inclined cracks or when the penetration depth is not negligibly small, two types of finite element simulation models were set up using the multiphysics simulator ANSYS [20]:  
* In the first model (simu.1), surface heat flux is directly applied to the crack sides and to the surface of the sample. The simulation calculates the temperature distribution during the heating pulse and in the cool-down phase after switching off the induction heating. * In the second model (simu.2), the inductive heating itself is also simulated, and the eddy current distribution is determined by the model. Due to the generated heat and taking into account the heat diffusion, the temperature distribution is calculated.