## 2 Materials and Methods
### Temperature Rise in Time Domain Due to the Static Heating Term
Indeed, many successful applications of crack detection in aeronautical or power-generation turbine components were reported [6]. Finally, for some materials like carbon-fibre-reinforced polymers (CFRP), \(\mu_{T}<<\delta\) is fulfilled for typical observation times in experiments. Equation (17) can be simplified to  
\[\Delta T(t)=\frac{H_{0}}{2}\mu_{0}\mu_{R}\omega\frac{t}{\rho c}. \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Material**} & \multirow{2}{*}{\(\sigma\) **in 10\({}^{6}\)**} & \multirow{2}{*}{\(\mu_{R}\)} & \multirow{2}{*}{\(\lambda\) **in**} & \multirow{2}{*}{\(\delta\) **in mm**} & \multirow{2}{*}{\(\mu_{T}\) **in mm**} & \multirow{2}{*}{**H\(\tau\) **in mm**} \\  & & & & & & & (100 kHz) & & **in K** \\ \hline \(\frac{\mu_{T}}{\delta}>>1\) & cast iron & 6.2 & 200 & 49 & 0.045 & 2.43 & 12.4 & 0.108 \\ nickel, pure & 14.62 & 100 & 90.7 & 0.042 & 3.03 & 3.9 & 0.146 \\ \hline \multirow{4}{*}{\(\frac{\mu_{T}}{\delta}>1\)} & silver, pure & 62.87 & 1 & 408 & 0.201 & 8.15 & 0.1 & 0.081 \\  & zinc, rolled & 16.24 & 1 & 113 & 0.395 & 4.06 & 0.4 & 0.021 \\ aluminium & 2014-T6 & 22.53 & 1 & 177 & 0.335 & 5.97 & 0.3 & 0.0357 \\ copper, pure & 60.09 & 1 & 401 & 0.205 & 6.83 & 0.1 & 0.0666 \\ \hline \multirow{4}{*}{\(\frac{\mu_{T}}{\delta}\cong 1\)} & Inconel 600 & 0.98 & 1 & 14.9 & 1.608 & 1.26 & 1.6 & 0.00157 \\  & stainless steel 316 & 1.33 & 1 & 13.4 & 1.380 & 1.18 & 1.5 & 0.00171 \\ titanium 6 AL-4V & 0.58 & 1 & 7.2 & 2.090 & 1.08 & 2.9 & 0.00103 \\ \hline \multirow{4}{*}{\(\frac{\mu_{T}}{\delta}<<1\)} & CFRP perp. to fibre & 0.0001 & 1 & 0.7 & 159 & 0.39 & 5.3 & 0.000005 \\  & SiC ceramic & 0.00005 & 1 & 80 & 225 & 3.36 & 3.5 & 0.000030 \\ silicon & 0.001 & 1 & 148 & 50 & 6.11 & 5.8 & 0.000243 \\ \hline \end{tabular}
\end{table} 
Table 1: Electrical conductivity \(\sigma\), magnetic permeability \(\mu_{R}\), thermal conductivity \(\lambda\), skin depth \(\delta\) at 100 kHz, thermal penetration depth \(\mu_{T}\) at 0.1 s, induction heating efficiency (the surface temperature rise in K after 0.1 s heating with 0.02 T induction field amplitude at 100 kHz), and ratio of thermal diffusion length to skin depth \(\mu_{2\omega}/\delta\) for selected materials. The surface temperature rises proportional to time, which is typical for a volume heat source. It is proportional also to the induction frequency. The induction heating efficiency does not reach that of magnetic steel, but is better than that of most alloys. Successful applications for CFRP have been reported [19; 20; 21; 22]. In Figure 2, the temperature contrast as a function of the normalised time \(t_{n}\) is shown. \(t_{n}\) is 1 when the thermal penetration depth \(\mu_{T}\) equals the skin depth \(\delta\). As discussed above, the slope in the double logarithmic representation approaches 1 for times \(t_{n}<<1\) and 0.5 for \(t_{n}>>1\). The total power density \(S\) in W/m\({}^{2}\) released in the sample can be obtained from the integration of the static power density term in Equation (16) from \(z=0\) to infinity. It is given by  
\[S=\int\limits_{0}^{\infty}q_{DC}(z)dz=\frac{{H_{0}}^{2}}{2}\sqrt{\frac{\omega \mu_{0}\mu_{R}}{2\sigma}} \tag{21}\]  
In all cases, higher magnetic permeability increases the signal, which explains the high induction efficiency in ferritic steel. For magnetic materials, Equations (3)-(6) are an approximation for small deviations of the magnetization from its equilibrium. It has been shown that static magnetic fields applied in addition to the high-frequency field can improve the thermographic contrast of cracks, when applied in a proper direction [23]. It should be mentioned that the electromagnetic skin depth as defined in Equation (7) is valid for a plane electromagnetic wave incident to a plane, electromagnetically thick conductor. The damping lengths will be somewhat different for rod and tube geometries, for thin plates, small components and under inductor wires. Even more complicated are the current paths in anisotropic conductors such as CFRP.