# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 4 Metal Slab in Flight in Fair and Stormy Weather
### Flight at the Tropopause in Fair Weather  
The aluminium 2024 plate used in many applications, including aeronautics, is often supplied with a thickness of \(L=4.26\,\mathrm{mm}\). The plate thickness used in aeronautics is usually 1 or 2 mm for fuselage panels and 2 or 3 mm for wing skins. A panel thickness  
\[L=2\,\mathrm{mm}=2\times 10^{-3}\,\mathrm{m}, \tag{76}\]  
is chosen. The same value is used for all four metals for comparison purposes. The coordinate \(x\) varies between (i) the outer surface in contact with the atmosphere \(x=0\) and (ii) the inner surface \(x=L\) corresponding to the wall of the passenger or cargo cabin. For flight at the tropopause at the altitude \(z=11\,\mathrm{km}\), in [18,19], the International Standard Atmosphere (ISA) the pressure is  
\[p_{0}=2.27\times 10^{4}\,\mathrm{Pa}=2.27\times 10^{4}\,\mathrm{kg}\,\mathrm{m}^{-1 }\,\mathrm{s}^{-2}, \tag{77}\]  
and the temperature is  
\[T_{0}=-56.5\,^{\circ}\mathrm{C}=216.5\,\mathrm{K}. \tag{78}\]  
The internal temperature is taken as the following comfortable value in an air-conditioned cabin:  
\[T_{1}=20\,^{\circ}\mathrm{C}=293\,\mathrm{K}. \tag{79}\]  
For flight in fair weather, the surface electric field is taken [20] as the average value on the surface of the Earth:  
\[E_{0}=100\,\mathrm{V}\,\mathrm{m}^{-1}=10^{2}\,\mathrm{kg}\,\mathrm{m}\, \mathrm{s}^{-3}\,\mathrm{A}^{-1}. \tag{80}\]  
The physical conditions during lightning storms [20,21] can be quite different from the average Earth conditions in fair weather. For example, the electric and magnetic fields in fair weather are the average Earth values, whereas, for a lightning storm, they should be calculated from the electric discharge. Lightning storms are associated with cumulonimbus clouds, which can extend over a wide altitude range from 0.5 to 16 km. Most lightning strikes on aircraft occur in low-level flight between the clouds and the ground. Lightning strikes on aircraft can also occur at high altitude near cumulonimbus clouds, although these are usually associated with storms, and are thus best avoided. Lightning strikes can occur up to 15 km from the clouds. The lightning strikes associated with thunderstorms can vary considerably from one event to another, and some indicative values are (a) a lightning strike can reach \(10^{8}\)-\(10^{9}\,\mathrm{V}\), that is up to 1 GV, with an average value of \(\Phi_{\mathrm{P}}=300\,\mathrm{MV}=3\times 10^{8}\,\mathrm{V}\); (b) the duration of a lightning flash is \(t_{\mathrm{f}}=6.5\times 10^{-5}\,\mathrm{s}\); (c) a lightning storm can consist of about \(N_{\mathrm{f}}=10\) flashes in close succession, lasting about \(\overline{t}=0.5\,\mathrm{s}\). Since the present theory is steady, an average voltage \(\bar{\Phi}=\Phi_{\mathrm{m}}t_{\mathrm{f}}N_{\mathrm{f}}/\overline{t}=3.9 \times 10^{5}\,\mathrm{V}\) is considered,  
\[\overline{\Phi}=3.9\times 10^{5}\,\mathrm{V}=3.9\times 10^{5}\,\mathrm{kg}\, \mathrm{m}^{2}\,\mathrm{s}^{-3}\,\mathrm{A}^{-1}, \tag{81}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Quantity** & **Symbol** & **Equation** & **Value** & **Unit** \\ \hline Thickness of slab & \(L\) & (76) & \(2.00\times 10^{-3}\) & \(\overline{\mathrm{m}}\) \\ External pressure & \(p_{0}\) & (77) & \(2.27\times 10^{4}\) & \(\mathrm{kg}\,\mathrm{m}^{-1}\,\mathrm{s}^{-2}\) \\ External temperature & \(T_{0}\) & (78) & 216.5 & \(\mathbf{K}\) \\ Internal temperature & \(T_{1}\) & (79) & 293 & \(\mathrm{K}\) \\ External electric field & & & & \\  — Fair weather & \(E_{0}\) & (80) & \(1\times 10^{2}\) & \(\mathrm{kg}\,\mathrm{m}^{2}\,\mathrm{s}^{-3}\,\mathrm{A}^{-1}\) \\ — Lightning storm & \(\overline{E}_{0}\) & (82) & \(1.95\times 10^{8}\) & \(\mathrm{kg}\,\mathrm{m}^{2}\,\mathrm{s}^{-3}\,\mathrm{A}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters for flight at the tropopause.