# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 4 Metal Slab in Flight in Fair and Stormy Weather
### Application to Aluminium, Copper, Titanium and Steel Plates
Table 2 indicates the physical properties derived from those in Table 1, namely, (i) the thermal stress coefficient (67); (ii) the decay time (34b); (iii) the decay length scale or penetration depth (35c); (iv) the effective elastic modulus (53). The basic and derived physical parameters in Tables 1 and 2, respectively, appear in the profiles as a function of the depth of the eight physical variables, namely, (i-iii) the electric field (35b), current (35d) and the magnetic field (37a)-(37b) \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & & & **A** & **C** & **S** & **T** & **Unit** \\ \hline Quantity & Symbol & Equation & Aluminium & Copper & Steel & Titanium & \\ \hline Coefficient of thermal stress & \(\alpha\) & (67) & \(\mathbf{4.7\times 10^{6}}\) & \(\mathbf{6.9\times 10^{6}}\) & \(\mathbf{5.2\times 10^{6}}\) & \(\mathbf{2.9\times 10^{6}}\) & kg\(\,\text{m}^{-1}\,\text{s}^{-2}\,\text{K}^{-1}\) \\ \hline Decay time & \(\theta\) & (34b) & \(2.2\times 10^{-19}\) & \(1.4\times 10^{-19}\) & \(1.2\times 10^{-20}\) & \(3.6\times 10^{-18}\) & s \\ \hline Penetration depth & \(\mathbf{l}\) & (35c) & \(\mathbf{6.7\times 10^{-11}}\) & \(\mathbf{4.2\times 10^{-11}}\) & \(\mathbf{3.6\times 10^{-12}}\) & \(\mathbf{1.1\times 10^{-9}}\) & \(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\bmbmbm }} }}}}}}}}}}}}}\) \\ \hline Effective elastic modulus & \(\
u\) & (53) & \(7.0\times 10^{10}\) & \(1.3\times 10^{11}\) & \(2.0\times 10^{11}\) & \(1.2\times 10^{11}\) & kg\(\,\text{m}^{-1}\,\text{s}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical properties derived from Table 1.  
\[\{\mathbf{E}(x),\mathbf{J}(\mathbf{x}),\mathbf{H}(\mathbf{x})\}=\exp\Bigl{(}- \frac{x}{l}\Bigr{)}\biggl{\{}E_{0}\mathbf{e}_{y},\chi E_{0}\mathbf{e}_{y},\sqrt {\frac{\varepsilon}{\mu}}E_{0}\mathbf{e}_{z}\biggr{\}}, \tag{68}\]  
for an external electric field \(E_{0}\); (iv) the temperature profile (49) with temperature at the outer (48a) and at the inner (48b) boundaries; (v) heat flux (50b), which takes the values at the outer and at the inner boundaries:  
\[\mathbf{G}(0) =\mathbf{e}_{\mathrm{x}}k\biggl{\{}\frac{T_{*}+T_{0}-T_{1}}{L}-T_ {*}\biggl{[}\frac{1}{L}\exp\Bigl{(}-\frac{2L}{l}\Bigr{)}+\frac{2}{l}\biggr{]} \biggr{\}}, \tag{69a}\] \[\mathbf{G}(L) =\mathbf{e}_{\mathrm{x}}k\biggl{\{}\frac{T_{*}+T_{0}-T_{1}}{L}-T_ {*}\biggl{(}\frac{1}{L}+\frac{2}{l}\biggr{)}\exp\Bigl{(}-\frac{2L}{l}\biggr{)} \biggr{\}}; \tag{69b}\]  
(vi) displacement (59), which vanishes at the inner boundary (57a) and takes the value  
\[u_{0}\equiv u(0)=-\frac{L}{2}(aL+bl)+\frac{bl^{2}}{4}\biggl{[}1-\exp\Bigl{(}- \frac{2L}{l}\Bigr{)}\biggr{]}-fL, \tag{70}\]  
at the outer boundary; (vii) strain (60), which takes the value  
\[e_{\mathrm{xx}}(0)=f=\frac{1}{\
u}[-p_{0}+\alpha(T_{0}-T_{\mathrm{ref}})]\] (71a) at the outer and \[e_{\mathrm{xx}}(L)=f+aL+\frac{bl}{2}\biggl{[}1-\exp\biggl{(}-\frac{2L}{l} \biggr{)}\biggr{]} \tag{71b}\]  
at the inner boundaries; (viii) stress (61), which equals minus the atmospheric pressure at the outer boundary (58b) and takes the value  
\[\tau_{\mathrm{xx}}(L)=-p_{0}-\frac{\chi l}{2c}\sqrt{\varepsilon\mu}E_{0}^{2} \biggl{[}1-\exp\biggl{(}-\frac{2L}{l}\biggr{)}\biggr{]} \tag{72}\]  
at the inner boundary. The approximation (62a) allows the omissions of (i) the first term in square brackets in (69a); (ii) the first term in curved brackets in (69b); (iii-v) the last terms in square brackets in (70), (71b) and (72).