# 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
The physical properties [16; 17] indicated in Table 1 are (i/ii) the [PERSON]'s modulus \(E\) and Poisson ratio \(\sigma\), which are the elastic moduli in [PERSON]'s law (21d); (iii) the thermal conductivity \(k\) in the heat flux (22b); (iv) the Ohmic electrical conductivity \(\chi\) in the electric current (22a); (v) the dielectric permittivity \(\varepsilon\) relating (21b) electric displacement \(\mathbf{D}\) to electric field \(\mathbf{E}\); (vi) the magnetic permeability \(\mu\) relating (21c) the magnetic induction \(\mathbf{B}\) to the magnetic field \(\mathbf{H}\); (vii) the mass density \(\rho\) for reference purposes, since it appears only in the inertia force in the unsteady momentum equation (17) but drops out in the steady case; (viii) the coefficient of thermal expansion  
\[\beta\equiv\frac{1}{3V}\bigg{(}\frac{\partial V}{\partial T}\bigg{)},\] (63a) where \[V\] is the volume that is distinct from the coefficient of thermal stress \[\alpha\equiv-\frac{\partial\tau_{\text{xx}}}{\partial T},\] (63b) which appears in ( 21d ) the direct Hooke law \[\tau_{ij}=-\alpha(T-T_{\text{ref}})\delta_{ij}+\frac{K}{1+\sigma}\bigg{(}e_{ ij}+\frac{\sigma}{1-2\sigma}e_{kk}\delta_{ij}\bigg{)}, \tag{63c}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Case** & & **A** & **C** & **S** & **T** & **Unit** \\ \hline Quantity & Symbol & Aluminium & Copper & Steel & Titanium & \\ \hline Young’s modulus & \(K\) & \(7.0\times 10^{10}\) & \(1.3\times 10^{11}\) & \(2.0\times 10^{11}\) & \(1.2\times 10^{11}\) & kg m\({}^{-1}\) s\({}^{-2}\) \\ \hline Poisson ratio & \(\sigma\) & \(3.3\times 10^{-1}\) & \(3.4\times 10^{-1}\) & \(2.9\times 10^{-1}\) & \(3.2\times 10^{-1}\) & – \\ \hline Linear expansion coefficient & \(\beta\) & \(2.3\times 10^{-5}\) & \(1.7\times 10^{-5}\) & \(11.1\times 10^{-5}\) & \(8.6\times 10^{-6}\) & K\({}^{-1}\) \\ \hline Thermal conductivity & \(k\) & \(2.4\times 10^{2}\) & \(4.0\times 10^{2}\) & \(5.2\times 10^{1}\) & \(2.2\times 10^{2}\) & kg m s\({}^{-3}\) K\({}^{-1}\) \\ \hline Ohmic electrical conductivity & \(\chi\) & \(3.8\times 10^{2}\) & \(6.0\times 10^{2}\) & \(7.0\times 10^{6}\) & \(2.4\times 10^{6}\) & kg\({}^{-1}\) m\({}^{-3}\) s\({}^{3}\) A\({}^{2}\) \\ \hline Magnetic permeability & \(\mu\) & \(1.3\times 10^{-6}\) & \(1.3\times 10^{-6}\) & \(1.3\times 10^{-4}\) & \(1.3\times 10^{-6}\) & kg m s\({}^{-2}\) A\({}^{-2}\) \\ \hline Dielectric permittivity & \(\varepsilon\) & \(8.5\times 10^{-12}\) \(\ast\) & \(8.5\times 10^{-12}\) \(\ast\) & \(8.5\times 10^{-14}\) \(\ast\) \whose relation is obtained subsequently for inclusion in the table of derived quantities (Table 2); (ix) the cross-coupling coefficient \(\eta\) between the heat flux (22b) and electric current (22a), which is assumed to be negligible, and is omitted from Table 1. Due to the lack of reliable values for the dielectric permittivity of metals, an estimated value in the SI system \(c=1\) is obtained assuming that the electromagnetic wave speed (35a) equals that in a vacuum  
\[c_{\text{em}}=3.00\times 10^{8}\,\text{m}\,\text{s}^{-1},\] (64a) so that the dielectric permittivity is calculated from the magnetic permeability \[\varepsilon=\frac{1}{\mu(c_{\text{em}})^{2}}=\frac{1.11\times 10^{-17}\,\text{m}^{ -1}\text{s}}{\mu}. \tag{64b}\]  
The relation between the coefficient of thermal expression (63a) and the coefficient of thermal stress (63b) is obtained in four steps: (i) the direct Hooke law (21d), equivalent to (63c), is used to relate the trace \(\tau_{ii}\) of the stress tensor, equal to \(-3p\), where \(p\) is the pressure, to the trace of the strain tensor \(e_{ii}\), equal to the dilatation \(\mathbf{\
abla}\cdot\mathbf{\upmu}\):  
\[-3p=\tau_{ii}=-\alpha(T-T_{\text{ref}})\delta_{ii}+\frac{K}{1+\sigma}\varepsilon _{ik}\Big{(}1+\frac{\sigma}{1-2\sigma}\delta_{ii}\Big{)}=-3\alpha(T-T_{\text{ ref}})+\frac{K}{1-2\sigma}\varepsilon_{ik}; \tag{65}\]  
(ii) substitution of (65) in the last term of the direct stress-strain Hooke law (63c), leading to the inverse strain-stress Hooke law:  
\[e_{ij}=\frac{1-2\sigma}{K}\alpha(T-T_{\text{ref}})\delta_{ij}+\frac{1+\sigma} {K}\tau_{ij}-\frac{\sigma}{K}\tau_{kk}\delta_{ij};\] (66a) (iii) the inverse Hooke law, which must be of the following form: \[e_{ij}=\beta(T-T_{\text{ref}})\delta_{ij}+\frac{1+\sigma}{K}\tau_{ij}-\frac{ \sigma}{K}\tau_{kk}\delta_{ij},\] (66b) where the thermal expansion coefficient ( 63a ) appears; (iv) comparing ( 66a ) \[V\]0 (66b ), which leads to the relation \[V\]1 (67) expressing the thermal stress coefficient ( 63b ) in terms of the coefficient of thermal expansion ( 63a ).