# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 2 Fundamental Equations of Dissipative Electromagnetic Thermoelasticity
### Fundamental Equations of Electricity, Magnetism, Energy and Momentum  
It is assumed in the equations that follow that a dot denotes the derivative with regard to time:  
\[\dot{A}\equiv\frac{\partial A}{\partial t}.\] (12a) The spatial derivatives with regard to the coordinates are denoted by \[\partial_{i}\equiv\frac{\partial}{\partial x_{i}}. \tag{12b}\]  
In the Maxwell equation for the electric field, where \(c=3.00\times 10^{8}\,\mathrm{m}\,\mathrm{s}^{-1}\) is the speed of light in a vacuum and \(e_{ijk}\) the permutation symbol, the constitutive relation is substituted for the magnetic induction (6c), leading to  
\[-e_{ijk}\partial_{j}E_{k}=\dot{B}_{i}=h_{i}\dot{T}+\vartheta_{ji}\dot{E}_{j}+ \mu_{ij}\dot{H}_{j}+\frac{1}{2}q_{ijk}\big{(}\partial_{j}\dot{u}_{k}+\partial_ {k}\dot{u}_{j}\big{)},\] (13a) where the relation between the strain tensor \[e_{ij}\] and displacement vector \[u_{i}\] is used: \[2e_{ij}=\partial_{i}u_{j}+\partial_{j}u_{i}. \tag{13b}\]  
The electric current density [13] is proportional to the electric field through the Ohmic electrical conductivity \(\chi_{ij}\) and to the temperature gradient through the thermoelectric coupling tensor \(\eta_{ij}\):  
\[J_{i}=\chi_{ij}E_{j}+\eta_{ij}\partial_{j}T.\] (14a) The Maxwell equation for the magnetic field becomes \[ce_{ijk}\partial_{j}H_{k}=\dot{D}_{i}+J_{i}=f_{i}\dot{T}+e_{ij}\dot{E}_{j}+ \vartheta_{ij}\dot{H}_{j}+\frac{1}{2}p_{ijk}\big{(}\partial_{j}\dot{u}_{k}+ \partial_{k}\dot{u}_{j}\big{)}+\chi_{ij}E_{j}+\eta_{ij}\partial_{j}T,\] (14b) where the constitutive equation for the electric displacement ( 6b ) and the diffusion relation for the electric field ( 14a ) are used. The heat flux \[G_{i}\] ( 13 ) with thermoelectric coupling corresponding to the electric current ( 14a ) is \[G_{i}=-k_{ij}\partial_{j}T+T\eta_{ij}E_{j},\] (15) where \[\partial_{i}\equiv\frac{\partial}{\partial x_{i}}. \tag{12b}\]0 is the Fourier thermal conductivity tensor. The energy equation becomes \[\partial_{i}\equiv\frac{\partial}{\partial x_{i}}. \tag{12b}\]1 (16) using the constitutive relation for the entropy ( 6a ) and the diffusion relation for the heat flux ( 15 ). The remaining fundamental equation is for the momentum. The momentum equation balances the inertia force,  
\[\partial_{i}\equiv\frac{\partial}{\partial x_{i}}. \tag{12b}\]2  
where \(\rho\) is the mass density and \(\dot{\mathbf{u}}\) the acceleration, against (i) the electric force; (ii) the magnetic force; (iii) the divergence of the stress tensor, where, in the electric force \(q\mathbf{E}\), the electric charge is given by the Maxwell equation \(q=\mathbf{\
abla}\cdot\mathbf{D}=\partial_{j}D_{j}\). Substituting the constitutive relations for the electric displacement (6b ), magnetic induction (6c ) and stress tensor (6d ) and the diffusion relation for the electric current (14a ) leads to\[ \begin{split}\rho\ddot{u}_{i}&=E_{i}\left[f_{j}\partial_{j }T+\varepsilon_{j}\partial_{j}E_{l}+\theta_{jl}\partial_{j}H_{l}+\frac{1}{2}p_{ jl}\partial_{j}(\partial_{l}u_{k}+\partial_{k}u_{l})\right]\\ &\quad+\frac{1}{c}\varepsilon_{ijk}\Big{(}\chi_{jl}E_{l}+\eta_{ jl}\partial_{l}T\Big{)}\left[h_{k}T+\theta_{mk}E_{m}+\mu_{km}H_{m}+\frac{1}{2}q_{ kmn}(\partial_{m}u_{n}+\partial_{n}u_{m})\right]\\ &\quad-\alpha_{ij}\partial_{j}T-p_{kij}\partial_{j}E_{k}-q_{kij} \partial_{j}H_{k}+\frac{1}{2}A_{ijkl}\partial_{j}(\partial_{k}u_{l}+\partial_ {l}u_{k}).\end{split}  \tag{18}\]  
The momentum Equation (18) is in quite general form because (i) it balances the inertia force against electromagnetic forces and internal stresses, as stated in (17); (ii) it uses the linear constitutive relations (6a)-(6d), including linear thermoelastic stresses, electromagnetic coupling and piezoelectric and piezomagnetic interactions.