# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 2 Fundamental Equations of Dissipative Electromagnetic Thermoelasticity
### Constitutive Relations for Heat, Electricity, Magnetism and Elasticity  
The first principle of thermodynamics [13] states that the change in internal energy \(\mathrm{d}U\) is the sum of the work \(\mathrm{d}W\) and the heat \(\mathrm{d}Q\):  
\[\mathrm{d}U=\mathrm{d}W+\mathrm{d}Q, \tag{1a}\]The latter is specified by the product of temperature \(T\) and the change of entropy dS:  
\[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]  
The work for piezoelectromagnetism,  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\] (2a) consists of three parts: (i) the elastic work [1; 2] of the stress tensor \[\tau_{ij}\] on the strain tensor \[\epsilon_{ij}\] : \[\mathrm{d}W_{\mathrm{u}}=\tau_{ij}\mathrm{d}e_{ij};\] (2b) (ii) the electric work [9] of the electric displacement \[\mathbf{D}\] on the electric field \[\mathbf{E}\] : \[\mathrm{d}W_{\mathrm{e}}=-\mathbf{D}\cdot\mathrm{d}\mathbf{E};\] (2c) (iii) the magnetic work of the magnetic induction \[\mathbf{B}\] on the magnetic field \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]0 : \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]1 (2d) Substituting (2b), (2c) and (2d) specifies the total work (2a), and adding (1b) leads to the internal energy according to (1a): \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]2 (3) The internal energy (3) may be replaced as a function of state by the free energy \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]3 (4a) whose differential is given by \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]4  
From (4b), it follows that the free energy has four independent variables, namely, the temperature, the electric and magnetic fields and the strain tensor,  
\[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]5 (5a) and the partial derivatives specify the dependent variables, respectively, entropy, electric displacement, magnetic induction and the stress tensor: \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]6  
Thus, each dependent variable is a function of all independent variables, leading to the following constitutive relations:  
\[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]7 \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]8 (6b) \[\mathrm{d}Q=T\mathrm{d}S. \tag{1b}\]9 (6c) \[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]0  
where the coefficients are the constitutive tensors specifying the properties of matter. Starting with the diagonal of (6a)-(6d), the constitutive tensors are (i) the specific heat at constant electric and magnetic fields and strain:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]1  
(ii) the dielectric permittivity tensor:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]2  
(iii) the magnetic permeability tensor:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]3  
(iv) the elastic stiffness tensor:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]4  
The six non-diagonal coupling effects have symmetries: (i) thermoelectric through the pyroelectric \(f_{i}\) vector:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]5  
(ii) thermomagnetic through the pyromagnetic \(h_{i}\) vector:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]6  
(iii) the electro-elastic effect through the piezoelectric \(p_{ijk}\) tensor:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]7  
(iv) the magneto-elastic effect through the piezomagnetic \(q_{ijk}\) tensor:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]8  
(v) the electromagnetic coupling tensor:  
\[\mathrm{d}W=\mathrm{d}W_{\mathrm{u}}+\mathrm{d}W_{\mathrm{e}}+\mathrm{d}W_{ \mathrm{m}},\]9  
(vi) the thermoelastic tensor:  
\[\tau_{ij}\]0  
The constitutive relations (Section 2.1) are substituted in the fundamental equations of electricity, magnetism, energy and momentum (Section 2.2).