# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 2 Fundamental Equations of Dissipative Electromagnetic Thermoelasticity
### Isotropic Media and Steady Fields  
In the case of isotropic media, that is, with properties independent of direction, the constitutive (6a)-(6d) and diffusive (14a), (15) equations can depend only on scalars, the identity matrix \(\delta_{ij}\) and the permutation symbol \(e_{ijk}\), and must still satisfy symmetries like (8a)-(11b). It follows that, in an isotropic medium, the following effects cannot exist: (i/ii) the pyroelectric (9a) and pyromagnetic (9b) effects, because they are specified by vectors that are isotropic only if they vanish:  
\[f_{i}=0=h_{i};\] (19a) (19b)  
(vi) the piezoelectric (10a) and piezomagnetic (10b) tensors that are of third order, which can be isotropic only if they are proportional to the third-order permutation tensor \(e_{ijk}\) that is skew-symmetric in all indices (\(i,j\), \(k\)), in contradiction with the symmetries in (10a) and (10b), respectively, so both must be zero:  
\[p_{ijk}=0=q_{ijk}; \tag{19b}\]  
(v) the electromagnetic coupling tensor (11a) relates (a) the electric field and displacement, which are polar vectors, to the (b) magnetic field and induction, which are axial vectors, and thus the electromagnetic coupling is a symmetric pseudo-tensor or oriented tensor [14] that changes sign \(\theta_{ij}\rightarrow-\theta_{ij}\) in a coordinate inversion relative to the origin \(\mathbf{x}\rightarrow-\mathbf{x}\) and thus cannot be isotropic and must vanish:  
\[\theta_{ij}=0. \tag{19c}\]  
Thus, there are no pyroelectric, pyromagnetic, piezoelectric, piezomagnetic or electromagnetic coupling effects in an isotropic medium. In contrast with (v), the (vi) dielectric permittivity (8a), (vii) magnetic permeability (8b), (viii) thermoelastic tensor (11b), (ix) Fourier thermal conductivity (15), (x) Ohmic electrical conductivity (14a) and (xi) thermoelectric coupling (14a) (15) are all symmetric absolute or non-oriented tensors relating polar vectors,and do not change sign by inversion of coordinates, so their isotropic form is the product of the identity matrix or Kronecker symbol \(\delta_{ij}\), shown by the following scalar:  
\[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\] (20a) where the dielectric permittivity is \[\varepsilon\], the magnetic permeability \[\mu\], the thermal stress \[\alpha\], the Fourier thermal conductivity \[k\], the Ohmic electrical conductivity \[\chi\]  and the thermoelectric coupling \[\eta\]. Similarly, the elastic stiffness tensor ( 8c ) in an isotropic medium must be a sum of products of identity matrices with the correct symmetries, and is thus of the following form: \[p_{ijk}=0=q_{ijk}; \tag{19b}\]0 (20b) Here, the Lame moduli of elasticity \[p_{ijk}=0=q_{ijk}; \tag{19b}\]1 may be replaced by the Young's modulus \[p_{ijk}=0=q_{ijk}; \tag{19b}\]2  and Poisson ratio \[p_{ijk}=0=q_{ijk}; \tag{19b}\]3, \[p_{ijk}=0=q_{ijk}; \tag{19b}\]4 (20c) in the usual form of the stiffness tensor, \[p_{ijk}=0=q_{ijk}; \tag{19b}\]5 (20d) relating strain to stress. Substitution of (19a), (19b), (19c), (20a) and (20d) in (6a)-(6d) leads to the constitutive relations for an isotropic material:  
\[p_{ijk}=0=q_{ijk}; \tag{19b}\]6 \[p_{ijk}=0=q_{ijk}; \tag{19b}\]7 (21b) \[p_{ijk}=0=q_{ijk}; \tag{19b}\]8 (21c) \[p_{ijk}=0=q_{ijk}; \tag{19b}\]9 \[\theta_{ij}=0. \tag{19c}\]0  
where \(\varepsilon\), \(\mu\), \(K\) and \(\sigma\) may be taken as constant for a homogeneous medium.