# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 1 Introduction  
The fundamental equations of elasticity [1, 2] are combined with electric and magnetic forces in, respectively, piezoelectricity and piezomagnetism [3], which are aspects of electromechanics [4], and the electrodynamics of continua [5] relevant to adaptive and morphing structures [6], which have many applications, including in aerospace engineering [7]. Allowing for the simultaneous presence of electric and magnetic fields in electromagnetism [8, 9] leads, in combination with elasticity, to the fundamental equations of piezoelectromagnetism [10]. These equations are further extended here to include coupled thermal and electrical diffusion, that is, Fourier heat conduction [11] and Ohmic electric currents [12] and their interaction through the Onsager reciprocity relations [13]. The purpose of this work is to present the first exact analytical one-dimensional solution for a steady anisothermal thermally and electrically conducting slab in the presence of the skin effect of surface electric charges/currents, with the associated electric and magnetic forcescausing elastic deformations, which are also affected by temperature gradients associated with Joule resistive heating.  
The general equations of dissipative electromagnetic thermoelasticity (Section 2) are obtained in two steps: (i) using the first principle of thermodynamics to obtain the constitutive equations for anisothermal elasticity in the presence of electromagnetic fields (Section 2.1); (ii) the constitutive equations from the first principle of thermodynamics are used together with the diffusion relations from the second principle of thermodynamics in the fundamental equations of electricity, magnetism, energy and momentum (Section 2.2). The resulting fundamental equations, three vector and one scalar, specify the displacement, electric and magnetic field vectors and temperature and are simplified for isotropic media in steady conditions (Section 2.3). A one-dimensional steady solution (Section 3) of the equations of electromagnetic thermoelasticity (Section 2) is obtained for a parallel-sided slab, with dependence only on the transverse coordinate, by considering (i) the electromagnetic skin effect on the metallic structures of external electric and magnetic fields mutually orthogonal and both parallel to the slab (Figure 1), and the associated Ohmic electric currents (Section 3.1); (ii) the thermal effect of Joule dissipation of electric currents by electric resistance, specifying the temperature and heat flux (Section 3.2); (iii) the elastic displacement, strain and stress due to the combination of external pressure and electric and magnetic fields with temperature gradients (Section 3.3).  
The numerical application (Section 4) of electromagnetic thermoelasticity (Section 2) to a slab in steady conditions (Section 3) considers (Tables 1 and 2) four types (Section 4.1) of materials: (i) aluminium (A), used for its lightness and good conductivity; (ii) titanium (T), used for its high strength in spite of its cost; (iii) steel (S), which is affordable and has good strength; (iv) copper (C), for its very high conductivity in spite of its weight. The simulations compare flight conditions (Table 3) at the tropopause in fair weather conditions (Table 4) and in a lightning storm (Section 4.2) and include (Section 4.3) (i) the profiles (Figures 3-10) as a function of the coordinate normal to the slab (Figure 1) of the electric and magnetic fields, temperature and heat flux and displacement, strain and stress; (ii) the values at the boundaries, that is, on the outer and inner sides of the slab (Table 5). The conclusion (Section 5) notes that, although there is a vast amount of literature on aircraft flight in adverse natural or man-made weather, like wind shears, turbulence and wakes, the combined electromagnetic and thermoelastic effects in lightning storms compared with fair weather have received less attention, and are the subject of the present paper, which may be the first which considers all effects combined.