# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 5 Conclusions
These conclusions are confirmed by the figures: (i) the electric and magnetic fields, respectively, in Figures 2 and 4, are largest for titanium, both in fair weather (left) and in a lightning storm (right); (ii) the temperature (Figure 5) increases from the outer wall at \"tropospheric\" temperature to the inner wall \"at room\" cabin temperature, except for in the case of titanium, only for the lightning storm, where the intense Joule heating causes the highest temperature in the thin skin region; (iii) the modulus of the heat flux (Figure 6) is again highest for titanium in a lightning storm; (iv) the modulus of the displacement (Figure 7) shows a greater difference between fair weather (left) and the lightning storm (right) for titanium; (v) the strain (Figure 8) increases from negative at the outer wall to positive at the inner wall, except for in the case of titanium in a lightning storm, for which the strain reduces from the peak temperature in the skin region to the lower temperature at the inner wall; (vi) the stress (Figure 9) equals the atmospheric pressure at the outer wall, and is almost uniform on a large scale (top right), with distinct variations for different metals in the skin region (bottom right). The basic study of aircraft flight performance and stability [23] considers an atmosphere at rest, supplemented by a vast amount of literature on the effects of adverse weather on the flight of aircraft, such as the altitude loss in wind shear [24] or the rolling motion induced by crossing wakes of other aircraft [25; 26]. The effects of lightning are usually countered by the Faraday cage effect of metallic aircraft structures or by embedding conducting wires like copper in composite structures. When the electrical insulation proves insufficient, the effects of lightning are considered mostly as concerns electric and electronic equipment, which can be severely damaged or totally disabled. The present paper considers the thermomechanical effects of a lightning storm compared with flight in fair weather, requiring an approach starting with the fundamental equations of thermoelasticity and electromagnetism: (i) the constitutive equations (6a)-(6d) relating (Section 2.1) entropy, electric displacement, magnetic induction and the stress tensor to temperature, electric and magnetic fields and the strain tensor are considered; (ii) in addition to the constitutive relations based on the first principle of thermodynamics, dissipation relations (14a) (15) between electric current and energy flux, and electric field and temperature gradient (Section 2.2) are used, satisfying the second principle of thermodynamics requiring entropy growth; (iii) the constitutive and dissipative relations are used (Section 2.2) in the electromagnetic field equations (13a) and (14b), energy equation (16) and momentum equation (18); (iv) the coupled system of one scalar and three vector equations for the temperature, displacement and electric and magnetic fields are simplified (Section 2.3) for an isotropic medium in a steady state; (v) the equations are solved analytically for an infinite parallel-sided slab with dependence only along the thickness (Figure 1) with displacement orthogonal to the sides and transverse electric and magnetic fields orthogonal to each other; (vi) this specifies the dependence on the thickness of the electric field, electric current and magnetic field (Section 3.1), temperature and heat flux (Section 3.2) and displacement, strain and stress (Section 3.3); (vii) the results are used to compare flight at the tropopause in fair weather (Section 4.2) and a lightning storm (Section 4.3) for a slab of four metals, namely, aluminium, copper, steel and titanium (Section 4.1). Conceptualisation, L.M.B.C.C. ; methodology, L.M.B.C.C. and M.J.S.S. ; software, M.J.S.S. ; formal analysis, L.M.B.C.C. and M.J.S.S. ; investigation, L.M.B.C.C. and M.J.S.S. ; data curation, L.M.B.C.C. and M.J.S.S. ; writing--original draft preparation, L.M.B.C.C. ; writing--review and editing, L.M.B.C.C. and M.J.S.S; visualisation, M.J.S.S.