Dichotomy method to solve the plasma expansion

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Development of a lightning strike to aircraft

In the following study, the usual denomination of lightning strike will refer to a lightning ash that involves an object on the ground or in the atmosphere [17], such as an aircraft. Naturally, lightning strikes are mostly due to CG ashes for the case of ground infrastructures, but also in the eld of aeronautics. Indeed, aircraft are mostly struck by lightning during the take-o and landing phases, because cruising aircraft can usually choose their path to avoid crossing dangerous thunderclouds. Therefore, even if IC ashes are the most observed in the atmosphere, -CG is the predominant type of ashes of interest regarding aircraft protection.
The average probability of a civil aircraft to be struck lies between once per 1000 or 10 000 hours of ight [19] [20]. This frequency depends on dierent parameters such as the local climate, the type of aircraft and the ight routine. One could think of such event to be the hazardous result of the interception of a lighting ash by the course of a plane, but this scenario only represents 10% of lightning strikes to aircraft. Most of the time, it is the presence of the aircraft that triggers a lightning ash. As for natural lightning  discharges, triggered lightning development involves at the beginning the formation of a small plasma around a sharp part, that geometrically increases the background electric- eld to values higher than the breakdown threshold. In this case, the eld enhancement is not provided by a small hydrometeor, but for example by the sharp winglets, noze or tail of an airliner as shown on Figure 1.5, showing a numerical simulation of the electric eld around an aircraft [21].

Normalized waves and aircraft zoning

Not all surfaces of an aircraft need to be designed to survive the same lightning threat. Indeed the lightning ash usually initiates on sharp areas, and is then swept backwards (cf Figure 1.7. ), towards areas that are in the path of the sweeping channel. Only some areas, at the rear of the aircraft, may receive the full energy of a ash, because the lightning arc can hang-on to them for the total ash duration. On the other hand, other areas will only experience a fraction of a lightning ash.
The proportion of the ash endured by any particular point therefore depends on the probability of initial attachment, sweeping and hanging on the skin of the aircraft. In order to optimize lightning protection and set the basis of systematic certication processes, standardized lightning current waveforms are dened together with lightning strike zones.
The lightning current waveforms are dened by a composition of current components. These standardized components are not intended to replicate specic lightning events, but are rather meant to correspond to upper bounds regarding the eects of lightning on aircraft. The main relevant parameters of a lightning strike regarding direct or indirect eects are the peak current amplitude, the action integral and the time duration. Standardized components are dened by combinations of extreme cases of these characteristics that may be observed on aircraft. Figure 1.13 illustrates four components of the current waveform with their corresponding maximum current level (kA), time duration (ms), action integral (A2 s) and transferred charge (C).

Brief history on electrical contacts science

The electrical contacts have been a critical matter since the early stages of the development of electricity and electrical circuits, but were not necessarily well understood or well investigated. Initially, the range of currents carried by electrical contacts was limited, from some fractions of an ampere to perhaps of few hundreds of amperes. The range of currents involved has greatly increased with new technologies, going from micro-amperes in microelectronics technologies up to mega-amperes in high power devices and switches. With the improvements of measuring tools and with the development of new solid matter theories came better knowledge on electrical contacts. Since the 1940s, a considerable amount of knowledge has been accumulated in electrical contact science. In 1952, the ASTM (American Society for Testing Materials) published the bibliography and abstracts on electrical contacts from 1835 to 1951 [35], and continued to do so until 1965. The recording of abstracts regarding electrical contacts was then continued by the Holm Conference Organization until nowadays [36]. However, only a few comprehensive books have been written on the subject. In 1940, Windred published Electrical Contacts [37], that treated the subject into details. Ragnar Holm published his rst book in 1941, in German, and continued to update his work until the publication of the widely cited book Electric Contacts: Theory and Application in 1967 [38], reprinted several times until 2000. A more recent book, Electrical Contacts: Principles and Applications [39], the fruit of the collaboration of many authors and edited by Paul G. Slade, is a very helpful tool to understand a broad variety of aspects regarding contact resistances.

Denitions and terminology

The contact resistance between two conducting surfaces is very well introduced by Holm [38]. The term electric contact refers to the junction, mechanically tightenable or releasable, between two conductors, when this junction can carry an electric current from one conductor to the other. Experimentally, a resistance between two points can be measured by placing probes in contact with these points, imposing a small current to go from one probe to the other, and measuring the voltage drop between the probes. The resistance is then dened by the ratio between the voltage drop and the current. The use of small continuous currents insure the linearity of electrical conduction laws, such that this ratio is independent on the current level. In the case of a single body conductor of constant section such as a cylinder, the resistance experimentally measured is proportional to the distance between the probes.

Holm’s formula for a at circular a-spot

In order to calculate the resistance of a conductor in static state, one usually assumes Ohm’s law that requires the current vector to be proportional the electric eld, that is the gradient of the potential function.
For a straight conductor of constant section, symmetry arguments make it easy to deduce the structure of equipotential surfaces. The current conservation is easily integrated in order to nd a relation between to total current owing through the conductor and the potential at its boundaries. The resistance of a conductor of section S, length l and conductivity is classically given by R = l S .
In the case of a constriction of the current lines as represented on Figure 1.17, the mathematical solution is much more complex. Using Smythe mathematical development, Holm’s show in [38] that for a perfectly at, circular and isolated a-spot of radius a, an analytical formula can be obtained, widely known in the literature as Holm’s formula 1.2. Rc = 1 2a (1.2).
The mathematical steps that lead to this formula are presented in Appendix A. The ideas and the method of this development can be interesting in order to fully apprehend the multiple phenomena that will be observed in this thesis:
Concentration of the current lines at the periphery of the a-spots.
Joule distribution in a non-at a-spot.
Interaction of the current lines in a cluster of a-spots.
To the author’s knowledge, Holm’s formula and method are used as a baseline model by most studies in the eld of electrical contacts. However, the hypotheses made to obtain this formula are too restrictive for a wide range of applications. Studies often aim to enlarge Holm’s method in order to model other aspects that do not appear in the case of a single and at a-spot between two homogeneous conductors. This is the spirit that lead this thesis, driven by the need to asses phenomena that may occur at the microscopic scale of contacts subject to very high and impulsive currents.

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Electrostatic study of a single a-spot

As a rst step, nite volume simulations have been performed to gain insight into the current distribution in a-spots, and to study the dependence of the electric resistance l = 500 µm. with the geometry. In this case the simulations are purely electrostatic, meaning that no heating dynamics or time-dependant phenomena are accounted for. The only equation solved in this case is the electrostatic current conservation equation 2.1: r j = 􀀀r r’ = 0 (2.1).
where ‘ (V) is the electric potential, j (A) the current density vector, and (Sm􀀀1) the electric conductivity. The current conservation equation is spacially discretized by means of a nite volume scheme, and the resulting sparse linear system is solved thanks to the SuperLU direct solver [57]. The 2D axisymmetric electrostatic simulations have been performed for a steel cylindrical a-spot ( 107 Sm􀀀1) for dierent values of its radius and thickness. The constriction of the current lines can be observed on Figure 2.4, for the case of a cylinder of radius a = 500 µm and thickness l = 500 µm. It can be noticed that the highest current densities take place at the border of the a-spot. The value of the maximum current density obtained is 5:2 109 Am􀀀2, whereas the mean current density given by the ratio of the total imposed current to the area of the a-spot gives 1:27 109 Am􀀀2. This is in adequation with Holm’s theory [38] that predicts a singular norm j of the current density vector j (Am􀀀2) at the rim (r = a) of a at cylindrical constriction, accordingly to equation 2.2: j(r) = I 2 a 1 p a2 􀀀 r2.

Mechanical tightening of a cylindrical a-spot

The strategy adopted here is to use a simple mechanical model that is compatible with both electric and thermal phenomena. The main drawback of spherical a-spot models is the large deformations that can be obtained at high tightening force, or when the temperature increases signicantly, that may change their shape. For this reason, any  model relying on the spherical a-spot hypothesis would fail after a too large mechanical crushing. On the other hand, cylindrical a-spots do not suer this limitation because the cylindrical geometry is preserved even for large mechanical deformations. The hypothesis of a uniform heating up to the boiling point and then a vaporization front propagating from the periphery to the center also conserves the cylindrical geometry. As a result, a simple mechanical model for cylindrical a-spots has been derived based on the following hypotheses:
A quasi-static mechanical equilibrium is assumed at all time: F = a2 , where F (N) is the tightening force, and (Pa) the mechanical stress.
The constraint follows an elastoplastic deformation law characterized by a yield limit y (Pa), a Young modulus E (Pa) and a creep exponent n.
The cylindrical shape is maintained during tightening.
In the elastic region ( < y), we neglect the eect of the Poisson coecient and the radius of the cylindrical a-spot is kept constant. The constraint is given by Hooke’s law: = E . The strain is simply given by = l=l0, where l0 is the initial thickness of the a-spot.
In the plastic region ( > y), the volume is conserved: the a-spot radius increases when its thickness decreases due to the tightening force. The constraint increases by creep.

Table of contents :

1 Overview of lightning strike to aircraft and sparking threat 
1.1 Overview of lightning strike to aircraft
1.1.1 Lightning and atmospheric electricity
1.1.2 Development of a lightning strike to aircraft
1.1.3 Eects of lightning strikes to aircraft
1.1.4 Sparking risk
1.2 Aircraft protection
1.2.1 Normalized waves and aircraft zoning
1.2.2 Experimental and numerical studies at Onera
1.3 Overview of contact resistance theories
1.3.1 Brief history on electrical contacts science
1.3.2 Denitions and terminology
1.3.3 Holm’s formula for a at circular a-spot
1.3.4 Electric contacts subject to large currents
2 Mult iphysical modeling of single a-spot contacts 
2.1 Geometric simplications
2.2 2D thermoelectric simulations
2.2.1 Electrostatic study of a single a-spot
2.2.2 Thermo-electric coupling
2.3 0D simplied model
2.3.1 Thermo-electric model
2.3.2 Mechanical tightening model
2.3.3 Mechanical tightening of a cylindrical a-spot
2.3.4 Thermo-mechanical coupling
2.4 Parametrical study
2.4.1 Single a-spot contact with constant thickness
2.4.2 Single a-spot contact with mechanical load
2.5 Conclusion
3 Multi physical modeling of multi-spot contacts 
3.1 Interactions between a-spots
3.1.1 Electrostatic interactions
3.1.2 Electrodynamic interactions
3.1.3 Mechanical interactions
3.2 0D multi-spot model
3.2.1 Equivalent circuit model
3.2.2 Two a-spots under constant tightening distance
3.2.3 Two micro-peaks under constant load
3.3 Conclusion
4 Realistic contact subject to a lightning wave 
4.1 Gaussian surface model
4.2 Electrostatic aluminium contact under increasing load
4.3 Realistic multi-spot contact dynamics
4.3.1 Multi-spot contact with constant length
4.3.2 Multi-spot contact under constant load
4.4 Conclusion
5 Sparking model around contacts 
5.1 Gas expansion model
5.1.1 Modelling of the conning media
5.1.2 Pressure equilibrium or plasma expansion speed limit
5.1.3 Current distribution in the plasma
5.1.4 Dichotomy method to solve the plasma expansion
5.2 Quotidian equation of state for metals
5.3 Eect of the plasma expansion on the resistance
5.4 A-spot subject to a D-wave for a constant connement pressure
5.5 A-spot subject to a D-wave with constant expansion volume: parametric study
5.6 Conclusion
6 Conclusion 
A Holm’s formula mathematical steps 
A.1 When can a set of surfaces be equipotentials
A.2 Semi-ellipsoids equipotentials
A.3 Current density on the constriction
A.4 Holm’s formula
B Résumé en Français 


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