Multiphysical modeling of multi-spot contacts

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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.

Electric contacts subject to large currents

In the a-spots, the current density, the Joule eect and the electric eld can be orders of magnitude higher than in the bulk material, and many physical phenomena occurring at  this microscopic scale are of crucial importance for a wide range of industrial applications and research elds, such as the micro-electronics industry [41] [42] [43] , high current connectors, switching devices, breakers technologies [44], railguns [45], or spot-welding process [46].
The main dierences between the many applications involving electric contacts rely in the nature of the materials, the surface treatment, the mechanical load on the contact and especially the current density level. Studies on electric contacts have been widely conducted in the Micro-Electro-Mechanical-System (MEMS) community, involving contacts subject to relatively small current density levels up to a few amperes, but during millions or billions of cycles of a few microseconds [47]. The performances of those systems strongly depend on their ability to sustain such large numbers of operating cycles with a constant electrical behaviour. The degradation and aging of these contacts is generally thought to be a consequence of a-spots failure, as heating and electromigration slowly degrade them until a usually fast breakdown [48]. On the other hand, applications related to welding, rail-guns, high current technologies and lightning stroke involve much higher current densities during much smaller time-scales, and the constraining physical phenomena may be very dierent. During the pulsed arc phase of a lightning stroke, the current can be as high as 200 kA on microsecond time-scales.

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2D thermoelectric simulations

A-spots in aeronautic contacts under lightning stroke conditions may be subject to a Joule heating large enough to heat the solid materials up to the boiling point on microsecond time-scales. The current density distribution, Joule heating, and related thermal phenomena are then very important regarding the contact dynamics at high intensity. A permanent-regime solution for the temperature eld in a at circular a-spot has been obtained in [38], following the so-called ‘􀀀 relation, where the equipotential lines and isotherms are merged. However, considering the time scales of interest during the pulsed arc phase of lightning stroke (1 µs), the thermal permanent regime hypothesis is irrelevant and a more detailed analysis of the unsteady current and temperature evolution in a single a-spot in those particular conditions is required.

Solving of the coupled thermo-electric temporal evolution

To solve the coupled electro-thermal phenomena in an a-spot subject to a lightning current, an explicit temporal scheme has been considered to successively solve the current distribution in the materials, and then the thermal phenomena related to Joule heating: At the beginning of each time-step, the volume internal energy e (Jm􀀀3) is supposed to be known from initial conditions or previous time-steps in each cell of the simulation domain. The temperature eld, the transport properties, such as electric and thermal conductivities, the thermal capacity, and the phases volume fractions are updated based on tabulated data. Then, the electrostatic current conservation equation with Seebeck eect 2.6 is solved to compute the potential distribution: r j = 􀀀r (r’ + SrT) = 0 .

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 Multiphysical 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 Multiphysical 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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