The Development of a Multi-Physics Approach for Modelling the Response of Aerospace Fastener Assemblies to Lightning Attachment William Paul Bennett1aStephen Timothy Millmore1and Nikolaos Nikiforakis1

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The Development of a Multi-Physics Approach for Modelling the

Response of Aerospace Fastener Assemblies to Lightning Attachment

William Paul Bennett,1, a) Stephen Timothy Millmore,1and Nikolaos Nikiforakis1

University of Cambridge,

Laboratory for Scientiﬁc Computing, Cavendish Laboratory, Department of Physics,

Cambridge, CB3 0HE, UK

This work is concerned with the development of a numerical modelling approach for studying the time-accurate re-

sponse of aerospace fasteners subjected to high electrical current loading from a simulated lightning strike. The elec-

tromagnetic, thermal and elastoplastic response of individual fastener components is captured by this method allowing

a critical analysis of fastener design and material layering. Under high electrical current loading, ionisation of gas

ﬁlled cavities in the fastener assembly can lead to viable current paths across internal voids. This ionisation can lead to

localised pockets of high pressure plasma through the Joule heating effect. The multi-physics approach developed in

this paper extends an existing methodology that allows a two-way dynamic non-linear coupling of the plasma arc, the

titanium aerospace fastener components, the surrounding aircraft panels, the internal supporting structure and internal

plasma-ﬁlled cavities. Results from this model are compared with experimental measurements of a titanium fastener

holding together carbon composite panels separated by thin dielectric layers. The current distribution measurements

are shown to be accurately reproduced. A parameter study is used to assess the internal cavity modelling strategy and

to quantify the relation between the internal cavity plasma pressure, the electrical current distribution and changes in

the internal cavity geometry.

Keywords: lightning strike; multi-physics; aerospace fastener; multi-material; ghost-ﬂuid method; Joule heating; struc-

tural response

I. INTRODUCTION

Composite materials now make up more than 50% of a

modern aircraft design by weight1. This is due to higher spe-

ciﬁc strength and better fatigue properties for high tension

components than conventional aluminium alternatives. Com-

posite materials also have lower electrical and thermal con-

ductivity than the supplanted aluminium, which can adversely

affect the response of aircraft components to lightning strikes.

The initial stages of a lightning strike results in a large current

ﬂow through the aircraft skin, which, for materials with low

electrical conductivity can result in large energy input through

Joule heating. In Carbon Fibre Reinforced Polymer (CFRP),

for example, the high energy input can result in ﬁbre fracture,

resin decomposition, delamination and thermal ablation.

The interaction of lightning with aircraft exterior surfaces

can be further complicated by the integration of conductive

components, such as metallic fasteners, with higher conduc-

tivity than the surrounding composite substrates. Aerospace

fasteners, used to join skin panels, are commonly manufac-

tured from titanium alloys that are lightweight, strong and

corrosion resistant. These can, however, act as a preferred

pathway for the current to access the internal airframe and em-

bedded carbon composite panels. High current ﬂow through

a fastener can arise either from a direct attachment of a light-

ning arc to the fastener head or indirectly as current conducted

from a remote attachment site. In addition to paint, panel and

sealant damage, the high current ﬂow in the metallic fastener

can cause a thermal ejection of hot particles (energetic dis-

charge) from the interfaces between fastener components. En-

a)Corresponding Author; wpb22@cam.ac.uk

ergetic discharge from fastener assemblies can represent a po-

tential threat in safety critical regions of an aircraft, such as in

integrated fuel tanks where signiﬁcant fuel vapour is present.

Chemartin et al.2outline three important mechanisms

through which a fastener can undergo energetic discharge.

The ﬁrst, termed ‘outgassing’, results from the current pas-

sage across small resistive gaps between the fastener and skin.

The formation of a plasma in the gap, and subsequent Joule

heating, increases the internal plasma pressure until a blow

out of sparks and hot gas occurs at the component interface.

Further information regarding the characteristics and causes

of outgassing (also known as pressure sparking) is provided

in the comprehensive review of Evans3. The second energetic

discharge mechanism outlined in Chemartin et al.2is thermal

sparking between contacting components, i.e., the fastener nut

and rib. Odam et al.4,5 suggests that thermal sparking occur

when a very high current is forced to cross a joint between two

conducting materials, which have imperfect mating between

their surfaces. This process is noted to be distinct from volt-

age sparking, which occurs when the voltage difference be-

tween two conducting materials is sufﬁcient to break down the

intervening medium, whether this is air or another dielectric

medium. The ﬁnal energetic discharge mechanism outlined

in Chemartin et al.2is edge glow. This is deﬁned by Revel

et al.6as consisting of a bright glow combined with strong

material ejections, and occurs on the edges of composite ma-

terials. Two mechanisms responsible for the presence of edge

glow are proposed in the available literature. The ﬁrst of these

occurs when the potential difference between adjacent com-

posite plies exceeds a threshold value, irrespective of whether

a pre-existing contact between the plies is present. The sec-

ond mechanism occurs when the power deposition into the

substructure is above a threshold power value that produces a

glow due to heating of sealant.

arXiv:2210.05360v1 [physics.comp-ph] 7 Sep 2022

These energetic discharge mechanisms can been distin-

guished using an appropriate experimental approach. Day and

Kwon7describe a method which analyses light over a narrow

spectral range using a spectrometer and can identify hot par-

ticle ejection, arcing and edge glow. Work by Haigh et al.8,

in contrast, applies two-colour spectroscopy to estimate the

temperature of sparks emitted using red/blue ratios, compar-

ing with baseline nickel and tungsten sparks. Later work by

Haigh et al.9uses photography to detect light sources that are

potential ignition hazards on a T-joint section with multiple

fasteners. Focusing speciﬁcally on outgassing events, Mulaz-

imoglu and Haylock10 relate sparking intensity to the fastener

material and geometry choice using energy dispersion spec-

trometer chemical analysis, and determine that the principle

constituent of the ejected particle debris in question is poly-

sulphide sealant, with small quantities of metallic droplets and

carbon ﬁbre particles. They surmise that the chemical compo-

sition of the debris mean that electrical arcing occurs between

the bolt and the CFRP hole surface. The ablated material is

then carried by hot gases during the outgassing ejection event

due to the arcing pressure. The microstructure of the result-

ing damage is analysed using scanning electron microscopy

and the outgassing characteristics that result from deliberate

design additions are analysed. These additions include the in-

troduction of metallic sleeve components, dielectric bolt head

coverings and bolt-line metallic meshes.

The wide range of potential fastener conﬁgurations, along

with the various mechanisms through which sparking can oc-

cur, mean that computational simulation can provide an ef-

ﬁcient and cost effective technique for rapidly exploring the

available parameter space. Computational techniques can also

provide a useful tool in the design of experimental testing for

proposed fastener conﬁgurations. Finite element methods, for

example, are a common, single-material approach to model

the effects of high current ﬂow through carbon composite sub-

strates. This is achieved through prescribing a current wave-

form along the upper surface, see, for example Ogasawara et

al.11, Abdelal and Murphy12, Guo et al.13, Dong et al.14, Wang

et al.15 and Liu et al.16 and for commercial software by Wang

and Pasiliao17, Kamiyama et al.18,19, Fu and Ye20 and Evans et

al.21. The prescription of a current waveform along the upper

surface of a composite material is perhaps most suited to cases

in which the damage to individual ply and resin layers is of di-

rect interest, since inter-ply loading and damage characteris-

tics can be efﬁciently modelled using modest computational

resources for comparison with experimental results. Mod-

elling a lightning strike using this approach in isolation can,

however, neglect the dynamic change in current and pressure

loading on the upper surface of the substrate by an evolving

plasma arc, and the non-linear feedback from these changes,

which in turn affect the arc behaviour.

To allow the evolution of the plasma arc to effect the

time-dependent substrate current and pressure loading, the

‘co-simulation’ approach couples two software packages or

approaches. Using this technique a magnetohydrodynamic

(MHD) code can be used to describe the evolution of temper-

ature, pressure and current density within the arc. The results

of running the MHD code are then fed as initial and boundary

conditions to a second simulation that models the mechan-

ical, thermal and electrodynamic evolution of the substrate.

Examples of this approach include Tholin et al.22, who cou-

ple two distinct software packages (Cèdre and Code–Saturn)

to model the plasma arc attachment to single material sub-

strates, and Millen et al.23 who couple two commercial soft-

ware packages (COMSOL Multiphysics and Abaqus FEA)

to model the mechanical loading and electromagnetic effects

on a carbon composite substrate. Kirchdoerfer et al.24,25 ap-

ply the co-simulation approach to aerospace fasteners, cou-

pling the results from COMSOL Multiphysics with a research

shock-physics code developed at Sandia National Laborato-

ries (CTH). The electric and magnetic ﬁelds, and current den-

sity, are solved in COMSOL and used to determine Joule heat-

ing effects. One-way coupling is then applied with the CTH

solver using an effective heating power, computed from the

modelled Joule heating, allowing the simulation of the ﬂuid-

structure interaction. This one-way coupled solution is used

to determine the temperature and pressure rise in an internal

cavity between a bolt, nut, and surrounding CFRP panels, with

the ﬁnal pressure rise being compared to experimental results.

The co-simulation approach results in a one-directional

coupling between materials where the substrate behaviour

does not inﬂuence the evolution of the arc. However, experi-

mental results, such as the optical measurements of Martins26,

indicate that changes in the electrical conductivity and sub-

strate shape can alter the arc attachment characteristics. This

work uses a multi-physics methodology introduced in Mill-

more and Nikiforakis27, to simulate a dynamic non-linearly

coupled system comprising the plasma arc, titanium aerospace

fastener components, surrounding aircraft panels and the in-

ternal supporting structure. The electromagnetic, thermal and

elastoplastic response of individual fastener components is

captured by this method, allowing a critical analysis of fas-

tener design and material layering. Dynamic feedback be-

tween the components is achieved in this multi-physics ap-

proach by simultaneously solving the governing hyperbolic

partial differential equations for each material in a single sys-

tem. The non-linear dynamic feedback between adjacent ma-

terials achieved by this approach provides a distinct improve-

ment over existing numerical methods for modelling lightning

strike attachment. The underlying numerical methods and im-

plementation used in this paper are outlined in Millmore &

Nikiforakis27, and extended in Michael et al.28 and Träuble

et al.29. Millmore and Nikiforakis compare numerical results

from the non-linear multi-physics approach used in this paper

with the optical measurements of Martins26, for a plasma arc

attachment to a single material substrate, and demonstrate that

the two-way interaction between the substrate and plasma is

accurately captured by this method.

The key aim of this work is to use this approach to model

the rise in pressure within an internal cavity between a tita-

nium fastener and a CFRP panel. The breakdown of air in this

cavity requires a strategy for deﬁning an internal plasma re-

gion, and the inﬂuence of parametric changes in the cavity ge-

ometry on the pressure rise through Joule heating can be stud-

ied. This mechanism is acknowledged by Chemartin et al.2

and Evans3as a major contributing factor in outgassing, hence

this paper focuses on this mechanism over thermal sparking

and edge glow. An overview of the multi-physics methodol-

ogy is given in section II and an assessment of the method-

ology for modelling lightning strikes on aerospace fasteners

is made in section III. The model is validated by comparing

to experimental measurements of a fastener holding together

carbon composite and aluminium panels, electrically isolated

from each other by a dielectric layer. The multi-physics

methodology is then exercised in section IV to investigate how

parametric changes in the design of an idealised fastener in-

ﬂuence the pressure loading, component temperature rise and

electrical current path characteristics. This study considers a

variety of fastener design and material layering choices, and

permits the pressure rise in conﬁned internal plasma regions

to be numerically quantiﬁed. Section V provides a summary

and an outlook to future work.

II. MATHEMATICAL MODEL DESCRIPTION

This section gives an overview of the mathematical model

used to simulate the response of aerospace fastener assemblies

to lightning attachment.

A. Plasma model

The plasma arc is described through a system of equa-

tions suitable for simulating the evolution and ionisation of

air caused by an electric discharge. This model must describe

the change in the chemical composition of the plasma under

the high temperatures of the arc. Additionally electromagnetic

effects can have a strong inﬂuence on the arc evolution. This

requires an MHD formulation which includes the effects of

current ﬂow in the arc through the Lorentz and Joule effects.

For lightning plasma, this system can be assumed to be under

local thermodynamic equilibrium22,26. The governing system

of equations is therefore given by

∂ ρ

∂t+∇·(ρv) = 0,(1)

∂

∂t(ρv) + ∇·(ρv⊗v) + ∇p=J×B,(2)

∂E

∂t+∇·[(E+p)v] = v·(J×B) + ηJ·J−Sr,(3)

−∇2A=µ0J.(4)

where ρis density, vis velocity, pis the pressure, Eis the

total energy, Jis the current density, Bis the magnetic ﬁeld,

ηis the resistivity of the plasma, Sris a term for the radiative

losses from a heated material, and Ais the magnetic vector

potential, related to the magnetic ﬁeld through B=∇×A.

The current density is governed by the continuity equation.

∇·J=−∇·(σ∇φ) = 0 (5)

where φis the electric potential and σ=1/ηis the electri-

cal conductivity of the plasma arc.

1. Equation of state

The system of equations (1)–(4) comprises 8 equations for

10 unknown variables, ρ,E,pand the vectors v,Band φ.

These are closed using equation (5) and an equation of state

which describes the thermodynamic properties of the sys-

tem. The equation of state is typically written in the form

p=p(e,ρ), where eis the speciﬁc internal energy, and is

related to the total energy through E=ρe+1

2ρv2. The equa-

tion of state of a plasma is complex, since its thermodynamic

properties depend strongly on the degree of ionisation.

To capture this behaviour, a tabulated equation of state for

air plasma is used in this paper. This was developed by Träu-

ble et al.29, based on the theoretical model of d’Angola et

al.30. This considers the 19 most important species present in

an air plasma at temperatures up to 60,000 K over a pressure

range of 0.01 <p<100 atm.

B. Elastoplastic model

In this work, the material substrate uses an elastoplastic

solid model described by Barton et al.31 and Schoch et al.32,33,

based on the formulation by Godunov and Romenskii34. The

plasticity model follows the work of Miller and Colella35. The

elastoplastic implementation used in the present work is de-

scribed in Michael et al.28, therefore only a brief overview is

provided here for completeness.

To account for the material deformation, the elastic defor-

mation gradient is deﬁned as

i j =∂xi

∂Xj(6)

which maps a coordinate in the original conﬁguration, Xi, to

its evolved coordinate in the deformed conﬁguration, xi. The

deformation gradient, along with the momentum, energy and a

scalar material history parameter, κ, give a hyperbolic system

of conservation laws

∂ ρui

∂t+∂

∂xk(ρuiuk−σik) = 0,(7)

∂ ρE

∂t+∂

∂xk(ρEuk−uiσik) = ηJiJi(8)

∂ ρFe

i j

∂t+∂

∂xkρukFe

i j −ρuiFe

k j=−ui

∂ ρFe

k j

∂xk+Pi j,(9)

∂ ρκ

∂t+∂

∂xi(ρuiκ) = ρ˙

κ.(10)

−∇2Ai=µ0Ji(11)

where σis the stress tensor, given by

σi j =ρFe

∂e

∂Fe

(12)

and eis the speciﬁc internal energy. The scalar material his-

tory, κ, tracks work hardening of the material through plastic

deformation. Source terms associated with the plastic update

are denoted P. The density is given by

ρ=ρ0

detFe(13)

where ρ0is the density of the initial, unstressed material and

the system is coupled with compatibility constraints on the

deformation gradient

∂ ρFji

∂xj=0 (14)

which ensure deformations of the solid remain physical.

1. Equations of state for elastoplastic materials

As with the plasma model, in order to describe the thermo-

dynamic properties of the model, and to close the system of

equations (7)–(11), an equation of state is required. A variety

of solid materials are used in aerospace fasteners, in particular,

aluminium, composite materials, titanium, dielectric coatings

and sealants. Additionally, experimental studies include an

electrode, which is typically tungsten or steel. This is used to

generate a plasma arc and evolution within this electrode does

not affect the simulation dynamics (and damage issues are not

an issue for simulation purposes). Therefore any conductive

metal is typically used as a substitute.

Aluminium is typically described through a Romenskii

equation of state36, for which the speciﬁc internal energy of

the metal is given by

e=K0

2α2Iα/2

3−12+cvT0Iγ/2

3(exp(S/cv)−1) +

2I(β/2)

3−I2

(15)

where

K0=c2

0−4

3b2

0,B0=b2

0(16)

and these are the bulk speed of sound and the shear wave

speed respectively, with c0being the sound speed, Sis the

entropy, cvis the heat capacity at constant volume and T0a

reference temperature. The constants α,βand γare related

to the non-linear dependence of sound speeds on temperature

and density, and must be determined experimentally for each

material. The quantities IKare invariants of the Finger strain

tensor G=F−TF−1, and are given by

I1=tr(G),I2=1

2h(tr(G))2−tr G2i,I3=det|G|.

(17)

The entropy is computed from the primitive variables and a

reference density ρ0,

S=cvlog





ρ−K0αρ

ρ0α((ρ)ρ0)α−1

γcvT0ρ

ρ0γ+1



.(18)

The parameters for aluminium, as used in Barton et al.37, are

given by

ρ0=2710 kg m−3,cv=900 J kg−1K−1

T0=300 K,b0=3160 m s−1

c0=6220 m s−1,α=1

β=3.577,γ=2.088.

(19)

Carbon composites are anisotropic materials, and thus have

a more complex equation of state. These are described in

the work of Lukyanov38, though their implementation in this

model is, at present, beyond the scope of this work. Ad-

ditional work is required within the elastoplastic model de-

scribed above to deal with material anisotropy. Following

Millmore and Nikiforakis27, an isotropic approximation to

CFRP can be made, which is suitable for modelling ‘with

weave’ and ‘against weave’ conﬁgurations due to the sym-

metries of the problem. This isotropic approximation uses the

equation of state as for aluminium, but with electrical conduc-

tivity values that approximate CFRP.

A Romenskii equation of state is used for modelling tita-

nium components due to the lack of readily available equa-

tions of state for this material. In the conﬁgurations consid-

ered in this work, electromagnetic effects dominate, and thus

this approximation does not have a signiﬁcant effect on the

evolution. The parameters for this equation of state are

ρ0=8030 kg m−3,cv=500 J kg−1K−1

T0=300 K,b0=3100 m s−1

c0=5680 m s−1,α=0.596

β=2.437,γ=1.563.

(20)

The dielectric coatings on aircraft skins are similarly com-

plex, often comprising several layers of material and equa-

tions of state for these materials are not openly available. For

such coatings used in this work, the plastic polymethyl methy-

lacetate (PMMA) is used, which has been widely studied due

to its use in improvised explosives39,40. For PMMA, a Mie-

Grüneisen equation of state is employed which directly relates

pressure and density through

p=ρ0c2

s(1−sη)1

1−sη−1,η=1−ρ0

ρ(21)

where c0and ρ0again refer to the speed of sound in the mate-

rial and the reference density, whilst sis a single experimen-

tally determined coefﬁcient. These quantities are given by

ρ0=1180 kgm−3,c0=2260 ms−1,s=1.82.(22)

For all materials, an electrical conductivity is also required

for use in equation (8). Over the temperature ranges observed

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TheDevelopmentofaMulti-PhysicsApproachforModellingtheResponseofAerospaceFastenerAssembliestoLightningAttachmentWilliamPaulBennett,1,a)StephenTimothyMillmore,1andNikolaosNikiforakis1UniversityofCambridge,LaboratoryforScienticComputing,CavendishLaboratory,DepartmentofPhysics,Cambridge,CB30HE,UKThiswo...

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The Development of a Multi-Physics Approach for Modelling the Response of Aerospace Fastener Assemblies to Lightning Attachment William Paul Bennett1aStephen Timothy Millmore1and Nikolaos Nikiforakis1

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