Multiscale Modelling of Aerospace Composites—A Perspective

Parvez Alam

(Institute for Materials and Processes, School of Engineering, University of Edinburgh, UK)

Multiscale Modelling of Aerospace Composites—A Perspective

Parvez Alam

(Institute for Materials and Processes, School of Engineering, University of Edinburgh, UK)

This paper introduces a series of approaches in the modelling of composites with complex multiscale structures and features. Fundamental theory underlying multiscale structures are brought to light and discussed. There can be considerable benefit in the representative application of multiscale modelling techniques to civil aircraft design and this aim of this paper is to provide essential perspectives from several fields of research that may be beneficial to aerospace composites modelling methodologies if applied appropriately.

molecular modelling; interface modelling; continuum modelling; multiscale modelling

1 Introduction

Aerospace composites are expected to perform under extreme mechanical and environmental conditions, and are often seen as lightweight replacements for metallic materials. Such composites are applied within civil aircraft as engine blades, brackets, propellers, wings and even as interior materials. Traditionally, aircraft parts have been modelled to address issues such as impaction and fatigue, with an aim to elucidate information on service lifetimes and failure behaviour. As such, single scale modelling has been the predominant approach and has given rise to date, to considerable advances in improved design and engineering. Nevertheless, single length scale modelling is not necessarily the most optimal route to analysis since composites and the components they are manufactured into are hierarchically structured at multiple length scales. Furthermore, recent developments in biological composites mechanics has drawn attention to the importance of structural hierarchy to properties such as fracture toughness and strength. Multiscale composites design and manufacture is now receiving significant attention, attracting governmental funding, and enthusing R&D engineers. The current pull towards multiscale materials design clarifies that there is a definitive need to develop standardisation procedures for multiscale materials with respect to testing, manufacture, safety and modelling. The objective of this paper is to provide a perspective as regards the multiscale modelling of composites for use within civil aircraft.

2 Scaling in engineered aerospace composites

Fibre reinforced composites (FRP) are base materials used in the engineering of civil aircraft parts. Traditionally, in the modelling of FRP, the smallest component is usually seen as the fibre, which is typically at the microscale. Yet, fibres are commonly surface treated by coating with polymeric materials (sizing), a process known as surface sizing. Surface sizing has the effect of protecting the fibre from degradation and increasing the compatibility of the fibre (adhesion) to its surrounding matrix, which will itself be a thermosetting or thermoplastic polymer. Surface sizing glass fibres commonly entails coating with silane-based polymers, whereas for carbon fibres, epoxies, phenolics or esters are more commonly used. The sizing itself is nevertheless, a molecular to nanoscale composite component and as such, its treatment within a model system is best not ignored. The sizing will not only change the topography of the fibre (varying thus the strength of interlocking between fibre and matrix), but will also increase the specific surface area (SSA), which in turn increases the number of secondary adhesive interaction sites between the fibre surface and the matrix material. These aspects at the molecular to nanoscales will affect the energy required to debond fibre and matrix, which affects the overall strength and fracture energy of the composite at its macroscale. A macroscale model thence, that does not incorporate the effect of molecular to nanoscale interactions at fibre-matrix interfaces, will be inherently flawed in its ability to predict strength and fracture. Multiscale modelling for an FRP is therefore a means by which interface phenomena can be more accurately represented and thus, incorporated within a macroscale model, circumventing thus, the need for correction factors applied to interface definitions, or indeed to entire models, for the sake of fitting.

As a result of extensive increases in computational power, improvements to parallelisation tools and developments in graphics processing unit (GPU) technologies, larger model systems can be simulated with greater relative ease. This has empowered molecular dynamics modelling and coarse grain modelling in a way that allows users to simulate massive clusters of atomistic systems, sometimes reaching up to hundreds of millions of atoms within one single model. As such, molecular to nanoscale sizing interactions with fibre surfaces and with matrix materials can currently be simulated fairly easily. Furthermore, detail from such simulations, such as interaction energies, can be incorporated into larger (macro) scale finite element models at interfaces to more accurately predict the global mechanical properties of FRP. Subsequent sections in this manuscript will detail basic theory in molecular to nanoscale modelling.

3 Atomic interactions

Atom-atom interface interactions and the way in which they are defined are essentially the cornerstones for atomistic modelling. Two predominating philosophies are routinely incorporated into atomistic modelling software. These include empirical as well as quantum mechanical representations. Both representations are currently indispensable in the more prominent atomistic modelling methods including Monte Carlo, Molecular Dynamics and the Lattice Energy Methods.

Empirical representations are borne essentially, from currently accepted empirical equations that have been derived to describe atomic separation. A benefit of this approach to the atom-atom interface is that it can be easily modified to the parameters of a specific equation. A classical example of an empirical model that is still used in simulations concerning certain ionic materials is the long range Coulomb Interaction Model with short range repulsion, Equation (1)[1]. Here, E is energy,ris the atom-atom spacing,Landηare correction factors that correspond to the empirical data set. An alternative, simpler model has greater functionality in that, unlike the Coulomb Interaction Model, harmonic functions such as torque and bond angle can be easily incorporated, Equation (2).

E(r) =Le(-r/η)

(1)

E=Lxη

(2)

Quantum mechanical models are different and are essentially probabilistic models. They have no reliance on empirical data and have no need for correction factors or simulation specific modifications to constitutive interatomic relations. Quantum mechanical models yield reliable simulation results and have particular benefit inabinitiosimulations where electrostatic optimisations of complex molecules may be necessary. Being a probabilistic approach, quantum mechanical models also have the added benefit of predicting the charge sharing effect of any deviation of a molecule, thus allowing for subtle variations in chirality, and isomerism to be modelled and compared with relative ease. In the quantum mechanical approach, electrons travel within electrical fields created by nuclei. The coinciding wave functions, Ψ, are shared between existing nuclear, vibrational, rotational and electronic wave functions.

Ψ = Ψelectronic+ Ψvibrational+ Ψrotational+ Ψnuclear

(3)

E=Eelectronic+Evibrational+Erotational+Enuclear

(4)

An alternative to using wave functions in the quantum mechanical methods is DFT (density functional theory)[2]. The fundamental difference between the two methods here is that DFT employs electron densities in the determination of interatomic activity. DFT uses the approximation shown in Equation (5), in which energy is a summation of both kinetic and potential energies. Here,Ukis the kinetic energy,Up-eis the potential energy of electrons arising through repulsion andUn-eis the potential energy of atomic electrons arising through attraction to the nucleus, which is positively charged.

E=Uk+Up-e+Un-e

(5)

4 Molecular interactions

Molecules are typically modelled from the atomistic principles previously described. When consideringmolecularinterfaces, there is a need to contextualise the specific interface under scrutiny. Considerable efforts have been made in this respect within the biological sciences. Molecular modelling methods such as MD and MC can be useful for modelling e.g. molecular folding mechanisms and intermolecular interactions. However, when modelling very large sets of molecules containing millions, perhaps billions of atoms, the need for structural and interfacial approximations becomes apparent. There are a few important articles that can be cited as useful templates to dealing with these inherently complex systems.

One such article[3]uses probabilistic functions in the form of Bayesian networks to distinguish interfacial characteristics between interacting molecules. The Bayesian network is very useful since it is essentially a steered probabilistic method representing variables and their conditional dependencies through a directed non-cyclic graph. The strength of the method is in its simplicity. Dependent variable outputs are eithertrueorfalse, allowing thus for the incorporation of numerous variables with relative ease. The Bayes hierarchical Bayesian network can most simplistically be written,

p(a,b|x) ∝p(x|a)p(a|b)p(b)

(6)

Here, a is a prior originating from the prior probabilityp(a) of the fundamental Bayesian formp(a|x) ∝p(x|a)p(a). The prior a depends on parameter b all of which relate to x data. A likelihood,p(a|b), and a new prior,p(b), replaces the prior probability,p(a) for the hierarchical form in Equation (6). Bradford and co-workers[3]distinguish between interacting and non-interacting patches at molecular interfaces by five variables. These are shown in Figure 1.

Jones and Thornton (1997)[4]consider a different approach to studying intermolecular interaction sites. They worked from a basis of molecular interfaces being hydrophobic[5-6], using proteins as examples. They studied essentially the same variables shown in Figure 1, but in discretepatchesdefined by a finite number of residues. A similar patch by patch method used by Preissner et al. (1998)[7]suggests that approximations based on molecular structure can be made for larger scale simulations. For low density systems simulations of structures are might employ Monte Carlo based approximations, Equation (7), such as have been used with LAMPPS open source software. This is a probabilistic (statistical) approach to approximating interface properties. In the equation, we have N configuration, of a system withξ1,ξ2,ξ3,ξ4…ξPwhereP(ξ) is a given probability distribution.

(7)

AbisectormethoddevelopedbySeongetal. (2011)[8]haspotentialforspeedingupmolecularclusterinterfaceinteractionpredictions.Thismethodcantakeessentially,largeclustersofatomsandbisectbetweenthemtoformaplanetraversingthroughEuclideanspaceyetconnectingsolelytheintersectingpoints.Thebisectorexistsinarealtimeenvironmentandasaconsequence,agreatvarietyofcomputationscaneasilybeappliedwithverylittlecostinmemory.

5 Nanomechanics

Whensurfacestructuresarenano-scale,theywillexhibitdimensionsbelow100nminatleastonedirection.Recentworkhasshownthatnanocrystalsincontactwithsoftamorphousphasesgiverisetosemi-crystallinemetastablestatesthatcantakebothamorphousandcrystallineformsdependingontheloadingconditions[9-10].Inhierarchicalmodelling,theremaybeaneedtotransitionfromthemoleculartonano,orevenmicronscales.Typicallyhigherscalemodelswilllackatomisticdetail.Yetdetailsinlowtohighscaletransitions,andinthebettercasescenario,specificpropertiesshouldbeincorporatedformoreappropriatemodelpredictions.Oneexampleisthenano-effect.Thenanoeffectcanbestbedescribedasrelatingtothebalancebetweenbulkandfreesurfaceproperties.Atthemolecularlevelsurface/molecularenergiesdominate,whereasatthemacroscale,thepropertiesofabodyofmatteraredeterminedbytheenergyofthebulk.Surfaceandbulkpropertiesareconsiderablydifferentandatthenano-scale,asafunctionofatomicvolume,thesurfacescontributeconsiderablytotheoverallpropertiesofthenanomaterialandcannotbeneglected.

Wemaybeginbyconsideringanano-objectasconsistingoftwophases,abulk,α,andasurface, β[11].Energetically,thesecanbeconsideredtoexistinastateofcontinuumwithaninterfaceseparatingeachphase.Theelasticsurfaceenergy[12]betweenthetwophases, Us,canberepresentedbyEquation(8).

Us= U - (Uα+ Uβ)

(8)

HereUistheinternalenergyofthesystemand(Uα+ Uβ)isthetotaluniformenergyinbothparts.Notably, (Uα+ Uβ)canbemodifiedtoincludeanyvariationsinenergyatthesurface.Itshouldbenotedthatthesurfaceisinacontinualstateof(molecular)motion.Whenthenanostructuresurfacedominatesthebulk,thismotioncanbeconsiderableandwork energyatthesurfaceneedstobeaccountedfor.ThiscanbedefinedaccordingtoEquation(9).

Ws= γdA

(9)

Here, Aistheareaofthesurfaceandγisthesurfacestress.Surfacedeformationofacompressiblematerialinducesavolumetric, V,changeineachphasesuchthat:

dVα= -dVβ

(10)

Thegeneralsurfacedeformationcanbeexpressedasastrain, εs:

εs= dA/A

(11)

andthesurfacestressisapproximatedfromShuttleworth(1950)[13]:

γ =Es+ ∂Es/∂εs

(12)

whereEsisthefreeenergyofthesurface.Differentapproximationsforthesurfacestresscanbemadefollowing[14]as:

whichcanalternativelybeexpressed:

-∂Es/∂E= q + (Es-γ) ∂εe/∂E

(14)

whereεeistheelasticsurfacedeformation, Eistheelectrodepotentialandqisthesurfacechargedensity.Withrespecttotheaboveequations,itiswellworthnotingthattodate,nonehaveyetbeenprovenexperimentally[15].NumerousflawscanbebroughttolightinboththeShuttleworthandtheCouchmanforms.Marichevsuggeststhatsincethe1stand2ndGokhsteinequationshavebeenconfirmedexperimentally, (see[16-17])thattheyratherbeusedasdeterminantsofsurfacetensionforsolidmaterials.

(∂γ/∂q)ε= (∂E/∂εe)Q-1stGokhsteineq.

(15)

whereQisthesurfacecharge.

-∂γ/∂q = q + ∂q/∂εe-2ndGokhsteineq.

(16)

Inananomaterialbothphasesαandβcanbeconsideredsolids.Itfollowsthen,thatwhenthesurfaceofαstrains,workisdoneattheinterfacebetweenαandβ.Duringdeformation,thetotalworkforasolid-solid interface, Ws-s,combinesthreeseparateworkterms.

Ws-s= Es-s(dAs-s) + γs-s(dAα)

+ γs-s(dAβ)

(17)

wherethetermEs-s(dAs-s)istheworkrequiredtocreateanewsurface, γs-s(dAα)istheworkrequiredtodeformphaseαabouttheinterfaceandγs-s(dAβ)istheworkrequiredtodeformphaseβabouttheinterfacethesameextentasα. Es-sistheinterfaceenergy, γs-sistheinterfacestress, As-sistheinterfaceareaandAαandAβaretheareasrelatedtodeformationinαandβsolidsrespectively.Thoughtotalworkalsoexistsinmicroandmacroscalematerialswithinterfaces,becausethebulkphasedominatesdimensionallyandwithregardstoproperties,thetotalworktermatananointerfacecaneffectivelybereducedto:

Ws-s= Es-s(dAs-s)

(18)

Thissimplificationispossiblebecause{γs-s(dAα) + γs-s(dAβ)}isnegligibleandthemagnitudeof{γs-s(dAα) + γs-s(dAβ)}relativeto{Es-s(dAs-s)}heightensasthedimensionsofeachphasecomesclosertothenanoscale.Thus,bothsurfaceandinterfacialeffectsdescribedherecannotbeignoredatthenanoscale.Indeed,surfacedominatingcharacteristicsatthenanoscalemaywellbeacauseformechanicallybeneficialphenomenasuchasmolecularpinning[18].

Thetotalmechanicalworkinacompositematerial, Wc,isasummationoftheworkdoneinthebulkmaterialofphasesαandβ, WαandWβrespectivelywiththeworkdoneatthesolid-solidinterfaces.Inmuchlarger-than-nanoscalematerials,

(Wα+ Wβ) >> Ws-s

(19)

andthereforeWc→ (Wα+ Wβ)

(20)

whileinnanostructuredmaterials,

(Wα+ Wβ) << Ws-s

(21)

thusWc→ Ws-s

(22)

6 Interfaces at higher length scales

Atthenanoscale,tractiveforcesmaystillbemodelledasmolecularsliding.However,asweincreasethelengthscale,molecularmodelsturnouttobeimpracticalinmanyways.Thereisanincreased(perhapsunnecessarilyso)CPU/GPUdemandalongsidememoryconsumption.Thereisalsomoredatathatrequiresconsiderableefforttoanalyse.Finallythereistheunfortunate,butoftenexperiencedscenariooflongrun-timemodelsthatturnouttohavebeendevelopedincorrectly.Thepredominatingquestionwouldbe:isthereanyaddedbenefitofscalinganatomisticmodelupwithmoreatoms,orcanwereachthesameconclusionsandinterpretationsfromourmodelbymakinguseofmoreappropriatemodellingmethodsforhigherscalematerials?Theauthorofthismanuscriptconsidersthatobservedincoherencybetweenlength-scalesmayinfactbeafunctionofinadequatelyorindeedinaccuratelydescribedinterfaces.Inthissectionwedescribesomesimplemeansbywhichlargerscalestructuralmodelscanbemademorerepresentativeofhierarchicalcomposites.

Energytransferandinterfacialmotionhavsalreadytosomeextentbeencoveredintheprevioussectiononnanomechanics.Neverthelessabovethenanoscale,workenergyattheinterfacetakesthefullformofEquation(17)withnofurthersimplification.Thisenergyisholisticinthat,thedeformationthatgivesrisetoitmaybeoneoracombinationofamultitudeofforms.Thesemayincludetypicallyfriction,viscoelasticorviscoplasticresistance,crackresistanceandtopographicalinterlocking.Itisworthmentioningthateachcharacteristicisessentiallyafunctionofmolecularlevelbehaviour.Thus,appropriateapproximationsmustbemadewhenapplyingmolecularmechanicalcomputationstomacro-scaleinterfaceproperties.Thereareanumberofgroundlevelrelationshipsthatarealreadyinusethatcanbemanipulatedtosemi-empiricalformsorfrombasismolecularcomputations.

Wecanstartsimplybyconsideringmechanicalenergytransferacrossaboundary.Assumingmolecularlevelcomputationsarecorrect,thataboundarysystemisincontinuum,andthattheenergyscalarfollowsfromthepreviousboundary,wecaneasilyreturntoEquation(17),whichaccountsforthethreephasesofα, βandinterface.OneimportantissuetoaddressinregardstoEquation(17)isthattheeffects from specificinterfacecharacteristicsneedtobeincludedfortheformtofunctionmorerealisticallyintransitionfrommolecularmodellingtomacro-scale(e.g.finiteelement)modelling.Whenfrictionisthedominantmodeofirrecoverableinterfacialdeformation,changestothisgeneralunderstandingofdeformationalworkaretobemodified.EnergywillbelostwhenmoleculesslideandthetermEs-s(dAs-s)requiresthus,modificationtoaccountfornon-linearreductionsinworkenergyasafunctionoftheinitialtransitionfromstoredworkenergytomolecularsliding, fs,andsubsequentkineticfriction, fd,wheremoleculesareincontinuousslidingmotion.Insuchacasetherefore,

Es-s(dAs-s) =f(fsN, fdN)

(23)

WhereNisthenormalforceactingbetweenphasesαandβ.Thisapproximationhasnumerouslimitationsincluding;anassumptionthatthenormalforceandfrictionalforcearelinearlyproportional,thatthecontactingareaisnotinastateofatomic saturation (i.eatom-atomcontactisnotincontinuum),andthattheoreticalmaterialscanfollowwhatisessentiallyanempiricalconstruction.Thegreatversatilityofthismodelhoweverisitssimplicityanditisthankstothesimplicitythatitcanbemodifiedviasemi-empiricalortheoreticalcorrections.

Suchcorrectionscanbetunedtothematerial propertiesoftheinterface.Keepingtheassumptionthatnormalandfrictionalforcesareproportional,alternativeforceapproximationsrelatedtoe.g.viscoelasticitycanbemade.TheimportantpointhereisthatthevariableNcanbemanipulatedrelativelypainlesslyinseriesformtoestablish(inthecaseofviscoelasticity)theforceperunittimeasafunctionofthestrainrate.Theseriesmaybeappliedfromanempiricallydetermined(orindeedcomputationallydetermined)datasets,orfromadirectmathematicalfunction,suchasinaPronyseries.

Defininganinterfacewithmaterialpropertiesandfractaltopographicalirregularityissomewhatmorechallengingandrequiresconsiderationofthelocaldeformationalcharacteristicsofindividualprotuberances,thewaybywhichtheprotuberanceaffectsforcetransferandinterfacemotion,andthesubsequentnon-linearvariationsasafunctionofprotuberanceshape.Discretisationisofcoursethenormalroutetodealingwithcomplexgeometry.Thissaid;fractalsurfacesareproblematicinthattheywillreachadimensionthatcannolongerberenderedandinterpolationfunctionsshouldberatherusedtodefinefractalinducedbehaviour.GoodderivationsofsuchfunctionscanbefoundinMistakdisandPanagouli(2003)[19].Inshort,thefractal effectincreasesfrictionatlowerordersbutasthefractalorderincreases,itseffectsonfrictionbecomeslessobviousthanitsprecedingorders.

7 Conclusions

Inthispaper,webrieflyhighlighttheimportanceofinterfaceconsiderationsinthehierarchicalmodellingofcomposites.Weconsiderthefundamentalsofinterfacemodellingfromtheatomisticlevel,throughthemolecularandthenanolevels,andfinishatthemacrolevelofmodelling.Theunderstandingandapplicationofjustifiableinterfaceconditionsateverylengthscaleiscriticaltothedevelopmentofstable,fastworking,effective,yetaccuratemodels.

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10.19416/j.cnki.1674-9804.2017.03.008