On the buckling and post-buckling of core-shell cylinders under thermal loading

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On the buckling and post-buckling of core-shell cylinders under thermal loading

Fan Xu, Radhi Abdelmoula, Michel Potier-Ferry

To cite this version:

Fan Xu, Radhi Abdelmoula, Michel Potier-Ferry. On the buckling and post-buckling of core-shell

cylinders under thermal loading. International Journal of Solids and Structures, Elsevier, 2017, 126-

127, pp.17 - 36. �10.1016/j.ijsolstr.2017.07.024�. �hal-01680246�

On the buckling and post-buckling of core-shell cylinders under thermal loading

Fan Xu

^a,

Radhi Abdelmoula

^b

, Michel Potier-Ferry

^c

a Institute of Mechanics and Computational Engineering, Department of Aeronautics and Astronautics, Fudan University, 220 Handan Road, Shanghai 200433, PR China

b Laboratoire des Sciences des Procédés et des Matériaux, LSPM, UPR CNRS 3407, Université Paris-Nord, 99 Avenue J.B. Clément, Villetaneuse 93430, France

c Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, LEM3, UMR CNRS 7239, Université de Lorraine, 7 Rue Félix Savart, BP 15082, 57073 Metz Cedex 03, France

Keywords:

Post-buckling Core-shell cylinder Bifurcation analysis Continuation technique Thermal wrinkling Therehasbeenastrongandrecentresearchactivitytoobtaintunablewrinklingpatternsinfilm/substrate

systems,whichproposestousegeometriccurvatureasacontrolparameter.Thispaperstudiescore-shell cylindricalsystemsunderthermalloads,withtheaimtodescribepossiblewrinklingmodes,bifurcation diagramsanddimensionlessparametersinfluencingtheresponseofthesystem.Inthecompanioncaseof axiallycompressedcore-shellcylinders,itwasestablishedthatinstabilitymodescanbeaxisymmetricor diamond-like,thepost-bucklingresponseofthesystemisgovernedbyasingledimensionlessparameter Cs,andthebifurcationbecomessupercriticalforasufficientlystiff core.Inthepresentcaseofcylindrical core-shellssubjectedtothermalloading,onefinds quitedifferentbuckling patterns,namedchurro-like modesthatarecharacterizedbyafastundulationinthecircumferentialdirection.Thereexistsanother curvature-relatedinfluencingparameterCt,andasubcriticaltosupercriticalbifurcationtransitionisob- served whenthe core stiffnessincreases. The problemis analyzedboththeoretically and numerically basedonfiniteelementcalculations.Lastly,theobtainedinstabilitymodesremainaboutthesameasin pureshellstructures,themaindifferencebeingthestabilizationofthepost-bifurcationbehavior.

1. Introduction

Quantitative characterization of wrinkling process for film/substrate systems has motivated considerable research interests duringpast few years for understanding and predicting pattern formation both in nature (Mahadevan and Rica, 2005;

Efimenko etal., 2005; Yin etal., 2009; Wang and Zhao, 2015;

Zhangetal.,2016;SáezandZöllner,2017)andinmodernindustry (Brauetal., 2011; Caietal., 2011;Cao andHutchinson, 2012; Sun etal., 2012; Zang etal., 2012; Xu etal., 2014; 2015a; 2015b; Fu and Cai, 2015; Xu and Potier-Ferry, 2016a; Huang etal., 2016).

Broad applications range from micro/nano-fabrication of flexible electronic devices with functional surface patterning (Bowden etal.,1998;Rogersetal.,2010;Li,2016),microlensarraysproduc- tion (Chan andCrosby,2006), adaptive aerodynamicdragcontrol (Terwagne etal., 2014), to themechanical property measurement of material characteristics (Howarter and Stafford, 2010). Most

previoustheoretical andcomputationalworksare concernedwith planarfilm/substrate systemsandsuggestthat wrinklingpatterns stronglydepend onappliedloading (Chen andHutchinson, 2004;

Huangetal.,2005; AudolyandBoudaoud,2008; Brauetal., 2011;

Caietal., 2011;CaoandHutchinson,2012;Xuetal., 2014;2015b).

Recent investigations revealed that wrinkling patterns may also varywiththesubstratecurvature(Yin etal., 2009; ChenandYin, 2010;Lietal.,2011;BreidandCrosby,2013;Terwagneetal.,2014;

Jiaetal., 2015) so that much attention has beenpaid to explore curvature effects on mode selection in non-planar film/substrate systems (Zhao etal., 2014;Patrício etal., 2014;Stoop etal., 2015;

XuandPotier-Ferry,2016b;Shaoetal.,2016),whichdemonstrates theimportantimpactsofthetopologicalconstraintsofcurvedge- ometryoninstabilitypatterns.Forinstance,aplanarfilm/substrate bilayer usually exhibits sinusoidal wrinkling patterns under uni- axial compression. Nevertheless, considering a core-shell soft cylinder subjected to axial compression, surface wrinkles may forminbothaxialandcircumferentialdirectionsduetocurvature effect, leading to the evolution from axisymmetric sinusoidal wrinklestonon-axisymmetricdiamond-likemode,whichdepends onone singledimensionless parameterC_s=(^Es/Ef)(^R/hf)^{3/2 that}

isafunction ofmodulus ratioandcurvatureofthe structure(Xu and Potier-Ferry, 2016b). This suggests that geometric curvature andmoduluseffectsplayakeyroleinpatternformationincurved surfaces,especiallythe core-shell cylinderthat appears to bethe simpleststructure with zero Gaussian curvature, which deserves furtherinvestigations.

Inhistory,thenonlinearbucklingandpost-bucklinganalysisof thin-walledcircularcylindricalshellsundervariousloadingcondi- tionshas receivedconsiderable attention(von Kármán andTsien, 1941; Koiter, 1945; Hutchinson and Koiter, 1970; Yamaki, 1984;

Bushnell, 1985), withthe most representative loadings beingthe axial compression and external pressure. Circular shell cylinders subjected to axial compression can be characterized by a high bucklingstress(

σ

^cr≈ 0.605Eh/R),short-wavelengthinstability pat- terns(w≈√

Rh) anda very strong imperfection-sensitivityasso- ciatedwithanextremelyunstablepost-bucklingbehavior.Theim- portanceofthisnonlinearpost-bifurcationpropertyhasbeenearly recognizedsince1940s(vonKármánandTsien,1941;Koiter,1945).

Inthesimplestbifurcationanalysis,manybucklingmodescoincide, includingaxisymmetricanddiamond-likeshapes.Theco-existence ofthesetwo pattern typesis typical ofcylindrical geometryand ithasbeenobservedboth inpureshell andcore-shell structures.

However,thepresenceofastiff core(C_s≥ 0.9)leadstosinusoidally deformedaxisymmetric patternsthat rarelyappearinpure shells withoutacore.Intheliteratureonshellbuckling,curvatureeffect is often accounted through the dimensionless Batdorf parameter Z=L²



1−

ν

²/(^Rh) ^{(Batdorf, 1947;Yamaki, 1984;Bushnell,1985;}

Abdelmoulaetal., 1992; Abdelmoula and Leger, 2008). There are alarge numberofreferencesonthe bucklingofcircularcylindri- calshells,namelyafew textbooks(Yamaki, 1984;Bushnell,1985;

Julien, 1991;Teng andRotter, 2006; Koiter, 2009), review papers (Hutchinson and Koiter, 1970; Arbocz, 1987; Teng, 1996) and a websitenamed‘ShellBuckling’(BushnellandBushnell,2015).

When a cylindricalshell is filled with a softcore, i.e. a core- shellsoftcylinder,short-wavelengthbifurcationmodescanbeper- sistently observed experimentally and numerically in both buck- ling and post-buckling stages, and the unstable post-bifurcation behaviorcan be stabilized by the presence ofa core(Zhao etal., 2014;XuandPotier-Ferry,2016b). Thereexistafewcontributions on the buckling analyses of a cylindrical shell filled with a soft core, which either study the stabilizing effects of soft cores on thebuckling strength of shells (Yao, 1962; Myint-U,1969) or fo- cuson axisymmetric instability modes(Karam andGibson,1995;

Yeetal., 2011; Wu etal., 2012). Core-shell cylinders subjected to axialcompressionhavebeenthoroughlystudiedtheoretically and numericallyfromaquantitative point ofview intherecentwork (XuandPotier-Ferry,2016b),wherethecriticalparameterCsdeter- minesa phasediagram ofaxisymmetric/diamond-likemode tran- sition. For a stiff core (Cs≥ 0.9), the buckling pattern is axisym- metricandpost-bifurcationsolutionsarestable;whereasforasoft core(C_s≤ 0.7), the bifurcated solution branch is often subcritical andthe associatedinstability modes tendto be diamond shaped after secondary bifurcations. The stabilization of post-bifurcation patternsturns out to be the major consequence ofthe presence ofacore.

Deviating from the axial compression case, circular cylindri- calshellsunderexternalpressureorthermalloadingwouldshow a totally different mechanical response with oscillations varying muchfasterincircumferentialdirectionthanintheaxialdirection.

Inotherwords,thewavelengthintheaxialdirectioncanbeglobal (x≈ L), while it is quite local in the circumferential direction (y x), in the order of y∼ L^1/2(Rh)^1/4 (Abdelmoula and Leger, 2008).Thisimpliesthat theexplicitexpressionofthiscriticalpa- rameterC_smaychangeitsformandhastobere-defined.Besides, this parameter would significantly affect supercritical/subcritical

post-bifurcation response of core-shell cylindrical structures and requirestobethoroughlystudied.Astheinstabilitymodeisnotlo- cal,theinfluenceofboundaryconditionsbecomesmoreimportant thanthat inthecaseofaxialcompression,whichinduces bound- arylayers(AbdelmoulaandLeger,2008) andsome strangeeffects of boundaryconditions (Sobel,1964). These features will be dis- cussedinthepresentcaseofcore-shellcylinders.

Thispaperaimsatexploring theoccurrenceandpost-buckling evolution of 3D wrinkling patterns in core-shell soft cylindri- cal structures subjected tothermalloading, throughapplying ad- vanced numerical methods from a quantitative standpoint. The- oretical analyses based on the well-known nonlinear Donnell–

Mushtari–Vlassov (DMV) shell formulations are first carried out toqualitativelydeterminedimensionlessparametersthatinfluence theinstabilitypatterns.Spatialpatternformationisthenquantita- tively investigatedbasedona nonlinear3D finiteelementmodel, associatinggeometricallynonlinearshellformulationsforthesur- face layer and linear elastic solids for the core, and a robust path-following continuation techniquecalledAsymptotic Numeri- cal Method (ANM) (Damil and Potier-Ferry, 1990; Cochelin etal., 1994; Cochelin, 1994; Cochelinetal., 2007). The same finite ele- ment procedurehas beenvalidated andapplied previously inXu etal.(2014); Xu and Potier-Ferry (2016b). Here we consider geo- metrically perfectcore-shell structures without anyimperfection.

The paperis organizedasfollows.In Section2, anonlinear finite elementmodelisconciselypresented.Section3isdevotedtosev- eral theoretical analyses within theframework ofthe DMV shell formulations and the Biot-Winklerassumption on the substrate.

First,onerecalls whythebucklingmodedependsstronglyonthe loadingtypeandoneexplainswhyithasconsequenceondimen- sional analyses. A simplified core-shell model, established in the samewayasinAbdelmoulaetal.(1992),allowsdeducingthebuck- lingstressandthepost-bucklingbehaviorofthesystem.Numerical calculationsare presentedinSection4,includingthe deformation shapeofbuckling patterns,bifurcation scenarioandtheinfluence ofleadingparameters.

2. Numericalmodel

The3Dcore-shellcylindricalsystemwillbeanalyzedbyanon- linearfiniteelementmodel,whichwasfirstintroducedinXuetal.

(2014) and then applied to hyperelastic film/substrate (Xu etal., 2015a)aswellascore-shellcylindricalsystem(XuandPotier-Ferry, 2016b). This finite element framework appears to be sufficiently versatileforthe presentcaseofcircularcylindricalgeometryand thermalloading.Inthismodel,thesurfacelayerisrepresentedby athinshellmodeltoallowlargerotations,whilethecoreismod- eled by smallstrain elasticity.Indeed, theconsidered instabilities aregoverned bynonlinear geometriceffectsforthestiff material, whilethe effectsare much smallerforthe softmaterial.Forpla- nar film/substrate systems, a thorough investigation on compari- sonbetweenfinitestrainhyperelasticmodelandsmallstrainelas- ticmodel,withrespectto awide rangeofYoung’s modulus, was carriedout inXuetal.(2015a). It demonstrates that deformation ofthesystemcan berelatively largeandfinitestrain constitutive lawshaveto betakenintoaccount, ifthestiffnessratioisrather small,e.g. E_f/Es≈ O(¹⁰)^{.Inmostcases of}film/substratesystems, i.e.E_f/Es O(¹⁰),smallstrain elastic models appearto besuffi- cient andarequalitatively orevenquantitatively equivalentto fi- nite strain hyperelastic models. This remains valid forcore-shell softcylindricalstructuresaswell,asseeninZhaoetal.(2014);Xu andPotier-Ferry (2016b). Therefore,in what follows,we consider Hookeanelasticityforthepotentialenergyofthesystemforsim- plicity.

Challengesinthenumericalmodeling ofsuch core-shellcylin- drical systems come from large aspect ratio (2

π

^R/hfO(¹⁰²),

L/hfO(¹⁰²)^{) and} radius/thickness ratio (R/hfO(¹⁰)^{), which} requireveryfinemeshesifemploying3Dbrickelementsforboth the surfaceand the core. Since finiterotations ofmiddle surface and small strains are considered in the surface layer, nonlinear shellformulationsarequitesuitable andefficientformodeling.In this part,a shell formulation incurvilinear coordinates,which is proventobesuitable forcore-shellcylinders(XuandPotier-Ferry, 2016b) and film/substratebilayers (Xu etal., 2014;2015a), is ap- plied. It is incorporated via the Enhanced Assumed Strain (EAS) concepttoimprovetheelementperformanceandtoavoidlocking phenomenasuchasPoissonthicknesslocking,shearlockingorvol- umelocking. Thishybrid shellformulationcan describe large ro- tationsandlargedisplacements,andhasbeensuccessivelyapplied tononlinearelasticthin-walledstructuressuchascantileverbeam, squareplate,cylindricalroofandcirculardeeparch(Zahrounietal., 1999;Boutyouretal.,2004).

Formulationsofgeometryandkinematicsofthe shellelement can be found in Xu etal. (2014); 2015a); Xu and Potier-Ferry (2016b).Thehybridshellformulationisderived fromathree-field variationalprinciplebasedontheHu–Washizufunctional(Büchter etal.,1994;Zahrounietal.,1999).The stationaryconditioncanbe writtenas

f



uf,

γ

,S



=



f



tS:

( γ

u+

γ



)

⁻¹₂^tS:D−1:S



d



, (1) whereDistheelasticstiffnesstensoroftheshell.The unknowns are, respectively, the displacement field u_f, the second Piola–

Kirchhoff stresstensorSandthecompatibleGreen–Lagrangestrain

γ

^{u.Theenhanced assumedstrain }

γ

^{,satisfies theconditionofor-}

thogonalitywithrespecttothestressfield.

A 8-node quadrilateral element with reduced integration is usedforthe7-parametershellformulation.Theenhancedassumed strain 

γ

^{neitherrequiresinter element}continuity,norcontributes tothetotalnumberofnodaldegreesoffreedom.Therefore,itcan be eliminated by condensation at the element level, which pre- serves theformal structureofa6-parametershelltheorywithto- tally48degreesoffreedomperelement.

Since the displacement, rotation and strain remain relatively small in the core, the linear isotropicelasticity theory can accu- rately describe the core (Xu and Potier-Ferry,2016b). Hence, the potentialenergyofthecorecanbeexpressedas

^s

(

^us

)

^=

s

1 2



_t

ε

^:Ls:

ε

^−t

ε

^:Ls:

ε

th



d



^{, (2)}

whereLsistheelasticmatrixofthecore.Thetotalstrainandther- malstrain arerespectivelydenotedas

ε

^and

ε

th.Inthispaper,8- node linear brick elements withreduced integration are applied todiscretizethecore, withtotally24degreesoffreedom oneach brickelement.Notethatthecoreissubjectedtothethermalload- ingthatcanbeexpressedas

ε

th=

α

^{TI with}

^{T<0, (3)}

where

α

^{, T and I denote the thermal expansion coefficient,} temperaturechangeandsecond-orderidentitytensor,respectively.

Thisthermalshrinkingloading

ε

thcanbecharacterizedbyaresid- ualstrain

ε

th=

ε

res=−

λ

^I,while

λ

^{isascalarloadparameterand}

onlynormalstrainsareconsideredforisotropicloading.

As the surface shell is bonded to the core, the displacement should be continuous at the interface (Xu etal., 2014; Xu and Potier-Ferry, 2016b). However, the shell elements and 3D brick elements cannot be simply joined directly since they belong to dissimilar elements. Therefore,additionalincorporatingconstraint equationshavetobeemployed.Here,Lagrangemultipliersareap- pliedtocouplethecorrespondingnodaldisplacementsincompat- ible meshesbetweentheshellandthecore(see Fig.1).Notethat using 8-node linearbrick element hereis only forcoupling con- venience,and20-node quadraticbrickelementwould beanother

Fig. 1. Sketch of coupling at the interface. The coupling nodes are marked by red color. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

goodcandidate,whilebothofthemfollowthesamecouplingstrat- egy.Consequently,thestationaryfunctionofthecore-shellsystem isgiveninaLagrangianform:

L

(

^uf,us,

)

=

f+

s+

nodei

i

u_f

(

ⁱ

)

− us

(

ⁱ

) 

, (4)

wherethedisplacementsoftheshellandthecoreare,respectively, denoted as u_f and us, while the Lagrange multipliers are repre- sentedby.From Eq.(4),three equationsareobtainedaccording to

δ

^uf,

δ

^usand

δ

:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

δ

f+

nodei

i

δ

^uf

(

ⁱ

)

=0,

δ

s−

nodei

i

δ

^us

(

ⁱ

)

=0,

nodei

δ

^iu_f

(

ⁱ

)

−

nodei

δ

^ius

(

ⁱ

)

=0.

(5)

A path-following continuation technique named ANM (Damil andPotier-Ferry,1990;Cochelinetal.,1994;Cochelin,1994;Coche- linetal.,2007)isappliedtosolvetheresultingnonlinearPDEs(5). TheANMisanumericalperturbationtechniquebasedonasucces- sionofhigh-order powerseriesexpansionswithrespecttoawell chosenpathparameter,whichappearsasanefficientcontinuation predictorwithoutanycorrectoriteration.Besides,one cangetap- proximationsofthesolutionpaththatareveryaccurateinsidethe radiusofconvergence.Inthispaper,themaininterestoftheANM is its ability to trace the post-buckling evolution on the equilib- riumpath andtopredictsecondary bifurcationswithoutanyspe- cialtool. Precisely,accumulationofsmallstepsin theANMisof- tenassociatedwiththeoccurrenceofabifurcation(Xuetal.,2014;

2015a;2015b;XuandPotier-Ferry,2016b).

3. Theoreticalanalyses

Inthissection,thebucklingofacore-shellcylinderunderther- mal loading is analyzed theoretically. The objective is to explain thedeformationshapeofthecorrespondingbucklingpatternsand tohighlightthedimensionlessquantitieswhichinfluencethebuck- ling and post-buckling response of the core-shell system. Previ- ousworksbyZhaoetal.(2014);XuandPotier-Ferry (2016b)have demonstrated that two types of modes can be observed when the cylindrical core-shell is subjected to an axial compression:

axisymmetricsinusoidalornon-axisymmetricdiamond-likeshape accordingtothevalueofonesingledimensionlessparameterCs= (^Es/Ef)(^R/hf)^{3/2.Infact,differentwrinklingshapeshavebeenob-} tainedfromthenumericalframeworkpresentedinSection2,with onlyonehalf-waveintheaxialdirectionbutshortwave-lengthin- stabilitycircumferentially(seeFig.2).Inotherwords,thebuckling

(a) (b) (c)

Fig. 2. Three representative instability patterns obtained from numerical calculations: (a) axisymmetric sinusoidal mode under axial compression; (b) non-axisymmetric diamond-like mode under axial compression; (c) churro-like mode under thermal loading.

Fig. 3. Geometry of core-shell cylindrical structure.

ofa core-shell cylinderunder thermalloads approximates tothe bucklingofa cylindricalshellunderexternalhydrostaticpressure that has been widely discussed in the literature (Batdorf, 1947;

Sobel,1964; AbdelmoulaandLeger, 2008; Timoshenko andGere, 1961;BrushandAlmroth,1975;Yamaki,1984;BažantandCedolin, 1991). Theexplanation issimple: thethermal loadinginduces an isotropic stress state in the pre-buckling stage (

σ

^x=

σ

^{y), but it}

iswell known that the critical buckling stress is generallymuch higherintheaxialcompressioncasethanintheexternalpressure situation,withthefollowingordersofmagnitude:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

| σ

x^c

|

E_f = 1



3

(

¹−

ν

²f

)

h_f

R ≈ 0.605h_f

R when

ν

f=0.3,

| σ

y^c

|

E_f =O

 

h_f R



3/2

R L



.

(6)

Thus, the circumferential stress

σ

y woulddestabilize the system much earlier than the axial stress

σ

^{x so that the} corresponding instabilitypatternshouldbesimilartotheexternalpressureload- ingcase. Inthesameway,itisknownthatthecriticalloadinthe lateralpressurecase(

σ

^x=0)isalmost thesameasinthehydro- staticpressurecase(

σ

^x=

σ

^y/2)(Batdorf,1947),whichconfirmsthe weakinfluenceoftheaxialstresswhen thecircumferentialstress iscompressive.

3.1.Core-shellDMVmodelwithafoundation

Weconsideranelasticcylindricalshellsupportedbyasoftcore, whichcan buckleunderexternal pressure p. Upon wrinkling,the shell elastically buckles to relax the compressive stress and the coreconcurrently deforms tomaintainperfectbonding atthe in- terface.Thecore-shellcylindricalsystemisconsideredtobethree- dimensionalandthegeometryisasshowninFig.3.Inthisstudy, we consider the case where the thickness of the shell is much

smallerthantheradiusofthecylinder,withR/hfO(¹⁰)^asusu- allyaccountedforthinshellmodels.Letxandybe axialandcir- cumferential coordinates, while z represents the radial direction coordinate.Thethicknessoftheshell,theradiusandthelengthof thesystemaredenotedbyh_f,R andL,respectively.Young’smod- ulusandPoisson’sratiooftheshellarerespectivelydenotedbyE_f and

ν

f,whileE_s and

ν

sarethe correspondingmaterialproperties forthe core.The sameframe will beadapted to thecomponents of thedisplacements u, v, w,and thestress function Ffor stress resultants.

TheDonnell-Mushtari-Vlassov(DMV)shell model(Brushand Almroth, 1975; Yamaki, 1984) that relies on the assumption of moderatelylargerotationsandvarioussimplificationsoncurvature istakenintoaccount.Itisnotasgeneralastheshellmodelusedin thenumericalcalculations,butitissufficientlyaccurate forbuck- lingmodeswithatleastfourorfivecircumferential waves.Itwill bealsosufficientforthequalitativeanalysisaimedinthissection.

Intheversiondescribingthemembranestressbyastressfunction, thegoverningequationsread

⎧ ⎪

⎨

⎪ ⎩

D

^2w−1 R

∂

^2F

∂

^{x2 − [F,w]=p,}

1

E_fh_f

^2F+1 R

∂

^2w

∂

^{x2 +12[w,w]=0,}

(7)

where ^{2 and D}=E_fh³_f/[12(¹−

ν

²f)^] respectively denote the bi- harmonicoperatorandflexuralrigiditiesoftheshell.Thenonlinear Monge-Ampèreoperatorisdefinedas

[a,b]=

∂

^2a

∂

^x2

∂

^2b

∂

^y2+

∂

^2a

∂

^y2

∂

^2b

∂

^{x2− 2}

∂

^2a

∂

^x

∂

^y

∂

^2b

∂

^x

∂

^{y. (8)}

ThestressfunctionFisrelatedtotheresultantstressby

Nx=

∂

^2F

∂

^{y2, Ny=}

∂

^2F

∂

^{x2, Nxy=−}

∂

^2F

∂

^x

∂

^{y. (9)}

Inacylindricalcore-shellsystem,thenormalpressurep(x,y)is afunctionofthenormalstressonthecoreboundary.Nevertheless, onecanavoidtheelasticsoliddescriptionofthecorebyconsider- inga Winkler-typeapproach: oneassumesthat thisnormalpres- sureisproportionaltothedeflectionw(x,y)atthesameplace.By consideringa sinusoidalload,Biot(Biot,1937;Allen, 1969)estab- lishedaclosedformoftheWinklerstiffnessratiothatdependson thehalfbucklingwavelengthwthatisaprioriunknown:

Ks= Es

(

³−

ν

s

)(

¹+

ν

s

)

2

π

w. (10)

Thetheoretical analyseswill beconductedwithin thisframework andthe core-shell systemis assumedto be governedby the fol-

lowingpartialdifferentialequations(PDEs):

⎧ ⎪

⎨

⎪ ⎩

D

^2w−1 R

∂

^2F

∂

^{x2 − [F,w]+Ksw=0,}

1

E_fh_f

^2F+1 R

∂

^2w

∂

^{x2 +}

1

2[w,w]=0. (11)

Various boundary conditions are concerned in the literature (Sobel,1964) andthe mostusedboundary conditionscan be ex- pressedasfollows:

C1:w=w_,x=u=

v

=0, S1:w=w_,xx=u=

v

=0, C2:w=w_,x=u=Nxy=0, S2:w=w_,xx=u=Nxy=0, C3:w=w_,x=

v

=Nx=0, S3:w=w_,xx=

v

=Nx=0, C4:w=w_,x=Nx=Nxy=0, S4:w=w_,xx=Nx=Nxy=0,

(12) in which C stands forclamped boundary conditions while S de- notessimplysupportedones.

3.2. Whyinstabilitypatternsunderthermalloadingdifferfromaxial loading?

For the sake ofcompleteness, we briefly re-discussthe linear bucklinganalysisofapure shell(Ks=0)underaxialcompression andlateralpressure,forrecallingtheconnectionamongloadtype, modeshapeandordersofmagnitude.Essentially,theseresultscan befoundinBatdorf(1947)andinseveraltextbooksonshellbuck- ling. Oneseeks bifurcationsfromuniformly biaxialloadingstates

σ

x⁰=0,

σ

y⁰=0,

σ

xy⁰ =0⇒F⁰

(

x,y

)

=1 2

 σ

y⁰x²+

σ

x⁰y²



. (13) The critical states are deduced from the linearized version of (11)fromF⁰(x,y)inEq.(13)andw⁰(^x,y)=0:

⎧ ⎪

⎨

⎪ ⎩

D

^2w−1 R

∂

^2F

∂

^{x2 − hf}

 σ

x⁰

∂

^2w

∂

^{x2 +}

σ

y⁰

∂

^2w

∂

^y2



=0, 1

Efhf

^2F+1 R

∂

^2w

∂

^{x2 =0.}

(14)

The twoequationsin (14)canbe classicallycombinedto getone singleeighth-orderpartialdifferentialequation(Batdorf,1947):

D

^4w+E_fh_f R²

∂

^4w

∂

^{x4 − hf}



σ

x⁰

∂

²

^2w

∂

^{x2 +}

σ

y⁰

∂

²

^2w

∂

^y2



=0. (15)

Classical double-sinusoidal modes are assumed as w(^x,y)= sin(^qxx+

ϕ

^x)^sin(^qyy+

ϕ

^y)^{. One seeks the critical stresses (as-} sumed negative)asthesmallest stressessuch that Eq.(15)hasa solution,whichleadstotheminimizationwithrespecttothewave numbers(qx,qy).Thus,onecangettherelationbetweenloadand wavenumbersfromEq.(15):

hf

 | σ

x^c

|

^q2x+

| σ

y^c

|

^q2y



=D



q²_x+q²_y



2

+E_fh_f R²

q⁴_x



q²_x+q²_y



2. (16)

Inthecaseofaxialcompression (

σ

y⁰=0), thebifurcation con- dition(16)canbeexpressedas

h_f

| σ

x^c

|

^=DA+E_R^fh₂^f1_A^{, A=}



qx+q²_y qx



2

. (17)

Theminimizationof

σ

x^c withrespecttowave numbersisstraight- forward.Therearelotsofminimalyingonthefamous“Koitercir- cle” (Koiter,1945)withtheequation:

A=



qx+q²_y qx



2

=



Efhf

DR² =



12

(

¹⁻

ν

²f

)

Rh_f . (18)

Thismeansthatthelinearstabilitytheorycanpredictaxisymmet- ric (qy=0) aswell asdiamond-like modes (qx=0, qy=0). A re- cent paper (Xu and Potier-Ferry, 2016b) has established numeri- cally that these two types of patternscan be observed forcore- shellcylindersunderaxialcompression.Notethatthewavelengths ofallthesemodesontheKoitercircleareveryshortascompared withtheradiusofthecylinder:

x= 2

π

qx =O



Rh_f



, y= 2

π

qy =O



Rh_f



. (19)

Thecriticalaxialstressistheminimalvalueof(17),whichleadsto thefamousformula(6).

Inthe caseoflateral pressure(

σ

x⁰=0), thebifurcation condi- tion(16)reads

hf

| σ

y^c

|

=D

(

^q2x+q²_y

)

² q²y

+E_fh_f R²q²y



1

1+q²_y/q²_x



2. (20) Since the critical stress predicted by (20) is an increasing func- tion of the axial wave number qx, the mathematical minimum withrespect to theaxial wave numberrequires qx=0, which is not permitted by boundary conditions. In fact, the axial wave- length is of the same order as the shell length (x=O(^L)^{), and} theminimizationof(20)withrespecttothecircumferentialwave numberqy leadsto thecriticalcircumferentialwavelengththat is generally smaller than the circle circumference 2

π

^{R but always}

largerthanthewavelength(19)intheaxialcompressioncase.Sim- pleformulaeforthecriticalcircumferentialstress andthecritical wavelengthy can beobtainedby consideringa givenaxialwave numberqx=O(¹/L)^{thatissmallerthanthe}circumferentialwave numberqy.Withthesesimplifications,Eq.(20)canbe writtenas

h_f

| σ

y^c

|

^=Dq2y+Efhf

R² q⁴_x

q⁶_y, (21)

andtheminimization withrespecttoqy isquitestraightforward:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

qy=

36



1−

ν

²



1/8 q¹_x^/2



Rh_f



1/4 ⇒y=2

π

qy =O



L^1/2



Rh_f



1/4



,

| σ

y^c

|

^{= 4}

[36

(

¹⁻

ν

²

)

^]3/4

E_fh³_f^/2qx

R^1/2 .

(22) Therefore, this simple and classical buckling analysis predicts bucklingpatternsunderlateralpressureloadingthatarequitedif- ferentfromthosefoundinaxialcompressioncase.Thecircumfer- ential wavelength y turns out to be smaller than the axial one (L^1/2(Rh_f)^1/4 L), andthecritical stress isgenerallymuch smaller thanintheaxialcompressioncase(h³_f^/2qx/R^1/2 hf/R).Thisstudy will be continued to account forthe effects of the corestiffness andboundaryconditions.

3.3.Asimplifiedanalysisforpressure-typebuckling

Inreality, the solutions ofthe nonlinear problems are not al- ways harmonics so that the previous analysis can be only qual- itative. Nevertheless, relatively simple critical and post-buckling solutions are available (Abdelmoula etal., 1992; Abdelmoula and Leger, 2008) based on the observation that the circumferential wavelength is much smaller than the cylindrical length. In this paper, this simplified approach is extended to core-shells gov- erned by system (11). First, to distinguish the tangent opera- tor at the bifurcation point and the nonlinear part, we modify Eq. (11) by introducing the additional stress function f(^x,y)=

F(^x,y)−1 2

 σ

y⁰x²+

σ

x⁰y²



:

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

D

^2w−1 R

∂

^2f

∂

^{x2 − hf}

 σ

x⁰

∂

^2w

∂

^{x2 +}

σ

y⁰

∂

^2w

∂

^y2



+Ksw− [f,w]=0,

1

E_fh_f

^2f+1 R

∂

^2w

∂

^{x2 +}

1

2[w,w]=0.

(23)

As shown in Fig.2, thermal or pressure loading generally leads to the churro-like mode that is characterized by faster varia- tions in the circumferential direction. This feature is the start- ingpointofasimplifiedapproachestablishedinAbdelmoulaetal.

(1992); AbdelmoulaandLeger(2008).It consistsinneglectingall theaxial derivatives(

∂

^/

∂

^x

∂

^/

∂

^{y), except forthecoupling terms}

(1/R)(

∂

^2/

∂

^{x2)duetothe crucialroleofthe curvatureinanyshell}

buckling problem, as well as the nonlinear terms [f, w] and [w, w]thatcontainthesamenumberofderivativesinbothdirections.

Thisleadstoasimplifiedsystem:

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

D

∂

^4w

∂

^{y4 −}

1 R

∂

^2f

∂

^{x2 − hf}

σ

y⁰

∂

^2w

∂

^{y2 +Ksw− [f,w]=0,}

1 E_fh_f

∂

^4f

∂

^{y4 +}

1 R

∂

^2w

∂

^{x2 +}

1

2[w,w]=0.

(24)

InAbdelmoula andLeger (2008), thissimplified model has been validatedby anasymptoticexpansion withrespecttoasmallpa- rameterrelatedtotheBatdorfparameter,i.e.1/√

Z≈



Rh_f/L.Each equation in (24) is reduced to the second order withrespect to

∂

^/

∂

^{xfromthestartingmodel(23)thatisofthefourthorder.This}

featuremeans thatthesolutionoftheinitialsystem(23)involves boundarylayersthathaveconsequencesonboundaryconditionsto beassociatedwiththesimplifiedsystem(24).

3.4.Dimensionalanalysis

Wenowseektheparametersinfluencingthebucklingbehavior ofcore-shellcylindricalsystemsunderthermalloading.Theinves- tigationis basedon the simplified model(24), knowingthat the transitionbetweenthesimplified system(24) andtheinitial one (23)isgovernedby theBatdorfparameter.Generally,withinnon- linearshell/platetheory,thedeflectionisassumedtobeoftheor- deroftheshell/platethickness.Accordingtothe previousdiscus- sion,the characteristic lengths are the structural length L inthe axialdirection andthe halfwavelength w inthe circumferential direction.Butthishalfbucklingwavelengthw isunknownandis involvedinthedefinitionoftheWinklerfoundationstiffnessKs,as seeninEq.(10).Nevertheless,theorderofitsmagnitudeisknown so that we can scale the circumferential variable “y” by another lengthL_ythat isofthesameorderofmagnitudeasw.Thisleads tothefollowingsimilitudeassumptions:

w=h_fw¯, x=L¯x, y=Ly¯y, f =D¯f, Ly=L^1/2



Rh_f



1/4

. (25) Nextonehastoreplacetheunknownwavelengthbydimensionless quantitiesandwe proposethenumberof circumferentialperiods

“n” that is an integer or thewave number ¯q associatedwiththe dimensionlessvariable ¯y.Theyareconnectedby

π

w = ¯q Ly= n

R. (26)

By substituting Eq. (25) into Eq. (24) and accounting for Eqs.

(10)and(26),oneobtains

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

∂

^4w¯

∂

^{¯y4 −}

∂

^2¯f

∂

^{¯x2 −}

^ ∂

^2w¯

∂

^{¯y2 +¯qCt∗w¯−}



Rh_f L²



1/2

¯f,w¯



=0,

1

12

(

¹−

ν

²_f

) ∂

^4¯f

∂

^{¯y4 +}

∂

^2w¯

∂

^{¯x2 +12}



Rh_f L²



1/2

[w¯,w¯]=0,

(27)

where the bracket operator is defined in (8), but with respect to the scaled dimensionless variables (^¯x,¯y)^{. Four} dimensionless parameters appear in (27). The first one is Poisson’s ratio

ν

f of the shell, whose influence should be rather limited. The second one,=

σ

y⁰h_fL²_y/D⇐⇒

σ

y⁰=E_fh³_f^/2/[12(¹−

ν

²f)^LR1/2],isthedi- mensionlessloadingparameter, i.e.thebifurcationparameterthat variestostudytheresponseofthesystem.Thethirdonecombines themodulusratioandthestructuralgeometryinvolvingcurvature:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

C_t^∗= 24

(

¹−

ν

²f

) (

³⁻

ν

^s

) (

¹⁺

ν

^s

)

^Ct, Ct= Es

E_f



Ly

h_f



3

= Es

E_f



L h_f



3/2



R h_f



3/4

.

(28)

ThisratioCtorC_t^∗isthemaincharacteristicdimensionlessparame- terofthecore-shellcylindricalsystem,whichimpliestheinfluence of the corestiffness with respect to the shell stiffness. It differs fromthe parameterCs introduced inXu andPotier-Ferry (2016b) since the instability modes are quite different between external pressure andaxial compression. The last parameter (Rh_f/L²)^1/2 in Eq. (27) recovers the Batdorf parameter, butone will see that it canbedropped asitappearsjustbeforethetwoquadraticterms.

Throughre-scalingtheunknownsofsystem(27)by

¯

w=w^∗/

η

, ¯x=x^∗, ¯y=y^∗, f =f^∗/

η

,

η

=



Rh_f/L²



1/2

, (29) onegetsanewversionofthemodelas

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

∂

^4w∗

∂

^{y∗4 −}

∂

^2f∗

∂

^{x∗2 −}

^ ∂

^2w∗

∂

^{y∗2 +¯qCt∗w∗− [f∗,w∗]=0,}

1

12

(

¹−

ν

²_f

) ∂

^4f∗

∂

^{y∗4 +}

∂

^2w∗

∂

^{x∗2 +12[w∗,w∗]=0.}

(30)

Thus,inwhatfollows,thebehaviorofthesystemwillbediscussed asafunctionoftheparameterC_t^∗definedin(28).

3.5. Boundarylayersandboundaryconditions

Thepresenceofboundarylayersisoneofthemainfeaturesof the buckling of cylindrical shells undercircumferential compres- sion(

σ

^y<0). Astheoretically explainedinAbdelmoulaandLeger (2008),theseboundary layers are associatedwithlarge valuesof the Batdorfparameter Z=L²



1−

ν

²_f/(^Rhf)^{, i.e.Z}≥ 100, whichis generallythe caseforthinshell buckling. The rapidvariationsin boundaryregions haveconsequences forthe boundaryconditions tobeappliedtothesimplifiedmodel(24).Toillustratethisstate- ment,letusconsiderfullfiniteelementcomputationsofcore-shell cylindersunderthermalloadingfortwo caseswiththesameBat- dorfparameterZ=954butwithdifferentdimensionlessparame- tersCt=0.0056andCt=266,respectively.Fig.4illustratestheax- ial profile of the deflection x→w(x,0) andits derivative around the first bifurcation to be representative of bifurcation modes. It shows a rapid variation of the slope w,x close to the boundary, which satisfies the clamped condition w_,x=0. In fact, it clearly appears that a smooth extrapolation of the slope w,x up to the boundary would satisfy w_,xx=0 but not w_,x=0. This boundary

Fig. 4. Buckling modes and the corresponding rotations obtained by the numerical model in Section 2 with boundary condition C 3 and Batdorf parameter Z = 954 : (a) C t = 0 . 0056 ; (b) C t = 266 . The size of the boundary layers remains the same since the same Batdorf parameter is taken into account.

Table 1

Boundary conditions (BCs) imposed on models.

BCs for initial problem (7) BCs for simplified problem (24) C 1, C 2, S 1, S 2 ( u = w = 0 ) w = w ,x = 0

C 3, C 4, S 3, S 4 ( N x = w = 0 ) w = w ,xx = 0

condition w_,xx=0 is the one to be imposed on the simplified model(24)thatdisregardsthe boundarylayers.Roughly,thisisa consequence ofboundary conditionson the axial force Nx andit followsthelinearversionofEq.(24b):

Nx=0⇒

∂

^2f

∂

^{y2 =0}

cf.(24−b)



_⇒

∂

^2w

∂

^{x2 =0. (31)}

Arigorousexplanation accountingforboundary layersisgivenin Abdelmoula andLeger(2008) aswell as a treatmentof the case u=0.Itwastheoreticallyfoundthatthewidthofboundarylayers isrelatedtotheBatdorfparameterandsatisfiesO(¹/√

Z)^.Indeed, boundarylayers showninFig.4havethesamesize.Theseresults for the boundary conditions associated withthe model (24) are summarizedinTable1.Notethatonlytheaxialboundary(u=0or Nx=0)hasastrongeffectonthebifurcationload,whichwasearly recognized bySobel(1964).Theother boundaryconditions(

v

₌0 or Nxy=0, w,x orw_,xx=0) offer a minorinfluenceand actonly ontheboundarylayers.Ofcoursethepersistenceoftheseconclu- sionsinthepost-bucklingregime andinthepresenceofthecore canbere-discussed.

3.6. Bifurcationanalysis

Bifurcationloadsareobtainedfromthelinearizedversionofthe dimensionlesssystem(30).Forsimplicity,weomitthesuperscripts stars.Thebifurcationmodesareassumedinaharmonicformalong thecircumferentialdirection:



w

(

^x,y

)

^=wˆ

(

^x

)

^cos

(

^qy

)

^,

f

(

x,y

)

=fˆ

(

^x

)

^cos

(

^qy

)

. (32) Hence,themodesintheaxialdirectionarethesolutionsofanor- dinarydifferentialsystemobtainedfromthelinearversionof(30):

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩



q⁴+



^q2+qCt^∗



wˆ−d²ˆf dx² =0, q⁴

12

(

¹−

ν

²f

)

^fˆ+ d²wˆ

dx² =0.

(33)

Through eliminating the stress function fˆ, one finds a deflection determinedbyafourth-orderdifferentialequation:

12



1−

ν

²f



d⁴wˆ dx⁴ +



q⁸+



^q6+q5Ct^∗



wˆ=0. (34)

This eigenvalue problem is well known (e.g. vibration modes of abeam) andithasbeenassociated withbucklingunderexternal pressureinAbdelmoulaandLeger(2008).Letusrecallthesmall- est eigenpair for x∈[−1/2,1/2]. In the case without axial force (Nx=w=w_,xx=0),thesolutionreads(<0):



_wˆ

₍

_x

₎

_=cos

₍ π

^x

)

,

| (

^q

) |

^=q2+12

⁽

¹⁻

ν

²f

) π

⁴

q⁶ +C_t^∗

q, (35)

or in the case where the axial displacement u is locked on the boundary(u=w=w_,x=0):



_wˆ

₍

_x

₎

_=Acos

₍ πρ

^x

)

+Bcosh

( πρ

^x

)

,

| (

^q

) |

^=q2+12

⁽

¹⁻

ν

²f

) ρ

⁴

π

⁴

q⁶ +C_t^∗

q, (36)

whereAandBareconstantsand

ρ

≈ 1.5056isthesmallestrootof tan(

πρ

/2)+tanh(

πρ

/2)=0.AsunderlinedinKaramandGibson (1995), the minimization of(35-b) or (36-b) withrespect to q is difficult since this leadsto an eighth-order polynomial equation.

Nevertheless,itisclearthatthecriticalstress(minimumof^(q)) dependsonlyontwodimensionlessparameters,i.e.Poisson’sratio

ν

foftheshellandthedimensionlessparameterC_t^∗.Thenumerical minimizationisquitestraightforwardasthepossiblevaluesofthe wavenumberarediscrete(q=nLy/R)(seeFig.5).

The values of the critical stress cr and the associated wave numberq_crarepresentedinTable2asafunctionofthecorestiff- nessparameterC_t^∗inthecaseof

ν

f=0.4.Theseresultscorrespond toexactsolutionsofthesimplified model(24)butnottothefull model (7). As expected, the wave number and the criticalstress increase withtherise ofcorestiffnessC_t^∗.Twogroupsof bound- ary conditions defined in Table1 are considered. For a very soft