Multi-phase-field modelling of the elastic and buckling behaviour of laminates with ply cracks

ContentslistsavailableatScienceDirect

Applied Mathematical Modelling

journalhomepage:www.elsevier.com/locate/apm

Multi-phase-field modelling of the elastic and buckling behaviour of laminates with ply cracks

Duc Hong Doan

^a,b,∗

, Thom Van Do

^c,∗∗

, Nguyen Xuan Nguyen

^d

, Pham Van Vinh

^c

, Nguyen Thoi Trung

^a,b

aDivision of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

bFaculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

cFaculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Viet Nam

dDepartment of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, Vietnam

a rt i c l e i nf o

Article history:

Received 4 September 2020 Revised 25 December 2020 Accepted 30 December 2020 Available online 12 January 2021 Keywords:

Buckling

Multiple transverse ply cracking laminates Finite element method

a b s t ra c t

The phase-field theoryisawell-known mathematical modelfor solving interfaceprob- lems,includingcrackproblemsinfracturemechanics.Inthisstudy,theformulaisderived byvariationalapproachesbasedontheReissner-Mindlin platekinematicsandthemulti- phase-fieldtheoryforsimulationofthebucklingphenomenonincrackedlaminates.Phase- field parametersaredefined independentlyindifferentplies oflaminatetocapture the crackbehavior ofeachply.Simulationiscarriedouttonumericallyinvestigatethe stiff- ness reductionand bucklingbehavior oftransverse crackedlaminatedcomposite plates.

Thispaperfocusesontheconsiderationoflaminatedcompositeplates,whichhaveacrack ineachlayer.Therefore,thisworkismorecomplicatedthanthecaseoftheplatehasone crackthroughouttheplatethickness.Thesignificantadvancementofthephase-field ap- proachforlaminatedcompositeplateswithcomplexcrackgeometriesisdemonstrated.

© 2021ElsevierInc.Allrightsreserved.

1. Introduction

Composite laminates are widely employed in various engineering fields because they have many advantagessuch as highstrength-to-weight andstiffness-to-weightratios,savingenergy,andlow productioncost.Manyof theirapplications can be mentioned such asaerospace, shipbuilding,automotive industry,wind power, andso on.In fact,thesecomposite structures arealwayssubjectedtovarioustypesofexternalloadswithhighintensity.However,theyare madeofdifferent reinforced materials andfibers, and thereforeoften appear cracks, especially cracks that do not penetrate the thickness of thestructure andalongthe angle-ply.The characteristics ofthistype ofcrackare oftendifficult toidentify, soifthey extend, theperformingabilityofthestructurecanbe reduced.Theanalysisofcracked structuresisasignificantchallenge forscientists.Tomodelcracks,ingeneral,some highlightapproachescanbeusedsuchasextendedfiniteelementmethod (XFEM), extended isogeometricanalysis (XIGA), meshfree method,generalized differential quadrature,andso on. Forthe XFEMapproach,itisbasedonthediscretemethodoffiniteelementmethod(FEM),theshapecanbeenrichedinalocation,

∗Corresponding author at: Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

∗∗Corresponding author.

E-mail addresses: doanhongduc@tdtu.edu.vn (D.H. Doan), thom.dovan@lqdtu.edu.vn (T. Van Do).

https://doi.org/10.1016/j.apm.2020.12.038 0307-904X/© 2021 Elsevier Inc. All rights reserved.

thus,itcanbeusedtosolvesimplecrackproblems.However,thecomputationalcostofthisapproachwillincreaserapidly for theproblems withmultiplecracks in differentsurfaces ofthe structure.The isogeometric analysis(IGA)is a method thatusesthesplinesfunctiontorepresentthegeometry,whichcangiveanapproximation.TheXIGAisdevelopedfromthe IGA andsome functionsaround the discontinuousdomain,soit can alsosimulatecracks.The meshfree methodis based ondiscontinuitythroughthecrackofthedisplacementfieldandasingularbehaviorofthestressesaroundthecrack,thus, thismethod alsocanbe employed toanalyzetheproblems relatedtocracks.The mentioned methodshavebeenapplied by scientistsinanalyzingproblemsrelatedtostaticanddynamiccracks [1,2,21,28,29].Amir andSoheil[2]usedtheXFEM toresearchthelinearbucklingbehaviorofcrackeduni-layercompositeplatesbasedonthe8-nodeelement.Thelaminated plateswithcrackswerealsoresearchedbySeifi andRanjbaran[3],thebuckling problemof theseplates werecarried outby usingthegeneralizeddifferentialquadraturemethodanddomaindecompositiontechnique.Amit[4]exploredthenonlinear bucklingresponseofdamagedlaminatedcompositeplatesbasedonthefiniteelementmethodandlayerwiseplatetheory.

However,mostoftheseapproachesanalyzedcracksthroughoutthethicknessofstructures,includingsomepublishedworks thatcomputedcompositeplatesconsideringsuchcracks.Thissometimesdoesnotproperlyreflecttheworkingprocessesof multi-layeredcompositestructures,becausethesestructuresaremadeupofdifferentreinforcementmaterials,andarranged inseparatelayers.Itispossiblethatcrackscanappearineachlayerduringfabrication,orduringtheworkingprocess,cracks ineachlayerwillappearwithoutpenetratingtheplatethickness.Theappearanceofcrackshasasignificantimpactonthe efficiencyofusingthem,sothecalculationstudyforthesestructuresisanurgentrequirementfromreality.

Recently,therehavebeenseveralcomputationalmodelsforcracksthatdonotpenetratethethicknessofthestructures [5-8].Lietal.[6]appliedLayerwisetheoriestheoryandXFEMtoanalyzelaminatedcompositeplateswiththedelamination phenomenon andthecracks occupiedina partofthethicknessofthe structure.Byusingtheblock elementsandLayer- wise theory,Lu et al.[7]investigated the displacement ofthe cracked sandwich plate. Liet al.[8] then used Layerwise theories andXFEMtheory to carry outthe free andforcedvibrationanalyses ofcracked FGMplates.Several workshave alsomentionedtheinvestigationsofbeamandplatecompositestructureswithmanycracks[9,10],Lietal.[9]exploredthe mechanicalbehaviorofcompositebeamwithmultiplecracksthatwereaffectedbyimpactloads.Yuanetal.[10]applieda noveltrans-scalesolutiontostudycompositeplateswithmultiplecracks.

The phase-fieldmethodis anewapproach todeal withfractureproblems,the advantageof thismethodisto change the crack domain (discontinuous domain) to become a continuous domain through a phase-field variable.This variable is continuously smooth from0to 1corresponding to thestate ofthe materialfromdestructive to non-destructive.Since its inception,it hasbeen highlyappreciatedby scientistsand iswidely appliedto analyze2D and3D cracked structures [15,16,18,19,21].Forbeamandplatestructures,thereisjustonepublicationapplyingthistheorytoexplorethemechanical behavior of the Euler- Bernoulli beam with multiple cracks subjected to bending load [11]. Thom et al. andDuc et al.

alsoappliedphase-fieldtheorytoinvestigatecrackedplateswithdifferentplatetheories[20,22-25].However,theprevious publicationsoftheauthorsonlyconsideredthecracks,whichwerethroughouttheplatethickness, sotherearenostudies tosimulatetheresponseofplateswithmulticracks,inwhicheachcracklocatesinonepartofthethicknessdirection.On the other hand, formultilayered compositeplates, each layer witha differentreinforcement angle mayappear cracks in onlyeach layer,therefore,theinvestigationofmechanicalresponsesofthesestructures isvery interesting,whichplaysan importantroleinusing,designing,andstudyingthesestructuresinengineeringpractice.

As mentionedabove,duetothecombinationofmanydifferentmaterials, structuresmadeofcompositematerials may crackinthedirectionoftheangle-plies,andforthisreason,severalstudieshavebeenconductedwithlaminatedcomposite structures that have cracks along theseangles [12,13,14]. Han et al. [12] used polynomial (second-order) to present the displacement oftheopen crack,thereby the stressaround thecrack canbe described. SinghandTalreja[13]studied the mechanical behavior of laminatedcomposite plates under tensileloads, which focused on the deformation and stiffness degradation in cracks by using analytical formulas andfinite element models, whichare integratedinto Ansys software.

CarraroandQuaresimin[14]usedananalyticalapproachtoexaminethedecreaseofstiffnessinperpendiculardirectionsto thecrackofthelaminatedcompositestructureswhenoneandmorecracksappearinthelayers.

Recently, therehave been some publications used highorder phase-fieldandmulti-phase-field to investigatecracked structures. Pengetal.[30] employed anovelphase-fieldmodeltostudyprogressivefailure incompositelaminatesunder tension load. Uditetal. [31]proposed a cohesive phase-fieldmodelto simulateintra-laminar fracture infiber-reinforced composites. Wu etal.[32]used afourth-order phase fieldandan efficientgradient smoothingmeshfree tomodelbrittle fractures.Chenandhisco-workers[33]studiedcrackkinkingandzig-zagcrackpropagationbasedonahigher-orderphase- field. MaandSun[34]employedmulti-phase-fieldfracture modelstosolvetwo phase-fieldmodelsdesignedandsimulate crackgrowthofstronglyanisotropicmaterialsinthebrittleregime.Deanetal.[35]alsousedamultiphase-fieldtoconsider triggeringintra-laminarcrackinginlongfiber-reinforcedcomposites.However,thesestudieshavenotprovidedanymodels andapproachestoanalyzethebucklingandfreevibrationproblems,whicharecommonissuesinpractice.Therefore,inthis work,thephase-fieldtheoryisemployedtoinvestigatethemulti-crackplatebasedonMindlinplatetheory,eachlayerhas onecrackthroughthethicknessofthelayeraccordingtotheangle-plyandnotthroughthetotalthicknessoftheplate.To developthistheory,manyphase-fieldvariablesareused,eachonedescribesthecrackofthelayer.

Thispaperisdividedinto4sections.Section2presentsbrieflysomefiniteelementformulationsoflaminatedcomposite platesbasedonthefirst-ordersheardeformationtheory.Thephase-fieldisemployedandshownclearlyinSection2tode- scribetheappearanceofcracksinthestructure.Section3isaboutnumericalresultsanddiscussionsoflaminatedcomposite plates.Somehighlightconclusionsarelistedinsection4.

Fig. 1. The geometric model of a cracked laminated plate.

2. Formulationforcompositeplatesbasedonfirst-ordersheardeformationtheory

ConsideracompositeplateunderaCartesiancoordinatesystemwiththethicknesshasshowninFig.1.

In thispaper,alaminatedplatewithdifferentlayers asshownin Fig.1isconsidered. Eachlayeroftheplatehasone crack along its angle-ply andthroughout the layer thickness. In order to describe all cracks, multi-phase-field theory is employedwithdifferentphase-filedvariablessq(q=1-N,Nisthenumberoflayers),eachvariablesqrepresentsthecrackof layerq.

As the plate isunder a compressive load inthe mid-plane, the energyof the structure includingthe energy ofeach crackedlayerisexpressedasfollows[37]:

^Pb = 1 2

_zq

zq−1

s²_q

ε

i jC^q_{i jkl}

ε

kldzd

+1 2

_zq

zq−1

s²_q

∇

^Tw

σ

^ˆ0

∇

^wdzd

+1

2

_zq

zq−1

s²_q

∇

^T

φ

^x

σ

^ˆ0

∇ φ

^xz2qdzd

+1 2

_zq

zq−1

s²_q

∇

^T

φ

^y

σ

^ˆ0

∇ φ

^yz2qdzd

+

_zq

zq−1

G_C^q

(

¹−sq

)

²

4l +l

| ∇

^sq

|

²

dzd

⁽¹⁾

wherethestrainsarethesameineachlayeranddefinedas:

ε

i j=1 2

ui,j+uj,i

(2)

andtheCauchystresstensor

σ

ijinlayerqiscomputedfromstrainas[38]

σ

i j^q=C_{i jkl}^q

ε

kl (3)

theprebucklingstresstensoris

σ

ˆ⁰⁼

σ

x⁰

τ

xy⁰

τ

xy⁰

σ

y⁰

(4)

Inthispaper,thephase-fieldapproachpresentedbyBourdinetal.[19]isused,thecrackismodeledbyanarrowregion andcontrolled bythephase-fieldvariablesq whichgetsvalue from0(total broken)to1(unbroken).When analyzingthe cracked plate, thepotential functionof eachply ismultipliedby thephase-fieldvariablesq.Indetail, inthenon-cracked

areaofeachply(s_q=1),theenergydoesnotchange.Incontrast,atthecrackedarea,duetothephase-fieldvariables_qvaries smoothlyfrom1to0correspondingtotherearandthecenterofthecrack,respectively,theenergyofthisareachangesas afunctionofthephase-fieldvariable.Atthesametime,theenergyreleasedatthecrackedareawillbeaddedtothewhole energyofthesystem. Thispartoftheenergyisthemainreasonleadingthecrackedply tobesofterincomparisonwith theplatewithoutanycracks[18,19,29].Therefore,inpractice,theenergyfunctionofthecrackedplyismostlikelysimilar tothatoftheplywithoutanycracksby usingthephase-fieldvariablessq,sotheaccuracyofthepresentapproachwillbe enhanced.Theinterpolationofvariables_q foreachelementinthecrackedarea iscarriedoutsimilarlytotheinterpolation ofotherdegreesoffreedomoftheelement,whichmakesthecrackedregionacontinuouszone,hence,itisconvenientfor computing. Note thatin Eq.(1)fortheproblemof crackedplates undercompressionloads,the crackwill beclosed, the phase-fieldmodelproposedinthisworkhasnotshownthispropertyclearly.However,accordingtothegroupofpublished works by the author [20,22] hasshown that when comparing this modelwith other computational methods as well as comparing with experiments, it isfound that the proposed model is still for acceptable results.Thereby, it can be seen that, for thebuckling problems ofplates underthe smalldeformation condition, the effect ofthe cracking characteristic whensubjectedtothecompressiveloadisnotobviousandcanbeignored.Therefore,tokeepthesolutionneat andavoid cumbersomecalculation, theauthors stilluse thismodeltocalculatelike thereferences[36,39], whichisalsothe model used byother authorstoinvestigatethecracked structures [39].The authorsalsofoundthat insome casessuchaspost- bucklingproblems,inwhichlargestrainistakenintoaccount,Eq.(1)needstobeadjustedtobetteraccommodatecracking closedwhensubjectedtocompressionloads,whichhasbeenproposedbyseveralotherauthors[37].

Derived fromphase-fieldtheory, G_C^q is thecritical energyreleaserate orsurfaceenergyin Griffith’s theory andl isa positiveregularizationconstanttoregulatethesizeofthefracturezone.Theeffectoftheparameterlonthecriticalbuckling loadwasinvestigatedandindicateddetailsin[22],wherelvariedfroma/800toa/100,thevalueoflhadasmalleffecton criticalloads,therefore,inthisstudyl=a/200waschosen.

BasedonMindlinplatetheory,thedisplacementfieldisexpressedas:

u

(

^x,y,z

)

^=u0

(

^x,y

)

^+z

φ

^x

(

^x,y

) v ₍

x,y,z

)

⁼

v

0

(

^x,y

)

^+z

φ

^y

(

^x,y

)

w

(

^x,y,z

)

^=w0

(

^x,y

)

⁽⁵⁾

whereu,v,warethedisplacementcomponentsinthex,y,zaxes,respectively.

ϕ

^x,

ϕ

^{yarethetransversenormalrotationsin}

thexz-andyz-planes;u₀,v₀,w₀aredisplacementsofthemiddlesurface.

Atthistime,theenergyEq.(1)willbewritteninthefollowingform:

^Pb

(

^u,s

)

^{= 1}₂

_zq

zq−1

s²_q

ε

^TpAq

ε

p+

ε

^TpBq

ε

b+

ε

^TbBq

ε

p+

ε

^TbDq

ε

b+

γ

s^THq

γ

s

d

+1 2

zq

zq−1

s²_q

∇

^Tw

σ

^ˆ0

∇

^wdzd

+1 2

zq

zq−1

s²_q

∇

^T

φ

x

σ

ˆ⁰

∇ φ

xz²_qdzd

+1 2

_zq

zq−1

s²_q

∇

^T

φ

^y

σ

^ˆ0

∇ φ

^yz2qdzd

+

_zq

zq−1

G_C^q

(

¹−sq

)

²

4l +l

| ∇

^sq

|

²

dzd

=

s²_q q

(

^u

)

^d

+

zq

zq−1

G^q_C

(

^1−sq

)

²

4l +l

| ∇

^sq

|

²

dzd

(6) wherethevectors

ε

^pand

ε

bcontainamembrane,bendingandtransversestrain,respectively.

ε

p=

u0,x

v

0,y

u_0,y+

v

0,x

ε

b=

φ

^x,x

φ

y,y

φ

^x,y+

φ

^y,x

γ

s=

φ

x+w0,x

φ

^y+w_0,y

(7)

wherethematricesAq,Bq,Dq,andHq arerespectivelytheextensional,bending-extensionalcoupling,bendingstiffnesscoef- ficientsandexplicitly,whicharegivenby

Aq=

_A

11 A12 A16

A12 A22 A26

A16 A26 A66

q

;Bq=

_B

11 B12 B16

B12 B22 B26

B16 B26 B66

q

;Dq=

_D

11 D12 D16

D12 D22 D26

D16 D26 D66

q

Hq=

H44 0 0 H55

q

(8) with

A_{i j} = N

q=1

Q¯_{i j}

zq−z_q−1

;B_{i j}=1 2

N

q=1

Q¯_{i j}

z²_q−z²_q−1

;D_{i j}=1 3

N

q=1

Q¯_{i j}

z³_q−z³_q−1

;i,j=1,2,6

H_{i j} = 5 6

N

q=1

Q¯_{i j}

zq−z_q−1

;i,j=4,5 (9)

hereinthequantitiesQ¯_{i j}aredeterminedfromthefiberdirection

θ

^{,Young’smodulusisparalleltoand}perpendiculartothe orientationofthefibersE₁,E₂,shearmodulusG₁₂,Poisson’sratios

ν

12,

ν

21areasinreference[17].

Thefirstvariationofthefunctional^Pb(^u,s)^isgivenby

δ

^Pb

(

^u,s,

δ

^u

)

⁼⁰

δ

^P_b

(

^u,s,

δ

^s

)

^{=0 (10)}

andthen,theformulationsforpre-bucklinganalysesofthecrackedplateareexpressedas

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

_zq

zq−1s²_q

ε

^TpAq

ε

^p+

ε

^TpBq

ε

b+

ε

^TbBq

ε

^p+

ε

^TbDq

ε

b+

γ

s^THq

γ

^s

d

+

_zq

zq−1s²_q

∇

^Tw

σ

^ˆ0

∇ ( δ

^w

)

^dzd

⁺z^zq^q−1s²_q

∇

^T

φ

^x

σ

^ˆ0

∇ ( δφ

^x

)

^z2qdzd

+

zq

zq−1s²_q

∇

^T

φ

y

σ

ˆ⁰

∇ ( δφ

y

)

^z2qdzd

=0

2sq q

(

^u

) δ

^sqd

+

zq zq−12G^q_C

(¹−sq)δ^sq

4l +l

∇

^sq

∇ ( δ

^sq

)

dzd

=0

(11)

Forthefracture problem,eachnode consistsofdisplacementdegreesandphase-fieldvariables,whendividingthecal- culateddomainintoelementswithnnodes,interpolationu_iandphase-fieldvariablesataninternal pointoftheelement accordingtothenodaldisplacementsu_ie={u_0i,v_0i,w_0i,

ϕ

xi,

ϕ

yi}^Tands_iatthenodesoftheelementisperformed.

u₀= n

i=1

N_iu_0i;

v

0= n

i=1

N_i

v

0i;

φ

^x=n

i=1

N_i

φ

xi (12)

φ

^y= n

i=1

Ni

φ

yi;w0= n

i=1

N_i^ww0i;sq= n

i=1

N_i^qsi (13)

whereN_iaretheshapefunctionsofthequadraticpolynomial,N_i^ware theshapefunctionsofthethird-degreepolynomial, andN_i^qaretheshapefunctionsofthefirst-degreepolynomial.Notethatinthiswork,toavoidtheshearlockingphenomenon, differentshapefunctionsareusedfordisplacementcomponentsandphase-fieldvariablestoobtainthebestconvergence.

Then:

ε

p=B1ue;

ε

b=B2ue;

γ

s=B3ue;w0=B4ue;

∂

^w0

∂

^{x =B5ue;}

∂

^w0

∂

^{y =B6ue (14)}

∂ φ

^x

∂

^{x =B7ue;}

∂ φ

^x

∂

^{y =B8ue;}

∂ φ

^y

∂

^{x =B9ue;}

∂ φ

^y

∂

^{y =B10ue;}

∇

^sq=Bq11sqe (15) whereB_iaredifferentialmatricesofshapefunctions,theyarecalculatedas:

B₁= n

i=1

⎡

⎣

∂^Ni

∂^x 0 0 0 0 0 ^∂_∂^N_yⁱ 0 0 0

∂^Ni

∂^y ∂^Ni

∂^x 0 0 0

⎤

⎦

;B₂= n

i=1

⎡

⎣

^{0 0 0}

∂^Ni

∂^x 0 0 0 0 0 ^∂_∂^N_yⁱ 0 0 0 ^∂_∂^N_y^{i ∂}_∂^N_xⁱ

⎤

⎦

(16)

B₃= n

i=1

0 0 ^∂_∂^N_xⁱ N_i 0 0 0 ^∂_∂^N_yⁱ 0 N_i

;B₄= n

i=1

0 0 N^w_i 0 0

(17)

B5= n

i=1

0 0 ^∂_∂^N_x^wi 0 0

;B6= n

i=1

0 0 ^∂_∂^N_y^wi 0 0

(18)

B7= n

i=1

0 0 0 ^∂_∂^N_xⁱ 0

;B8= n

i=1

0 0 0 ^∂_∂^N_yⁱ 0

(19)

B₉= n

i=1

0 0 0 0 ^∂_∂^N_xⁱ

;B10= n

i=1

0 0 0 0 ^∂_∂^N_yⁱ

;B^q₁₁= n

i=1

_∂Nq

∂^xi

∂^Ni^q

∂^y

(20)

Sincethen,Eq.(11)iswritteninthematrixformasfollows:

s^T_qe

_zq

zq−1

2

N_s^q

T

q

(

^u

)

^Nqsdz

s_qed

+

zq

zq−1

2G^q_C

⎡

⎣

Ns^qs_qe−s^T_qe

Ns^q

T

N^qss_qe

4l +s^T_qel

B^q₁₁

T

B^q₁₁

⎤

⎦

dzd

^{=0 (21a)}

K^e+

λ

crK^e_G

ue=0 (21b)

inwhich,theelementstiffnessmatrixK^e,geometricelementstiffnessmatrixK^e_G,andelementnodaldisplacementvectorue

aredefinedas:

K^e=

3

q=1

s²_q

B^T₁AqB^T₁+B^T₁BqB^T₂+B^T₂BqB₁+B^T₂DqB₂+B^T₃HqB₃

d

K^e_G=

3

q=1

z_q−z_q−1

s²_q

B^T₅B^T₆

σ

ˆ^0[B5B₆]d

+

3

q=1

1 3

z³_q−z_q³₋₁

s²_q

B^T₇B^T₈

σ

ˆ^0[B7B₈]d

+

3

q=1

1 3

z³_q−z³_q−1

s²_q

B^T₉B^T₁₀

σ

ˆ^0[B9B₁₀]d

ue= n

i=1

u_ie (22)

NotethatEq.(21a)couldbewritteninthefollowingform:

s^T_qe

_z

q

zq−1

2

N_s^q

T

q

(

^u

)

^Nqsd

^dz

s_qe+s^T_{qe zq}

zq−1

2G^q_C

−

N_s^q

T

N_s^q 4l +l

B^q₁₁

T

B^q₁₁

d

^dz

s_qe

+

_z

q

zq−1

2G^q_C

N^q_s 4l

dzd

s_qe=0 (23)

orintheshortformas:

s^T_qe

_zq

zq−1

2

N_s^q

T

q

(

^u

)

^Nqsd

^dz

+s^T_qe zq

zq−1

2G^q_C

−

N^q_s

T

N_s^q 4l +l

B^q₁₁

T

B^q₁₁

d

^dz

=−

_zq

zq−1

2G_C^q

N_s^q 4l

dzd

(24)

Forthewholestructure,wehavethefollowinggeneralequation:

e

s^T_qe

_z

q

zq−1

2

N^q_s

T

q

(

^u

)

^Ns^qd

^dz

+s^T_{qe zq}

zq−1

2G_C^q

−

N_s^q

T

N_s^q 4l +l

B^q₁₁

T

B^q₁₁

d

^dz

=−

e

_z

q

zq−1

2G_C^q

N_s^q 4l

dzd

(25a)

e

K^e+

λ

^cr

e

K^e_G u=0 (25b)

Eq.(25a)hasavariables^T_q=!

e

s^T_qe,bysolvingthisequation,weobtainphase-fields_qatthecracks.

Eq.(25b)isusedtofindcriticalloads

λ

^{croftheplate,however,duetostiffnessmatricesKe},K^e_Gdependingonphase-field variable sq, therefore, firstly, phase-field sq variable of layer q (q=1-N) according to the corresponding mesh as shown in Figs. 2-4(assumingthat the platecontainsthree layers)through Eq.(25a)is processed.Then thevalues ofsq willbe substituted inEq.(22)todetermine thestiffnessmatricesthroughthegeneralmeshforthewholeplate(see Fig.5)with thespecifiedsqvalues.Next,thecriticalloadsandbucklingmodeshapesofthestructurecanbeobtainedfromEq.(25b).

Tocalculates_qoflayerq,thecrackshapeineachlayerisdefinedbysolvingEq.(25a)withfunction _q(u)basedonthe coordinatesofthiscrackasfollow[18,19,29]

q

(

^u

)

=BG^q_C

4l.Hq

(

^x

)

^(26a)

where Hq

(

^x

)

⁼

1 i f x≤cqand⁻₂^l0 ≤y≤^l₂⁰

0 else (26b)

herein cq isthe crack lengthin layer q,the higherthe scalarB’s magnitudeis the better,in thiswork its can be gotten B= 10³,seedetaillyinTable2.Normally, G^q_C isa materialparameteranddeterminedthrougha materialhomogenization

Fig. 2. The crack model in layer 1 of the composite plate.

Fig. 3. The crack model in layer 2 of the composite plate.

Fig. 4. The crack model in layer 3 of the composite plate.

Fig. 5. The general model of three cracks of the composite plate with three layers.

Table 1

The present results are compared with those of other results for buckling load of composite plate.

Angle-ply This work [26] [27]

1200 elements 1500 elements 5500 elements 70 0 0 elements

(0 ⁰, 0 ⁰, 0 ⁰) 2.381 2.370 2.363 2.362 2.39 2.391

(15 ⁰, -15 ⁰, 15 ⁰) 2.451 2.442 2.426 2.425 2.45 2.448

(30 ⁰, -30 ⁰, 30 ⁰) 2.579 2.565 2.559 2.558 2.57 2.578

(45 ⁰, -45 ⁰, 45 ⁰) 2.651 2.643 2.629 2.627 2.64 2.649

(0 ⁰, 90 ⁰, 0 ⁰) 2.398 2.386 2.364 2.362 2.39 2.394

Fig. 6. The geometric model of a plate with an edge crack.

processbasedontheG^q_C valueofthematerialcomponents.Inthiswork,G^q_C isdetermineddependingonthematerialprop- ertiesandithasimportantimplicationsincrackpropagationproblemsduetothevalueofG^q_Cdeterminestheloadcapacity before thecrackspreads. However,inthisproblem, theauthors donot takeinto accountthe propagationcapacityof the crack,andG_C^qisusedonlyinEq.(26)todeterminetheshapeofthecrackaccordingtothemethodsproposedintheworks [18,19,29]. Accordingly, _q(u) ischosen asEq.(26a) toensure _q(u)>> G^q_C. Thisselection guarantees s_q =0 inthe area definedbyHq=1inEq.(26b).NotethatEq.(26b)isonlyusedtodefinestraightcracks,however,forcrackswithcomplex shapes,itisnecessarytousetheLevelsetfunctionasshowninthereference[29].

Remark:Throughoutthisproposedtheory,inordertobeconvenient,theformulaofonecrackisestablishedintheangle- plydirectionofeachlaminatedlayer.Nevertheless,theexpansionoftheproblemwithmultiplecracksandthedirectionof crackscanbefreelychangedbydefiningthefunctionHqinEq.(26b).

3. Numericalresults

Herein, in the following investigations, some boundary conditions are used and described as follows: S represents a simplysupportededge,Crepresentsaclampededge,andFrepresentsafreeedge.So,afully4-edgesimplysupportedplate isSSSS,andafully4-edgeclampedsupportedplateisCCCC.

Theplateissimplysupportedatx=0,a:v₀=w=

ϕ

^y=0 Theplateissimplysupportedaty=0,b:u₀=w=

ϕ

^x=0 Theplateisclampedatanyedges:u0=v0 =w=

ϕ

x =

ϕ

y=0 Theplateisfreeatanyedges,nodegreesoffreedomarefixed.

3.1. Comparison

A laminatedsquare platehasthe dimension: length a,width b witha = b = 10m, the thickness[26,27] h= 0.06m, this plate is fully simply supported (SSSS). The plate material parameters of E-glass/epoxy are employed E₁/E₂ = 2.45, G₁₂ =0.48E₂,

ν

12=0.23,

ν

21=

ν

12E₂/E₁;theplateissubjectedtothecompressiveloadinthex-axis.Thebucklingcritical loadisdefinedas[26,27]

k=

λ

^cra2/

π

^{2D0 (27)}

whereD₀ =E₁h³/(12(1−

ν

12

ν

21))

The buckling criticalloads ofthe three-layer compositeplatein thiswork are comparedwith thoseof themesh-free method[26,27]aspresentedinTable1.Itcanbeseenthatbothtwomethodshavegoodagreement.Inthiscomparison,a meshwithanumberofelementsfrom1200to7000elementsisused.

Next,thecriticalbucklingtemperaturerise(CBTR)ofacrackedplatedepictedinFig.6isexamined,thisplatehasfour boundaryconditions:fullyclamped(CCCC),full4-edgesimplysupported(SSSS),2edgesaresimplysupportedandtheother two edgesare free(SFSF), thetwooppositesidesare clampedandthe tworemainingedges aresimplysupported (SCSC).

ThisplateismadefromAl₂O₃(Young’smodulusE=380GPa,Poisson’sratio0.3,thecoefficientofthermalexpansion74.10⁻⁷

Fig. 7. The model geometry of a cracked composite plate.

Table 2

Comparison of the CBTR of a rectangular Al 2O 3plate ( c / a = 0.5, a = 2m, b = 1m, h / b = 0.01).

Method The boundary conditions

CCCC SCSC SSSS SFSF

XIGA [28] 17.036 14.415 6.225 1.960

This work

B = 10 ³ 1200 elements 17.521 14.763 6.316 1.987

1500 elements 17.396 14.619 6.237 1.979

5500 elements 17.180 14.529 6.170 1.960

7000 elements 17.179 14.526 6.168 1.959

B = 10 ⁴ 7000 elements 17.178 14.525 6.164 1.950

B = 10 ⁴ 7000 elements 17.178 14.525 6.164 1.950

1/°C)witha=2m,b=1m,thicknessh=b/100,cracklengthc=a/2.ThepresentresultsandthedatafromXIGAmethod [28] are compared. The similarity betweenthose data can be seen in Table 2. The mesh asin the comparison above is also used, fromthe computed results basedon this mesh,it found that the meshhas 7000elements that have a good convergence,soforthecomingsections,thismeshisused.ThevalueofparameterBin(26a)increasesgraduallyfrom10³ to10⁵,whenB=10³ hasensuredtherequiredprecisionsothatinthiswork,itsvaluewaschosen.

Finally,the comparisonofthiswork andexperimentapproachby Seifi. etal.[40] forthehomogenousplatewithone crackiscarriedout. Considerasquareplatewitha=240mm,theplatethickness12mm,andPoisson’sratio0.33.Theplate is underthecompressive loadattwo opposite clamped edges,the remaining edgesare free. Thecomparative resultsare presentedinTable3,itcanbeseenthatalargestdifferencebetweenthisworkandexperimentapproachis8.97%.Thisis explainedthat theplatematerialismadenotpure andthecrackshapeisnot perfect.Therefore,itcanbeconcludedthat theproposedtheoryandmathematicalmodelarereliable.

3.2. Numericalanalysis

Basedontheproposedtheoryandcalculationprogramwiththehighaccuracytoinvestigatetheeffectofseveralparam- etersonthebucklingofcrackedcompositeplates,acrackedcompositeplatewiththeratioa/b=1,thethicknessh=a/100is nowexamined.Thisplateismadefromathree-layerE-glass/epoxywithmaterialcharacteristicsandcriticalbucklingloads describedintheabovesection(section3.1).TheplateiscompressedalongtheOxaxisasshowninFig.7,andthebuckling criticalloadisdefinedinEq.(27).

Fig. 8. Buckling mode shapes of cracked plate ( a = b = 0.2m, h = b /100, c / a = 0.8, axial compression).

Table 3

Critical buckling load of cracked plate of present results are compared with experimental results.

Crack length ( c/a ) Crack inclination ( α) Experimental buckling load (N) [40] This work (N) % Error

0.1 0 1627 1773 8.97

0.3 1531 1537 0.39

0.5 1317 1205 8.50

0.1 30 ⁰ 1651 1782 7.94

0.3 1551 1626 4.83

0.5 1396 1413 1.21

0.1 60 ⁰ 1674 1798 7.40

0.3 1660 1756 5.78

0.5 1636 1723 5.31

3.2.1. Bucklingofcrackedcompositesquareplate

First ofall, theeffectofthelengthofthecrackonthebucklingofthisplateiscarriedout.Wesuppose thattheplate hasdifferentangle-pliesamonglayers,theboundaryconditionisSSSS.Eachlayerhasacrackinthecenteroftheplate,and thelengthofeachcrackisrepresentedasc.Bychangingcsothatc/a=0.1-0.8,d=0.5a,thegottennumericalresultsare presentedinTables4and5.NoticethatthebucklingloadiscomputedasEq.(27).

It can be observedthat when increasing thecrack length,the criticalloadof theplatedecreases (caused by thearea of energy loss) when the compressed load increases.In consideration of the angle-plyorientations, the composite plate (45⁰,-45⁰,45⁰)hasthehighestcriticalload,whichcanbeexplainedthatthecombinationofthelayersofmaterialsmakes

Fig. 9. Buckling modes of cracked (0 ⁰, 30 ⁰, 0 ⁰) plate ( a = b = 0.2m, h = H /100, c / H = 0.5, compressed along Ox axis.

Table 4

Effect of crack length on the critical buckling loads with various angle-ply orientations for the composite plate (axial compression).

Angle-ply c/a

0.1 0.3 0.5 0.6 0.8

Two layers (0 ⁰, 0 ⁰) 2.33442 2.21871 2.07616 2.01489 1.93345

(15 ⁰, -15 ⁰) 2.39736 2.36844 2.35099 2.34730 2.34985

(30 ⁰, -30 ⁰) 2.52789 2.50200 2.48515 2.47888 2.47607

(45 ⁰, -45 ⁰) 2.60281 2.57977 2.56029 2.55534 2.54846

(0 ⁰, 90 ⁰) 2.30687 2.28426 2.27022 2.26713 2.26794

Three layers (0 ⁰, 0 ⁰, 0 ⁰) 2.33431 2.21853 2.07612 2.01484 1.93343

(15 ⁰, -15 ⁰, 15 ⁰) 2.40625 2.36771 2.34039 2.33288 2.33013

(30 ⁰, -30 ⁰, 30 ⁰) 2.54082 2.50792 2.48216 2.46969 2.46082

(45 ⁰, -45 ⁰, 45 ⁰) 2.61174 2.58287 2.55881 2.54726 2.53155

(0 ⁰, 90 ⁰, 0 ⁰) 2.34716 2.32169 2.30598 2.30573 2.30531

this platestiffer.Fig. 8presents thefirst fivebuckling mode shapesof thecracked composite platewiththree angle-ply types.Itcanfindthattheangle-plyalsohasagreateffectoncriticalformsofthecrackedcompositeplate.Formode1,the bucklingmodeshapechangesslightlyaccordingtotheangle-ply,however,bucklingmodeshapeschangesclearlyinhigher frequencies.Thereasonisthatwhenthecracksandangle-plieslocateindifferentpositionswillcausethesurfacetorelease energyatthosecracksaccordingtothedirectionoftheexternalforce.