Vibrations of asymptotically and variationally based Uflyand-Mindlin plate models

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Vibrations of asymptotically and variationally based Uflyand-Mindlin plate models

I. Elishakoff, F. Hache, Noël Challamel

To cite this version:

I. Elishakoff, F. Hache, Noël Challamel. Vibrations of asymptotically and variationally based Uflyand- Mindlin plate models. International Journal of Engineering Science, Elsevier, 2017, 116, pp.58-73.

�10.1016/j.ijengsci.2017.03.003�. �hal-01693909�

Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models

I. Elishakoff

^a,∗

, F. Hache

^a,b

, N. Challamel

^b

aDepartment of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33481-0991, USA

bUniversité de Bretagne Sud (UBS), Institut de Recherche Dupuy de Lôme (IRDL), Centre de Recherche, Rue de Saint Maudé, BP92116, 56321 Lorient cedex, France

Inthispaper,weprovidealternativeUflyand–Mindlin’splateequationstakingintoaccount rotaryinertiaandsheardeformation,basedonbothasymptoticexpansionandvariational arguments.TheaimistoderivetruncatedversionsofUflyand–Mindlin’sequations,specif- icallywithoutthefourthorderderivativetermwithrespecttotime.Thetruncatedversion ofUflyand–Mindlin’splatemodelmaybederivedstartingfromthree-dimensionalelastic- ityequations,byusingasymptoticargumentsbasedonexpansionofdisplacementswith respecttoasmallgeometricalparameter.Thisexpansionmethodalsoleadstoaproper identificationoftheshearcorrectionfactor.Itisshownthatsuitablymodifiedvariational derivationleadstoanadditionaltermwhichisshowntobenegligiblefordetermination ofthefundamental naturalfrequency oftheall-round simplysupportedplates, butmay contributesignificantlyinestimationofhighernaturalfrequencies.Itisargued thatthe proposedversionofUflyand–Mindlin’splateequationsissimplerandmoreconsistentthan theoriginalUflyand–Mindlinequations.Likewise,itisadvantageousovertheequationthat stemsfromneglectingthefourthordertimederivativeinoriginalUflyand–Mindlinequa- tions.The twoalternativetruncated modelsserve asintermediate theoriesbetweenthe classicalplatetheoryandtheoriginalUflyand–Mindlintheorytheirusefulnessdepending ontheproblemathand.

1. Introduction

Initiated by Germain(1826) andcorrected by Lagrange (1828), the Classical Plate Theoryor German-Lagrange theory established the governing partial differential equations describing the mechanical behavior of thin plates in vibrations (Reismann,1988).AsexplainedbyVentselandKrauthammerintheirmonograph(Ventsel&Krauthammer,2001).“Cauchy (1828)andPoisson(1829)werefirsttoformulatetheproblemofplatebendingonthebasisofgeneralequationsoftheory ofelasticity”.Afewyearslater,Navier(1823)studiedthetheoryforaflexuralrigidityfunctionofthethicknessoftheplate.

Then, Kirchhoff (1850) broughtmanyadditionalresultsabouttheory ofthinplates.Accordingto Leissa,in hisforwardto the book ofLiew, Xiang, Kitipornchai,and Wang(1998), “a plateis typically considered to be thinwhen theratio ofits thicknesstorepresentativelateraldimension(e.g.,circularplatediameter,square plateside length)is1/20orless.Infact, mostplatesusedin practicalapplications satisfythiscriterion. Thisusually permitsoneto useclassicalthinplatetheory

∗ Corresponding author.

E-mail addresses: elishako@fau.edu (I. Elishakoff),fhache2014@fau.edu (F. Hache),noel.challamel@univ-ubs.fr (N. Challamel).

toobtainafundamental(i.e.,lowest)frequencywithgoodaccuracy”.However,theclassical platetheory maysignificantly overestimatehigherfrequencies.Inthelastcentury,lotofefforts havebeenmadetodescribethebehaviorofthickplates.

AsLiewetal.(1998),mention,Reissner(19441945)andNavier(1823)introduced“atheoryofplatesthattakesaccountof sheardeformationonly” inadditiontoclassicaleffects(seealsoKirchhoff,1850).

In 1921, Timoshenko (1921) published his study of vibrations of beams and introduced his governing differential equation that take into account both shear deformation and rotary inertia. The beamequations derived by Timoshenko areidenticaltotheonesofBresse(1859) thatarecorrectedbya shearcorrectionfactorwhichmaydifferfromunity. The Uflyand–Mindlinplatetheory,alsolabelledasthick platetheory(Mindlin,1951;Uflyand,1948),constitutesanextensionof theclassical Kirchhoff-Lovetheory bytakingintoaccount sheardeformation androtary inertiaandthus representingthe two-dimensionalanalogueoftheBresse–Timoshenkobeamtheory.

ThecheckonGoogleScholaroftheterm“Uflyand–MindlinPlate” yields29500hitsattesting theenormouspopularity ofthistheory.Thereisa definitivemonographdevotedto Uflyand–Mindlinplates,by Liewetal.(1998). Theinaccuracies describedbyLeissa(1969)arelargelyeliminatedbyuseoftheUflyand–Mindlintheory,foritdoesincludetheeffectsofad- ditionalplateflexibilityduetosheardeformation,andadditionalplateinertiaduetorotations(supplantingthetranslational inertia).Botheffectsdecreasethefrequencies.TherearestillothereffectsnotaccountedforbytheUflyand–Mindlintheory (e.g.stretching inthethicknessdirection,warpingofthenormaltothemidplane), butthesearetypicallyunimportantfor thelowerfrequenciesuntilverythick platesareencountered.ItappearsinstructivetoquoteHerrmann(1974):“Aboveall, Mindlin’sworkismotivatedbyaconcernforphysicalreality.Hisanalyticalstudiesalwaysbeginwithanintense desireto explainandinterpret,inmathematicalterms,observedbutpoorlyunderstoodphysicalphenomena”.

Overtheyears,manyresearchersattemptedto providedifferentderivationofUflyand–Mindlinplateequations.Oneof themisbasedonan asymptoticapproachconsidering athree-dimensionalproblemandreducing ittoatwo-dimensional problem (Vashakmadze, 1999). The use of asymptotic methods to validate a model has been used in the literature for beams (Berdichevsky & Kvashina, 1974) andsome attempt haves been performed for plates (Berdichevsky, 1973). Thus, Widera(1970),withoutanyassumptionaboutthedisplacementsoverthethicknessoftheplateandneglectingtheeffectof rotaryinertia andshear deformations,derived asetofequationsforthedeterminationofthein-planedisplacements, the samethan fortheclassicalthinplatetheory.Oneoftheaimsofthepresentpaperistoderiveasymptotically aversionof theUflyand–Mindlinplatemodelthrough apowerseriesexpansion. Inparallelofthisapproach,manyarticles havebeen publishedintheliteraturededicatedtothevariationalderivationofUflyand–Mindlin’s(Uflyand,1948;Mindlin,1951)plate equations.

Among them, one should mention the definitive monographs by Liew etal. (1998) or Wang, Reddy,and Lee (2000) andnumerousreferenceslisted there(see forinstanceBrunelle & Roberts,1974;Brunelle, 1971; Sharma,Sharda,& Nath, 2005).Elishakoff (1994)andFalsone,Settineri,andElishakoff (2014,2015)suggestedtoutilizetruncatedversionofUflyand–

Mindlin’s(Mindlin, 1951) equation, neglecting the fourthorderderivative intime. In thispaper,we presenta variational derivationoftruncatedUflyand–Mindlin’sequationbasedonslopeinertia.Itturnsoutthatanadditionaltermappears.We conductcomparisonoffourtheories:(a)classicalplatetheory,(b)Uflyand–Mindlin’s(Mindlin,1951;Mindlin,Schacknow,&

Dereciewicz,1956) original theory,(c)Elishakoff (1994)truncatedsetofequations,(d)variationallyderived truncatedset.

Whereaswe refrainfromjudgingthesuperiorityoftheabovemethods,we emphasizethatforlower rangeoffrequencies thelattersetatleastleadstosimilarresultsinamuchsimplerformulation,inadditionofbeingvariationallyderivable.

2. RecapitulationoforiginalandtruncatedUflyand–Mindlin’stheories 2.1. OriginalUflyand–Mindlinplatetheoryviatheequilibriumequations

Theplateisreferredtoax,y,z-systemofCartesiancoordinates.Assumingthatthefacesoftheplateareundernormal pressuresq₁andq₂,theboundaryconditionsare:

τ

^xz

z=±h 2

=

τ

^yz

z=±h 2

=0 (1a)

σ

^z

z=h 2

=−q1(x,y,t);

σ

^z

z=−h 2

=−q2(x,y,t) ^(1b)

Thebendingandtwistingmomentsandthetransverseshearingforcesaredefinedasfollows:

_M

x

My

Myx

= h/2

−h/2

σ

^x

σ

^y

τ

^yx

zdz;

Qx

Qy

= h/2

−h/2

τ

^xz

τ

yz

dz (2)

Foranisotropicmaterialonegets

Mx=D(x+

ν

y)^;My=D(y+

ν

x)^{;Myx= D}₂(¹⁻

ν

)yx;Qx=

κ

^Ghxz;Qy=

κ

^Ghyz (3)

Fig. 1. Rotations of a transverse normal about the y axis.

where D=Eh³/12(¹−ν²) ^{is the plate’s flexural rigidity, h the thicknessof the plate,} ν ^{the Poisson’s ratio,}κ ^{the shear}

coefficient,Gtheshearmodulusofelasticityandx,y,yx,xz,yztheplate-strainscomponentsdefinedasfollows:

(x,y,yx)^=12h−3

h 2

−^h2

ε

^x

ε

y

γ

^yx

zdz;

xz

^yz

=h^{−1 h}₂

−^h2

γ

^xz

γ

^yz

dz (4)

Theusualplate-strain-displacementrelationshipsarethefollowing:

ε

^x

ε

^y

γ

^yx

=

⎛

⎜ ⎝

∂^u

∂^x

∂v∂^y

∂v∂^x+^∂_∂^u_y

⎞

⎟ ⎠

^;

γ

^xz

γ

^yz

=

_∂u

∂^z+^∂_∂^w_x

∂v∂^z+^∂_∂^w_y

(5)

IntheMindlin(1951)platetheory,thedisplacementcomponentsareassumedtobegivenby:

u=z

ψ

^x(x,y,t);

v

=z

ψ

^y(x,y,t);w=w(x,y,t) ⁽⁶⁾ ψ^{x and}ψ^{y arethebendingrotations ofatransversenormalaboutthexandyaxes,}respectively,asshowninFig.1.It worthnothingthattheKirchhoff-Loveplatetheorycanberecoveredbysettingψx=−∂^w/∂^xandψy=−∂^w/∂^y.

Substituting Eq. (6) into Eq. (5) and then substituting in the resulting equation in Eq. (4), the plate-displacements componentsbecome:

^x=

∂ ψ

^x

∂

^{x ,y=}

∂ ψ

^y

∂

^{y ,yx=}

∂ ψ

^y

∂

^{x +}

∂ ψ

^x

∂

^{y , xz=}

ψ

^x+

∂

^w

∂

^x,yz=

ψ

^y+

∂

^w

∂

^{y (7)}

SubstitutionofEq.(7)intoEq.(3)leadsto:

Mx =D

∂ ψ

^x

∂

^{x +}

ν ∂ ψ

^y

∂

^y

;My=D

∂ ψ

^y

∂

^{y +}

ν ∂ ψ

^x

∂

^x

;Myx=D 2(¹⁻

ν

)

∂ ψ

^y

∂

^{x +}

∂ ψ

^x

∂

^y

Qx =

κ

^Gh

ψ

^x+

∂

^w

∂

^x

;Qy=

κ

^Gh

ψ

^y+

∂

^w

∂

^y

(8)

Thedynamicequilibriumequationsofthree-dimensionalelasticityread:

∂ σ

^x

∂

^{x +}

∂ τ

^yx

∂

^{y +}

∂ τ

^zx

∂

^{z =}

ρ ∂ ∂

^2tu2

∂ τ

yx

∂

^{x +}

∂ σ

y

∂

^{y +}

∂ τ

zy

∂

^{z =}

ρ ∂

²

v

∂

^t2

∂ τ

^zx

∂

^{x +}

∂ τ

^yz

∂

^{y +}

∂ σ

^z

∂

^{z =}

ρ ∂ ∂

^{2tw2 (9)}

Multiplicationbyzandintegrationovertheplatethicknessprovide,usingEq.(2),asystemofthreeequations:

∂

^Mx

∂

^{x +}

∂

^Myx

∂

^{y −Qx=}

ρ

^h3

12

∂

²

ψ

^x

∂

^t2

∂

^Myx

∂

^{x +}

∂

^My

∂

^{y −Qy=}

ρ

^h3

12

∂

²

ψ

y

∂

^t2

∂

^Qx

∂

^{x +}

∂

^Qy

∂

^{y +q=}

ρ

^h

∂

^2w

∂

^{t2 (10)}

whereq=q₂−q₁,istheresultantpressure.SubstitutingEq.(8)intoEq.(10),theequationsofmotionbecome:

D 2

(¹⁻

v

₎

∇

²

ψ

x+(¹⁺

v

₎

∂

²

ψ

x

∂

^{x2 +}

∂

²

ψ

y

∂

^x

∂

^y

−

κ

^2Gh

ψ

y+

∂

^w

∂

^x

=

ρ

^h3

12

∂

²

ψ

x

∂

^t2

D 2

(¹−

v

₎

∇

²

ψ

^y+(¹+

v

₎

∂

²

ψ

x

∂

^x

∂

^y+

∂

²

ψ

^y

∂

^y2

−

κ

^2Gh

ψ

^y+

∂

^w

∂

^y

=

ρ

^h3

12

∂

²

ψ

^y

∂

^t2

κ

^2Gh

∇

^2w+

∂ψ

^x

∂

^{x +}

∂ψ

^y

∂

^y

+q=

ρ

^h

∂

^2w

∂

^t2

(11)

FromEq.(11),agoverningequationofthedeflectionisobtained.DifferentiatingthetwofirstequationsofEq.(11),with respecttoxandy,respectively,andaddingtheseequations,oneobtains,setting=∂ψ^x/∂^x+∂ψ^y/∂^y

D

∇

²⁻

κ

^2Gh−

ρ

^h3

12

∂

²

∂

^t2

=

κ

^2Gh

∇

^{2w (12)}

where∇^{2 is the Laplace operator.} Substituting thisequation inthe first of the equationsof motions Eq.(11) yields the governingdifferentialequationwhichisthetwo-dimensionalanalogueofTimoshenko’sbeamequation

D

∇

²⁻

ρ

^h3

12

∂

²

∂

^t2

∇

²⁻

ρ κ

^G

∂

²

∂

^t2

w+

ρ

^h

∂

^2w

∂

^{t2 =}

1−D

∇

²

κ

^{Gh +}

ρ

^h3

12KG

∂

²

∂

^t2

q (13)

Withoutanyexternalload,thisequationisreducedto:

D

∇

⁴⁺

ρ

^h

∂

^2w

∂

^{t2 −}

ρ

h³

12+ D

κ

^G

∂

²

∂

^t2

∇

^2w+

ρ

^2h3

12 1

κ

^G

∂

^4w

∂

^{t4 =0 (14)}

or:

D

∇

^4w+

ρ

^h

∂

^2w

∂

^{t2 −}

ρ

^h₁₂³

1+12 h³

D

κ

^G

∂

²

∂

^t2

∇

^2w+

ρ

^2h3

12 1

κ

^G

∂

^4w

∂

^{t4 =0 (15)}

2.2.DerivationoftheoriginalUflyand-–Mindlinplatemodelfromthevariationalprinciple

ItappearsinstructivetoprovidethevariationalderivationofUflyand–Mindlin’sequationaspresentedbyMindlin(1951) himselfandLiewetal.(1998).Thepotentialenergyisgivenby

V=

V

Wdxdydz (16)

whereVisthevolumeoccupiedbytheplate,Wisthestrainenergydefinedasfollows:

W= 1

2(

σ

^x

ε

^x+

σ

^y

ε

^y+

σ

^z

ε

^z+

τ

^xy

γ

^xy+

τ

^yz

γ

^yz+

τ

^zx

γ

^zx) ⁽¹⁷⁾

SubstitutionofEq.(5)intoEq.(17)yields 2W =

σ

x

∂

^u

∂

^x+

σ

y

∂ v

∂

^y+

σ

z

∂

^w

∂

^{z +}

τ

xy

∂ v

∂

^x+

∂

^u

∂

^y

+

τ

yz

∂

^w

∂

^{y +}

∂v

∂

^z

+

τ

zx

∂

^w

∂

^{x +}

∂

^u

∂

^z

(18)

DefiningtheresultoftheintegrationofWoverthicknessasW¯: W¯ =

Wdz (19)

UsingEqs.(6)and(19), 2W¯ =Mx

∂ ψ

^x

∂

^{x +My}

∂ ψ

^y

∂

^{y +Myx}

∂ ψ

^y

∂

^{x +}

∂ ψ

^x

∂

^y

+Qx

∂

^w

∂

^{x +}

ψ

^x

+Qy

∂

^w

∂

^{y +}

ψ

^y

(20)

or,

2W¯ =Mxx+Myy+Myxyx+Qxxz+Qyyz (21)

SubstitutingEq.(8)intoEqs.(21)and(16),thepotentialenergyinthefollowingformissetas:

V =

W¯dxdy=

1 2

D

∂ ψ

^x

∂

^{x +}

∂ ψ

^y

∂

^y

2

−2(¹−

ν

)

∂ ψ

^x

∂

^x

∂ ψ

^y

∂

^{y −14}

∂ ψ

^x

∂

^{y +}

∂ ψ

^y

∂

^x

2

+

κ

^Gh

∂

^w

∂

^{x +}

ψ

^x

2

+

∂

^w

∂

^{y +}

ψ

^y

2

dxdy (22)

Theexpressionofthekineticenergyisthefollowing:

T=

V

ρ

2

∂

^u

∂

^t

2

+

∂ v

∂

^t

2

+

∂

^w

∂

^t

2

d

v

(23)

Usingtheexpressionofthedisplacementandintegratingoverthethickness

T=1 2

ρ

^h

∂

^w

∂

^t

2

+

ρ

^h3

12

∂ ψ

^x

∂

^t

2

+

∂ ψ

^y

∂

^t

2

dxdy (24)

where^{istheareaofthe}mid-surfaceoftheplate.

AccordingtotheHamilton’sprinciple:

δ

^t

ti

^dt=0 (25)

wheretheLagrangian^isgivenby:

^=T−V=1₂

ρ

^h3

12

∂ ψ

^x

∂

^t

2

+

∂ ψ

^y

∂

^t

2

+

ρ

^h

∂

^w

∂

^t

2

dxdy

−

κ

^Gh

2

∂

^w

∂

^{x +}

ψ

^x

2

+

∂

^w

∂

^{y +}

ψ

^y

2

dxdy

−1 2

D

∂ ψ

x

∂

^{x +}

∂ ψ

y

∂

^y

2

−2(¹−

ν

)

∂ ψ

x

∂

^x

∂ ψ

y

∂

^{y −14}

∂ ψ

x

∂

^{y +}

∂ ψ

y

∂

^x

2

dxdy (26)

Oneobtains:

_t

ti

−D

∂ ψ

^x

∂

^{x +}

v ∂ ψ

^y

∂

^y

∂δψ

^x

∂

^{x +}

∂ ψ

^y

∂

^{y +}

v ∂ ψ

^x

∂

^x

∂δψ

^y

∂

^y

−D(¹−

v

₎

2

∂ ψ

^x

∂

^{y +}

∂ ψ

^y

∂

^x

∂δψ

^x

∂

^{y +}

∂δψ

^y

∂

^x

−

κ

^Gh

∂

^w

∂

^{x +}

ψ

^x

∂ δ

^w

∂

^{x +}

δψ

^x

+

∂

^w

∂

^{y +}

ψ

^y

∂ δ

^w

∂

^{y +}

δψ

^y

+

ρ

^h

∂

^w

∂

^t

∂ δ

^w

∂

^t

+

ρ

^h3

12

∂ ψ

^x

∂

^t

∂δψ

^x

∂

^{t +}

∂ ψ

^y

∂

^t

∂δψ

^y

∂

^t

dxdydt=0 (27)

Integratingbypartresultsin:

t ti

D

∂

²

ψ

^x

∂

^x2

δψ

^x+

∂

²

ψ

^y

∂

^y2

δψ

^y+

ν ∂

²

ψ

^x

∂

^x

∂

^y

δψ

^y+

μ∂

²

ψ

^y

∂

^x

∂

^y

δψ

^x

+D(¹⁻

ν

) 2

∂

²

ψ

^x

∂

^y2

δψ

^x+

∂

²

ψ

^y

∂

^x2

δψ

^y+

∂

²

ψ

^x

∂

^x

∂

^y

δψ

^y+

∂

²

ψ

^y

∂

^x

∂

^y

δψ

^x

−

κ

^Gh

ψ

^x

δψ

^x−

∂ψ

^x

∂

^x

δ

^w

+

ψ

^y

δψ

^y−

∂ψ

^y

∂

^y

δ

^w

+

∂

^w

∂

^x

δψ

^x−

∂

^2w

∂

^x2

δ

^w

+

∂

^w

∂

^y

δψ

^y−

∂

^2w

∂

^y2

δ

^w

−

ρ

^h

∂

^2w

∂

^t2

δ

^w−

ρ

^h3

12

∂

²

ψ

^x

∂

^t2

δψ

^x+

∂

²

ψ

^y

∂

^t2

δψ

^y

dxdydt

dxdydt

− t

ti

D

∂ψ

^x

∂

^{x +}

ν ∂ψ ∂

^yy

δψ

^xdy−

∂ψ

^y

∂

^{y +}

ν ∂ψ ∂

^xx

δψ

^ydx

+D(¹⁻

μ

) 2

(

δψ

y−

δψ

x)

∂ψ

^x

∂

^{y dy−(}

δψ

x−

δψ

y)

∂ψ

y

∂

^{x dx}

+

κ

^Gh

ψ

x+

∂

^w

∂

^x

dy−

ψ

y+

∂

^w

∂

^y

dx

δ

^w

dt=0.

(28)

Fig. 2. Rectangular coordinates, and normal and tangential directions.

where^{istheboundarypath.Bygroupingthetermsintheforegoingfunctionalwithrespecttothevariationterms,} t

ti

D

∂

²

ψ

^x

∂

^{x2 +}

ν ∂

²

ψ

^y

∂

^x

∂

^y

+D(¹⁻

ν

) 2

∂

²

ψ

^x

∂

^{y2 +}

ν ∂

²

ψ

^y

∂

^x

∂

^y

−

κ

^Gh

ψ

^x+

∂

^w

∂

^x

−

ρ

^h3

12

∂

²

ψ

^x

∂

^t2

δψ

^x

+

D

∂

²

ψ

^y

∂

^{y2 +}

ν ∂

²

ψ

^x

∂

^x

∂

^y

+D(¹⁻

ν

) 2

∂

²

ψ

^y

∂

^{x2 +}

ν ∂

²

ψ

^x

∂

^x

∂

^y

−

κ

^Gh

ψ

^y+

∂

^w

∂

^y

−

ρ

^h3

12

∂

²

ψ

^y

∂

^t2

δψ

^y

+

κ

^Gh

∂ψ

^x

∂

^{x +}

∂

^2w

∂

^{x2 +}

∂ψ

^y

∂

^{y +}

∂

^2w

∂

^y2

−

ρ

^h

∂

^2w

∂

^t2

δ

^w

dxdydt

− _t

ti

D

∂ψ

^x

∂

^{x dy+}

ν ∂ψ ∂

^yydy

−D(¹−

ν

) 2

∂ψ

^x

∂

^{y dx+}

∂ψ

^y

∂

^{x dx}

δψ

^x

+

−D

∂ψ

^y

∂

^{y dx+}

ν ∂ψ ∂

^xxdx

+D(¹−

ν

) 2

∂ψ

^x

∂

^{y dy+}

∂ψ

^y

∂

^{x dy}

δψ

^y

+

κ

^Gh

ψ

^xdy+

∂

^w

∂

^xdy−

ψ

^ydx−

∂

^w

∂

^ydx

δ

^w

dt=0

(29)

Equatingthecoefficientsofthevariationtermstozeroforthefunctionalovertheplatearea,Eq.(11)areobtained,and thus,thegoverningdifferentialequationisestablishedvariationally.

Forboundaryconditions,thelineintegralofEq.(29)issettozeroandrewrittenas:

t ti

D

∂ψ

^x

∂

^{x +}

ν ∂ψ ∂

^yy

δψ

^xdy−D

∂ψ

^y

∂

^{y +}

ν ∂ψ ∂

^xx

δψ

^ydx+D(1−

ν

) 2

∂ψ

^x

∂

^{y +}

∂ψ

^y

∂

^x

δψ

^ydy

− D(¹⁻

ν

) 2

∂ψ

^x

∂

^{y +}

∂ψ

^y

∂

^x

δψ

^xdx+

κ

^Gh

ψ

^x+

∂

^w

∂

^x

δ

^wdy−

κ

^Gh

ψ

^y+

∂

^w

∂

^y

δ

^wdx

dt=0

(30)

SubstitutingEq.(8)intoEq.(30): _t

ti

[Mxx

δψ

^xdy−Myy

δψ

^ydx+Mxy

δψ

^ydy−Mxy

δψ

^xdx+Qx

δ

^wdy−Qy

δ

^{wdx]dt=0 (31)}

Accordingtothe Frenet–Serretformulas, thesubscriptsnandsdenotingthe normalandtangentialdirections,respec- tively(seeFig.2):

dx=−sin

θ

^ds;dy=cos

θ

^ds;

ψ

^x=

ψ

^ncos

θ

−

ψ

^ssin

θ

;

ψ

^y=

ψ

^nsin

θ

+

ψ

^scos

θ

;Qn=Qxcos

θ

+Qysin

θ

Mnn=Mxxcos²

θ

^+Myysin2

θ

^+2Mxysin

θ

^cos

θ

^;Mnn=(^Myy−Mxx)^sin

θ

^cos

θ

^+Mxy

cos²

θ

^−sin2

θ

(32) So,substitutingEq.(32)intoEq.(31),itbecomes:

_t

ti

[Mnn

δψ

^n+Mns

δψ

^s+Qn

δ

^{w]dsdt=0 (33)}

Hence,attheboundaryoftheplate:

Mnn=0 Mns=0 Qn=0

or

ψ

ⁿ

ψ

s

w

arespeci fied

2.3.TruncatedUflyand–Mindlinplatetheory

In his paper, Elishakoff (1994) stated that “the original Mindlintheory isinconsistent inthe sense that it takes into accountsecondaryeffectoftheinteractionbetweenthesheardeformationandrotaryinertia”.Consequently,thelastterm

in Eq.(15), the one with the fourthorder derivative with respect to time, must not appear andhe proposed to reduce Eq.(15)to:

D

∇

^4w+

ρ

^h

∂

^2w

∂

^{t2 −}

ρ

₁₂^h3

1+12 h³

D

κ

^G

∂

²

∂

^t2

∇

^{2w=0 (34)}

This truncated equation is directly derivable from equilibrium considerations, by replacing ∂²ψ^x/∂^{t2 and} ∂²ψ^y/∂^t2

in Eq. (11) by ∂^3w/∂^x∂^{t2 and} ∂^3w/∂^y∂^t2, respectively, as shown by Elishakoff et al. This process is an extension for platesofthe oneused by Elishakoff etal.(Elishakoff &Livshits, 1984; Elishakoff &Lubliner,1985;Elishakoff et al.,2012; Elishakoff, Kaplunov, & Nolde, 2015; Elishakoff, 2009) to obtain the truncated version of the Bresse–Timoshenko beam model.

Eq.(34)isalsoobtainablebyasymptoticargumentsfromthree-dimensionalelasticity,followingtheworkofBerdichevsky (1973) and Kaplunov (1996), using the reduction method, in which, the displacement is expanded in an infinite series of powers of the thickness coordinate (Widera, 1970) and approximateequations are derived, introducing an error that becomessmallerbyincreasingtheorderoftheasymptoticexpansion.

Thethree-dimensionalequilibriumequationsforaplatearewrittenasfollows(Widera,1970):

(

λ

+G)

⎛

⎝

∂∂^x

∂∂^y

∂∂^z

⎞

⎠ θ

+

∂

^w

∂

^z

+G

∇

²+

∂

²

∂

^z2

_u

v

w

=

ρ ∂ ∂

^t22

_u

v

w

(35)

whereλ^istheLamé coefficientandθ ^and∇^{2 aredefinedas}

θ

=

∂

^u

∂

^x+

∂ v

∂

^y;

∇

²=

∂

²

∂

^x2+

∂

²

∂

^{y2 (36)}

Thestressvanishingonthefreesurfacesz=±h/2

σ

z

x,h 2

=

σ

z

x,−h 2

;

σ

yz

x,h 2

+

σ

yz

x,−h 2

=0 (37)

Thedisplacementsolutionsisdevelopedinapowerasymptoticexpansion:

θ

^=∞

k=0

θ

k(^x,y,t)^zk;w=∞

k=0

w_k(^x,y,t)^{zk (38)}

Substitutingintheseequations,itbecomes ∞

n=1

2(

λ

+2G)ⁿ

h

2

2n−1

w2n+

λ

h 2

2n−1

θ

2n−1=0

∇

^2w0+∞

n=1

h 2

2n

∇

^2w2n+(²ⁿ⁻¹)

h

2

2n−2

θ

²ⁿ−1

=0

G

∇

²⁻

ρ ∂ ∂

^t22

w2n+(

λ

^+2G)(²ⁿ⁺¹)(²ⁿ⁺²)^w2n+2+(

λ

^+G)(²ⁿ⁺¹)

θ

²ⁿ+1=0 (

λ

^+G)²ⁿ

∇

^2w2n+

(

λ

^+2G)

∇

²⁻

ρ ∂ ∂

^t22

θ

2n−1+G2n(²ⁿ⁺¹)

θ

2n+1=0 (39)

wherec²=G/ρ^.

Considerthedimensionlessvariables:

θ

¯^n=Ln

θ

^{n;w¯n=Ln−1wn;}

∇

^¯2=L2

∇

^{2;h¯ =}₂^h_L^{;t¯= htc}_2L2 =h¯tc

2L (40)

Thethreeequationsarere-expressedas:

∞ n=0

2(

λ

+2G)(ⁿ+1)^w¯2n+2+

λ θ

^¯2n+1

_¯

h²ⁿ=0

∇

¯^2w¯0+^∞

n=0

_¯

h²

∇

¯^2w¯2n+2+(²ⁿ⁺¹)

θ

^¯2n+1

_¯

h²ⁿ=0

¯

wN+2=− G

(

λ

^+2G)(^N+1)(^N+2)

∇

¯^2−h¯2

∂

²

∂

^t¯2

¯

wN− (

λ

^+G)

(

λ

^+2G)(^N+2)

θ

^¯N+1;N=0,2,4,6,...