A nonlinear shell model of Koiter's type

(1)

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A nonlinear shell model of Koiter’s type

Philippe G. Ciarlet, Cristinel Mardare

To cite this version:

Philippe G. Ciarlet, Cristinel Mardare. A nonlinear shell model of Koiter’s type. Comptes

Ren-dus Mathématique, Elsevier Masson, 2018, 356 (2), pp.227-234. �10.1016/j.crma.2017.12.005�.

�hal-01727395�

(2)

Contents lists available atScienceDirect

C. R.

Acad.

Sci.

Paris,

Ser. I

www.sciencedirect.com

Mathematical problems in mechanics/Differential geometry

A

nonlinear

shell

model

of

Koiter’s

type

Un modèle non linéaire de coques de type Koiter

Philippe

G. Ciarlet

a

,

Cristinel Mardare

b

a_Department_of_Mathematics,_City_University_of_Hong_Kong,₈₃_Tat_Chee_Avenue,_Kowloon,_Hong_Kong b_Sorbonne_{Universités,}_Université_{Pierre-et-Marie-Curie,}_Laboratoire_{Jacques-Louis-Lions,}_Paris,_France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received14December2017

Acceptedafterrevision14December2017 Availableonline5January2018 PresentedbyPhilippeG.Ciarlet

Wedefineanewtwo-dimensionalnonlinearshellmodel“ofKoiter’stype”thatcanbeused forthemodelingofanytypeofshellandboundaryconditionsandforwhichweestablish anexistencetheorem.Themodelusesaspecificthree-dimensionalstoredenergyfunction ofOgden’stypethatsatisfiesalltheassumptionsofJohnBall’sfundamentalexistence the-oreminthree-dimensionalnonlinearelasticityandthatisadaptedheretothemodeling ofthinnonlinearlyelasticshellsbymeansofspecificdeformationsthatarequadraticwith respecttothetransversevariable.

r

é

s

u

m

é

Nous déﬁnissonsun nouveau modèle bidimensionnel non linéaire de coques « de type Koiter»quipeutêtreutilisépourlamodélisationdetouttypedecoqueetdeconditions auxlimitesetpourlequelnousétablissonsunthéorèmed’existence.Cemodèleutiliseune densitéd’énergiedetypeOgdensatisfaisanttoutesleshypothèsesduthéorèmed’existence fondamentaldeJohnBallenélasticitétridimensionnellenonlinéaireetquiestadaptéeicià lamodélisationdescoquesnonlinéairementélastiquesmincesaumoyendedéformations particulières,quisontquadratiquesenlavariabletransverse.

1. Preliminaries

Greek indicesandexponents varyintheset

{

1

,

2

}

,Latinindicesandexponents varyintheset

{

1

,

2

,

3

}

,andthe sum-mationconventionforrepeatedindicesandexponentsisusedinconjunctionwiththeserules.Boldfacelettersareusedto designatevectorandmatrixﬁelds.

Thethree-dimensionalEuclideanspaceisdenoted

E

3.Theinnerproduct,theexteriorproduct,andthenorm,in

E

3 are respectivelydenoted

·

,

∧

,and

|

· |

.Thespaceofrealn

×

n matricesisdenoted

M

n _and_the_Frobenius_norm_in

_M

n_is_denoted

·

.A matrixin

M

n _with_components_g

i j isdenoted

(

gi j

)

,theﬁrstindex(herei)indicatingtherowinthematrix.

E-mailaddresses:[email protected](P.G. Ciarlet),[email protected](C. Mardare). https://doi.org/10.1016/j.crma.2017.12.005

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228 P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 227–234

Adomain in

R

2 isaboundedandconnectedopensubsetof

R

2 whoseboundaryisLipschitz-continuousinthesenseof Neˇcas[15].

Givenanopensubset

ω

of

R

2_,_a_ﬁnite_dimensional_real_space

_Y

_,_any_p

_≥

_{1 and}_any_integer_m

_≥

_0,_the_notation

_C

m

₍

_ω

_;

_Y)

_, resp. Wm,p

₍

_ω

_;

_Y)

_,_denotes_the_space_of

_Y

_-valued_ﬁelds_with_components_in

_C

m

₍

_ω

₎

_,_resp._in_the_Sobolev_space_Wm,p

₍

_ω

₎

_.

Given an open subset

ω

of

R

2, we let y

= (

yα

)

denote a generic point in

ω

and we let

∂

α

:= ∂/∂

yα and

∂

αβ

:=

∂

2

/∂

yα

∂

yβ.Animmersion from

ω

into

E

3 is amapping

ψ

∈

C

1

(

ω

;

E

3

)

such thatthetwo vectorﬁelds

∂

α

ψ

:

ω

→ E

3 are

linearlyindependentateachpointof

ω

.Theimage

ψ (

ω

)

of

ω

by

ψ

isasurface (withboundary)in

E

3_.

Givenanimmersion

ψ

∈

C

2

₍

_ω

_;

_E

3

₎

_,_we_let

a3

(ψ )

:=

∂

1

ψ

∧ ∂

2

ψ

|∂

1

ψ

∧ ∂

2

ψ

|

,

a_αβ

(ψ )

:= ∂

α

ψ

· ∂

β

ψ,

bαβ

(ψ)

:= ∂

αβ

ψ

·

a3

(ψ ),

and a

(ψ )

:=

det

(

aαβ

(ψ ))

;

the functionsaαβ

(ψ )

andbαβ

(ψ)

respectivelydenotethecovariantcomponentsoftheﬁrstfundamentalform andthoseof

thesecondfundamentalform alongthesurface

ψ (

ω

)

.

Givenanimmersion

θ

∈

C

2

(

ω

;

E

3

₎

_considered_as_ﬁxed,_we_let_(for_brevity)

a3

:=

a3

(θ ),

aαβ

:=

aαβ

(θ ),

bαβ

:=

bαβ

(θ ),

and a

:=

a

(θ ),

and,givenanyarbitraryimmersion

ψ

∈

C

2

(

ω

;

E

3

₎

_,_we_let

Gαβ

(ψ)

:=

1

2

(

aαβ

(ψ )

−

aαβ

)

and Rαβ

(ψ )

:=

bαβ

(ψ )

−

bαβ

respectively denote the covariant components of the changeofmetrictensor and those of the changeofcurvaturetensor

betweenthesurfaces

θ (

ω

)

and

ψ (

ω

)

.

Givenadomain

ω

⊂ R

2_,_a_“small”_parameter

_ε

_>

_0,_and_an_immersion

_θ

_∈

_C

2

₍

_ω

_;

_E

3

₎

_,_we_let

:=

ω

× ]−

ε

,

ε

[

and

∈

C

1

(

; E

3

)

denotetheextensionofthemapping

θ

deﬁnedby

(

y

,

x3

)

:= θ(

y

)

+

x3a3

(

y

)

at each y

∈

ω

and each x3

∈ [−

ε

,

ε

].

Thenonecanshowthat,if

ε

>

0 issmallenough,det

∇

>

0 in

and

isinjectiveover

;cf.Theorem 4.1-1in[5]. Weletx

= (

xi

)

with

(

xα

)

= (

yα

)

∈

ω

andx3

∈ [−

ε

,

ε

]

denoteagenericpointin

,welet

∂

i

:= ∂/∂

xi,andwelet

gi j

:= ∂

i

· ∂

j

and

(

gkl

)

:= (

gi j

)

−1

respectivelydenotethecovariantandcontravariantcomponentsofthemetrictensorﬁeld associatedwiththemapping

. Finally,welet Ai jkl

:= λ

gi jgkl

+

μ

(

gikgjl

+

gilgjk

)

in

,

and aαβσ τ

:=

4

λ

μ

λ

+

2

μ

a αβ_aσ τ

₊

₂

_μ

₍

_aασ_aβτ

₊

_aατ_aβσ

₎

_in

_ω

_,

respectivelydenotethecontravariantcomponentsofthethree-dimensional,andtwo-dimensional,elasticitytensor associated

withanelasticmaterialwithLaméconstants

λ

and

μ

.If3

λ

+

2

μ

>

0 and

μ

>

0,bothtensorsareuniformlypositive-deﬁnite, inthesensethatthereexist twoconstantsC0

>

0 andc0

>

0 dependingon

λ

and

μ

suchthat

C0

i,j

(

ti j

)

2

≤

Ai jkl

(

x

)

tklti j for all x

∈

and all symmetric 3

×

3 tensors

(

ti j

),

and

c0

α,β

(

sαβ

)

2

≤

aαβσ τ

(

y

)

sσ τsαβ for all y

∈

ω

and all symmetric 2

×

2 tensors

(

sαβ

)

;

(cf. Theorems 3.9-1 and 4.4-1 in [5]). Note, however,that the Laméconstants ofall known elastic materials satisfy the strongerassumptions

λ

>

0 and

μ

>

0.

(4)

2. Abriefreviewofthemathematicalmodelingofnonlinearlyelasticshells

Anonlinearlyelasticshell isathree-dimensionalelasticbodywhosereferenceconﬁguration isoftheform

()

with

:=

ω

× ]−

ε

,

ε

[,

where

ε

>

0 and

: → E

3 isdeﬁnedintermsofan immersion

θ

∈

C

2

(

ω

;

E

3

₎

_as_in_Sect.₁_._Then _S

_{= θ(}

_ω

₎

_and₂

_ε

_>

₀

respectivelydesignatethemiddlesurface andthethickness oftheshell.Notethatweassumeforsimplicitythatthethickness oftheshellisconstant.

Thedeformation undergonebysuchashellsubjectedtoappliedbody andsurfaceforces andtospeciﬁcboundaryconditions

canbecomputedbyusingeitherathree-dimensionalmodel,i.e.wheretheshellisconsideredasathree-dimensionalbody,ora

two-dimensionalmodel,i.e.wheretheshellisassimilatedtoitsmiddlesurface.

The objective of thisNote is to propose a newtwo-dimensionalnonlinearshellmodel thathas the advantage over the existing onesthattheassociated minimizationproblemhasatleastone solutionforanytype ofgeometryofthemiddle surface andany type ofboundary conditions(Theorem 4below),while beingat thesame time “formallyasymptotically equivalent”(inaspeciﬁcsense;cf.Theorem 3)totheclassicalnonlinearKoitermodel,whichisoneofthemostcommonly usednonlinearshellmodels,butforwhichnoexistencetheoremisavailableintheliterature.

Tobeginwith,wegiveabriefdescriptionoftwoavailablenonlinearshellmodels.Foradetailedaccount,see,e.g.,[3,4]

andthereferencestherein.

Thethree-dimensionalnonlinearshellmodel assertsthatthedeformationofashellwith

( ˆ

)

− asitsreferenceconﬁguration shouldminimizethetotalenergy

ˆ

I

( ˆ

)

:=

ˆ

W

(

x

ˆ

, ˆ

∇ ˆ(ˆ

x

))

d

ˆ

x

− ˆ

L

( ˆ

),

where

ˆ := ()

andW

ˆ

: ( ˆ)

−

× M

3₊

→ R

denotesthe storedenergyfunction oftheelasticmaterial constitutingthe shelland

L

ˆ

isa linear functionalthat takesintoaccount theapplied forces,over aset ofadmissibledeformations (aprecisedeﬁnitionofthisset will be givenlater; forthetime being, we simply mentionthat such deformationsmust be orientation-preserving, inthe sensethatdet

ˆ∇ ˆ(ˆ

x

)

>

0 ateachx

ˆ

∈ ( ˆ)

−).

We willassume herethat thereferenceconﬁguration

( ˆ

)

− ofthe shellis anaturalstate (i.e.stressfree) andthatthe elasticmaterialconstitutingtheshellisisotropic,homogeneous,andsatisﬁestheaxiomofframe-indifference.Thenonecanshow (cf.,e.g.,[3,Theorem4.5-1])that,inthiscase,thefollowingTaylorexpansionmustholdateach

ˆ

x

∈ ( ˆ)

− andeach F

∈ M

3

+:

ˆ

W

(

ˆ

x

,

F

)

=

λ

2

(

tr E

)

2

₊

_μ

_E

2

₊

_O

₍

_E

3

₎

_{with E}

_:=

1 2

(

F T_F

₋

_I

_),

where

λ

>

0 and

μ

>

0 aretheLaméconstants ofthegivenelasticmaterial.

WhiletheLaméconstantsaregenerallyknownforeachelasticmaterial,theremainder

O(

E

3

₎

_is_not._As_a_consequence,

severalcompetingexpressionsofW exist

ˆ

intheliterature.Amongthevariousexamplesofstoredenergyfunctionsavailable intheliterature,ofparticularrelevancetothispaperisthatproposedbyCiarletandGeymonatin[7],viz.,

ˆ

W

(

F

)

:=

a

F

2

+

b

CofF

2

+

c

(

det F

)

2

−

d log

(

det F

)

+

e at each F

∈ M

3₊

,

wheretheconstantsa

>

0,b

>

0,c

>

0,d

>

0,ande

∈ R

,arechosenintermsoftwogivenLaméconstants

λ

>

0 and

μ

>

0 insuch awaythatthe principalpartof W with

ˆ

respectto E

:= (

FTF

−

I)/2 is“governed”bytheLaméconstants

λ

and

μ

, accordingtotheaboveformulafortheTaylorexpansionof W

ˆ

(

ˆ

x

,

F

)

.Thisstoredenergyfunctionisinadditionpolyconvex

andbecomesinﬁniteasdet F

→

0+,sothatthecorresponding totalenergy

ˆI

possessesatleastaminimizer inaspeciﬁcsetof admissibledeformations,accordingtothelandmarkexistencetheoremofBall[1].

The two-dimensionalnonlinearshellmodelofW.T.Koiter assertsthat the deformationofthemiddlesurface ofa shell with

()

asitsreferenceconﬁgurationisasuﬃcientlyregularimmersion

ψ

:

ω

→ E

3 _that_minimizes_the_total_{energy (cf.}_[13]₎

j

K

(ψ )

:=

ε

2

ω

aαβσ τGσ τ

(ψ )

Gαβ

(ψ)

+

ε

2 3a αβσ τ_R σ τ

(ψ )

Rαβ

(ψ)

√

a d y

− (ψ),

wherethefunctionsaαβ_{σ τ ,} _Gα

β

(ψ)

,Rαβ

(ψ )

,and

√

a,arethosedeﬁnedinSect.1,and

(5)

where

L

ˆ

is thelinear functional appearing in theexpression of the total energy

ˆI

above and

KL is the Kirchhoff–Love

deformationassociatedwith

ψ

,viz.,themapping

KL

: → E

3deﬁnedby

KL

(

·,

x3

)

:= ψ +

x3a3

(ψ )

in

ω

at each x3

∈ [−

ε

,

ε

].

Thetwo-dimensionalnonlinearshellmodelofW.T.Koiterisoneofthemostcommonlyusedtwo-dimensionalnonlinear shellmodelsincomputationalmechanics,inspiteofthefactthatithasnotyetbeenjustiﬁedbyanexistencetheorem(and mostlikelywillneverbe).

Ourobjectivehereistodeﬁneatwo-dimensionalnonlinearshellmodel thatissimilartoKoiter’smodel (Theorem 3),while beinginadditionjustiﬁedbyanexistencetheorem (Theorem 4).

It is worthwhile mentioningthat the two-dimensionalnonlinear shellmodel deﬁnedinthispaper canbeusedforany typeofshell, without any restriction on the geometry of the surface or on the boundary conditions, by contrast with the two-dimensional nonlinear shellmodels that have beenjustiﬁed so far in the literature by existence theorems: the

membrane-dominatedmodel (seeLe Dret&Raoult[14]),theﬂexural-dominatedmodel (seeCiarlet&Coutand[6]andFriesecke, James,Mora&Müller[10]),andthetwo-dimensional modelofKoiter’stypeforsphericaland“almostsphericalshells” ob-tainedbyR.Bunoiuandtheauthorsofthepresentpaper[2,8].

Detailsofthoseproofsthatareonlybrieﬂysketchedherewillappearinaforthcomingpaper[9].

3. Anewstoredenergyfunction

The starting pointof the deﬁnitionof ournew nonlinearshell model (Sect.4) is the followingstoredenergyfunction,

whichisdifferentfrom,butofthesametypeas,thestoredenergyfunctionproposed in[7]:botharepolyconvex andthe principalpartoftheirTaylorexpansionsintermsofE

:= (

FF

−

I

)/

2 issimilarlygovernedbytheLaméconstants

λ

and

μ

. Theorem1.Givenconstants

λ

and

μ

thatsatisfy3

λ

+

μ

>

0 and

μ

>

0,deﬁnethefunctionG

: ]

0

,

∞[ → R

by

G

(

t

)

= (λ −

2

μ

3

)

√

t

− (λ +

μ

3

)

log

√

t

− (λ +

μ

12

)

for each t

>

0

.

ThenthestoredenergyfunctionW

ˆ

: M

3₊

→ R

deﬁnedby

ˆ

W

(

F

)

:=

μ

12

tr

(

FF

)

2

+

G

(

det

(

FF

))

for each F

∈ M

3₊

,

hasthefollowingproperties:

(i)W satisﬁes

ˆ

theaxiomofframeindifference:

ˆ

W

(

R F

)

= ˆ

W

(

F

)

for all F

∈ M

3₊and all R

∈ O

3₊

;

(ii)W becomes

ˆ

inﬁniteunderinﬁnitecompressionordilatation,inthesensethat

lim

det F→0+

ˆ

W

(

F

)

= +∞

and lim

det F→+∞W

ˆ

(

F

)

= +∞;

(iii)W is

ˆ

coercive,inthesensethat

ˆ

W

(

F

)

≥

μ

₁ 16

F

4

₋

₁₆

_{for all F}

_{∈ M}

3 +

;

(iv)theprincipalpartofW with

ˆ

respecttothestraintensorE isgovernedbytheLaméconstants

λ

and

μ

,inthesensethat

ˆ

W

(

F

)

=

λ

2

(

tr E

)

2

₊

_μ

_E

2

₊

_O

₍

_E

3

_),

_{where E}

_:=

1 2

(

F _F

₋

_I

₎

_;

(v)W is

ˆ

polyconvex,inthesensethat

ˆ

W

(

F

)

= W(

F

,

det F

)

for all F

∈ M

3₊

,

where

W

: M

3

× ]

0

,

∞[ → R

istheconvexfunctiondeﬁnedby

W(

F

,

t

)

:=

μ

12

F

4

_{+ (λ −}

2

μ

3

)

t

− (λ +

μ

3

)

log t

− (λ +

μ

12

)

for all

(

F

,

t

)

∈ M

3

_{× ]}

₀

_,

_∞[.

₂

(6)

Sketchoftheproof. Theproofsofparts(i),(ii),and(v)arestraightforward.

Part(iii)isprovedbycombiningthefollowingestimates,satisﬁedbyallmatricesF

∈ M

3 +:

ˆ

W

(

F

)

=

μ

12

F

4

₊

_G

₍₍

_{det F}

₎

2

₎

_≥

μ

12

F

4

₋

_μ

_{det F}

₊

μ

4 and det F

≤

1 3

√

3

F

3

_.

Part(iv)isastraightforwardconsequenceofthefollowingrelations,satisﬁedbyallmatricesF

∈ M

3 +:

FF

=

I

+

2E

,

tr

(

FF

)

=

3

+

2 tr E

,

det

(

FF

)

=

1

+

2 tr E

+

2

(

tr E

)

2

−

2 tr

(

E2

)

+

8 det E

.

Remark1. The assumptions 3

λ

+

μ

>

0 and

μ

>

0 made in Theorem 1 are thus slightly stronger than those used for establishing theuniform positive-deﬁnitenessof thethree-dimensional andtwo-dimensional elasticitytensors, viz.,3

λ

+

2

μ

>

0 and

μ

>

0 (cf.Sect.1);butthey aresigniﬁcantlyweakerthanthosethat weremadein[7],viz.,

λ

>

0 and

μ

>

0, however.

2

Thenexttheorem,together withTheorem 1(iv),showsthatthestoredenergyfunctiondeﬁnedinTheorem 1isindeed analternativetothestoredenergyfunctionproposedin[7].

Theorem2.Let

ˆ

beadomainin

R

3_,_let

ˆ

0beanon-emptyrelativelyopensubsetoftheboundaryof

ˆ

,letW

ˆ

: M

3₊

→ R

bethe

storedenergyfunctiondeﬁnedinTheorem 1,let

L :

ˆ

W1,4

_{( ˆ}

_;

_E

3

₎

_{→ R}

_be_a_continuous_linear_form,_and_let

ˆ

0

∈

W1,4

( ˆ

;

E

3

)

bea

mappingthatsatisﬁesdet

ˆ∇ ˆ

0

>

0 a.e. in

ˆ

.

DeﬁnethesubsetU

( ˆ

;

E

3

)

ofW1,4

_{( ˆ}

_;

_E

3

₎

_and_the_functional

_{J :}

ˆ

_U

_{( ˆ}

_;

_E

3

₎

_{→ R}

_{∪ {+∞}}

_by

U

( ˆ

; E

3

)

:= { ˆ ∈

W1,4

( ˆ

; E

3

)

;

det

ˆ∇ ˆ >

0 a.e. in

ˆ

and

ˆ = ˆ

0on

ˆ

0

},

ˆ

J

( ˆ

)

:=

ˆ

W

( ˆ

∇ ˆ)

d

ˆ

x

− ˆ

L

( ˆ

)

at each

ˆ ∈

U

( ˆ

; E

3

).

Thenthereexists

ˆ

∗

∈

U

( ˆ

;

E

3

₎

_such_that

ˆ

J

( ˆ

∗

)

=

inf

ˆ∈U( ˆ;E3₎

ˆ

J

( ˆ

).

2

Sketchoftheproof. Itiseasytoseethat

CofF

≤

F2 and det F

≤

1 3

√

3

F

3 _{for each F}

_{∈ M}

3 +

.

ThenweinferfromthecoercivitypropertyofW established

ˆ

inTheorem 1(iii)that

ˆ

W

(

F

)

≥

μ

34

F

4

₊

_CofF

2

_{+ |}

_{det F}

_|

4/3

₋

₁₆

_μ

_{for each F}

_{∈ M}

3 +

.

Together with theother propertiesof W established

ˆ

inTheorem 1, this showsthat the functional

J

ˆ

and set U

( ˆ

;

E

3

)

satisfy all theassumptions ofJohn Ball’sfundamentalexistence theorem (see[1]), whichthus ensures theexistence ofa minimizer

ˆ

∗ of

J

ˆ

overthesetU

( ˆ

;

E

3

)

.

2

We now establish that the stored energy function of Theorem 1, in addition to being suited for modeling three-dimensionalnonlinearelasticbodies, isalso equallywell suited formodelingtwo-dimensionalnonlinearlyelasticshells inthe followingsense:itshowsthatthetwo-dimensionalstrainenergyobtainedfromW by

ˆ

integratingacrossthethicknessthe correspondingstrainenergyrestrictedtoaspeciﬁcclassofdeformationscoincides“towithintheﬁrstorder”withKoiter’sstrain energy.

Theorem3.Given

ε

>

0,anopensubset

ω

of

R

2_,_and_an_immersion

_θ

_∈

_C

3

₍

_ω

_;

_E

3

₎

_,_let

:=

ω

× ]−

ε

,

ε

[

and

ˆ := (),

(7)

Givenanyimmersion

ψ

∈

C

3

(

ω

;

E

3

₎

_,_deﬁne_the_mapping

˜

KL

∈

C

1

(

;

E

3

)

by

˜

KL

(

·,

x3

)

= ψ +

x3

1

−

λ

a αβ

λ

+

2

μ

Gαβ

(ψ )

−

x3 2 Rαβ

(ψ )

a3

(ψ )

in

ω

at each x3

∈ [

ε

,

ε

].

LetW

ˆ

: M

3₊

→ R

bethestoredenergyfunctiondeﬁnedinTheorem 1.Then

1 2

ε

−ε

ˆ

W

( ˆ

∇( ˜

KL

◦

−1

))

◦

dx3

=

1 4

aαβσ τG_{σ τ}

(ψ )

G_αβ

(ψ )

+

ε

2 3a αβσ τ_R σ τ

(ψ )

Rαβ

(ψ )

+ δ(

ε

, ψ )

in

ω

,

where

δ(

ε

,

ψ)

∈

C

0

₍

_ω

₎

_is_of_a_higher_order_than_the_term

_aαβ_{σ τ Gστ}

_{(ψ )}

_Gα β

(ψ )

+

ε

2

3aαβσ τ Rστ

(ψ )

Rαβ

(ψ )

above,inthesense thatiteitherdependsatleastcubicallyonGαβ

(ψ )

and

ε

Rαβ

(ψ )

,orquadratically,butthenwithamultiplicativefactorof

ε

toapower

≥

1.

2

Sketchoftheproof. Theproofisbasedonthefollowingthreeobservations. First,givenanymapping

∈

C

1

(

;

E

3

)

,Theorem 1(iv)impliesthat

ˆ

W

( ˆ

∇( ◦

−1

))

◦ =

1 2

i j

₍₎

_E i j

()

+ δ

1

((

Ei j

()))

in

,

where

i j

()

:=

Ai jklEkl

()

and Ei j

()

:=

1 2

(∂

i

· ∂

j

− ∂

i

· ∂

j

),

and

δ

1 isafunctionthatdependsonthematrixﬁeld

(

Ei j

())

atleastcubically.

Second, givenanyimmersion

ψ

∈

C

3

₍

_ω

_;

_E

3

₎

_,_we _infer_from_the_speciﬁc _deﬁnitions_of_the_mapping

˜

KL intermsof

ψ

giveninthestatementofthetheorem,andofthefunctionsAi jkl andaαβσ τ givenintermsof

θ

inSect.1,that

Eαβ

( ˜

KL

)

=

Gαβ

(ψ )

−

x3Rαβ

(ψ )

+ δ

2

(

x3

, ψ )

in

,

αβ

( ˜

KL

)

=

1 2a αβσ τ

_G σ τ

(ψ )

−

x3Rσ τ

(ψ )

+ δ

3

(

x3

, ψ )

in

,

3i

( ˜

KL

)

=

i3

( ˜

KL

)

= δ

4

(

x3

, ψ )

in

,

where

δ

2,

δ

3,and

δ

4,arefunctionsthateitherdependatleastcubicallyonthematrixﬁelds

(

Gαβ

(ψ ))

and

(

x3Rαβ

(ψ ))

,or

quadratically,butthenwithamultiplicativefactorofx3 toapower

≥

1 (recallthat

|

x3

|

≤

ε

andthat

ε

isthehalf-thickness

of the shell, which may be chosen as smallas we please). Consequently, there exists a function

δ

5 satisfying the same

propertiesasthefunctions

δ

2

,

δ

3

,

δ

4 abovesuchthat

i j

( ˜

KL

)

Ei j

( ˜

KL

)

=

1 2a αβσ τ

_G σ τ

(ψ )

−

x3Rσ τ

(ψ )

Gαβ

(ψ )

−

x3Rαβ

(ψ)

+ δ

5

(

x3

, ψ )

in

.

Third, ε

−ε aαβσ τ

Gσ τ

(ψ )

−

x3Rσ τ

(ψ)

Gαβ

(ψ )

−

x3Rαβ

(ψ )

dx3

=

2

ε

aαβσ τGσ τ

(ψ)

Gαβ

(ψ )

+

ε

2 3 a αβσ τ_R σ τ

(ψ )

Rαβ

(ψ)

in

ω

.

2

Remark2.The vectorﬁelds

˜

KL deﬁnedinTheorem 3 arequadratic withrespectto thetransversevariable x3,whilethe

Kirchhoff–Lovedeformations

KL appearing in the deﬁnition ofthe two-dimensional nonlinear shell modelof W.T. Koiter

(Sect.2)areonlyaﬃne withrespecttox3.ThereasonwhyKirchhoff–Lovedeformations

KL cannotbeusedinTheorem 3

insteadofthevectorﬁelds

˜

KL isthatthenormalstress

33

(

KL

)

associatedwithaKirchhoff–Lovedeformationisnotof

anorderlowerthanthatofthetangentialstresstensorﬁeld

(

αβ

₍

KL

))

,asitshould,accordingtoJohn[11,12];indeed,

33

(

KL

)

:=

A33klEkl

(

KL

)

= λ

gσ τEσ τ

(

KL

)

isofthesameorderas

αβ

(

KL

)

:=

AαβklEkl

(

KL

)

=

Aαβσ τEσ τ

(

KL

)

= (λ

gαβgσ τ

+

2

μ

gασgβτ

)

Eσ τ

(

KL

),

(8)

4. Atwo-dimensionalnonlinearmodelofKoiter’stype

Wearenowinapositiontodefine ournewnonlinearshellmodel andtojustifythismodelbyanexistencetheorem.Note that this new model,defined in part (ii) of the next theorem, is two-dimensional sinceits unknown is the matrix field

(ψ,

η

,

ζ )

:

ω

→ M

3 _formed_by_a_vector_ﬁeld

_ψ

_:

_ω

_{→ E}

3_that_governs_the_deformation_of_the_middle_surface_of_the_shell_and

bytwocolinearvectorﬁelds

η

:

ω

→ E

3 _and

_ζ

_:

_ω

_{→ E}

3_that _govern_the_deformation_of_the_ﬁbers_orthogonal_to_the_middle

surfaceoftheundeformedshell,allofwhichbeingthusdeﬁnedoverthetwo-dimensional domain

ω

. Theorem4.Given

ε

>

0,anopensubset

ω

of

R

2,andanimmersion

θ

∈

C

3

(

ω

;

E

3

₎

_,_let

:=

ω

× ]−

ε

,

ε

[

and

ˆ := (),

wheretheimmersion

: → E

3isdeﬁnedasinSect.1intermsof

θ

.Let

γ

0beanon-emptyrelativelyopensubsetoftheboundary

of

ω

,let

L :

ˆ

W1,4

( ˆ

;

E

3

)

→ R

beacontinuouslinearform,andletG

: ]

0

,

∞[ → R

bethefunctiondeﬁnedinTheorem 1.

(i)DeﬁnethesubsetU

(

;

E

3

)

ofW1,4

₍

_;

_E

3

₎

_{J :}

_U

₍

_;

_E

3

₎

_{→ R}

_{∪ {+∞}}

_by

U

(

; E

3

)

:= { ∈

W1,4

(

; E

3

)

;

det

∇ >

0 a.e. in

and

=

on

γ

0

× ]−

ε

,

ε

[},

J

()

:=

_μ

12

tr

(

Ci_j

())

2

+

G

(

det

(

Ci_j

()))

√

g dx

− ˆ

L

(

◦

−1

)

at each

∈

U

(

; E

3

),

where gi j

:= ∂

i

· ∂

j

,

g

:=

det

(

gi j

), (

gkl

)

:= (

gi j

)

−1

,

and Cij

()

:=

gik

(∂

k

· ∂

j

).

Thenthereexists

∗

∈

U

(

;

E

3

)

suchthat

J

(

∗

)

=

inf

∈U(;E3₎

J

().

(ii)DeﬁnethesubsetV

(

ω

;

M

3

₎

_of_W1,4

₍

_ω

_;

_M

3

₎

_j

_:

_V

₍

_ω

_;

_M

3

₎

_{→ R}

_{∪ {+∞}}

_by

V

(

ω

; M

3

)

:= {(ψ,

η

, ζ )

∈

W1,4

(

ω

; M

3

)

;

η

∧ ζ =

0 a.e. in

ω

and

η,ζ

∈

U

(

; E

3

)

},

j(ψ ,

η

, ζ )

:=

J

(

η,ζ

)

at each

(ψ ,

η

, ζ )

∈

V

(

ω

; M

3

),

where

η,ζ

(

y

,

x3

)

:= ψ(

y

)

+

x3

η

(

y

)

+

x23

ζ (

y

)

at each

(

y

,

x3

)

∈ .

Thenthereexists

(ψ

∗

,

η

∗

,

ζ

∗

)

∈

V

(

ω

;

M

3

₎

_such_that

j(ψ

∗

,

η

∗

, ζ

∗

)

=

inf

(ψ,η,ζ )∈V(ω;M3₎

j(ψ ,

η

, ζ ).

2

Sketchoftheproof. Part(i)ofthetheoremisastraightforwardconsequenceofTheorem 2with

U

( ˆ

; E

3

)

:= { ◦

−1

; ∈

U

(

; E

3

)

}

and

J

ˆ

( ˆ

)

:=

J

( ˆ

◦ )

for each

ˆ ∈

U

( ˆ

; E

3

).

Part(ii)ofthetheoremisprovedbyshowingthattheset

V

( ˆ

; E

3

)

:= {

η,ζ

◦

−1

;

η,ζ

(

·,

x3

)

:= ψ +

x3

η

+

x23

ζ

in

ω

,

x3

∈ ]−

ε

,

ε

[, (ψ,

η

, ζ )

∈

V

(

ω

; M

3

)

}

is sequentially weakly closed in the space W1,4

_{( ˆ}

_;

_E

3

₎

_and _that _the _restriction _of _the _above _functional

_J

ˆ

_to _the _set

V

( ˆ

;

E

3

)

issequentially weaklysemi-continuous. Tothisend, we note that the storedenergyfunction of thefunctional

ˆ

J

is precisely the function W deﬁned

ˆ

in Theorem 1, so that we can use John Ball’s arguments in [1]to establishthe existenceofaminimizer.

2

Acknowledgements

Theworkdescribedinthispaperwas substantiallysupportedbyagrantfromtheResearchGrantsCounciloftheHong KongSpecialAdministrativeRegion,China[ProjectNo.9042222,CityU11300315].

(9)

References

[1]J.Ball,Convexityconditionsandexistencetheoremsinnonlinearelasticity,Arch.Ration.Mech.Anal.63(1977)337–403. [2]R.Bunoiu,P.G.Ciarlet,C.Mardare,Existencetheoremforanonlinearellipticshellmodel,J.EllipticParabolicEqu.1(2015)31–48. [3]P.G.Ciarlet,MathematicalElasticity,VolumeI:Three-DimensionalElasticity,North-Holland,Amsterdam,1988.

[4]P.G.Ciarlet,MathematicalElasticity,VolumeIII:TheoryofShells,North-Holland,Amsterdam,2000.

[5]P.G.Ciarlet,AnIntroductiontoDifferentialGeometrywithApplicationstoGeometry,Springer,Dordrecht,TheNetherlands,2005. [6]P.G.Ciarlet,D.Coutand,Anexistencetheoremfornonlinearlyelastic“ﬂexural”shells,J.Elast.50(1998)261–277.

[7]P.G.Ciarlet,G.Geymonat,Surlesloisdecomportementenélasticiténonlinéairecompressible,C.R.Acad.Sci.Paris,Ser.II295(1982)423–426. [8]P.G.Ciarlet,C.Mardare,AmathematicalmodelofKoiter’stypeforanonlinearlyelastic“almostspherical”shell,C.R.Acad.Sci.Paris,Ser.I354(2016)

1241–1247.

[9] P.G.Ciarlet,C.Mardare,Anexistencetheoremforatwo-dimensionalnonlinearshellmodelofKoiter’stype,inpreparation.

[10]G.Friesecke,R.D.James,M.G.Mora,S.Müller,Derivationofnonlinearbendingtheoryforshellsfromthreedimensionalnonlinearelasticityby Gamma-convergence,C.R.Acad.Sci.Paris,Ser.I336(2003)697–702.

[11]F.John,Estimatesforthederivativesofthestressesinathinshellandinteriorshellequations,Commun.PureAppl.Math.18(1965)235–267. [12]F.John,Reﬁnedinteriorequationsforthinelasticshells,Commun.PureAppl.Math.24(1971)583–615.

[13]W.T.Koiter,Onthenonlineartheoryofthinelasticshells,Proc.K.Ned.Akad.Wet.,Ser.B,Phys.Sci.69(1966)1–54.

[14]H.LeDret,A.Raoult,Themembraneshellmodelinnonlinearelasticity:a variationalasymptoticderivation,J.NonlinearSci.6(1996)59–84. [15]J.Neˇcas,LesMéthodesDirectesenThéoriedesEquationsElliptiques,Masson,Paris,1967.Englishtranslation:DirectMethodsintheTheoryofElliptic