a new linear shell model for shells with little regularity

J ElastDOI 10.1007/s10659-014-9469-2

A New Linear Shell Model for Shells with LittleRegularity

Josip Tambaca

Received: 27 March 2013© Springer Science+Business Media Dordrecht 2014

Abstract In this paper, a new model of linearly elastic shell is formulated. The model isof Koiter’s type capturing the membrane and bending effects. However, the model is wellformulated for shells with little regularity, namely the shells whose middle surface is param-eterized by a W 1,∞ function.

Unknowns in the model are the displacement u of the middle surface of the shell andthe infinitesimal rotation ω of the shell cross-section. The existence and uniqueness of thesolution of the model has been proved for (u, ω) ∈ H 1 × H 1 with the help of the Lax-Milgram lemma.

The model is analyzed asymptotically with respect to the small thickness of the shell.It is shown that the model asymptotically behaves just as the membrane model, the flexu-ral model, and the generalized membrane model in each regime. In this way the model isjustified.

Since the model is well formulated for shells whose middle surface is with corners, wecompare the model with Le Dret’s model of folded plates. It turns out that in the regimeof the flexural shells the models are the same. The differential equations of the model arederived. They imply that the model is as a special case of the Cosserat shell model with asingle director for a particular constitutive law.

Keywords Linear elasticity · Shell model · Koiter model · Little regularity · Justification ·Junctions · Folded shells

Mathematics Subject Classification 74K25 · 74K30 · 74K15 · 74B05

1 Introduction

Modeling of elastic shells is very intensively studied area because of its wide area of ap-plications. There are many different models in use (membrane, flexural, Koiter, Budansky-Sanders and many others). One of the most important issues of the mathematical theory

J. Tambaca (B)Department of Mathematics, University of Zagreb, Zagreb, Croatiae-mail: [email protected]

mailto:[email protected]

J. Tambaca

of elasticity is the relation of the solutions of these models with the solution of three-dimensional elasticity for the appropriate problem, see [9]. In a series of papers Ciarlet,Lods and Miara derived and justified, in a mathematically rigorous way, the following linearmodels: the membrane model in [10], the flexural model in [13], the Koiter model in [11]and the generalized membrane model in [12]. Justification for the membrane, the generalizedmembrane and the flexural models is obtained directly from the three-dimensional linearizedelasticity. In fact, it is shown that the solution to a problem of three-dimensional linearizedelasticity for a thin shell-like body of thickness ε, in an appropriate topology, converges toa solution of the model, when the thickness of the shell ε tends toward zero. However, theKoiter model contains the membrane and flexural terms, which are of different order withrespect to the thickness ε and thus can not be justified in this way. Therefore, the authorsanalyzed the behavior of solutions to the Koiter model, and found that in each of the regimes(membrane, bending and generalized membrane) the solution tends towards the solution ofthe appropriate model (membrane, bending and generalized membrane, respectively).

The smoothness of the middle surface plays important role in the above mentionedderivation of the shell models. All the models are derived in the local coordinates of theshell, defined by its covariant or contravariant bases. Therefore C3 smoothness of the middlesurface is assumed in order to have all the tensors (metric tensor, curvature tensor, Christof-fell symbols) of the shell well defined as they appear as the coefficients in the model, see(4.1). The main idea of relaxing the smoothness assumption of the middle surface is to tryto write the model in the canonical coordinates using the vector displacement of the midsur-face and not its contravariant components. This idea is applied in the case of the Koiter shellmodel in [6] and in the case of the Naghdi shell model in [7] for W 2,∞ middle surfaces. TheKoiter model is again considered in [4] where the vector of infinitesimal rotation is usedin the formulation of the model. The existence and uniqueness of the solution was provedfor W 1,∞ shells whose normal vector is in W 1,∞ too, see also [20]. A similar idea has beenapplied in the case of curved rods in [26], but with a different definition of the infinitesimalrotation vector. It allowed a formulation of the curved rod model for W 1,∞ rods. Inspiredby this different definition of the infinitesimal rotation vector ω, which accounts the rotationof the cross-section around its axes, the flexural shell model is in [27] formulated for shellswith W 1,∞ middle surface. The existence and uniqueness of the solution of this formulationof the flexural shell model is shown for the unknown (u, ω) in a subset of H 1 × H 1; here uis a displacement of the middle surface of the shell.

It is also noted in [27] that the same approach with the infinitesimal vector ω as anunknown can not be applied in the case of the Koiter model. Therefore in this paper weformulate a new linearly elastic shell model of the Koiter type, see (3.1). The main featuresof the model are:

(i) the model is formulated for the unknown (u, ω) in a subset of H 1(ω;R3)×H 1(ω;R3)

(ω ⊂ R2 open, bounded with Lipschitz boundary), where the existence and uniqueness

of the solution is proved (see Theorem 3.3),(ii) the model is well defined for the middle surface parameterized by ϕ ∈ W 1,∞(ω;R3),

(iii) the energy of the model contains the membrane and flexural terms which are of differ-ent order with respect to the thickness ε of the shell,

(iv) for smooth geometry the solution of the model in each of the regimes (membrane,generalized membrane and flexural) tends to the solution of the corresponding shellmodel, when thickness ε tends to zero (see Sect. 5).

Modeling of junctions of plates/shells or models of folded plates/shells attracts a greatattention in last years, see [1, 14, 16–19, 23–25, 28]. Since the new model is well formu-lated for surfaces parameterized by ϕ ∈ W 1,∞(ω;R3) models of folded plates or shells are

A New Linear Shell Model for Shells with Little Regularity

inherently built into the new model. It turns out that for inextensional displacements themodel (3.1) is the same as the Le Dret’s model of folded plates given in [16–18]. Moreover,the formulated model (3.1) for the geometry of the middle surface which is piecewise C3

and for inextensional displacements coincides with the model in [1]. We also note that thefunction space of inextensional displacements DI (S) from [14] is isomorphic to the spaceVF (ω) and coincides with the function space of inextensional displacements which belongto V u

K(ω) (defined in (3.6)).We also derive the differential equations of the model. This enables us to recognize the

formulated model as a special case of the special Cosserat shell model with a single directorfor a particular constitutive law. Moreover, for regular solutions, we are able to deduce thejunction conditions. As in the case of rods, the junction conditions at the junction faces aregiven by:

(i) kinematical junction conditions: the displacement u and infinitesimal rotation ω coin-cide at junction faces,

(ii) dynamic junction conditions: equilibrium of contact forces and contact couples at junc-tion faces.

At the end of the paper we also propose a Naghdi type shell model which is well formulatedfor W 1,∞ midsurfaces. Its properties are left for the future work.

As a rule through the paper we take: Greek indices take their value in the set {1,2}, whileLatin belong to the set {1,2,3}. The summation convention is used. σ(A) denotes the set ofeigenvalues of the matrix A.

2 Geometry

Let ω ⊂R2 be an open bounded and simply connected set with Lipschitz-continuous bound-

ary γ . Let y = (yα) denote a generic point in ω and let ∂α := ∂/∂yα . Let ϕ : ω → R3 be an

injective mapping of class C3 such that the two vectors

aα(y) = ∂αϕ(y)

are linearly independent at all points y ∈ ω. They form the covariant basis of the tangentplane to the 2-surface

S = ϕ(ω)

at ϕ(y). The contravariant basis of the same plane is given by the vectors aα(y) defined by

aα(y) · aβ(y) = δαβ .

We extend these bases to the base of the whole space R3 by the vector

a3(y) = a3(y) = a1(y) × a2(y)

|a1(y) × a2(y)| .

The first fundamental form, or the metric tensor in covariant Ac = (aαβ) or contravariantAc = (aαβ) components of the surface S are given respectively by

aαβ = aα · aβ, aαβ = aα · aβ .

J. Tambaca

The second fundamental form of the surface S, also known as the curvature tensor, in co-variant Bc = (bαβ) or mixed components B = (bβ

α) are given respectively by

bαβ = a3 · ∂βaα = −∂βa3 · aα, bβα = aβσ bσα,

while the Christoffel symbols Γ σ are given by

Γ σαβ = aσ · ∂βaα = −∂βaσ · aα.

We will also use the notation Γ 3αβ for bαβ . The area element along S is

√ady, where a :=

det Ac . We also define the covariant derivatives bσβ |α and the covariant components cαβ of

the third fundamental form of the surface S by

bσβ |α = ∂αb

σβ + Γ σ

ατ bτβ − Γ τ

βαbστ , cαβ = bσ

αbσβ = ∂αa3 · ∂βa3.

Lemma 2.1 The following symmetries hold

aαβ = aβα, aαβ = aβα, bαβ = bβα, cαβ = cβα, Γ σαβ = Γ σ

βα.

The change from the basis to basis is done using aα = aασ aσ ,aα = aασ aσ . Moreover, onehas

|a1 × a2| =√

a11a22 − a212 = √

det Ac = √a,

|a1 × a2| =√

a11a22 − (a12

)2 = √det Ac,

∂α

√a = √

a(Γ 1

α1 + Γ 2α2

),

a1 × a3 = −√aa2, a2 × a3 = √

aa1,

a1 × a2 = det Ac√

aa3, a1 × a3 = −det Ac√

aa2, a2 × a3 = det Ac√

aa1,

∂αaβ = Γ iαβai = Γ σ

αβaσ + bαβa3, ∂αa3 = −bσαaσ = −bασ aσ ,

∂αaβ = −Γ

β

αiai = −Γ β

ασ aσ + bβαa3,

∂αbβη = −bασ Γ σβη + a3 · ∂αβaη, ∂αΓ

σβη = −Γ σ

ατΓτβη + bσ

αbβη + aσ · ∂αβaη.

Proof The proof of some statements can be found in [27] and [9]. �

3 New Shell Model

In this section we formulate a new shell model and prove existence and uniqueness of thesolution of the model. The shell model is defined for the middle shell parameterized by aϕ ∈ W 1,∞(ω;R3) function. Note that for such parametrization the vectors of the covariantand contravariant bases are well defined and belong to L∞(ω;R3). In the following sectionswe analyze the relation of the model with other shell models.

Let the part of the boundary γ0 ⊂ ∂ω be of positive length. Let us define the functionspaces:

VN(ω) = H 1γ0

(ω;R3

) × H 1γ0

(ω;R3

) = {(v, w) ∈ H 1

(ω;R3

)2 : v|γ0 = w|γ0 = 0},


VK(ω) ={(v, w) ∈ VN(ω) : w = 1√

a

((∂2v · a3)a1 − (∂1v · a3)a2

+ 1

2(∂1v · a2 − ∂2v · a1)a3

)},

VF (ω) = {(v, w) ∈ VN(ω) : ∂α v + aα × w = 0, α = 1,2

}.

All three spaces when equipped with the norm

‖(v, w)‖VN (ω) = (‖v‖2H 1(ω;R3)

+ ‖w‖2H 1(ω;R3)

)1/2

are Hilbert spaces.In the notation (u, ω) ∈ VN(ω), u is the displacement vector of the middle surface of

the shell, while ω is the infinitesimal rotation of the cross-section (segment, in undeformedconfiguration perpendicular to the middle surface), see Sect. 6.1. The subspace VK(ω) cor-responds to the set of unknowns in which the infinitesimal rotation ω is uniquely determinedby the displacement u in a way that the deformed cross-section remains approximately per-pendicular to the deformed middle surface (within the linear theory). Thus we may say thatthe shell is unshearable if the model is stated on VK(ω). The set VF (ω) is a subspace ofinextensional displacements in VK(ω) (see Corollaries 4.3 and 4.4), again within the lin-ear theory. Thus we may say that the shell is unshearable and inextensible if the model isstated on VF (ω). The subscripts of these function spaces suggest that they correspond to theNaghdi, the Koiter and the flexural shell type models.

The main focus of this paper is at the following shell model: find (u, ω) ∈ VK(ω) suchthat

ε

∫

ω

QCmQT[∂1u + a1 × ω ∂2u + a2 × ω

] · [∂1v + a1 × w ∂2v + a2 × w]√

adx

+ ε3

12

∫

ω

QCf QT ∇ω · ∇w√

adx =∫

ω

f ε · v√adx, (v, w) ∈ VK(ω). (3.1)

Here, the shell thickness is ε, f ε is the surface force density, while the elasticity tensors Cm,Cf : M3,2(R) → M3,2(R) are given by

CmC · D = 4λμ

λ + 2μ(I · C)(I · D) + 4μAcCAc · D + Bmc · d,

Cf C · D = aA([

0 −11 0

]C

)·[

0 −11 0

]D + aBf c · d,

where we have used the notation Q = [a1 a2], Q = [a1 a2 a3

]and

C =[

CcT

], D =

[DdT

]∈ M3,2(R), C,D ∈ M2(R), c,d ∈ R

2,

the matrices Bm,Bf ∈ M2(R) are assumed to be positive definite and the elasticity tensor Ais given by

AD = 4λμ

λ + 2μ

(Ac · D

)Ac + 4μAcDAc, D ∈ M2(R),

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where λ and μ are the Lamé coefficients. We assume that 3λ + 2μ,μ > 0. For E,D ∈Sym (R2) one has

AE · D = 4λμ

λ + 2μ

(Ac · E

)Ac · D + 4μAcEAc · D

= 4λμ

λ + 2μ

(Ac · E

)Ac · D + 2μ

(AcEAc · D + AcET Ac · D

)

= 4λμ

λ + 2μaαβeαβaστ dστ + 2μ

(aασ eστ a

τβdαβ + aασ eτσ aτβdαβ

).

Thus, when applied at symmetric matrices the tensor A is the same as the elasticity operatorthat appears in the classical shell theories.

Let us denote the bilinear forms related to the model (3.1) by

Bm

((u, ω), (v, w)

) :=∫

ω

QCmQT[∂1u + a1 × ω ∂2u + a2 × ω

]

· [∂1v + a1 × w ∂2v + a2 × w]√

adx,

Bf

((u, ω), (v, w)

) := 1

12

∫

ω


adx.

These forms are exactly the terms appearing in the formulation of the shell model (3.1).The term εBm((u, ω), (u, ω)) on VK(ω) corresponds to the extensibility of the shell and isexactly the membrane energy of the shell (see Lemma 4.5). The term ε3Bf ((u, ω), (u, ω))

on VF (ω) is exactly the energy in the flexural shell model (see Lemma 4.7 and Remark 4.8).The equation in (3.1) can also be considered on VF (ω) and on VN(ω). The flexural shellmodel (see [13]) is obtained if VF (ω) is used, see Remark 4.8. A Naghdi type model isobtained if VN(ω) is used instead of VK(ω) in (3.1), see Sect. 6.5 for some more details.

In the sequel we also assume that Ac,Ac and a are uniformly positive definite, i.e., that

ess infy∈ω

σ(Ac(y)

), ess inf

y∈ωσ(Ac(y)

), ess inf

y∈ωa(y) > 0. (3.2)

Note that these conditions are fulfilled in the smooth case, i.e., for ϕ ∈ C3(ω;R3) such that∂1ϕ(y), ∂2ϕ(y) are linearly independent for all y ∈ ω. Note also that Ac = A−1

c .

Lemma 3.1 The tensors Cm and Cf are positive definite, i.e., there are cm, cf > 0 such that

CmD · D ≥ cmD · D, Cf D · D ≥ cf D · D, D ∈ M3,2(R).

Proof In this proof we use that 3λ + 2μ,μ > 0, that Bm,Bf are positive definite and thatAc,Ac, a are uniformly positive definite (see (3.2)).

Let E : Sym(R2) → Sym(R2) be given by

EE = 4λμ

λ + 2μ(I · E)I + 4μE.

The eigenvalues of this operator are 4μ,8μ,4μ3λ+2μ

λ+2μ. Since 3λ + 2μ,μ > 0 then λ + 2μ is

also positive. Thus the tensor E is positive definite, i.e., there is cE > 0 such that EE · E ≥cEE · E for all E ∈ Sym(2).


Since Ac and Ac are positive definite and AcAc = I then√

Ac and√

Ac exist and aresymmetric and mutually inverse. For D ∈ M3,2(R) we compute

CmD · D = 4λμ

λ + 2μ

(I · √AcD

√Ac

)2 + 4μ√

AcD√

Ac · √AcD√

Ac + Bmd · d

= E(√

AcD√

Ac) + Bmd · d ≥ cE

√AcD

√Ac · √AcD

√Ac + Bmd · d

≥ cEAcD√

Ac · D√

Ac + Bmd · d≥ cE ess infσ(Ac)D

√Ac · D

√Ac + minσ(Bm)d · d

≥ cE ess infσ(Ac)AcDT · DT + minσ(Bm)d · d≥ cE ess infσ(Ac) ess infσ

(Ac

)D · D + minσ(Bm)d · d

≥ min{cE ess infσ(Ac) ess infσ

(Ac

),minσ(Bm)

}D · D.

To prove that Cf is positive definite, we first show that A is such. For D ∈ M2(R) wehave

AD · D = E(√

AcD√

Ac) ≥ cE

√AcD

√Ac · √AcD

√Ac

= cEAcD√

Ac · D√

Ac ≥ cE ess infσ(Ac

)D

√Ac · D

√Ac

= cE ess infσ(Ac

)AcDT · DT ≥ cE

(ess infσ

(Ac

))2D · D.

Using this estimate we compute

Cf D · D = aA([

0 −11 0

]D

)·[

0 −11 0

]D + aBf d · d

≥ cE ess infσ(Ac

)2a

[0 −11 0

]D ·

[0 −11 0

]D + a minσ(Bf )d · d

≥ cE ess infσ(Ac

)2aD · D + a minσ(Bf )d · d

≥ min{cE ess infσ

(Ac

)2,minσ(Bf )

}ess infaD · D. �

Lemma 3.2 The following estimate holds: there is CN > 0 such that for all (v, w) ∈ VN(ω)

‖(v, w)‖VN (ω) ≤ CN

(∥∥[∂1v + a1 × w ∂2v + a2 × w

]∥∥2

L2(ω;M3,2(R))

+ ‖∇w‖2L2(ω;M3,2(R))

)1/2.

Proof The proof is simple and direct. Let CP denotes the constant from the Poincaré typeinequality ‖v‖H 1(ω;R3) ≤ CP ‖∇v‖L2(ω;M3,2(R)) on H 1

γ0(ω;R3).

‖(v, w)‖2VN (ω) ≤ C2

P

(‖∇v‖2L2(ω;M3,2(R))

+ ‖∇w‖2L2(ω;M3,2(R))

)

≤ C2P

(‖∂1v‖2L2(ω;R3)

+ ‖∂2v‖2L2(ω;R3)

+ ‖∇w‖2L2(ω;M3,2(R))

)

≤ 2C2P

(‖∂1v + a1 × w‖2L2(ω;R3)

+ ‖a1 × w‖2L2(ω;R3)

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+ ‖∂2v + a2 × w‖2L2(ω;R3)

+ ‖a2 × w‖2L2(ω;R3)

+ ‖∇w‖2L2(ω;M3,2(R))

)

≤ 2C2P

(‖∂1v + a1 × w‖2L2(ω;R3)

+ ‖∂2v + a2 × w‖2L2(ω;R3)

+ C‖w‖2L2(ω;R3)

+ ‖∇w‖2L2(ω;M3,2(R))

)

≤ 2C2P

(‖∂1v + a1 × w‖2L2(ω;R3)

+ ‖∂2v + a2 × w‖2L2(ω;R3)

+ CC2P ‖∇w‖2

L2(ω;M3,2(R))+ ‖∇w‖2

L2(ω;M3,2(R))

).

This implies the statement of the lemma. �

The consequence of Lemmas 3.1 and 3.2 is the existence and uniqueness of the solutionof the shell model (3.1).

Theorem 3.3 The model (3.1) has unique solution.

Proof Due to Lemmas 3.1 and 3.2 the bilinear form εBm + ε3Bf on the left hand side of(3.1) is positive definite, i.e., there is c > 0 such that:

εBm

((v, w), (v, w)

) + ε3Bf

((v, w), (v, w)

)

≥ εcm

∥∥QT[∂1v + a1 × w ∂2v + a2 × w

]∥∥2

L2(ω;M3,2(R))+ ε3

12cf ‖QT ∇w‖2

L2(ω;M3,2(R))

≥ c(∥∥[

∂1v + a1 × w ∂2v + a2 × w]∥∥2

L2(ω;M3,2(R))+ ‖∇w‖2

L2(ω;M3,2(R))

)

≥ c‖(v, w)‖2VN (ω).

The H 1 continuity of the bilinear form and the right hand side is obvious. Thus the existenceand uniqueness of the model (3.1) is an easy consequence of the Lax-Milgram lemma. �

Remark 3.4 Note that positivity of the bilinear form on the left hand side of (3.1) is provenon the space VN(ω). Thus the existence and uniqueness of the solution also holds on thebigger space VN(ω). By the same argument the solution exists and is unique if we replaceVK(ω) by VF (ω) in (3.1).

Remark 3.5 The model (3.1) seems arbitrary due to unspecified matrix functions Bm and Bf .However, we will show that on the function space VK(ω) the matrix Bm is applied at zerovectors. However, on VN(ω) these terms are related to the transverse shear of the cross-sections and we can use the classical Naghdi shell model to conclude Bm = μAc, see(Sect. 6.5 and [22]). Arbitrariness still remains in Bf .

Remark 3.6 The model (3.1) can be reformulated by inserting ω into the variational equation(3.1). For that we define the operator ω : H 1(ω;R3) → L2(ω;R3) by

ω(u) = 1√a

((∂2u · a3)a1 − (∂1u · a3)a2 + 1

2(∂1u · a2 − ∂2u · a1)a3

)

and then instead of VK(ω) define the function space

V uK(ω) = {

u ∈ H 1(ω;R3

) : ω(u) ∈ H 1(ω;R3

), u|γ0 = ω(u)|γ0 = 0

}. (3.3)


It is easy to show that V uK(ω) with the norm v �→ (‖v‖2

H 1 + ‖ω(v)‖2H 1)

1/2 is a Hilbert space.Moreover, V u

K(ω) is isometrically isomorphic with the space VK(ω). Note also that VK(ω)

can be viewed as the graph space for the operator ω. Now the model (3.1) can be written inan equivalent way on V u

K(ω): find u ∈ V uK(ω) such that

Bm

((u, ω(u)

),(v, ω(v)

)) + ε2Bf

((u, ω(u)

),(v, ω(v)

)) = L((

v, ω(v)))

, v ∈ V uK(ω).

4 Relation to the Other Shell Models

In this section we relate the terms appearing in the model (3.1) with usual shell strains.Let ϕ ∈ C3(ω;R3) and v ∈ H 1(ω) × H 1(ω) × H 2(ω) and let us define matrix functions

γ and � by

γαβ(v) = 1

2(∂αvβ + ∂βvα) − Γ σ

αβvσ − bαβv3 = 1

2(∂αvβ + ∂βvα) − Γ i

αβvi,

�αβ(v) = ∂αβv3 − Γ σαβ∂σ v3 + bσ

β

(∂αvσ − Γ τ

ασ vτ

) + bσα

(∂βvσ − Γ τ

βσ vτ

)(4.1)

+ bσα |βvσ − cαβv3.

These are the classical linearized change of metric and linearized change of curvature ten-sors. They are usually used to define some linear shell theories, e.g., the membrane shellmodel (see [10]), generalized membrane shell equations (see [12]), the flexural shell model(see [13]) and the Koiter shell model (see [11, 15]). The term γ measures the membrane en-ergy of the shell, while the term � measures the flexural energy of the shell. However, thesefunctions need a C3 parametrization of the middle surface of the shell to be well defined.The components of v = (v1, v2, v3) are covariant components of the displacement vector,i.e., the displacement vector is given by v = via

i = Qv.However, Blouza and Le Dret in [6] showed that linearized change of metric and lin-

earized change of curvature can be written in the following way

γαβ(v) = 1

2(∂α v · aβ + ∂β v · aα), (4.2)

�αβ(v) = (∂αβ v − Γ σ

αβ∂σ v) · a3. (4.3)

These expressions allow the flexural shell model (see [13]) and the Koiter shell model (see[11, 15]) to be well defined for the middle surface parameterized in W 2,∞. Moreover themembrane part γ (v) is well defined for W 1,∞ middle surfaces. Thus in the sequel when werefer to γ and � we assume they are defined by (4.2) and (4.3).

Moreover, Anicic, Le Dret and Raoult in [4] improved the formulation using

θ(v) = (∂1v · a3)a1 + (∂2v · a3)a

2.

For C3 geometry they showed that

γαβ(v) = 1

2(∂α v · aβ + ∂β v · aα),

�αβ(v) = ∂αθ(v) · aβ − ∂β v · ∂αa3.

J. Tambaca

The Koiter model is therefore well formulated for ϕ ∈ W 1,∞(ω;R3) such that a3 ∈W 1,∞(ω;R3) (therefore for G1 shells as well, see Le Dret, [20]). The existence is thenproved in the space

{v ∈ H 1

0

(ω;R3

) : θ(v) ∈ H 10

(ω;R3

)}

which is very similar to V uK(ω).

In [6, Remark after Lemma 13] it was noted that the following lemma holds.

Lemma 4.1 Let ϕ ∈ W 1,∞(ω;R3). Then the following are equivalent

(i) (v, w) ∈ VK(ω)

(ii) (v, w) ∈ VN(ω) and [a1 a2 ]γ (v) = [∂1v + a1 × w ∂2v + a2 × w ].

Proof For (v, w) ∈ VN(ω) the equation in (ii) is equivalent to (we multiply the equation byQ−1 = [a1 a2 a3]T )

[γ (v)

0

]= [

a1 a2 a3]T [

∂1v + a1 × w ∂2v + a2 × w]

=⎡⎣

∂1v · a1 ∂2v · a1 + √aa3 · w

∂1v · a2 − √aa3 · w ∂2v · a2

∂1v · a3 + √aa2 · w ∂2v · a3 − √

aa1 · w

⎤⎦ ,

According to (4.2) γ (v) is symmetric. Therefore the above equation is equivalent to

1

2(∂2v · a1 + ∂1v · a2) = ∂2v · a1 + √

aa3 · w = ∂1v · a2 − √aa3 · w,

∂1v · a3 + √aa2 · w = 0,

∂2v · a3 − √aa1 · w = 0,

i.e.,

√aa3 · w = 1

2(∂1v · a2 − ∂2v · a1),

√aa2 · w = −∂1v · a3,

√aa1 · w = ∂2v · a3,

This is equivalent to the relation of v and w in VK(ω). �

Now we easily obtain another characterization.

Corollary 4.2 Let ϕ ∈ W 1,∞(ω;R3). Then the following are equivalent

(i) (v, w) ∈ VF (ω)

(ii) (v, w) ∈ VK(ω) and γ (v) = 0.

This lemma implies that VF (ω) is the function space of inextensional displacements, i.e.,the space on which the flexural shell model is given, see [13, 27].


Corollary 4.3

VF (ω) ⊂ VK(ω) ⊂ VN(ω).

Now it is obvious that (see also [6])

Corollary 4.4 Let (v, w) ∈ VK(ω). Then

∂α v + aα × w = 0, α = 1,2 ⇐⇒ γ (v) = 0.

Let us now, using Lemma 4.1, compute the membrane part of the problem (3.1). Firstnote that

QT Q =[

Ac

0 0

].

This actually implies that in the model (3.1) Bm is applied at zero vector. Now, usingAcAc = I, we obtain

εBm

((u, ω), (v, w)

)

= ε

∫

ω

QCmQT Qγ (u) · Qγ (v)√

adx

= ε

∫

ω

4λμ

λ + 2μ

(I · Acγ (u)

)(I · Acγ (v)

) + 4μAcAcγ (u)Ac · Acγ (v)√

adx

= ε

∫

ω

4λμ

λ + 2μ

(Ac · γ (u)

)(Ac · γ (v)

) + 4μAcγ (u)Ac · γ (v)√

adx

= ε

∫

ω

Aγ (u) · γ (v)√

adx,

and we conclude that the first term on the left hand side in (3.1) εBm((u, ω), (v, w)) is equalto the membrane term in the classical Koiter’s shell theory, see [15].

Lemma 4.5 Let ϕ ∈ W 1,∞(ω;R3) and (u, ω) ∈ VK(ω). Then (in the sense of (4.2))

εBm

((u, ω), (v, w)

) = ε

∫

ω

Aγ (u) · γ (v)√

adx.

Let us now explore the linearized change of curvature �(v). From Lemma 4.1 one has

∂1v + a1 × w = γ11(v)a1 + γ21(v)a2, ∂2v + a2 × w = γ12(v)a1 + γ22(v)a2,

i.e.,

∂α v + aα × w = γσα(v)aσ .

According to (4.3), for ϕ ∈ W 2,∞(ω;R3), one can write

�αβ(v) = (∂αβ v − Γ σ

αβ∂σ v) · a3

= (−∂β(aα × w) + ∂β

(γσα(v)aσ

) + Γ σαβaσ × w − Γ σ

αβγτσ (v)aτ) · a3

= (−∂βaα × w − aα × ∂βw + ∂βγσα(v)aσ + γσα(v)∂βaσ

J. Tambaca

+ Γ σαβaσ × w − Γ σ

αβγτσ (v)aτ) · a3

= −a3 × (Γ σ

αβaσ + bαβa3) · w − a3 × aα · ∂βw + γσα(v)∂βaσ · a3

+ Γ σαβa3 × aσ · w

= −a3 × aα · ∂βw + γσα(v)(−Γ σ

βτaτ + bσ

β a3

) · a3

= −a3 × aα · ∂βw + bσβ γσα(v).

Therefore

√a

[−a2 · ∂1w −a2 · ∂2w

a1 · ∂1w a1 · ∂2w

]=

[�11(v) − bσ

1 γσ1(v) �12(v) − bσ2 γσ1(v)

�21(v) − bσ1 γσ2(v) �22(v) − bσ

2 γσ2(v)

]

= �(v) − γ (v)

[b1

1 b12

b21 b2

2

]= �(v) − γ (v)BT .

To compute ∂αw · a3 we again start from

∂α v + aα × w = Qγ (v)eα, α = 1,2.

The equation for α = 1 we scalar multiply by a2 and for α = 2 we scalar multiply by a1 toobtain

∂1v · a2 + a2 × a1 · w = Qγ (v)e1 · a2, ∂2v · a1 + a1 × a2 · w = Qγ (v)e2 · a1.

Therefore

∂1v · a2 − √aa3 · w = γ21(v), ∂2v · a1 + √

aa3 · w = γ12(v).

We differentiate the first equation with respect to the second variable and the second equationwith respect to the first variable and obtain

∂2γ21(v) = ∂12v · a2 + ∂1v · ∂2a2 − ∂2√

aa3 · w − √a∂2a3 · w − √

aa3 · ∂2w

= ∂1(∂2v · a2) − ∂2v · ∂1a2 + ∂1v · ∂2a2 − ∂2√

aa3 · w − √a∂2a3 · w

− √aa3 · ∂2w,

∂1γ12(v) = ∂12v · a1 + ∂2v · ∂1a1 + ∂1√

aa3 · w + √a∂1a3 · w + √

aa3 · ∂1w

= ∂2(∂1v · a1) − ∂1v · ∂2a1 + ∂2v · ∂1a1 + ∂1√

aa3 · w + √a∂1a3 · w

+ √aa3 · ∂1w.

Therefore

√aa3 · ∂1w = ∂1γ12(v) − ∂2γ11(v) + ∂1v · ∂2a1 − ∂2v · ∂1a1

− √a(Γ 1

11 + Γ 212

) 1

2√

a(∂1v · a2 − ∂2v · a1) − √

a∂1a3 · w

= ∂1γ12(v) − ∂2γ11(v) + ∂1v · (Γ σ21aσ + b21a3

) − ∂2v · (Γ σ11aσ + b11a3

)

− 1

2

(Γ 1

11 + Γ 212

)(∂1v · a2 − ∂2v · a1) + b11∂2v · a3 − b12∂1v · a3


= ∂1γ12(v) − ∂2γ11(v) + ∂1v · (Γ 121a1 + Γ 2

21a2

) − ∂2v · (Γ 111a1 + Γ 2

11a2

)

− 1

2

(Γ 1

11 + Γ 212

)(∂1v · a2 − ∂2v · a1)

= ∂1γ12(v) − ∂2γ11(v) + Γ 121γ11(v) − Γ 2

11γ22(v) + (Γ 2

12 − Γ 111

)γ12(v).

Similarly

√aa3 · ∂2w = −∂2γ12(v) + ∂1γ22(v) − ∂2v · ∂1a2 + ∂1v · ∂2a2 − ∂2

√aa3 · w − √

a∂2a3 · w= −∂2γ12(v) + ∂1γ22(v) − ∂2v · (Γ σ

12aσ + b12a3

) + ∂1v · (Γ σ22aσ + b22a3

)

− √a(Γ 1

21 + Γ 222

) 1

2√

a(∂1v · a2 − ∂2v · a1) + (b21∂2v · a3 − b22∂1v · a3)

= −∂2γ12(v) + ∂1γ22(v) − ∂2v · (Γ 112a1 + Γ 2

12a2

) + ∂1v · (Γ 122a1 + Γ 2

22a2

)

− (Γ 1

21 + Γ 222

)1

2(∂1v · a2 − ∂2v · a1)

= −∂2γ12(v) + ∂1γ22(v) − Γ 212γ22(v) + Γ 1

22γ11(v) + (Γ 2

22 − Γ 121

)γ12(v).

Therefore we have proved the following lemma.

Lemma 4.6 Let ϕ ∈ W 2,∞(ω;R3) and (v, w) ∈ VK(ω) sufficiently smooth. Then

√a

[−a2 · ∂1w −a2 · ∂2w

a1 · ∂1w a1 · ∂2w

]= �(v) − γ (v)BT ,

√a

[a3 · ∂1w

a3 · ∂2w

]=

[∂1γ12(v) − ∂2γ11(v) + Γ 1

21γ11(v) − Γ 211γ22(v) + (Γ 2

12 − Γ 111)γ12(v)

−∂2γ12(v) + ∂1γ22(v) − Γ 212γ22(v) + Γ 1

22γ11(v) + (Γ 222 − Γ 1

21)γ12(v)

]

= Q−1 div

(Qγ (v)

[0 −11 0

]).

We now compute Bf .

Bf

((u, ω), (v, w)

)

= 1

12

∫

ω


adx

= 1

12

∫

ω

aA([

0 −11 0

]QT ∇ω

)·[

0 −11 0

]QT ∇w

√adx

+ 1

12

∫

ω

aBf

[a3 · ∂1ω

a3 · ∂2ω

]·[

a3 · ∂1w

a3 · ∂2w

]√adx

= 1

12

∫

ω

aA[−a2 · ∂1ω −a2 · ∂2ω

a1 · ∂1ω a1 · ∂2ω

]·[−a2 · ∂1w −a2 · ∂2w

a1 · ∂1w a1 · ∂2w

]√adx

+ 1

12

∫

ω

aBf

[a3 · ∂1ω

a3 · ∂2ω

]·[

a3 · ∂1w

a3 · ∂2w

]√adx.

Thus, using Lemma 4.6 we obtain the following lemma.

J. Tambaca

Lemma 4.7

Bf

((u, ω), (v, w)

)

= 1

12

∫

ω

aA(�(u) − γ (u)BT

) · (�(v) − γ (v)BT)√

adx

+ 1

12

∫

ω

Q−T Bf Q−1 div

(Qγ (u)

[0 −11 0

])· div

(Qγ (v)

[0 −11 0

])√adx.

Remark 4.8 The consequence of Corollary 4.2 and Lemma 4.7 is that the energy ε3Bf ofthe model (3.1) on the space VF (ω) is the classical flexural shell energy since γ (u) = 0.Moreover the function space VF (ω) is equivalent to the classical function space for theflexural shell model when the geometry is smooth. Therefore, the flexural shell model canbe obtained from the variational equation (3.1) when it is taken on the smaller space VF (ω).Thus the flexural shell model can be formulated for W 1,∞(ω;R3) middle surfaces. This hasbeen already proven in [27].

Remark 4.9 Lemma 4.7 shows that on the space VK(ω) the energy ε3Bf ((u, ω), (u, ω)) isslightly different from the flexural energy of the Koiter shell model in two aspects.

The first aspect is that in the Koiter shell model the density of the flexural energy is givenby (ε3/12)a3/2A�(u) ·�(u). Instead of the term �(u) we have here the term �(u)−γ (u)BT .Koiter in [15] notices that this change produces “an equivalent shell theory”. Moreover thetensor �(v) − γ (v)BT has already been introduced in [2, 3] and noted as better adapted tomeasuring the variations of the principal curvatures of the midsurface. This strain is verysimilar to the Budiansky-Sanders (see [8, 9]) which is given by

�BS(u) = sym(�(u) − γ (u)BT

) = �(u) − 1

2

(γ (u)BT + Bγ (u)

).

This implies that the Budiansky-Sanders change of curvature tensor �BS can be written by

�BS(v) = √a

[−∂1w · a2 1

2 (∂1w · a1 − ∂2w · a2)12 (∂1w · a1 − ∂2w · a2) ∂2w · a1

].

This is well defined for a W 1,∞ geometry as well.The second aspect is an additional term in the energy in the second line of Bf from

Lemma 4.7. This term is with derivatives of γ (u) and thus measures gradient of the changeof metrics (gradient of γ ). For plates, or parts of the middle surface with constant normal itis also expressed by

∂αω · a3 = ∂α(ω · a3).

Thus it can be seen as measuring the gradient of the twist of the cross-section around its axes,since ω · a3 = (∂1u · a2 − ∂2u · a1)/(2

√a) is the twist of the normal line (see Sect. 6.1).

5 Asymptotic Properties of the Model

In this section we consider asymptotics of the shell model (3.1) with respect to the thicknessof the shell in three regimes. The first one is leading to the flexural shell model, the second


one to the membrane shell model and the third one to the generalized membrane model ofthe first kind. This is the same behavior as for the Koiter shell model, see [9, 11]. What weactually prove is that the solution of the model (3.1) under suitable assumptions convergeweakly to the solution of the particular model. Strong convergence could be also obtainedin a classical way.

In this way we prove that the model (3.1) is as good as the Koiter model in the sensegiven in the papers [11, 12].

5.1 Flexural Shell Model

In this subsection we assume that the surface force density is of the form

f ε = ε3f

and analyze the behavior of the solution of (3.1) when ε tends to zero. Now, the model canbe written by: find (uε

, ωε) ∈ VK(ω) such that

1

ε2Bm

((uε

, ωε), (v, w)

) + Bf

((uε

, ωε), (v, w)

) =∫

ω

f · v√adx, (v, w) ∈ VK(ω).

(5.1)

We insert (v, w) = (uε, ωε

) in this equation and use Lemma 3.2 to obtain the a priori esti-mate: for ε ≤ 1 one has

min {cf , cm}C2

N

∥∥(uε

, ωε)∥∥2

VN (ω)≤ 1

ε2Bm

((uε

, ωε),(uε

, ωε)) + Bf

((uε

, ωε),(uε

, ωε))

≤ C‖f ‖L2(ω;R3)‖uε‖VN (ω).

Therefore the families

∥∥(uε

, ωε)∥∥

VN (ω),

1

ε2Bm

((uε

, ωε),(uε

, ωε))

are bounded and there exists a function (u, ω) ∈ VK(ω) and a sequence ((uεj , ωεj ))j ⊂VK(ω) such that

uεj ⇀ u weakly in H 1(ω;R3

),

ωεj ⇀ ω weakly in H 1(ω;R3

),

Bm

((uεj , ωεj

),(uεj , ωεj

)) → 0.

The last convergence implies

∂αuεj + aα × ωεj → 0 strongly in L2

(ω;R3

), α = 1,2.

Therefore

∂αu + aα × ω = 0, α = 1,2,

which is equivalent to (u, ω) ∈ VF (ω) by Corollary 4.4. Now taking the limit in (5.1) for(v, w) ∈ VF (ω) we obtain that the limit (u, ω) ∈ VF (ω) satisfies

Bf

((u, ω), (v, w)

) =∫

ω

f · v√adx, (v, w) ∈ VF (ω). (5.2)

J. Tambaca

For smooth geometry this problem coincides with the classical flexural shell model (seeLemma 4.7, Remark 4.8 and [27]). The uniqueness of the solution of the flexural shell modelimplies that the whole family (uε

, ωε) is convergent.

Thus in the “flexural” regime the shell model (3.1) behaves as the “flexural” shell model.

5.2 Membrane Shell Model

In this subsection we consider the behavior of the model (3.1) in the elliptic membrane shellregime. This means that middle surface is elliptic parameterized by ϕ ∈ C3(ω;R3), thatthe shell is clamped at all of the boundary, i.e., γ0 = ∂ω and assume that the surface forcedensity is of the form f ε = εf .

Let us denote

VM(ω) = H 10 (ω) × H 1

0 (ω) × L2(ω),

‖v‖VM(ω) = (‖v1‖2H 1(ω)

+ ‖v2‖2H 1(ω)

+ ‖v3‖2L2(ω)

)1/2.

Elements of this space are vectors with contravariant components of physical quantities, i.e.,v such that v = Qv. From the inequality of Korn’s type on an elliptic surface [9, Theorem2.7-3] (or [11, (2.6)], [10, (4.3)]) Lemma 4.5 implies that there is Ce > 0 such that

‖v‖2VM(ω) ≤ C2

e Bm

((v, w), (v, w)

), (v, w) ∈ VK(ω). (5.3)

Note also that, due to Lemma 4.5, for (v, w), (u, ω) ∈ VK(ω) one has

Bm

((u, ω), (v, w)

) =∫

ω

Aγ (u) · γ (v)√

adx ≤ C‖u‖VM(ω)‖v‖VM(ω)

and thus the bilinear form is continuous with respect to the norm on VM(ω). Moreover thisalso implies that the function v → ‖γ (v)‖L2(ω;M2(R)) is a norm on VM(ω) equivalent to theclassical norm. Thus with this norm the space VM(ω) is complete. In the sequel when wewrite the bilinear form Bm we consider it is given using γ ′s.

We now analyze the behavior of the solution of (3.1) when ε tends to zero. Now, themodel can be written by: find (uε

, ωε) ∈ VK(ω) such that

Bm

((uε

, ωε), (v, w)

) + ε2Bf

((uε

, ωε), (v, w)

) =∫

ω

f · v√adx, (v, w) ∈ VK(ω). (5.4)

We insert (v, w) = (uε, ωε

) in this equation and use (5.3) to obtain the a priori estimate: forε ≤ 1 one has

1

C2e

‖uε‖2VM(ω) ≤ Bm

((uε

, ωε),(uε

, ωε)) + ε2Bf

((uε

, ωε),(uε

, ωε))

≤ Ca‖f ‖L2‖uε‖L2(ω;R3).

Since uε = Quε it follows that ‖uε‖L2(ω;R3) ≤ C‖uε‖VM(ω), where C is independent of ε (itdepends on Ac, i.e., on the geometry of the surface). This implies uniform bounds for the


families

‖uε‖VM(ω), Bm

((uε

, ωε),(uε

, ωε))

, ε2Bf

((uε

, ωε),(uε

, ωε))

.

The definition of Bm and Bf and weak compactness of VM(ω) and L2 then implies that thereare u ∈ VM(ω), σ f ∈ L2(ω;M3,2(R)) and sequences, still indexed by ε, such that

uε ⇀ u weakly in VM(ω),

γ(uε

)⇀ γ (u) weakly in L2

(ω;M2(R)

),

ε∇ωε⇀ σ f weakly in L2

(ω;M3,2(R)

).

We now fix (v, w) ∈ VK(ω) and using these convergences we let ε → 0 in (5.4) to obtain

∫

ω

Aγ (u) · γ (v)√

adx =∫

ω

f · v√adx, (5.5)

since f · v = f · v, where the vector f = QT f is a vector of contravariant components ofthe force density. This equation holds for every such pair of test functions (v, w) ∈ VK(ω).Due to the smoothness of the geometry it implies that (5.5) holds for all v from

V = {v ∈ H 1

0

(ω;R3

) : v = Qv ∈ V uK(ω)

}.

Since V contains H 20 (ω;R3) it is a dense subset of VM(ω). Thus we obtain that (5.5) holds

for v ∈ VM(ω). Therefore u satisfies the membrane shell model. The uniqueness of thesolution of the of the membrane model implies that the whole family uε is convergent.

Thus in the “elliptic membrane” regime the shell model (3.1) behaves as the “membrane”shell model.

5.3 Generalized Membrane Shell Model

In this subsection we consider the behavior of the model (3.1) in the generalized membraneshell regime (we follow [11, Sect. 7.2] and [12]). This means that for the given geometryVF (ω) = {0} but we are not in the case of elliptic membrane shell. We again assume themiddle surface is parameterized by ϕ ∈ C3(ω;R3). We first substitute ω in the model as inRemark 3.6 and write the model on V u

K(ω) in the equivalent formulation: find u ∈ V uK(ω)

such that

Bm

((u, ω(u)

),(v, ω(v)

)) + ε2Bf

((u, ω(u)

),(v, ω(v)

)) = L((

v, ω(v)))

, v ∈ V uK(ω).

Since VF (ω) = {0} in this case, by Lemma 4.1, the function

v → ‖v‖V u#KM

(ω) = ∥∥γ (v)∥∥

L2(ω;M2(R))

is a norm on V uK(ω). However, with this norm V u

K(ω) might not be complete, so we define

V u#KM(ω) = completion of V u

K(ω) in ‖ · ‖V u#KM

(ω).

J. Tambaca

Additionally, the generalized membrane shell model is derived in [12] for the (admissible)surface force density given by

Lε(v) = εL(v) = ε

∫

ω

F · γ (v)√

adx,

for a F ∈ L2(ω;Sym(2)). Therefore L is continuous with respect to ‖ · ‖V u#KM

(ω) and

|L(v)| ≤ C‖v‖V u#KM

(ω) ≤ CBm

((v, ω(v)

),(v, ω(v)

))1/2, v ∈ V u

K(ω).

Note also that the bilinear form (see Lemma 4.5)

Bum(u, v) = Bm

((u, ω(u)

),(v, ω(v)

)) =∫

ω

Aγ (u) · γ (v)√

adx

is continuous with respect to this norm on V uK(ω). Therefore there exist L# and Bu#

m whichare unique continuous extensions of L and Bm from V u

K(ω) to V u#KM(ω), respectively.

Now the model (3.1) is given by

Bum

(uε

, v) + ε2Bf

((uε

, ωε(uε

)),(v, ω(v)

)) = L(v), v ∈ V uK(ω). (5.6)

We insert v = uε into this equation to obtain

Bum

(uε

, uε) + ε2Bf

((uε

, ω(uε

)),(uε

, ω(uε

))) = L(uε

) ≤ CBum

(uε

, uε)1/2

.

This implies uniform bounds for the families

Bum

(uε

, uε), ε2Bf

((uε

, ω(uε

)),(uε

, ω(uε

)))

and therefore uniform bounds for

‖uε‖V u#KM

(ω),∥∥ε∇ω

(uε

)∥∥L2(ω;M3,2(R))

.

Since V u#KM(ω) and L2(ω;M3,2(R)) are complete there exist u ∈ V u#

KM(ω) and σ ∈ L2(ω;M3,2(R)) and a sequence in the family {ε, ε > 0}, still denoted by ε, such that

uε⇀ u weakly in V u#

KM(ω), ε∇ω(uε

)⇀ σ weakly in L2

(ω;M3,2(R)

).

Fix now v ∈ V uK(ω) and let ε → 0 in (5.6). Since bilinear form Bu

m is continuous on V u#KM(ω)

and since ε2∇ω(uε) → 0 in L2(ω;M3,2(R)) we obtain

Bu#m (u, v) = L#(v), v ∈ V u

K(ω).

Since V uK(ω) is dense in V u#

KM(ω) we obtain the generalized membrane shell model: findu ∈ V u#

KM(ω) such that

Bu#m (u, v) = L#(v), v ∈ V u#

KM(ω).

Let us now assume that we are in the case of the “generalized membrane shell of thefirst kind”, i.e., that ‖ · ‖V u#

KM(ω) is a norm on the space H 1

γ0(ω;R3). Let us denote by V #

M(ω)


completion of H 1γ0

(ω;R3) with respect to ‖ · ‖V u#KM

(ω) norm. Note that

H 2γ0

(ω;R3

) = {v ∈ H 2

(ω;R3

) : v|γ0 = 0,∇v|γ0 = 0} ⊂ V u

K(ω) ⊂ H 1γ0

(ω;R3

).

Since H 2γ0

(ω;R3) is dense in H 1γ0

(ω;R3) in the H 1 norm and since ‖v‖V u#KM

(ω) ≤ ‖v‖H 1(ω;R3)

on H 1γ0

(ω;R3) the set H 2γ0

(ω;R3) is also dense in H 1γ0

(ω;R3) with the norm ‖ · ‖V u#KM

(ω).

Therefore V uK(ω) is dense in H 1

γ0(ω;R3) with the norm ‖ · ‖V u#

KM(ω). Therefore the comple-

tions of these two spaces with respect to ‖ · ‖V u#KM

(ω) are the same, i.e., V u#KM(ω) = V #

M(ω).

Therefore u ∈ V #M(ω) satisfies

Bu#m (u, v) = L#(v), v ∈ V #

M(ω). (5.7)

For smooth geometry, Q ∈ C2(ω;R3), v ∈ H 1γ0

(ω;R3) if and only if v = Qv ∈ H 1γ0

(ω;R3).Since for smooth geometry the norm of the completion is the same ‖ · ‖V u#

KM(ω) one has

v ∈ V #M(ω) ⇐⇒ v = Qv ∈ V #

M(ω).

The space V #M(ω) is the space function for the generalized membrane shell of the first kind.

Moreover Bu#m and L# coincide with the functionals as defined in [12].

Thus in the “generalized membrane shell of the first kind” regime the shell model (3.1)behaves as the “generalized membrane shell of the first kind” model (5.7).

Remark 5.1 In the model (5.7) the elliptic membrane model is also contained. Namely, dueto the inequality (5.3) the norm ‖ · ‖V u#

KM(ω) is equivalent to the VM(ω) norm on V u

K(ω).

Moreover, for smooth geometry QT V uK(ω) = V u

K(ω) is dense in VM(ω). Therefore the com-pletions of these spaces are the same, i.e., since VM(ω) is complete, V u#

KM(ω) = VM(ω).

6 Further Properties

6.1 Interpretation for ω

The choice of ω should be commented. In [21] authors justify the Kirchhoff-Love hypothe-ses. Geometrically, hypotheses state that a normal to the initial (undeformed) middle sur-face, viewed as a material segment of the shell, remains normal and unstretched duringthe deformation. Linearization of this assumption leads to the following form of the three-dimensional displacement of the thin shell

uKL

(x, xε

3

) = u(x) − xε3

((∂1u(x) · a3(x)

)a1(x) + (

∂2u(x) · a3(x))a2(x)

).

Here x = (x1, x2) and the superscript ε is added to x3 to stress that the variable x3 runsthrough a segment [−ε/2, ε/2] for small ε.

Note now that (u, ω) ∈ VK(ω) implies that u and ω are related through

ω = 1√a

((∂2u · a3)a1 − (∂1u · a3)a2 + 1

2(∂1u · a2 − ∂2u · a1)a3

). (6.1)

A simple calculation shows that using ω the above Kirchhoff-Love displacement can alsobe written by

uKL(x, xε

3

) = u(x) + xε3 ω(x) × a3(x).

J. Tambaca

The two components of the vector ω actually correspond to the infinitesimal rotation ofthe normal to the surface around vectors a1 and a2. Note that the third component of thevector ω(x) in the direction of a3(x) is not participating in the Kirchhoff-Love displacement.However this component

∂1u · a2 − ∂2u · a1

has a clear interpretation of the infinitesimal rotation around the normal to the surface a3.Thus the vector ω is actually the vector of the infinitesimal rotation of the cross-section ofthe shell. Moreover, the condition in the space VK(ω), (6.1), now states that the cross-sectionof the shell after deformation remains approximately (within the scope of the linear theory)perpendicular to the deformed shell. A similar conclusion is drawn in [6] too.

6.2 VK(ω) Revisited

The condition (u, ω) ∈ VK(ω) poses certain regularity on u at points where the geometry issmooth since

ω = 1√a

((∂2u · a3)a1 − (∂1u · a3)a2 + 1

2(∂1u · a2 − ∂2u · a1)a3

)∈ H 1

(ω;R3

).

This can be also viewed from the equivalence (u, ω) ∈ VK(ω) if and only if u ∈ V uK(ω),

where V uK(ω) is defined in (3.3).

Let ϕ be piecewise C2. Then there is a family Di ⊆ ω, i = 1, . . . ,m of disjoint opensets such that ω = Int

⋃m

i=1 Di . Let v be such that v|Di∈ H 2

0 (Di;R3). Then v ∈ V uK(ω) and

therefore (v, ω(v)) ∈ VK(ω). Thus VK(ω) is not empty.At points where the geometry is not smooth we obtain junction conditions. Let p be a

parametrization of an injective Lipschitz curve in ω and let p0 be an element of the curve.Let ωp0 ⊂ ω is a small enough open neighborhood of p0 such that the curve p splits it intwo open sets ω+

p0and ω−

p0each from one side of p. Let (u+

, ω+) and (u−

, ω−) denote the

restriction of the solution of the shell model (3.1) in ω+p0

and ω−p0

. Then the trace theoremimplies that

(u+

, ω+) = (u−

, ω−)on p. (6.2)

This implies that the two shell parts from different sides of the curve p move simultaneously.Moreover, (6.2) implies that the cross–sections rotate simultaneously as well. In a sensethese are junction conditions (kinematical) on p at corners of the undeformed surface.

Let us consider a simple example of two plates connected at the right angle. Theparametrization is given by

ϕ : 〈0,1〉 × 〈−1,1〉 → R3, ϕ(x1, x2) =

{(x1, x2,0)T , x2 < 0

(x1,0, x2)T , x2 ≥ 0.

Then ϕ ∈ W 1,∞(〈0,1〉 × 〈−1,1〉;R3). This skeleton is made of two plates. We computevectors of covariant basis for each plate of the skeleton

a−1 = e1, a−

2 = e2, a−3 = e3, a− = 1,

a+1 = e1, a+

2 = e3, a+3 = −e2, a+ = 1,


with a notation that superscript ± corresponds to the sign of x2. From the junction conditions(6.2) for infinitesimal rotations we obtain

1√a+

((∂2u

+ · a+3

)a+

1 − (∂1u

+ · a+3

)a+

2 + 1

2

(∂1u

+ · a+2 − ∂2u

+ · a+1

)a+

3

)

= 1√a−

((∂2u

− · a−3

)a−

1 − (∂1u

− · a−3

)a−

2 + 1

2

(∂1u

− · a−2 − ∂2u

− · a−1

)a−

3

)

on 〈0,1〉 × {0}.

Using the calculated values of geometrical quantities we obtain

(−∂2u+ · e2

)e1 + (

∂1u+ · e2

)e3 − 1

2

(∂1u

+ · e3 − ∂2u+ · e1

)e2

= (∂2u

− · e3

)e1 − (

∂1u− · e3

)e2 + 1

2

(∂1u

− · e2 − ∂2u− · e1

)e3 on 〈0,1〉 × {0}.

This implies

−∂2u+ · e2 = ∂2u

− · e3,

1

2

(∂1u

+ · e3 − ∂2u+ · e1

) = ∂1u− · e3,

∂1u+ · e2 = 1

2

(∂1u

− · e2 − ∂2u− · e1

),

on 〈0,1〉 × {0}. In addition the conditions related to displacement on 〈0,1〉 × {0} are givenby u+ = u−, so all junction conditions on 〈0,1〉 × {0} are given by

u+ = u−,

−∂2u+2 = ∂2u

−3 ,

1

2

(∂1u

+3 − ∂2u

+1

) = ∂1u−3 , (6.3)

∂1u+2 = 1

2

(∂1u

−2 − ∂2u

−1

).

6.3 Relation to the Folded Plate Model

In this subsection we compare the model from the previous subsection of two joined plateswith the models of folded plates obtained by Le Dret (see [16, 17] or [18]). Since the modelsconsidered there are the classical plate models we are in the flexural shell regime, i.e., thesolution is approximately inextensional displacement (or the solution of (3.1) is close to thesolution of the flexural shell model according to Sect. 5.1). Thus the functions u+ and u−

satisfy

∂αu+ + a+

α × ω+ = ∂αu− + a−

α × ω− = 0, α = 1,2.

This implies that besides the classical plate equation and the above junction conditions (6.3)one has

∂1u± · a±

1 = ∂1u± · a±

2 + ∂2u± · a±

1 = ∂2u± · a±

2 = 0.

J. Tambaca

Fig. 1 Deformation of folded plates with both plates fixed (left) and only horizontal plate fixed (right)subjected to the constant load in −e3 direction (greatest magnitude is colored in white)

This implies

∂1u+1 = ∂1u

+3 + ∂2u

+1 = ∂2u

+3 = 0, ∂1u

−1 = ∂1u

−2 + ∂2u

−1 = ∂2u

−2 = 0

and hence (u+1 , u+

3 ) and (u−1 , u−

2 ) are two-dimensional infinitesimal rigid displacements, i.e.,there are a+,a− ∈R

2 and b+, b− ∈R such that

[u+

1u+

3

]= a+ + b+

[x2

−x1

],

[u−

1u−

2

]= a− + b−

[x2

−x1

].

6.3.1 The Case of Both Plates Clamped

If both plates are clamped at both x2 = ±1 then a+ = a− = 0 and b+ = b− = 0. Thuslongitudinal displacements for both plates are equal to zero. Thus the classical plate equationfor both transversal displacements u+

2 and u−3 remains with the junction conditions, that

follow from (6.3), which are given by

u+2 = 0, u−

3 = 0, −∂2u+2 = ∂2u

−3 .

The obtained junction conditions and the model is the same as the model in [17].

6.3.2 The Case of Only One Plate Clamped

If only the part of the boundary given by x2 = −1 is clamped we have that a− = 0 andb− = 0. Therefore u−

1 = u−2 = 0. Therefore at the junction from (6.3) we obtain

u+1 = 0, u+

2 = 0, u+3 = u−

3 ,

−∂2u+2 = ∂2u

−3 ,

1

2

(∂1u

+3 − ∂2u

+1

) = ∂1u−3 , ∂1u

+2 = 0.

This effectively implies the following conditions for the unknown functions u−3 and u+

2 atthe junction

u+2 = 0, u−

3 = a+3 − b+x1, −∂2u

+2 = ∂2u

−3 .

The obtained junction conditions and the model is the same as the model from [16] and [19].Deformation of L shaped plate for both boundary conditions can be seen in Fig. 1.


6.4 Differential Formulation of the Model

In this subsection we derive the differential equations of the model under the assumptionthat the solution of the shell model (3.1) is regular enough. Let us first denote

M = ε3

12QCf QT , m = M∇ω,

(6.4)N = εQCmQT , nsym = [

nsym1 nsym

2

] = N[∂1u + a1 × ω ∂2u + a2 × ω

].

The superscript sym comes from the fact that

nsym1 · a3 = nsym

2 · a3 = nsym1 · a2 − nsym

2 · a1 = 0.

Thus the tensor nsym has only three relevant components. Then we define the tensor n =[n1 n2] ∈ M3,2(R) by its components

n1 · a1 = nsym1 · a1, n2 · a2 = nsym

2 · a2,

n1 · a2 + n2 · a1 = nsym1 · a2 + n1 · a1,

n1 · a2 − n2 · a1 = − 1

adiv (

√am) · a3, n1 · a3 = 1

adiv (

√am) · a2,

n2 · a3 = − 1

adiv (

√am) · a1.

The last three equations can also be written by

1√a

div (√

am) + aα × nα = 0. (6.5)

Using this notation the model (3.1) can be written by

∫

ω

nsym · [∂1v + a1 × w ∂2v + a2 × w]√

adx +∫

ω

m · ∇w√

adx

=∫

ω

f ε · v√adx, (v, w) ∈ VK(ω).

Now we make partial integration in the flexural term in the equation to obtain

∫

ω

nsym · [∂1v + a1 × w ∂2v + a2 × w]√

adx −∫

ω

div (√

am) · wdx

=∫

ω

f ε · v√adx, (v, w) ∈ VK(ω) ∩ (

H 10

(ω;R3

) × H 10

(ω;R3

)).

Next we insert w = ω(v) and use Lemma 4.1(ii) to obtain

−∫

ω

div (√

am) · 1√a

((∂2v · a3)a1 − (∂1v · a3)a2 + 1

2(∂1v · a2 − ∂2v · a1)a3

)dx

+∫

ω

nsym · Qγ (v)√

adx =∫

ω

f ε · v√adx, v ∈ H 1

0

(ω;R3

) ∩ V uK(ω).

J. Tambaca

Using the definition of n and the fact that γ (v) is symmetric we obtain

−∫

ω

((∂2v · a3)(−n2 · a3) − (∂1v · a3)n1 · a3

+ 1

2(∂1v · a2 − ∂2v · a1)

(n2 · a1 − n1 · a2

))√adx

+∫

ω

QT n · γ (v)√

adx =∫

ω


0

(ω;R3

) ∩ V uK(ω).

Therefore

∫

ω

(∂1v · n1 + ∂2v · n2)√

adx =∫

ω


0

(ω;R3

) ∩ V uK(ω).

According to the note in Sect. 6.2 at each domain D of smooth geometry ϕ the H 20 (D;R3)

functions are in H 10 (ω;R3) ∩ V u

K(ω). Thus at each point of smooth geometry we obtain thatthe following differential equation holds

div (√

an) + √af ε = 0. (6.6)

Therefore from (6.6), (6.5) and (6.4) we obtain that the shell model is given by

div (√

an) + √af ε = 0,

div (√

am) + √aaα × nα = 0,

m = M∇ω, (6.7)

nsym = N[∂1u + a1 × ω ∂2u + a2 × ω

],

ω = 1√a

((∂2u · a3)a1 − (∂1u · a3)a2 + 1

2(∂1u · a2 − ∂2u · a1)a3

).

The first and the second equation are the equations of equilibrium of forces and moments,where n and m denote the force and couple stress tensors. The last three equations arethe constitutive relations. More precisely, the last equation is a material restriction of un-shearability. Only a part (nsym) of the stress tensor n is related to the strains. The remainingcomponents are the Lagrange multipliers related to the unshearability. This form of the equa-tions coincides with a special Cosserat shell model with single director, see [5]. However,a particular constitutive law is given here.

Let p denotes a parametrization of an injective Lipschitz curve in ω. As in Sect. 6.2p splits the domain in two parts denoted by superscripts + and −. Now two partial integra-tions performed above imply the following dynamic junction conditions:

√a+n+ν+ + √

a−n−ν− = 0,√

a+m+ν+ + √a−m−ν− = 0 on p, (6.8)

where ν± denotes the unit outer normal for the ± domain, respectively. These junctionconditions can be seen as the equilibrium of the contact force and couple at the junction.They should be accompanied to the kinematical junction conditions (6.2).


6.5 A Naghdi Like Model

The Naghdi model is a two-dimensional shell model, see [22]. The unknowns of the prob-lem, similarly to our formulation of the Koiter shell model, are the displacement of thepoints of the shell midsurface and the rotation field of the normal to the midsurface (classi-cally not including twist). The energy of Naghdi’s shell model is consisting of three parts:the membrane energy, the flexural energy and the transverse shear energy. The first twoterms are the same as in the energy of the Koiter shell model. Namely, in contrast to theKoiter shell model, the cross-section is allowed to change the angle with respect to the de-formed middle surface, but this change is “penalized” in the transverse shear energy term.In this spirit we propose a shell model of the Naghdi type valid for W 1,∞ middle surfaces:find (u, ω) ∈ VN(ω) such that

ε

∫

ω

QCmQT[∂1u + a1 × ω ∂2u + a2 × ω

] · [∂1v + a1 × w ∂2v + a2 × w]√

adx

+ ε3

12

∫

ω


adx =∫

ω

f ε · v√adx, (v, w) ∈ VN(ω). (6.9)

The model is given by the same equation as in (3.1), but the function space is different.We do not force that ω is expressed in terms of the derivatives of the displacement, i.e.,the cross–sections do not need to remain perpendicular to the deformed middle surface.This difference is hidden in ∂αu + aα × ω. Have in mind that in the model of the Koitertype these two vectors have only three relevant components. According to Remark 3.4 thesolution of (6.9) exists and is unique. Properties of this model are left for the future work.However, note that the classical Naghdi shell model can help us to determine the elasticitytensor Bm = μAc since this coefficient appears in the transverse shear energy.

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a new linear shell model for shells with little regularity

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