A new approximate model of nonlinearly elastic flexural shell and its numerical computation

A New Approximate Model of Nonlinearly ElasticFlexural Shell and its Numerical ComputationXiaoqin Shen,1 Kaitai Li,2 Can Li1

1Department of Applied Mathematical, School of Sciences, Xi’an University ofTechnology, Xi’an 710054, People’s Republic of China

2Department of Computational Mathematical, School of Mathematics and Statistics,Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

Received 29 May 2013; accepted 17 October 2013Published online 18 November 2013 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/num.21834

In this article, we construct a two-dimensional model for the nonlinearly elastic flexural shell using differen-tial geometry and tensor analysis under the assumption that flexural energy is dominant, that is, the metric ofmiddle surface remains invariant. We conduct a numerical experiment for special shell—a portion of cylindershell, which is applied to the forces along the opposite direction of the normal vector. The displacementsdistribution of all points in the middle surface when the shell deforms is obtained. Numerical experimentresults are consistent with the theory, which proves the validity of the proposed model. We then comparethe proposed model and Ciarlet’s model with 3D model, which proves that the proposed model is moreapproximate to 3D model than Ciarlet’s. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq30: 1727–1739, 2014

Keywords: bending energy; flexural energy; nonlinearly flexural shell

I. INTRODUCTION

A classical shell model includes two parts:

1. Giving the following generic form in a given two-dimensional (2D) domain ω (i. e., threecomponents of the displacement ζ(x) = (ζ i , i = 1, 2, 3) on the middle surface S satisfies awell-defined variational formulation):

Correspondence to: Shen, Xiaoqin; School of Sciences, Xi’an University of Technology, Xi’an 710054, People’s Republicof China (e-mail: [email protected])Contract grant sponsor: Research Grants Council of the Hong Kong Special Administrative Region; contract grant number:9041738-CityU 100612Contract grant sponsor: National Natural Science Foundation of China; contract grant number: NSFC 11101330, NSFC11202159, NSFC 61004122Contract grant sponsor: Program of “New Star of Youth on Science and Technology” in Shaanxi Province; contract grantnumber: 2013KJXX-34Contract grant sponsor: Program of Science and Technology, Xi’an City; contract grant number: CXY1341(4)Contract grant sponsor: Education Office Foundation of Shaanxi Province; contract grant number: 2013JK0581

© 2013 Wiley Periodicals, Inc.

1728 SHEN, LI, AND LI

ε£M(ζ , η) + ε3£B(ζ , η) = L(η), ∀η ∈ V

where £B represents the bending (flexural) energy, £M represents the membrane energy,L(η) is the external virtual work associated with η, 2ε is the thickness of the shell, and Vis the test function space, which is a vector Sobolev space.

2. Giving the approximation expression of the displacement u(x1, x2, x3) at any point in theshell based on the following kinematic assumption:

u∗(x, ξ) = ζ(x) + ξT1(ζ ) + ξ 2T2(ζ ).

where x = (x1, x2), x3ε = ξ , and T1, T2 are two smooth functions.

Several classical shell models are available.Koiter proposed a 2D model for linearly and nonlinearly elastic shell called Koiter model,

which takes the following form [1, 2]:

⎧⎪⎨⎪⎩

Find η ∈ VK(ω) = {η = (ηi) ∈ H 1(ω) × H 1(ω) × H 2(ω); ηi = ∂nη3 = 0 on γ } such that

ε

4

∫ ∫ωaαβστ (aστ (η) − aστ )(aαβ(ζ ) − aαβ)

√adx

+ ε3

3

∫ ∫ωaαβστ (bστ (η) − bστ )(bαβ(ζ ) − bαβ)

√adx = ∫ ∫

ωP · ζ

√adx, ∀ζ ∈ VK(ω)

where aαβστ are the contravariant components of the 2D elasticity tensor of the shell, aαβ(η)−aαβ

and bαβ(η) − bαβ are the covariant components of the changes of metric and curvature tensorsthat are associated with a displacement field η of the middle surface S, and the given functionsP ∈ L2(ω) account for the applied body forces. In addition, Greek indices α, β, σ , τ take theirvalues from the set {1, 2}, that is, 16 contravariant components exist for the 2D elasticity andcurvature tensors.

The linearity and the nonlinearity of the model of Koiter’s type depend on the form ofaαβ(η) − aαβ and bαβ(η) − bαβ .

Several years later, Ciarlet et al. [3–9] established the appropriate mathematical foundationsof the 2D theory of shells. They provided a mathematical justification of the 2D shell theoriesusing asymptotic methods with thickness as the “small” parameter. They proposed 2D modelsfor linear and nonlinear shells. In particular, they classified the shell into “membrane” shell andflexural shell.

Ciarlet recently proposed another approach to study the model of Koiter’s type by introducingthe changes of metric and curvature tensors as new unknown variables [10–12], which emphasizedthe importance of two basic forms.

In this article, we consider the change of the curvature tensor when the original middle surfaceS deforms and becomes S(η). These factors are very important for constructing the nonlinearlyflexural elastic shell proposed by Ciarlet.

Let

Rαβ(η) := bαβ(η) − bαβ

Ciarlet replaced

Rαβ(η) = 1√a(η))

∂αβ(θ + ηiai){a1(η) ∧ a2(η)} − bαβ

Numerical Methods for Partial Differential Equations DOI 10.1002/num

NONLINEARLY ELASTIC FLEXURAL SHELL MODEL 1729

with

R�

αβ(η) := 1√a∂αβ(θ + ηia

i){a1(η) ∧ a2(η)} − bαβ

He then provided the approximate full expressions of Rαβ(η) and proposed the model ofnonlinearly elastic flexural shell (in Chapter 10 of [13]).

In this article, we give a new approximately expressions of the change of the curvature tensor,which is more exact than that given by Ciarlet. Thus, we can construct a new approximate modelof nonlinearly elastic flexural shell model, which is better than Ciarlet’s model in theory. Then weconduct the numerical experiment for special shell—a portion of cylinder shell, which is appliedto the forces along the opposite direction of the normal vector. Numerical experiment results provethe validity of our model. At the same time we likewise compare our model and that of Ciarletwith 3D model, which proves that our model is more approximate to 3D model than Ciarlet’s.

II. PRELIMINARIES

In the following, Latin indices and exponents: i, j , k · · · take their values from the set {1, 2, 3},whereas Greek indices and exponents: α, β, γ , · · · take their values from the set {1, 2}. In addi-tion, Einstain’s repeated index summation convention is systematically used. The Euclidean scalarproduct and the exterior product of a, b ∈ R3 are noted by a · b and a ∧ b, respectively.

Let ω be an open, bounded, connected subset of R2, the boundary γ of which is Lipschitz-continuous, the set ω being locally on one side of γ . Let y = (yα) denote a generic point in theset ω, and let ∂α := ∂/∂yα . Let there be given an injective mapping θ ∈ c2(ω; R3), such that thetwo vectors

eα(y) := ∂αθ(y) (1)

are linearly independent at all points y ∈ ω. These two vectors thus span the tangent place to thesurface

S := θ(ω) (2)

at the point θ(y), and the unit vector

n(y) = e3(y) := e1(y) ∧ e2(y)

|e1(y) ∧ e2(y)| (3)

is normal to S at the point θ(y). These vectors ei(y) constitute the covariant basis at the pointθ(y), whereas the vectors ei(y) are defined by the relations

ei(y) · ej (y) = δij . (4)

These relations constitute the contravariant basis at the point θ(y) ∈ S, where δij is the Kro-

necker symbol(noted that e3(y) = e3(y) and that the vector eα(y) are also in the tangent plane toS at θ(y)).

The convariant and contravariant components aαβ and aαβ of the metric tensor of S, the Christof-

fel symbol∗

�σαβ on S, and the covariant and mixed components bαβ and bβ

α of the curvature tensor



of S are then defined as follows (the explicit dependence on the variable y ∈ ω is henceforthdropped):

aαβ = eα · eβ , aαβ = eα · eβ , aαβaβλ = δαλ (5)

∗�σ

αβ = eσ · eαβ , bαβ = n · eαβ , bβα = aβσ bσα (6)

where eαβ := ∂βeα .The determinant of metric tensor is

a := det(aαβ) (7)

In addition, the permutation tensor is given as follows:

εαβ =

⎧⎪⎨⎪⎩

√a,

−√a,

0,

εαβ =

⎧⎪⎨⎪⎩

1√a, (α, β) : even permutation of (1, 2),

− 1√a, (α, β) : odd permutation of (1, 2), 0,

otherwise,

(8)

and the covariant derivative on S is defined as [14]

∗∇αbστ = ∂bσ

τ /∂yα + ∗�σ

αβbβτ −

∗�β

ατbσβ (9)

where∗∇ is an affine connection.

We introduce the elastic tensor on S

aαβστ = 4λμ

λ + 2μaαβaστ + 2μ(aασ aβτ + aατaβσ ) (10)

III. MAIN RESULTS

Declaration: In this article, we only discuss the nonlinearly flexural shell, that is, the manifold

MF (ω) = {η = (ηi) ∈ W 2,4(ω); η = ∂νη = 0 on γ0, aαβ(η) − aαβ = 0 in ω} �= {0}.(11)

The condition MF (ω) �= {0} indicates that there exist nonzero displacement fields ηiai of the

middle surface S that are admissible and inextensional.Consider a nonlinearly elastic shell with middle surface S, subjected to a boundary condition

of place along a portion of its lateral face with θ(γ0) as its middle curve, where

γ0 ∪ γ1 = γ , γ0 ⊂ γ . (12)

Let aαβ and aαβ(η) be metric tensors of S = θ(ω) and S(η) = (θ + η)(ω), respectively, whereS = θ(ω) is the middle surface of shell before deformation and S(η) = (θ + η)(ω) is the middlesurface of the shell after deformation.

In Theorem 1 of [15], we know

aαβ(η) − aαβ = 2γαβ(η) + ϕαβ(η).



where γαβ(η), ϕαβ(η) are defined as follows:⎧⎪⎪⎨⎪⎪⎩

γαβ(η) := 12 (aβσ

0∇αησ + aαλ

0∇βηλ), ϕαβ(η) := aij

0∇αηi

0∇βηj

a3α = aα3 = 0, a33 = 10∇αη

σ = ∗∇αησ − bσ

αη3,0∇αη

3 = ∗∇αη3 + bαλη

λ

(13)

Based on the above assumption, we have

2γαβ(η) = 0, ϕαβ(η) = 0. (14)

Let

q−2 := a(η)

a,

from Lemma 1 of [15], we know

q−2 = 1 + 2γ0(η) + π(η)

where γ0(η) and π(η) are defined as follows:⎧⎪⎨⎪⎩

γ0(η) = aαβγαβ(η), ϕ0(η) = aαβϕαβ(η)

εαλεβσ γαβ(η)γλσ (η) = 1adet(γαβ(η)), εαλεβσϕαβ(η)ϕλσ (η) = 1

adet(ϕαβ(η))

π(η) = ϕ0(η) + 2εαλεβσ γαβ(η)ϕλσ (η) + 2adet(γαβ(η)) + 1

2adet(ϕαβ(η)) = o(|η|2)

Thus, under the declaration, we derive

γ0(η) = 0, π(η) = 0. (15)

Considering that the metric of middle surface remains invariant, we should only determine thechange of the curvature tensor to construct the model of nonlinearly elastic flexural shell.

Given that

bαβ = n · eαβ , (16)

we should determine the change of the unit normal vector first.We then consider the change of the outside unit normal vector after deformation when the

original middle surface S deforms and becomes S(η).

Theorem 1. Let n and n(η) be the outward unit normal vectors of S = θ(ω) and S(η) =(θ +η)(ω), respectively, where θ is an injective mapping θ ∈ C2(ω, R3). Then following formulathus hold

n(η) = (1 + det(0∇αη

β))n + (−δαν + εαβεσν

0∇βησ )

0∇αη3eν (17)

Proof. Because

n(η) = 1

2εαβ(η)eα(η) ∧ eβ(η) (18)



From Expression (16) of [15] and εαβ(η) =√

a

a(η)εαβ = εαβ , we have

n(η) = 1

2εαβ((δλ

α + 0∇αηλ)eλ + 0∇αη

3n) ∧ ((δσβ + 0∇βη

σ )eσ + 0∇βη3n)

= 1

2εαβ[(δλ

α + 0∇αηλ)(δσ

β + 0∇βησ )eλ ∧ eσ

+ (δλα + 0∇αη

λ)0∇βη

3eλ ∧ n + (δσβ + 0∇βη

σ )0∇αη

3n ∧ eσ ] (19)

here we use n ∧ n = 0. From

eα ∧ eβ = εαβn, n ∧ eα = εαβeβ , εαβεαλ = δ

β

λ (20)

Then

n(η) = 1

2εαβ[ελσ (δλ

α + 0∇αηλ)(δσ

β + 0∇βησ )n

+ εσν(δσβ + 0∇βη

σ )0∇αη

3eν − ελν(δλα + 0∇αη

λ)0∇βη

3eν]

= 1

2εαβ{ελσ [δλ

αδσβ + δλ

α

0∇βησ + δσ

β

0∇αηλ + 0∇αη

λ0∇βη

σ ]n

+ [εσν(δσβ + 0∇βη

σ )0∇αη

3 − ελν(δλα + 0∇αη

λ)0∇βη

3]eν}

= 1

2{[2 + 2

0∇βηβ + εαβελσ

0∇αηλ

0∇βησ ]n+

+ [(−δαν + εαβεσν

0∇βησ )

0∇αη3 − (δβ

ν + εαβελν

0∇αηλ)

0∇βη3]eν}

From

εαβ0∇βη

λ0∇αη

3 = −εβα0∇βη

λ0∇αη

3 = −εαβ0∇αη

λ0∇βη

3 (21)

n(η) is then rewritten as

n(η) = (1 + 0∇βηβ + 1

2εαβελσ

0∇αηλ

0∇βησ )n

+ (− 0∇νη3 + 1

2εαβεσν

0∇βησ

0∇αη3 − 1

2εαβελν

0∇αηλ

0∇βη3)eν

= (1 + 0∇βηβ + 1

2εαβελσ

0∇αηλ

0∇βησ )n

+ (− 0∇νη3 + εαβεσν

0∇βησ

0∇αη3)eν

Notably,0∇βη

β = ∗∇βηβ − 2Hη3 = γ0(η) = 0, and εαβελσ

0∇αηλ

0∇βησ = 2det(

0∇αηβ). So

n(η) = (1 + det(0∇αη

β))n + (−δαν + εαβεσν

0∇βησ )

0∇αη3eν

The proof is completed.�



We provide the new approximate expression of change of the curvature tensor when the originalmiddle surface S deforms and becomes S(η).

Theorem 2. Under the assumption of Theorem 1, let bαβ and bαβ(η) be the curvature tensors ofS = θ(ω) and S(η) = (θ + η)(ω), respectively, where θ is an injective mapping θ ∈ C2(� , R3).The following formula thus holds

bαβ(η) − bαβ = ραβ(η) + ψαβ(η), (22)

where ραβ(u) and ψαβ(η) are defined as follows

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

ραβ(η) := 12 [

∗∇β

0∇αη3 + ∗∇α

0∇βη3 + bβλ

0∇αηλ + bαλ

0∇βηλ]

ψαβ(η) := N 2αβ(η) + N 3

αβ(η)

N 2αβ(η) := bαβdet(

0∇αηβ) − �σ

αβ(η)0∇σ η3 + (ενμεσλ − δμ

σ δνλ)

�

�λαβ

0∇νησ

0∇μη3,

N 3αβ(η) := (ραβ(η) +

�

�λαβ

0∇λη3)det(

0∇αηβ) + (�σ

αβ(η) +�

�λαβ

0∇λησ )ενμεστ

0∇νητ

0∇μη3,

�σαβ(η) := 1

2 [∗∇β

0∇αησ + ∗∇α

0∇βησ − bσ

β

0∇αη3 − bσ

α

0∇βη3]

(23)

Proof. Let eαβ = ∂αeβ . From Gaussian and Weingarten formulae, we have

eαβ =∗

�λαβeλ + bαβn,

When the middle surface deforms, eαβ becomes

eαβ(η) = (eα + ηα)β = eαβ + ∂βηα =∗

�λαβeλ + bαβn + ∂βηα (24)

From (15) of [15], we have

∂βηα = ∂β(0∇αη

σ eσ + 0∇αη3n)

= 0∇αησ (

∗�λ

σβeλ + bσβn) + ∂β

0∇αησ eσ + ∂β

0∇αη3n + 0∇αη

3(−bλβeλ)

= [ 0∇αησ

∗�λ

σβ + ∂β

0∇αηλ − 0∇αη

3bλβ]eλ + [ 0∇αη

σ bσβ + ∂β

0∇αη3]n

Given that

∂β

0∇αηλ = ∗∇β

0∇αηλ −

∗�λ

σβ

0∇αησ + ∗

�σαβ

0∇σ ηλ

and

∂β

0∇αη3 = ∗∇β

0∇αη3 +

∗�λ

αβ

0∇λη3,



we have

∂βηα = (∗∇β

0∇αηλ + ∗

�σαβ

0∇σ ηλ − 0∇αη3bλ

β)eλ + (∗∇β

0∇αη3 +

∗�λ

αβ

0∇λη3 + 0∇αη

σ bσβ)n (25)

Substituting (25) into (24), we obtain

eαβ(η) = (∗∇β

0∇αηλ + ∗

�σαβ

0∇σ ηλ +∗

�λαβ − 0∇αη

3bλβ)eλ

+ (∗∇β

0∇αη3 +

∗�λ

αβ

0∇λη3 + 0∇αη

σ bσβ + bαβ)n (26)

In a similar manner, we can obtain eβα(η), hence

eαβ(η) = 1

2(eαβ(η) + eβα(η))

Set

ραβ(η) := 1

2[ ∗∇β

0∇αη3 + ∗∇α

0∇βη3 + bβλ

0∇αηλ + bαλ

0∇βηλ]

�σαβ(η) := 1

2[ ∗∇β

0∇αησ + ∗∇α

0∇βησ − bσ

β

0∇αη3 − bσ

α

0∇βη3]

Then

eαβ(η) = (∗

�λαβ(δ

σλ + 0∇λη

σ ) + �σαβ(η))eσ + (bαβ + ραβ(η) + ∗

�σαβ

0∇σ η3)n (27)

Given that bαβ(η) = eαβ(η) · n(η), from (21) and (27), we have

bαβ(η) = (1 + 1

2det(

0∇αηβ))(bαβ + ραβ(η) + ∗

�σαβ

0∇σ η3)

+ (−δμν + εμτ εσν

0∇τ ησ )

0∇μη3(∗

�λαβ(δ

νλ + 0∇λη

ν) + �ναβ(η))

= bαβ + ραβ(η) + [N 2αβ(η) + N 3

αβ(η)]= bαβ + ραβ(η) + ψαβ(η) (28)

where N 2αβ(η), N 2

αβ(η), and ψαβ(η) are defined by (23).Finally, (28) can be rewritten as

bαβ(η) − bαβ = ραβ(η) + ψαβ(η)

The proof is completed.�

Remark. If η is sufficiently small,

bαβ(η) − bαβ = ραβ(η) + h.o.t ,

where ψαβ(η) = h.o.t .



FIG. 1. Cylinder shell with applied body forces.

Theorem 2. shows that the change of the curvature tensor comprises two parts: one is the lin-earized term ραβ(η), and the other is the nonlinearized polynomial ψαβ(η) = O(|η|3) with respectto η.

From Theorems 1 and 2, we can directly infer the following theorem:

Theorem 3. Under the assumption of Theorems 1 and 2, the new approximation model ofnonlinearly elastic flexural shell is as follows:{

Find η ∈ VK(ω) = {(ηi) ∈ W 2,4(ω); η = ∂νη = 0 on γ0, aαβ(η) − aαβ = 0 in ω} such thatε3

3

∫ ∫ωaαβστ (ραβ(η) + ψαβ(η))(ραβ(ζ ) + ψαβ(ζ ))

√adx = ∫ ∫

ωp · ζ

√adx, ∀ζ ∈ VK(ω)

(29)

where ραβ(η) and ψαβ(η) are defined by (23).

IV. NUMERICAL EXPERIMENT

To satisfy the basic assumptions of nonlinearly flexural shell, we assume that the middle surfaceS of shell is a portion of cylinder (as shown in Fig. 1), whose reference configuration is

θ(y1, y2) = (r cos y1, r sin y1, y2), (30)

where r > 0 is a constant, 0 ≤ y1 ≤ π and 0 ≤ y2 ≤ 2.Then

e1 = ∂1θ = (−r sin y1, r cos y1, 0), e2 = ∂2θ = (0, 0, 1).

Hence, the covariant and contravariant components of the metric tensor are given by

(aαβ) =[r2 00 1

], (aαβ) =

[r−2 00 1

]



FIG. 2. Mesh. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Let

a3α = aα3 = 0, a3α = aα3 = 0, a33 = a33 = 0. (31)

The unit normal vector to S is

�n = e3 = e1 × e2

|e1 × e2| = (cos y1, − sin y1, 0). (32)

Thus,

(bαβ) =[−r 0

0 0

], (bαβ) =

[−r−3 00 0

]

(cαβ) =[

1 00 0

], (cαβ) =

[r−4 00 0

]

Let the thickness be 2ε = 0.0005m, and Lamé parameter be λ = 2×105MPa, μ = 1×106MPa.Suppose the shell is applied to the forces P along the opposite direction of the normal vector. Theshell is subjected to a boundary condition of place along a portion of its lateral face with θ(γ0) asits middle curve. The boundary conditions are strong clamping, that is, η = φνη = 0, on γ0.

In theory, the shell will deform when subject to body forces. In particular, the deformationonly exists in the component η3 of η under the assumption that the flexural energy is dominantbecause the applied forces are along the outside normal vector. Thus, the components η1 and η2

of η will remain invariant.We let p1,0 = p2,0 = 0, p3,0 = 100MPa. We use the finite element method to do the numerical

experiments for our model [(28) in Theorem 3] and for that of Ciarle (as discussed in Chapter 10of [13]). The integration domain w = [0, π ] × [0, 2] is partitioned into triangles. The mesh is 20× 5 (Fig. 2). We use the P2-element (continuous piecewise quadratic, as shown in [16]) in everyunit.

The two graphs in Figs. 3 and 4 show the numerical results of the new model and that of Ciarlet,respectively. Each graph includes three parts, where (a) shows the deformations distributions of η1

on S, (b) shows the deformations distributions of η2 on S, (c) shows the deformations distributionsof η3 on S. From (a) and (b) of Figs. 3 and 4, we know that η1 and η2 have no deformations,which are consistent with the theory and practice. From (c) of Figs. 3 and 4, it is obvious that the



FIG. 3. Numerical results of Ciarlet’s model. [Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

FIG. 4. Numerical results of the proposed model. [Color figure can be viewed in the online issue, whichis available at wileyonlinelibrary.com.]

deformation of η3 on S becomes smaller as y2 becomes bigger. Moreover, the deformation on Sis nearly equal when y2 is fixed.

In addition, we derive the deformation distribution of all points (in Fig. 5) in the nonlinearlyflexural shell by the approximation expression u(x, ξ) = η(x) − (∂αη

3 + bαβηβ)eαξ .

The numerical experiment shows that the deformation is radial-symmetric when y2 is fixed,that is, the deformation of all points in the same radius is nearly equal; a smaller radius results insmaller deformation. Thus, the deformation at the bottom surface is the largest, and that on thetop surface is the smallest, which is consistent with the theory and thus proves the validity of theproposed model.

We also compare the two models with 3D model (chapter5 of [14]) in Table I. It is true thatour proposed model executes longer than that of Ciarlet, which is secondary. Moreover, the datesevidently show that the proposed model is closer to the 3D results compared with the model byCiarlet, which is the key.

FIG. 5. Deformation distribution of all points in the shell. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]



TABLE I. Datas of η3 in two models compared with 3D model.

Model Ciarlet’s model The proposed model 3D model

y1 = π/2, y2 = 0 6.35403e-014 4.12082e-014 4.11336e-014y1 = π/2, y2 = 0.5 5.13125e-014 3.66734e-014 3.6611e-014y1 = π/2, y2 = 1 2.64605e-014 1.71192e-014 1.7088e-014CPU(s) 0.313s 0.656s 1.562s

V. CONCLUSIONS

In this article, we propose a new approximation model of nonlinearly elastic flexural shell. Theproposed model is more exact than that proposed by Ciarlet. We conduct a numerical experimentfor special shell—a portion of cylinder shell, which is applied to the forces along the oppositedirection of the normal vector. We obtain the displacements distribution of all points in the middlesurface when the shell deforms. The numerical experiment shows that the deformation is radial-symmetric when y2 is fixed, that is, the deformation of all points in the same radius is nearly equal;a smaller radius results in smaller deformation. Thus, the deformation on the bottom surface isthe largest, whereas that at the top surface is the smallest, which is consistent with the theoryand thus proves the validity of the proposed model. In addition, we compare two models with 3Dmodel. The comparison proves that the proposed model is more approximate to 3D model thanthat by Ciarlet’s.

References

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A new approximate model of nonlinearly elastic flexural shell and its numerical computation