Some solutions to the asymptotic bending problem of non-inhibited shells.

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Some solutions to the asymptotic bending problem of non-inhibited shells.

Daniel Choï

To cite this version:

Daniel Choï. Some solutions to the asymptotic bending problem of non-inhibited shells.. 2005. �hal-

00004347�

non-inhibited shells.

Daniel Cho¨ı

Laboratoire de Math´ematiques Nicolas Oresme , Universit´e de Caen 14032 Caen, France

Abstract

We study the asymptotic limit bending problem of thin linearly elastic shells as the thickness goes to zero.

Such asymptotic bending problem makes sense whenever bendings are admissible. We present two alternate expressions of the change of curvature tensor giving new formulations for the asymptotic bending problem. In some case of cylinders and hyperbolic surfaces, we are able present solutions, eventually analytic. Such solutions can constitute tests for the membrane locking.

Keywords : non inhibited shell; asymptotic bending problem; inextensional displacement; membrane locking

1 Introduction

The natural trend for a very thin elastic shell is to perform bendings as it is shown by the asymptotic analysis of linear thin shells. Unlike the bendings in beams problems, bending deformation on a surface is not always admissible: it depends on the boundary conditions and the geometry [8]. Thin shells are thus classified as [with bendings] inhibited or not-inhibited [12, 4]. This classification is also referred as membrane dominated and bending dominated [7]. These two very distinct asymptotic behaviors lead to strong difficulties in the numerical finite element approximation of very thin elastic shells: boundary layers, propagation and reflexion of singu- larities, sensitivity within the inhibited case and numerical (membrane) locking in the non-inhibited or bending dominated case, see [10, 11, 6, 4, 9, 12].

In this paper, we study the limit problem of thin non-inhibited elastic shells, the asymptotic bending problem, making sense whenever the set of kinematically admissible infinitesimal bending is not reduced to trivial bendings (rigid displacements).

With alternative formulations, the asymptotic bending problem simplifies greatly in some cases of cylinder and hyperbolic shells and allow an easy resolution, analytically, eventually.

Such solution can be used as a test, as in [5] , to the membrane locking that appears in the finite element approximation of very thin elastic shell problems [10, 6, 4, 1].

2 The asymptotic bending problem

In this paper, we employ the Einstein convention of summation on repeated upper or lower indices. The Greek (resp. Latin) indices range over {1, 2} (resp. {1, 2, 3}). The partial derivatives with respect to variables x α are denoted in lower indices preceded by a comma. We use overarrow to indicate space vectors. The variables x 1 , x 2

lives in bounded domain Ω ⊂ R ² .

Let S be a surface given by a map (Ω, ~ r), Ω ∈ R ² with

~

r = x(x 1 , x 2 )~ e 1 + y(x 1 , x 2 )~ e 2 + z(x 1 , x 2 )~ e 3 . (1)

Daniel Cho¨ı Asymptotic bending limit problem Third MIT Conference

The vectors (~ e ₁ , ~ e ₂ , ~ e ₃ ) constitutes a Cartesian basis of space R ³ and the direction ~ e ₃ will be referred as the vertical direction. The coefficients of the first and second fundamental form are

a αβ = ~a α .~a β and b αβ = ~a 3 .~a α,β , (2)

where the tangent vectors ~a α = ~ r ,α and the unit normal vector ~a 3 constitute the covariant basis on S. The covariant basis is associated with its dual basis: the contravariant basis (~a ¹ , ~a ² , ~a ³ ). We also have the coefficients a ^αβ = ~a ^α .~a ^β defining the inverse matrix to (a αβ ).

The Christoffel’s symbols are

Γ ^µ _αβ = ~a ^µ .~a _α,β . (3)

We let ε denote the thickness so that the shell the midsurface of which is S, is defined as the set {~ r(x 1 , x 2 ) + x 3 ~a 3 , x 3 ∈ [−ε/2, ε/2]} ⊂ R ³ .

Let us consider a shell problem. The mechanical problem in the space of kinematically admissible displace- ments − →

V and with external applied load L ^ε , writes ( find ~ u ^ε ∈ − →

V such that a ^ε (~ u ^ε , ~ v) = L ^ε (~ v), ∀~ v ∈ − →

V ,

(4)

where a ^ε denotes the deformation energy bilinear form which depends on the linearized variations of the funda- mental forms of the surfaces, especially the symmetric tensors ρ _αβ and γ _αβ , respectively the change of curvature tensor and the membrane strain tensor. ρ _αβ and γ _αβ are the linearized variations of the coefficients a _αβ and b _αβ . Classically, we have the following asymptotic behavior valid for thin shells linear models such as Koiter’s or Naghdi’s [12, 7, 4]:

Theorem 2.1. If the (scaled) load L ^ε (~ v) depends on ε as L ^ε (~ v) = ε ³ L(~ v), then the solutions ~ u ^ε of (4) converge towards the solution ~ u ⁰ of the asymptotic limit bending problem



 

 

find ~ u ⁰ ∈ − →

G such that Z

S

A ^αβλµ

12 ρ αβ (~ u)ρ λµ (~ v), = L(~ v ^∗ ), ∀~ v ^∗ ∈ − → G ,

(5)

where − →

G is the set of kinematically admissible infinitesimal bendings (also called inextensional displacement).

In the case of homogeneous and isotropic material, A ^αβλµ = E

2(1 + ν )

a ^αλ a ^βµ + a ^αµ a ^βλ + 2ν

1 − ν a ^αβ a ^λµ

,

where E is the Young’s modulus and ν the Poisson ratio.

3 Resolution of the asymptotic bending problem

The limit bending problem (5) is a constrained problem, thus one can try to solve it with a mixed formulation.

But at least two alternate formulations of the asymptotic bending problem, leading to important simplifications,

permits its resolution in specific cases. One first is based on the notion of infinitesimal rotation field associated

with an inextensional displacement and has been developed in [5] exhibiting some solutions for hyperbolic sur-

faces. A second is based on a new expression of the tensor ρ αβ with only one component of the displacement,

in the case of an inextensional bending. Both methods take advantage of properties of infinitesimal bendings, for

which we recall some properties.

The infinitesimal bendings or inextensional displacements are displacements leaving the linearized variation of the first fundamental form, that is the intrinsic metric, unchanged. This variation is given by the membrane strain tensor

γ _αβ (~ u) = 1

2 (~ u _,α .~ r _,β + ~ r _,α .~ u _,β ) . (6)

Definition 3.1. A displacement ~ u is an inextensional displacement if γ _αβ (~ u) = 0.

Any inextensional displacement ~ u defines an unique vector field ~ ω, called associated infinitesimal rotation field satisfying the relations :

~

u _,1 = ~ ω ∧ ~a ₁ and ~ u _,2 = ω ~ ∧ ~a ₂ . (7)

Let us introduce the notation w ~ α = w ^µ _α ~a µ for the partial derivatives of a rotation field ~ ω :

~

w α = ~ ω ,α .

It has been shown that such vector fields are tangent to the surface and their components satisfy a partial differ- ential system, the characteristic of which coincides with the asymptotic lines of the surface [5]:

w ¹ _1,2 + Γ ¹ _2λ w ^λ ₁ = w ¹ _2,1 + Γ ¹ _1λ w ^λ ₂ (8)

w ² _1,2 + Γ ² _2λ w ^λ ₁ = w ² _2,1 + Γ ² _1λ w ^λ ₂ (9)

b 12 w ¹ ₁ + b 22 w ² ₁ = b 11 w ¹ ₂ + b 12 w ² ₂ (10)

w ¹ ₁ + w ² ₂ = 0. (11)

3.1 The tensor of curvature variation for inextensional displacements

We give here two alternative expressions for the tensor of curvature variation int the case of inextensional dis- placements. One is based on the associated rotation field, and the second is based on only one cartesian compo- nent of the displacement.

Proposition 3.2. Let ~ u be an inextensional displacement on the considered surface S and let ρ _αβ (~ u) be the tensor of curvature variation. If we denote by w ^λ _α the contravariant components of (~ w ₁ , ~ w ₂ ), the partial derivatives of the associated rotation field ~ ω, we have :







ρ ₁₁ (~ u) = −w ₁ ² √ a ρ ₂₂ (~ u) = +w ₂ ¹ √

a ρ ₁₂ (~ u) = ¹ ₂ (w ¹ ₁ − w ² ₂ ) √

a.

(12)

Obtaining (12) is quite straightforward writing the variation of the coefficients of the second fundamental form b αβ and replacing the derivatives of an inextensional displacements with its associated rotation field.

Another expression is based on the possibility to reconstruct a surface for a given first fundamental form and one component of its mapping [8]. Transposing this property to inextensional displacements gives the following expression of ρ αβ :

Proposition 3.3. For any infinitesimal bending ~ u = (u ₁ , u ₂ , u ₃ ), in Cartesian components, the linearized change of curvature tensor ρ αβ (~ u) depends only on the surface coefficients and one component of the displacement :

ρ αβ (~ u) = ρ αβ (u 3 ) = N

u 3 | αβ + N (z ,λ u 3,µ a ^λµ )z| αβ

(13)

with N = _1−z ¹

,λ

z

a

and z| αβ = z ,αβ − Γ ^ν _αβ z ,ν .

These two expressions of ρ αβ (12) and (13) allow simplifications and resolution of the asymptotic bending

problem (5) on specific cases.

Daniel Cho¨ı Asymptotic bending limit problem Third MIT Conference

3.2 Case of a hyperbolic paraboloid

Let S be a hyperbolic paraboloid and be defined by the mapping (1) with

x = x 1 , y = x 2 , z = x 1 x 2 . (14)

The covariant and contravariant basis and various coefficients of the surface are :

~a 1 = (1, 0, y) a 11 = 1 + y ² a ¹¹ = 1 + x ²

a b 11 = 0 Γ ¹ ₁₂ = y a

~a 2 = (0, 1, x) a 12 = xy a ¹² = xy

a b 12 = 1

√ a Γ ² ₁₂ = x a

~a 3 = 1

√ a (−y, −x, 1) a 22 = 1 + x ² a ²² = 1 + y ²

a b 22 = 0 Γ ^β _αα = 0 a = 1 + x ² + y ²

and the elasticity coefficients for homogeneous and isotropic elastic shells are A ¹¹¹¹ = E 1 + x ² 2

12(1 − ν ² )a ² A ¹¹²² = E x ² y ² + νa

12(1 − ν ² )a ² A ²²²² = E 1 + y ² 2

12(1 − ν ² )a ² .

The simplifications gives easily the set of associated rotation fields and thus the set of inextensional displacements on a hyperbolic paraboloid [5] :

Proposition 3.4. For any inextensional displacement ~ u on the hyperbolic paraboloid S, there is a unique couple (φ ~ u2 , φ ~ u1 ) ∈ L ² y × L ² x , such that modulo a rigid displacement, we have

~ u(x, y) = R(φ ~ u2 , φ ~ u1 ) = [yΦ ~ u1 (x) − Ψ ~ u2 (y)] ~ e 1 + [Ψ ~ u1 (x) − xΦ ~ u2 (y)] ~ e 2

− [Φ _{~ u1} (x) − Φ _{~ u2} (y)] ~ e 3 , (15) where the functions Φ ~ uα and Ψ ~ uα are defined with φ ~ uα by quadrature

Φ ~ uα (x) = Z x

0

φ ~ uα (z)(x − z)dz and Ψ ~ uα (x) = Z x

0

φ ~ uα (z)z(x − z)dz, with ρ 11 (~ u) = − √

aφ _{~ u1} , ρ 12 (~ u) = 0, and ρ 22 (~ u) = √ aφ _{~ u2} .

As an example, if we set Ω = [0, 1] ² and suppose the hyperbolic paraboloid is clamped along the boundary x = 0. This condition imposes that φ ~ u1 = 0 in (15). The asymptotic bending problem simplifies then to a one dimensional problem: with a unitary localized vertical loading on the point (0,1), we have



 

 

Find φ _{x~ u} ∈ L ² x ([0, 1]) such that Z 1

0

αφ _{~ u} φ _{~ v} dx = Φ _{~ v} (1) = Z 1

0

φ _{~ v} (x)(1 − x)dx ∀φ _{~ v} ∈ L ² _x ([0, 1]), (16) where

α(x) = Z 1

0

A ¹¹¹¹ (x, y)a √

ady, (17)

Giving immediately the solution φ ~ u (x) = 1 − x

α(x) and u 3 (1, 0) = Z 1

0

(1 − z) ² α(z) dz.

With E = 1 and ν = 1/3, we obtain u 3 (1, 0) = 3.5534.

3.3 Case of a cylinder

Let us assume now that S is a cylinder. Let S be defined by the mapping (1) with

x = x ₁ , y = x ₂ , z = ϕ(x ₁ ). (18)

The curve (x 1 , 0, ϕ(x 1 )) is a directrix and the curves (straight lines) at x 1 = const. are the generatrix. The covariant and contravariant basis and various coefficients of the surface are :

~a 1 = (1, 0, ϕ ⁰ (x 1 )),

~a ₂ = (0, 1, 0),

~a ₃ = 1

a (−ϕ ⁰ (x ₁ ), 0, 1),

a 11 = a, a 12 = 0, a ₂₂ = 1,

b 11 = ϕ ⁰⁰

√ a b ₁₂ = 0, b 22 = 0,

Γ ¹ ₁₁ = ϕ ⁰ ϕ ⁰⁰ a , Γ ² ₁₁ = 0, Γ ^α _β2 = 0,

a ¹¹ = 1 a , a ¹² = 0, a ²² = 1, with a = 1 + (ϕ ⁰ ) ² . The elasticity coefficients are

A ¹¹¹¹ = E

12a ² (1 − ν ² ) , A ¹¹¹² = 0, A ¹²¹² = E a(1 + ν) .

In the following we shall suppose that ϕ ⁰⁰ doesn’t vanishes, in other words that S is not a plane. Then, it can be shown that the displacement u ₃ is necessarily a polynomial in x ₂ of degree 1 :

u 3 = g(x 1 ) + x 2 h(x 1 ),

where g and h are arbitrary functions of x 1 . Then,

ρ ₁₁ = N

(u _3,11 − Γ ¹ ₁₁ u _3,1 + N z _,1 u _3,1 a ¹¹ (z _,11 − Γ ¹ ₁₁ z _,1 )), ρ ₁₂ = N

u _3,12 , ρ ₂₂ = 0.

Furthermore, as we have

N a ¹¹ ϕ ⁰ (ϕ ⁰⁰ − Γ ¹ ₁₁ ϕ ⁰ ) − Γ ¹ ₁₁ = 0, we state:

Proposition 3.5. Let ~ u be an infinitesimal bending on a cylinder (18), then u 3 = g(x 1 ) + x 2 h(x 1 ) and ρ ₁₁ (~ u) = N

(g ⁰⁰ + x ₂ h ⁰⁰ ) ρ ₁₂ (~ u) = N

h ⁰ ρ ₂₂ (~ u) = 0.

Actually, in the case of inextensional displacement, it is possible to construct the components u 1 and u 2 from u 3 by quadratures from the equations of the bending system. Thus it is possible to express L(~ u) = L(u 3 ). The asymptotic bending problem (5) then writes as a variational problem of two functions of one variable :



 

 

Find u ₃ = g(x ₁ ) + x ₂ h(x ₁ ) ∈ G ~ Z

S

1 12

ρ 11

2ρ ₁₂ T

A ¹¹¹¹ 0 0 A ¹²¹²

T ρ ^∗ ₁₁ 2ρ ^∗ ₁₂

= L(u ^∗ ₃ ) ∀u ^∗ ₃ = g ^∗ (x 1 ) + x 2 h ^∗ (x 1 ) ∈ G. ~

(19)

It is then possible for any cylinder shells, to reduce the constrained problem to a one dimensional differential problem.

As an example, we consider Ω = [−

√ 2 2 ,

√ 2

2 ] × [−0.5, 0.5] and the cylinder defined with z = p 1 − x ² ₁ , clamped on the boundary x ₁ = −

√ 2

2 . It is easy to find that such configuration admits infinitesimal bending:

all vertical displacements u ₃ = g(x ₁ ) + x ₂ h(x ₁ ) satisfying the boundary conditions g(0) = g ⁰ (0) = h(0) = h ⁰ (0) = 0 define admissible infinitesimal bendings. This geometry is also referred as a Scoderlis-Lo roof. The shell is submitted to a localized vertical force − →

F = F ~ e ₃ on the point p ₃ = (

√ 2 2 , 0.5,

√ 2 2 ).

In this case, we don’t have the exact analytical solution but the reduced asymptotic bending problem (5) is very simple, and it suffices to implement a standard one dimensional finite element to solve it. We then obtain the value of the vertical component of the displacement on p 3 by numerical approximations, independent of the thickness.

With E = 1, ν = 1/3, F = 1, we have u 3 (p 3 ) = 2.1168107.

Daniel Cho¨ı Asymptotic bending limit problem Third MIT Conference

4 Concluding remarks and membrane locking

We only gave two examples here, but it is not difficult to construct more, with different shapes and loadings.

These solutions can constitute tests for the membrane locking: we compare with numerical results given by finite element schemes at given mesh for different thicknesses (going to 0).

The membrane locking in finite element computation of thin shells is the deterioration of the approximation for a fixed mesh when the thickness goes to 0, [2]. We have shown in [6] that any conformal finite element method is subject to membrane locking. Actually, the proof is also valid for some non-conformal methods such as DKT, but for non-conformal methods such as proposed in [1], including MITC [4] , a further analysis is needed. In our opinion, as convergence need consistency, such methods should not be strictly locking-free, in the same way are the conformal methods, even though they seem to perform very well with respect to membrane locking [3].

In the particular case of localized loading, the deformation energy is given by the displacement at the loading point. It is easy then to compare the deformation energy : the asymptotic bending energy is necessarily smaller than the deformation energy for a given thickness. If present, the locking can then be detected on the specific case.

References

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[2] I. Babuska and M. Suri. On locking and robustness in the finite element methods. SIAM Journal of Numerical Analysis, 29:261–293, 1992.

[3] K.-J. Bathe, A. Iosilevich, and D. Chapelle. An evaluation of the mitc shell elements. Computers & Structures, 75:1–30, 2000.

[4] D. Chapelle and K.-J. Bathe. The finite element analysis of shells – Fundamentals. computational fluid and solid mechanics. Springer, 2003.

[5] D. Cho¨ı. Computations on thin non-inhibited hyperbolic elastic shells. benchmarks for membrane locking. Journal Mathematical Methods Applie Sciences, 22(15):1293–1321, 1999.

[6] D. Cho¨ı, F. J. Palma, E. Sanchez-Palencia, and M. A. Vilari˜no. Remarks on membrane locking in the finite element computation of very thin elastic shells. Mod´elisation Math´ematique et Analyse Num´erique, 32(2):131–152, 1998.

[7] P. G. Ciarlet. Mathematical Elasticity Volume III : Theory of shells. Studies in Mathematics and its Applications. North Holland, 2000.

[8] G. Darboux. Th´eorie g´en´erale des surfaces, volume 4. Gauthier-Villars, 1896.

[9] P. Karamian, J. Sanchez-Hubert, and E. S. Palencia. Non-smoothness in the asymptotics of thin shells and propagation of singularities. hyperbolic case. International Journal of Mathematical Computational Science, 22(1):81–90, 2002.

[10] J. Pitk¨aranta. The problem of membrane locking in finite elements analysis of cylindrical shells. Numerische Mathe- matik, 61:523–542, 1992.

[11] J. Pitk¨aranta, Y. Leino, O. Ovaskainen, and J. Piila. Shell deformation states and the finite element method: A benchmark study of cylindrical shells. computer methods in applied mechanics and engineering, 128:81–121, 1995.

[12] J. Sanchez-Hubert and E. Sanchez-Palencia. Coques ´elastiques minces : Propri´et´es asymptotiques. Masson, 1997.