Asymptotic analysis of nonlinear elastic plates

C. R. Mecanique 330 (2002) 581–586

Asymptotic analysis of nonlinear elastic plates Djamal Ahmed Chacha Département de Maths-Infor, Université de Ouargla, 30000 Ouargla, Algérie Received 28 May 2002; accepted 1 July 2002 Note presented by Évariste Sanchez-Palencia.

Abstract

We study in this Note the asymptotic analysis of nonlinear elastic plates with varying thickness. We suppose that the material moduli of the plates are anisotropic and nonhomogeneous, and the plates are submitted to body forces, to surfaces forces on the lower and upper faces and to pressure forces on the lateral boundary such that the displacements remains plane. To cite this article: D.A. Chacha, C. R. Mecanique 330 (2002) 581–586.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS solids and structures / plate / shallow shell / anisotropic / asymptotic analysis

Analyse asymptotique des plaques élastiques non linéaires Résumé

Nous étudions dans cette Note l’analyse asymptotique des plaques élastiques non linéaires d’épaisseur variable. Nous supposons que les constituants du matériau sont anisotropes et non homogènes. Les forces appliquées sont de type volumiques, surfaciques sur la face inférieure et supérieure ; sur le bord latéral on impose des forces de pression horizontales de sorte que les déplacements demeurent plans. Pour citer cet article : D.A. Chacha, C. R. Mecanique 330 (2002) 581–586.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS solides et structures / plaque / coque peu-profonde / anisotrope / analyse asymptotique

Version française abrégée On considère dans le cadre de l’élasticité non linéaire une plaque élastique tridimensionnelle non homogène et anisotrope dont l’épaisseur varie d’une façon non symétrique par rapport au plan moyen en général. On suppose que l’épaisseur dépend d’un petit paramètre ε qui sera destiné à tendre vers zéro. On s’intéresse dans ce travail à l’étude du comportement asymptotique des équations d’équilibre et des lois de comportement d’une telle plaque lorsque ε tend vers zéro, sachant que les forces appliquées sont de types : volumiques dans le domaine de la plaque, surfaciques sur les faces inférieure et supérieure ; sur le bord latéral on impose des forces de pressions tels que les déplacements soient plans. On utilise, d’une façon formelle, la méthode de développement asymptotique mixte, dite de Hellinger–Reissner, appliquée à la formulation variationnelle du problème posé dans un domaine de référence. En utilisant les mises à l’échelle (the scaling) des forces, des contraintes et du champ de déplacements en suivant Ciarlet et Destuynder [2], Ciarlet [1], ou on obtient au premier ordre significatif que le champ de déplacements est de type Kirchhoff–Love, ce qui est devenu classique pour la théorie des plaques en utilisant les méthodes E-mail address: [email protected] (D.A. Chacha).  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés S 1 6 3 1 - 0 7 2 1 ( 0 2 ) 0 1 5 0 4 - 8 /FLA

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asymptotiques, aussi la formulation variationnelle du problème bidimensionnel des plaques élastiques non linéaires, non homogènes et anisotropes d’épaisseur non constante. On obtient les cas particuliers suivants : • Si l’épaisseur de la plaque est constante et la surface supérieure et inférieure sont planes et les forces volumiques et surfaciques sont verticales alors on obtient la formulation variationnelle du problème bidimensionnel des plaques élastiques, non linéaires, non homogènes et anisotropes de von Karman. Ainsi, on généralise le travail de [3,4] au cas des plaques de von Karman anisotropes et non homogènes. • Si l’épaisseur de la plaque est constante, et les surfaces supérieure et inférieure sont courbées, et les forces volumiques et surfaciques sont verticales, alors on obtient la formulation variationnelle du problème bidimensionnel des coques élastiques peu profondes (shallow shell), non linéaires, non homogènes et anisotropes de Marguerre–von Karman. Ainsi, on généralise le travail de [5] au cas des coques peu profondes de Marguerre–von Karman anisotropes et non homogènes.

1. Setting of the problem Before proceeding, we introduce some notations. We shall write x = (x1 , x2 , x3 ) for the current point in R3 space and x = (x1 , x2 ) for that of R2 space. We shall underline and overline the vectors of physical entities of R3 and R2 space, respectively, using, for example, u = (u1 , u2 , u3 ) and u = (u1 , u2 ), respectively 1 for displacement vector and H 1 = (H 1 )3 and H = (H 1 )2 respectively, for Sobolev space, and double underline a second-order tensor like σ . Latin indices will usually range from 1 to 3 and Greek ones (except for ε) from 1 to 2. The convention of summation of repeated indices is applied. Moreover, the following symbols of differentiation will be used: ∂iε = ∂/∂xiε , ∂i = ∂/∂xi , ∂ij2 = ∂ 2 /∂xi ∂xj . Let ω is a bounded domain of R2 with smooth boundary γ . We define the functions: • φα : x ∈ ω → φα (x) ∈ R3 (α = 1, 2), two functions which will be serve to describe the lower and upper surfaces of the plates; • h : x ∈ ω → h(x) ∈ R∗+ with h(x) = 12 (φ2 − φ1 )(x); • S : x ∈ ω → S(x) ∈ R3 with S(x) = 12 (φ1 + φ2 )(x). We suppose in the following that: h(x)
h0 > 0,

∀x ∈ ω,

φα (x) ∈ W 2,∞ (ω),

α = 1, 2

(1)

Let ε be a small parameter (0  ε < 1) tending to zero, determining the lenght scale of the thickness. The three-dimensional geometry of the plate is defined by     
 ε = x ε ∈ R3 ; x ε = x1ε , x2ε ∈ ω, εφ1 x ε < x3ε < εφ2 x ε (2) We shall denote by 1ε and 2ε the lower and the upper faces of the plate and by 0ε the lateral ones:  
αε = x ε ∈ R3 ; x ε ∈ ω, x3ε = εφα x ε (α = 1, 2)
ε  ε   ε 3 ε ε 0 = x ∈ R ; x ∈ γ , εφ1 x < x3 < εφ2 x ε

(3) (4)

Our goal in the sequel is the study of the asymptotic analysis of a nonlinearly elastic anisotropic nonε homogeneous plates of variable thickness occupying the set  . We suppose that the elastic moduli (aijε kl ) of the plates satisfy the following conditions:  ε ε ∞ ε   aij kl (x ) ∈ L ( ) ε = aε aijε kl = ajεikl = aklij (5) klj i   ∃c > 0, a ε τ τ
cτ τ , ∀τ = τ ij ij ij ji ij kl kl ij The plate is subjected to three kinds of applied forces:

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• volumic forces throughout ε , of density f ε ; • surface forces on the upper and lower faces 1ε ∪ 2ε , of density g ε ; • surface forces on the lateral face 0ε , for which only the resultant p ε along the boundary γ of ω is know. Notice that the functions pαε are defined only on γ . We suppose that the forces acting on the plate satisfy:     2 f ε ∈ L2 ε , g ε ∈ L2 1ε ∪ 2ε , p ε ∈ L (γ ) (6) The boundary conditions involving the displacement uε are uε1 , uε2 , independent of x3 ,

uε3 = 0

on 0ε

We can see [1] (p. 72) for more details on the boundary conditions (7). The problem then consists in studying the asymptotic behavior (ε ↓ 0) of the problem (P ε ):  ε ε −∂j (σij + σkjε ∂kε uεi ) = fiε in ε    ε ε ε ε ε ε ε ε    (σij + σkj ∂k ui )nj = gi on 1 ∪ 2  ε   1 εφ 2 ε ε ε ε ε ε ε P εh εφ1 (σαβ + σkβ ∂k uα )νβ dx3 = pα on γ     uε independent of x3ε on 0ε    αε u3 = 0 on 0ε where



ε σijε = aijε kl Ekl (uε ) the stress tensor

Eijε (uε ) = 12 [∂iε uεj + ∂jε uεi + ∂iε uεm ∂jε uεm ] the nonlinear strain tensor

(7)

(8)

(9)

nε is the exterior unit normal to 1ε ∪ 2ε and ν ε is the exterior one to γ . This problem is similar to that proposed in [1] (p. 71) but for nonhomogeneous anisotropic plates with varying thickness. The variational formulation of the problem (P ε ) is  Find (uε , σ ε ) ∈ V ε × ) ε such that   
εφ2 ε ε  ε 1 ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε (σij + σkj ∂k ui )∂j vi dx = ε f v dx + 1ε ∪2ε g v d + 2 γ εφ1 vα dx3 pα dγ , ∀v ∈ V    (Aσ ε ) τ ε dx ε − τ ε eε (uε ) dx ε − 1 τ ε ∂ ε uε ∂ ε uε dx ε = 0, ∀τ ε ∈ ) ε ij ij

ε

ε ij ij

2 ε ij i

l j l

(10) where



V ε = {v ε ∈ W 1,4 (ε ); v3ε |0ε = 0 and vαε |0ε is independent of x3ε } ) ε = {τ ε = (τijε ); τijε = τjεi , τijε ∈ L2 (ε )}

(11)

The mapping A : S 3 → S 3 , S 3 is the space of symmetric tensors of order 3, is defined by: (AX)ij = bij kl Xkl , ∀X = (Xij ) ∈ S 3 , the functions bij kl are the compliances. The mapping A−1 : S 3 → S 3 is defined by: (A−1 Y )ij = aij kl Ykl , ∀Y = (Yij ) ∈ S 3 , the functions aij kl are the elastic stifnesses. They enjoy the symmetry properties (5). 2. Variational formulation of (P ε ) on reference domain We define a problem equivalent to problem (P ε ) but now posed over a domain which does not depend on ε. Accordingly, we let  = ω×]−1, +1[, We define a

0 = ∂ω×]−1, +1[,

1 = ω × {−1},

2 = ω × {+1}

(12)

C 1 -diffeomorphism F ε : ε

F ε : x = (x1 , x2 , x3 ) ∈  → x ε = (x1ε , x2ε , x3ε ) ∈  ,

where xαε = xα and x3ε = ε(S + x3 h)

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With the spaces V ε , ) ε of (11), we associate the spaces

V = {v ∈ W 1,4 (), v3 |0 = 0 and vα |0 is independent of x3 } ) = {τ = (τij ); τij = τj i , τij ∈ L2 ()}

(13)

We follow Ciarlet and Destuynder [2], and claim that the displacements, stresses, as well as the interior and exterior forces scale as follows:  ε ε ε ε 2   uα (x ) = ε uα (x; ε), u3 (x ) = εu3 (x; ε)    v ε (x ε ) = ε2 vα (x), v ε (x ε ) = εv3 (x) α 3 (14) ε (x ε ) = ε 2 σ (x; ε), σ ε (x ε ) = ε 3 σ (x; ε), σ ε (x ε ) = ε 4 σ (x; ε) σ  αβ α3 33 αβ α3 33     τ ε (x ε ) = ε2 τ (x), τ ε (x ε ) = ε3 τ (x), τ ε (x ε ) = ε4 τ (x) αβ α3 33 αβ α3 33  ε ε ε 2 ε 3   fα (x ) = ε fα (x), f3 (x ) = ε fα (x) (15) gαε (x ε ) = ε3 gα (x), g3ε (x ε ) = ε4 g3 (x)   ε ε 2 pα (x ) = ε pα (x) L EMMA 1. – Let us put ψ(x) = ϕ ◦ F ε (x) for all ϕ defined in ε . So we have:       ϕ dx ε = ε hψ dx, ϕ d ε = 5ε ψ d, ϕ dt ε = ε ε



1ε ∪2ε

1 ∪2

0ε

hψ dt

0

1 + ε2 [(∂1 φα )2 + (∂2 φα )2 ] on α . Moreover, we have   1   (∂α ϕ)ε = ∂α ψ − ∂α (S + x3 h) ∂3 ψ h   (∂ ϕ)ε = 1 ∂ ψ 3 3 εh P ROPOSITION 2. – Let (u(ε), σ (ε)) ∈ V × ) construct from one solution (uε , σ ε ) ∈ V ε × ) ε of problem (10) via the scaling formulas (14), (15). Then (u(ε), σ (ε)) solve the following problem:   ∀τ ∈ ) : a0 (σ , τ ) + εa1 (σ , τ ) + ε2 a2 (σ , τ ) + ε3 a3 (σ , τ ) + ε4 a4 (σ , τ ) (16) 1 1  + A(τ , u) + B(τ , u, u) + ε2 C(τ , u, u) = 0 2 2 2 ∀v ∈ V : A(σ , v) + B(σ , u, v) + ε C(σ , u, v) = F (v) (17)

where 5ε = 5εα =

where ∀σ , τ ∈ ) and ∀u, v ∈ V ,   a0 (σ , τ ) = hbαβγ δ σγ δ ταβ dx, a1 (σ , τ ) = 2 h[bαβγ 3 σγ 3 ταβ + bα3γ δ σγ δ τα3 ] dx    a2 (σ , τ ) = h[bαβ33 σ33 ταβ + 4bα3γ 3σγ 3 τα3 + b33γ δ σγ δ τ33 ] dx    a3 (σ , τ ) = 2 h[bα333σ33 τα3 + b33γ 3σγ 3 τ33 ] dx, a4 (σ , τ ) = hb3333σ33 τ33 dx     A(τ , u) = − hτij Hij (u), B(τ , u, v) = − hτij Ii3 (u)Ij 3 (v) dx    C(τ , u, v) = − hτij Iiλ (u)Ij λ (v) dx     1 F (v) = − hf v dx − 5ε gv d − hpα vα dt 2 0  1 ∪2

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(18) (19) (20)

(21) (22)

Pour citer cet article : D.A. Chacha, C. R. Mecanique 330 (2002) 581–586

 1 H (v) = eαβ (v) − 2h [5α ∂3 vβ + 5β ∂3 vα ]    αβ  5α 1 1 Hij (v) = Hα3 (v) = H3α (v) = 2 [ h ∂3 vα + ∂α v3 − h ∂3 v3 ]     H33 (v) = 1 ∂3 v3 h

Iαj (v) = ∂α vj − h1 5α ∂3 vj , Iij (v) = 5α = ∂α S + x3 ∂α h I3j (v) = h1 ∂3 vj ,

(23)

(24)

3. Asymptotic analysis We postulate that the scaled displacement and stress (u(ε), σ (ε)) can be written as         u(ε), σ (ε) = u0 , σ 0 + ε u1 , σ 1 + ε2 u2 , σ 2 + · · ·

(25)

Substituting expansion (25) into (16), (17) and equating the terms of the same order with respect to ε. We obtain at order ε0 , ε1 and ε2 respectively:

∀τ ∈ ) : a0 (σ 0 , τ ) + A(τ , u0 ) + 12 B(τ , u0 , u0 ) = 0  0 P ∀v ∈ V : A(σ 0 , v) + 12 B(σ 0 , u0 , v) = F (v)

∀τ ∈ ) : a0 (σ 1 , τ ) + a1 (σ 0 , τ ) + A(τ , u1 ) + 12 B(τ , u0 , u1 ) + 12 B(τ , u1 , u0 ) = 0  1 P ∀v ∈ V : A(σ 1 , v) + B(σ 1 , u0 , v) + B(σ 0 , u1 , v) = 0  2 1 0   ∀τ ∈ ) : a0 (σ , τ ) + a1 (σ , τ ) + a2 (σ , τ )  2  P + A(τ , u2 ) + 12 B(τ , u1 , u1 ) + 12 B(τ , u0 , u2 ) + 12 B(τ , u2 , u0 ) + 12 C(τ , u0 , u0 ) = 0    ∀v ∈ V : A(σ 2 , v) + B(σ 2 , u0 , v) + B(σ 1 , u1 , v) + C(σ 0 , u0 , v) = 0 P ROPOSITION 3. – Assume that (u0 , σ 0 ) solution of problem (P 0 ) belonging to V × ) and ∂3 u03 ∈ C 0 (). Then u0 is a (scaled) Kirchhoff–Love displacement field, i.e., there exist function ξ defined on the middle surface ω of the reference plate such that: u03 (x1 , x2 , x3 ) = ξ3 (x1 , x2 )

∀x ∈ 

(26)

u0α (x1 , x2 , x3 ) = ξα (x1 , x2 ) − x3 h∂α ξ3 (x1 , x2 )

∀x ∈ 

The function ξ is solution of the variational problem 1

Find ξ ∈ H = H (ω) × H02 (ω) such that c(ξ , ζ ) = F (ζ ), ∀ζ ∈ H, where   µν µ ν 1ν ν c(ξ , ζ ) = hCαβγ δ (x)>γ δ (ξ )>αβ (ζ ) dx + hCαβγ δ (x)>γ δ (ξ )∂α ξ3 ∂β ζ3 dx ω

ω

  +1    +1   f dx3 ζ dx − h2 x3 fα dx3 ∂α ζ3 dx F (ζ ) = h −1 −1 ω ω     1    g + g 2 ζ dx + h gα1 − gα2 ∂α ζ3 dx + hpα ζα dγ + 

ω

µν

Cαβγ δ (x) =



ω

+1

−1

γ

t µ+ν−2 cαβγ δ (x1 , x2 , t) dt

cαβγ δ (x) = aαβγ δ (x) − aαβi3 (x)dij (x)aj 3γ δ (x)

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where (cαβγ δ ) is the inverse of (bαβγ δ ) and d = (dij ) is the inverse of (ai3j 3 ) matrix.   1 1 1 >αβ (η) = eαβ (η) + ∂α η3 ∂β η3 + (∂α S∂β η3 + ∂β S∂α η3 ), >2αβ (η) = −h∂αβ η3 2 2 0 are given by and the stresses σαβ  

 1 0 σαβ = cαβγ δ (x) Hγ δ (u0 ) + Iγ 3 (u0 )Iδ3 (u0 ) = cαβγ δ (x) >1γ δ (ξ ) + x3 >2γ δ (ξ ) 2

(27)

Remarks. – • If the plate is isotropic and homogeneous with variable thickness we have   2µλ cαβγ θ (x) = δαβ δγ θ + µ(δαγ δβθ + δαθ δβγ ) λ + 2µ where λ and µ are the Lamé constants. • If the plate is isotropic and homogeneous with constant thickness (φ2 = +1, φ1 = −1), we have

 1 3 0 = cαβγ θ (x) ϒγ1 θ (ζ ) + x3 ϒγ2 θ (ζ ) = n0αβ + x3 m0αβ σαβ 2 2 where 1 ϒγ2 θ (ζ ) = −∂γ θ ζ3 ϒγ1 θ (ζ ) = eγ θ (ζ ) + ∂γ ζ3 ∂θ ζ3 , 2    +1 4µλ 0 1 1 n0αβ = σαβ dx3 = 2cαβγ θ ϒγ1 θ (ζ ) = (ζ )δαβ + 4µϒαβ (ζ ) ϒθθ λ + 2µ −1  +1 2 4µλ 4 0 2 2 ϒθθ x3 σαβ dx3 = cαβγ θ ϒγ2 θ (ζ ) = (ζ )δαβ + µϒαβ (ζ ) m0αβ = 3 3(λ + 2µ) 3 −1 Moreover, if we have fα = gα = 0, we obtain the variational formulation of the von Karman plates problem, which is the same result as proposed in [1,3,4]. • If the lower and upper faces of the plate are of the same geometrical form (φ2 = S + 1, φ1 = S − 1), this is a shallow shell, we suppose that is homogeneous and isotropic and fα = gα = 0. Thus we obtain the variational formulation of the Marguerre–von Karman shallow shell problem which is the same as proposed in [1,5]. References [1] P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures, An Asymptotic Analysis, Masson, Paris, 1990. [2] P.G. Ciarlet, P. Destuynder, A justification of a nonlinear model in plate theory, Comp. Methods Appl. Mech. Engrg. 17/18 (1979) 227–258. [3] P.G. Ciarlet, A justification of the von Karman equations, Arch. Rational Mech. Anal. 73 (1980) 349–389. [4] P.G. Ciarlet, P. Rabier, Les équations de von Karman, Lecture Notes in Math., Vol. 826, Springer-Verlag, Berlin, 1980. [5] P.G. Ciarlet, J.C. Paumier, A justification of the Marguerre–von Karman equations, Comput. Mech. 1 (1986) 177– 202.

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