Cécile Poutous

Jean-Marie Thomas

Robert Luce

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1207–1212http://france.elsevier.com/direct/CRASS1/Mathematical Problems in MechanicsAsymptotic modelling of a fluid–structure couplingin the case of a prestressed inflated orthotropic membrane shellRobert Luce a, Cécile Poutous a,b, Jean-Marie Thomas aa Laboratoire de mathématiques appliquées, UMR 5142, Université de Pau et des Pays de l’Adour, BP 1155, 64013 Pau cedex, Franceb Centre de recherche de l’armée de l’air, CReA-EOAA BA701, 13661 Salon de Provence, FranceReceived 31 March 2008; accepted 5 May 2008Available online 16 October 2008Presented by Philippe G. CiarletAbstractIn this Note, we consider an inflated orthotropic linearly elastic generalized membrane shell submitted to an outer surfaceperturbation. We obtain the strong convergence towards the solution of a well-posed “2D” problem of the mean value in themembrane thickness 2ε of the “3D” scaled displacements, as ε approaches zero. To cite this article: R. Luce et al., C. R. Acad.Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméModélisation asymptotique d’un problème de couplage fluide–structure dans le cas d’une membrane orthotrope gonfléeprécontrainte. Dans cette Note, on considère une coque en membrane en flexion pure inhibée, gonflée et précontrainte, que l’onsoumet à des perturbations extérieures. On établit, quand l’épaisseur 2ε de la coque tend vers zéro, la convergence forte de lavaleur moyenne dans l’épaisseur du déplacement « 3D » normalisé vers la solution d’un problème « 2D » bien posé. Pour citer cetarticle : R. Luce et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeOn considère une structure souple gonflée précontrainte composée de deux pièces d’un tissu orthotrope fixées surune armature rigide. Le volume intérieur V ainsi délimité est gonflé sous pression Π par un gaz parfait, créant ainsi uneprécontrainte, supposée connue, 0. On applique à ce système en équilibre une perturbation surfacique extérieure h.Le fluide intérieur génère un couplage entre les déplacements, et donc entre les contraintes, des deux pièces de tissu.On suppose que du fait de la précontrainte, la déformation se fait à volume constant et que l’on reste dans le cadrede l’élasticité linéaire. On suppose également que la géométrie et les conditions d’encastrement sont telles que lematériau se comporte comme une coque en membrane généralisée linéairement élastique de type 1, c’est à dire enflexion pure inhibée (voir définition dans [2] p. 262 ou dans [7]). Enfin, on suppose que la densité h est admissible ausens donné dans [5] et que la forme linéaire liée au couplage est continue par rapport à la norme membranaire.E-mail addresses: robert.luce@univ-pau.fr (R. Luce), cpoutous@cr-ea.net (C. Poutous), jean-marie.thomas@univ-pau.fr (J.-M. Thomas).1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.09.0261208 R. Luce et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1207–1212Fig. 1. An example of prestressed system.Soit un domaine ω = ω+ ∪ ω− de R2, de paramétrisation de la surface moyenne des tissus S = S+ ∪ S− encoordonnées curvilignes choisies le long des directions principales d’orthotropie. On considère les cartes associéesrespectives θ+ et θ− (θ± :ω± → R3 ∈ C3(ω±;R3)) injectives, telles que, pour α ∈ {1,2}, les vecteurs a±α = ∂αθ±(y)sont orthogonaux en tout point y ∈ ω±. On leur associe la famille de coques de surface moyenne S = θ(ω̄), etd’épaisseur 2ε. L’encastrement se traduit par des conditions aux limites de type Dirichlet homogène posées sur unemême portion du bord latéral θ±(γ ±0 ). En faisant tendre l’épaisseur vers zéro, on se ramène à un problème variationnelposé sur S, bien posé puisque h appartient au dual de l’espace où on cherche les déplacements (voir [1], p. 129 ou [2]).En partant de la démonstration donnée dans [3], on élargit les résultats de convergences fortes au cas d’un matériauorthotrope avec un couplage fluide structure. On prouve que la solution u(ε) de (1), problème variationnel normalisé3D associé où figure le terme de couplage, converge fortement dans un espace obtenu par complétion, et que savaleur moyenne dans l’épaisseur, u(ε), converge fortement également, dans un espace complété qui prend en comptele couplage, vers l’unique solution ζ du problème variationnel (2) posé cette fois-ci sur une surface. Dans le modèleasymptotique obtenu, l’équation variationnelle est identique à celle que l’on obtiendrait sans le couplage, mais l’espacedans lequel on cherche la solution est un sous-espace fermé de l’espace naturel pour les coques en membranes.1. Notations and 3D modellingIn this Note, Greek indices take their values in {1,2}, whereas Latin indices belong to {1,2,3}, the repeated indexsummation convention is used and δij is the Kronecker symbol.Assumption 1. Let ω± be domains in R2 (open, bounded, connected subsets with a Lipschitz-continuous boundary,the sets ω± being locally on one side of their boundary), let θ± :ω± → R3 ∈ C3(ω̄;R3) be injective mappings suchthat the vectors a±α := ∂αθ±(y) are orthogonal at each point y ∈ ω±. Let the surfaces S± := θ±(ω±) representthe middle surfaces of two pieces of an orthotropic material which orthotropy directions are a±α in the surface anda±3 := a±1 ∧a±2|a±1 ∧a±2 |in the thickness. Let 2ε be the thickness of the material and Ω±ε := S± × ]−ε, ε[ be the volumeoccupied by the shells. Let S±ε := S± × {ε} be the outer surfaces and S±−ε := S± × {−ε} be the inner surfaces. Let∂ω± = γ ±0 ∪ γ ±1 , Γ ±0,ε := θ±(γ ±0 ) × ]−ε, ε[ and Γ ±1,ε := θ±(γ ±1 ) × ]−ε, ε[. We suppose that the shells are clampedalong Γ ±0,ε in a prestressed equilibrium state, that the inflating pressure is Π , the prestress ±0 .Remark 1. Let us insist on the fact that the orthotropy of the material imposes the choice of the surface parameterscontrary to the usual isotropic case where the choice is free.We pertubate the outer conditions with a surface density force h ∈ L2(S±−ε). This perturbation modifies the ge-ometry of the shells and thereby the inner pressure of Π and the stresses which become ±. We let n denote theouter unit normal vector to the current surface and A denote the elasticity tensor in an orthotropy orthonormal baseR. Luce et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1207–1212 1209(this is the natural base to underline the symmetry properties of the tensor and to cancel most of the coordinates). Itscontravariant components areAijkl = 0 apart from Aiijj = λij and Aijij = Aijji = μij when i = jwhere λij = λji and μij = μji are positive constants which only depend on the material (see [4] for the calculus).The equilibrium equations in the reference configuration (prestressed state) are:⎧⎪⎪⎪⎨⎪⎪⎪⎩−div(± − ±0 ) = 0 in Ω±ε ,(± − ±0 )n = h on S±ε ,(± − ±0 )n = −Πn on S±−ε,(± − ±0 )n = 0 on Γ ±1,ε.Assumption 2. Elasticity laws can be linearized (expected effect of the prestress).Let u±ε denote the displacement between the prestressed configuration and the deformed configuration. Then,± − ±0 = A : e±ε(u±ε)where e±ε(u±ε) = 12(∇u±ε + ∇u±ε T ).The inflating gas is perfect so the inner volume variation V and the inner pressure variation Π are linked byΠ = −VVΠ . The volume variation and the displacement field satisfy V = − ∫S±−ε u±ε · n ds. Lastly, the clampingcondition reads u±ε = 0 on Γ ±0,ε .Since the unknowns are defined on sets that vary with ε, we naturally transform the 3D variational problemsequivalent to the former PDE and boundary conditions system into problems posed over a set that does not dependon ε through the bijection (y, x3) → (y, εx3) and thus obtain: u(ε) ∈ V(Ω), ∀v ∈ V(Ω)∫ΩAijkl(ε)ek‖l(ε;u(ε))ei‖j (ε;v)√g(ε)dx + 1ε∫Γ −u(ε) · n√g(ε)dy ∫Γ −v · n√g(ε)dy = ∫Γ +hi±vi√g(ε)dy. (1)The functions hi± are in L2(Γ +) and independent of ε, Γ + := ω × {1}, Γ − := ω × {−1}, Ω := ω × ]−1,1[, Γ0 =γ0 × ]−1,1[ and V(Ω) := {v = (vi) ∈ H1(Ω);v = 0 on Γ0},  : ω̄ × [−ε, ε] → Ω±ε is the canonical extension of θand thus verifies (y, x3) := θ(y) + x3a3 and det(g1 ,g2,g3) > 0 ((gi )i := (∂i)i is the covariant base of the set(ω̄ × [−ε, ε]) whereas (g1,g2,g3) is the contravariant base such that gj · gi = δji ). Then, g(ε) denotes the scaledfunction of gε := det(gi · gj )ij and Aijkl(ε) are the contravariant components of the scaled 3D elasticity tensor A(ε).They satisfy the change of bases and coordinates formulaAijkl(ε)(y, x3) =(∣∣gi∣∣∣∣gj ∣∣∣∣gk∣∣∣∣gl∣∣Aijkl)(y, εx3),the symmetry conditions Aijkl(ε) = Ajikl(ε) = Aklij (ε) and the first order developments Aijkl(ε) = Aijkl(0) + O(ε).The order symbol is meant with respect to the norm ‖w‖0,∞,Ω̄ := sup{|w(x)|, x ∈ Ω̄}. Moreover,Aαβσ3(ε) = Aα333(ε) = Aαβσ3(0) = Aα333(0) = 0.For any vector field v = (vi) ∈ H1(Ω), the scaled linearized strains ei‖j (ε;v) = ej‖i (ε;v) ∈ L2(Ω) are defined byeα‖β(ε;v) := 12 (∂βvα + ∂αvβ) − Γ pαβ(ε)vp, eα‖3(ε;v) := 12 ( 1ε ∂3vα + ∂αv3) − Γ σα3(ε)vσ and e3‖3(ε;v) := 1ε ∂3v3 withΓpij (ε) being the scaled 3D Christoffel symbols obtained from Γε,pij := gp · ∂igj .Proposition 1. The scaled variational problems (1) are well posed.Proof. The bilinear form Bε(u,v) :=∫ΩAijkl(ε)ek‖l (ε;u)ei‖j (ε;v)√g(ε)dx is a norm over V(Ω) equivalent (in-equality of Korn) to the usual norm. The coupling term Cε(u,v) := 1ε∫Γ − u3√g(ε)dy∫Γ − v3√g(ε)dy is bicontinuousand increases the positivity of the operator. The second member Lε(u) :=∫+ hi±vi√g(ε)dy is continuous. Γ1210 R. Luce et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1207–1212In this Note, we establish that the coupling term remains bounded and we prove the strong convergence of thesequences (u(ε))ε and (u(ε))ε := ( 12∫ 1−1 u(ε)dx3)ε as ε approaches zero. The asymptotic 2D model is very close tothe orthotropic membrane model obtained without the coupling.2. Main resultsIn a computing perspective, we are interested in the convergence of (u(ε))ε towards the solution of a variationalproblem posed over a surface. Analytical preliminaries (we let ε approach zero in (1) when v = εw and w is an arbi-trary function in the space V(Ω)) suggest to consider the 2D elasticity tensor A(ω) which contravariant componentsareaαβστ := 2∣∣aα∣∣∣∣aβ ∣∣∣∣aσ ∣∣∣∣aτ ∣∣(Aαβστ − Aαβ33A33στA3333)and to consider, for any vector field (ηi) in V(ω) := {η = (ηi) ∈ H1(ω);η = 0 on γ0}, the 2D linearized changeof metric tensor of covariant components γαβ(η) := 12 (∂βηα + ∂αηβ) − Γ σαβησ − bαβη3 where bαβ := a3 · ∂αaβ arethe covariant components of the curvature tensor and Γ σαβ := aσ · ∂αaβ are the surface Christoffel symbols. Let usintroduce the semi-norm | · |Mω defined by |η|Mω := {∑α,β |γαβ(η)|20,ω}1/2.Remark 2. At this step, let us underline that the orthotropy does not affect the curvature tensor nor the semi-norm, itonly changes the elasticity tensor.Assumption 3. The geometry and the boundary conditions are such that | · |Mω is a norm over the space V(ω) which isnot equivalent to the norm ‖ · ‖1,ω .Hence we let V#M(ω) be the completion of V(ω) with respect to | · |Mω and we associate to | · |Mω the 3D normover V(Ω) defined by |v|MΩ = {|∂3v|20,ω + (|v̄|Mω )2}1/2. The completion of V(Ω) with respect to | · |Ω is V#M(Ω).Last, we let aαβ := aα · aβ be the covariant components of the metric tensor and a := det(aαβ). The bilinear formBM(ζ ,η) :=∫ωaαβστ γστ (ζ )γαβ(η)√a dy is obviously continuous with respect to | · |Mω so we let B#M denote theunique continuous extension of the bilinear form BM from V(ω) to V#M(ω). When LM(η) :=∫ωhiηi√a dy andCM(η) :=∫ωη3√a dy are | · |Mω -continuous, we let L#M and C#M be their unique continuous extension from V(ω)to V#M(ω).Assumption 4. The coupling term CM is | · |Mω -continuous and there exist functions hαβ = hβα ∈ L2(ω) such that forall η ∈ V(ω), ∫ωhiηi√a dy = ∫ωhαβγαβ(η)√a dy (i.e. the surface density of forces (hi) is admissible in the sensegiven in [5]).Remark 3. For instance, if f = (0,0,1) is admissible, then CM is | · |Mω -continuous.Theorem 1. Suppose Assumptions 1, 2, 3 and 4 are satisfied. There exists ζ in V#M0(ω) := {η ∈ V#M(ω),C#M(η) = 0}and u in V#M(Ω) such thatu(ε) −→ u in V#M(Ω) as ε → 0 and u(ε) −→ ζ in V#M(ω) as ε → 0and the limit ζ satisfies this variation of the scaled 2D variational problem of a linearly elastic generalized membraneshell of the first kind:{ζ ∈ V#M0(ω), ∀η ∈ V#M0(ω),B#M(ζ ,η) = L#M(η).(2)To prove this theorem, we use the same pattern as in [3] and [5]. We only develop the proof of the parts where thereis a difference due to the orthotropy (part (iii)) or to the coupling (parts (ii), (v), (vii) and (viii)).R. Luce et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1207–1212 1211Proof. Part (ii). We need to add new results to part (ii): there exists a subsequence still denoted (u(ε))ε such thatlimε→0 CM(u(ε)) = limε→0∫Γ −u3(ε)√g(ε)dΓ = 0.Combining the uniform positive definiteness of A(ε), the positivity of Cε(u(ε),u(ε)), Assumption 4, the variationalproblem (1) and [5] (ii)’s bounding |Lε(u(ε))|  c√∑i,j ‖ei‖j (u(ε))‖20,Ω , we obtain∑i,j∥∥ei‖j (u(ε))∥∥20,Ω  cBε(u(ε),u(ε))  c(Bε(u(ε),u(ε)) + Cε(u(ε),u(ε))) = cLε(u(ε)) c√∑i,j∥∥ei‖j (u(ε))∥∥20,Ω .Therefore the sequences (ei‖j (u(ε)))ε and (Cε(u(ε),u(ε)))ε are bounded. Consequently,∂3u3(ε) = εe3‖3(u(ε)) −→ 0 in L2(Ω) and limε→0∫Γ −u3(ε)√g(ε)dΓ = limε→0√ε√Cε(u(ε),u(ε)) = 0.Moreover,limε→0 CM(u(ε)) = limε→0∫ωu3(ε)√g(ε)dy = limε→012∫Ωu3(ε)√g(ε)x3=0 dx= limε→012∫∂Ω(x3 − 1)u3(ε)√g(ε)x3=0n3 ds − limε→012∫Ω(x3 − 1)∂3u3(ε)√g(ε)x3=0 dx.Hence the result holds since n3 = 0 on ∂ω × [−1,1] and n3 = −1 on Γ −.Part (iii). With the same technique as in [3] it becomes: eα‖3 = 0, e3‖3 = −λα3λ33 aααeα‖α .Part (v). It is changed into: There exists a constant cc ∈ R such that for all η ∈ V(ω),∫ωaαβστ eσ‖τ γαβ(η)√a dy =∫ωhαβγαβ(η)√a dy − ccCM(η).Indeed, let v3 be in H 1(ω) such that v3 = 0 on γ0 and∫ωv3√a dy = 0. For v = (0,0, v3) (which is in V(Ω)) we haveCε(u(ε),v) = 1ε∫Γ −u3(ε)√g(ε)dΓ∫Γ −v3√g(ε)dΓ = Lε(v) − Bε(u(ε),v).Let ε approach zero, thenLε(v) −→ LM(v),∫Γ −v3√g(ε)dΓ −→ CM(v) and Bε(u(ε),v) −→ ∫ωaαβστ eσ‖τ bαβv3 dy.These limits are in R, hence there exists cc ∈ R such that limε→0 1ε∫Γ − u3(ε)√g(ε)dΓ = cc and the result holds.Part (vii). To establish the strong convergence of (ei‖j (u(ε)))ε towards ei‖j in L2(Ω) announced in part (vii) weproceed in two steps. First, we prove thatlimε→0∑i,j∥∥ei‖j (u(ε)) − ei‖j∥∥20,Ω  c(limε→0 Lε(u(ε)) − 2∫ωAijkl(0)ek‖l ei‖j√a dy).Indeed, from the uniform positive definiteness of A(ε) we have∑i,j∥∥ei‖j (u(ε)) − ei‖j∥∥20,Ω c(Bε(u(ε),u(ε)) − 2∫ Aijkl(ε)ek‖l(u(ε))ei‖j√g(ε)dx +∫Aijkl(ε)ek‖lei‖j√g(ε)dx).Ω Ω1212 R. Luce et al. / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1207–1212Thereby, as for all (i, j), ei‖j = ei‖j (they are independent of x3), we havelimε→0∫ΩAijkl(ε)ek‖l(u(ε))ei‖j√g(ε)dx = limε→0∫ΩAijkl(ε)ek‖lei‖j√g(ε)dx = 2∫ωAijkl(0)ek‖l ei‖j√a dy.Moreover,Bε(u(ε),u(ε)) = Lε(u(ε)) − Cε(u(ε),u(ε))andlimε→0 Cε(u(ε),u(ε)) = ( limε→01ε∫Γ −u3(ε)√g(ε)dΓ)(limε→0∫Γ −u3(ε)√g(ε)dΓ)= cc × 0 = 0.So, if they exist, limε→0 Bε(u(ε),u(ε)) = limε→0 Lε(u(ε)) and letting ε approach zero we obtain the announcedinequality. Second, letting η = u(ε) and ε approach zero in part (v) we prove thatlimε→0 Lε(u(ε)) = 2∫ωAijkl(0)ek‖l ei‖j√a dy.Indeed, as limε→0 CM(u(ε)) = 0 and Lε(u(ε)) = Lε(u(ε)), we obtainlimε→0 Lε(u(ε)) = limε→0∫ωaαβστ eσ‖τ γαβ(u(ε))√a dy.We conclude thanks to the weak convergence of (γαβ(u(ε)))ε towards eα‖β and noticing that2∫ωAijkl(0)ek‖l ei‖j√a dy =∫ωaαβστ eσ‖τ eα‖β√a dy.Part (viii). It becomes : the limit ζ ∈ V#M(ω) found in part (vii) is in V#M0(ω) and satisfies the equationsB#M(ζ ,η) = L#M(η), for all η ∈ V#M0(ω).With the same technique as in [3], we prove that ζ , which existence has already been proved, satisfies the ill-posedproblemζ ∈ V#M0(ω), B#M(ζ ,η) = L#M(η) − cc C#M(η), ∀η ∈ V#M(ω).Thereby, it satisfies the announced well posed problem. 3. ConclusionFrom architecture to aeronautics industry, from anatomy to pneumatics, the scope of application of inflated struc-tures is very wide. This Note is the generalization to any kind of linearly elastic generalized membrane shells of thefirst kind of results obtained in [6] when studying the behaviour of the outer envelop of an airship. The generalizationhas supposed to broaden some assumptions: for instance the continuity in the energy norm of the coupling term wasproved whereas it is assumed in this Note.References[1] D. Chapelle, K.J. Bathe, The Finite Element Analysis of Shells Fundamentals, Springer, 2003.[2] P.G. Ciarlet, Mathematical Elasticity, vol. III, Theory of Shells, North-Holland, Amsterdam, 2000.[3] P.G. Ciarlet, V. Lods, Asymptotic analysis of linearly elastic shells: Generalized membrane shells, J. Elasticity 43 (1996) 147–188.[4] J. Coirier, Mécanique des milieux continus, Dunod, Paris, 2007.[5] R. Luce, C. Poutous, J.M. Thomas, Weakened conditions of admissibility of surface forces applied to linearly elastic membrane shells, C. R.Acad. Sci. Paris, Ser. I 344 (2007) 721–726.[6] C. Poutous, Modélisation asymptotique et analyse numérique d’un problème de couplage fluide–structure, Thèse de doctorat, Université de Pauet des Pays de l’Adour, 2006.[7] J. Sanchez-Hubert, E. Sanchez-Palencia, Coques élastiques minces. Propriétés asymptotiques, Masson, Paris, 1997.

Asymptotic modelling of a fluid–structure coupling in the case of a prestressed inflated orthotropic membrane shell

Luce, Robert

Poutous, Cécile

Thomas, Jean-Marie

Numérisation de Documents Anciens Mathématiques

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2008.09.026.pdf

Asymptotic modelling of a fluid–structure coupling in the case of a prestressed inflated orthotropic membrane shell

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