International audienceAbstract In this Note, we consider an inflated orthotropic linearly elastic generalized membrane shell submited to an outer surface perturbation. We obtain the strong convergence towards the solution of a wellposed ”2D” problem of the mean value in the membrane thickness 2ε of the ”3D” scaled displacements, as ε approaches zero

Luce, Robert

Poutous, Cécile

Thomas, Jean-Marie

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HAL Id: inria-00341893https://hal.inria.fr/inria-00341893Submitted on 26 Nov 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Asymptotic modelling of a fluid-structure coupling in thecase of a prestressed inflated orthotropic mebrane shellRobert Luce, Cécile Poutous, Jean-Marie ThomasTo cite this version:Robert Luce, Cécile Poutous, Jean-Marie Thomas. Asymptotic modelling of a fluid-structure couplingin the case of a prestressed inflated orthotropic mebrane shell. Comptes rendus de l’Académie dessciences. Série I, Mathématique, Elsevier, 2008. ￿inria-00341893￿Asymptotic modelling of a fluid-structure coupling in the case of aprestressed inflated orthotropic membrane shellRobert Luce(a), Cécile Poutous(a) (b), Jean-Marie Thomas(a)robert.luce@univ-pau.fr, cpoutous@cr-ea.net, jean-marie.thomas@univ-pau.fr(a)Laboratoire de Mathématiques Appliquées, UMR 5142, Université de Pau et des Pays de l’Adour, BP 1155, 64013 Pau Cedex, France,(b)Centre de Recherche de l’Armée de l’Air, CReA-EOAA BA701 F-13661 Salon de ProvenceAbstractIn this Note, we consider an inflated orthotropic linearly elastic generalized membrane shell submited to an outer surfaceperturbation. We obtain the strong convergence towards the solution of a wellposed ”2D” problem of the mean valuein the membrane thickness 2ε of the ”3D” scaled displacements, as ε approaches zero.RésuméModélisation asymptotique d’un problème de couplage fluide-structure dans le cas d’une membraneorthotrope gonflée précontrainte. Dans cette Note, on considère une coque en membrane en flexion pure inhibée,gonflée et précontrainte, que l’on soumet à des perturbations extérieures. On établit, quand l’épaisseur 2ε de la coquetend vers zéro, la convergence forte de la valeur moyenne dans l’épaisseur du déplacement ”3D” normalisé vers la solutiond’un problème ”2D” bien posé.1 Version française abrégéeOn considère une structure souple gonflée précontrainte composée de deux pièces d’un tissu orthotrope fixées sur unearmature rigide. Le volume intérieur V ainsi délimité est gonflé sous pression Π par un gaz parfait, créant ainsi uneprécontrainte, supposée connue, Σ0. 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 qu’on reste dans le cadre del’élasticité linéaire. On suppose également que la géométrie et les conditions d’encastrement sont telles que le matériause comporte comme une coque en membrane généralisée linéairement élastique de type 1, c’est à dire en flexion pureinhibée (voir définition dans [Ciarlet, 2000] p262 ou dans [Sanchez-Hubert, Sanchez-Palencia, 1997] ). Enfin, on supposeque la densité h est admissible au sens donné dans [Luce-Poutous-Thomas, 2007] et que la forme linéaire liée au couplageest continue par rapport à la norme membranaire.Soit un domaine ω = ω+∪ω− de R2, de paramétrisation de la surface moyenne des tissus S = S+∪S− en coordonnéescurvilignes choisies le long des directions principales d’orthotropie. On considère les cartes associées respectives θ+ et θ−(θ± : ω± −→ R3 ∈ C3(ω±; R3)) injectives, telles que, pour α ∈ {1, 2}, les vecteurs a±α = ∂αθ± (y) sont orthogonaux entout point y ∈ ω±. On leur associe la famille de coques de surface moyenne S = θ (ω) , et d’épaisseur 2ε. L’encastrementse traduit par des conditions aux limites de type Dirichlet homogène posées sur une même portion du bord latéralθ±(γ±0). En faisant tendre l’épaisseur vers zéro, on se ramène à un problème variationnel posé sur S, bien posé puisqueh appartient au dual de l’espace où on cherche les déplacements (voir [Chapelle-Bathe, 2003], p129 ou [Ciarlet, 2000]).En partant de la démonstration donnée dans [Ciarlet-Lods, 1996], on élargit les résultats de convergences fortes au casd’un matériau orthotrope avec un couplage fluide structure. On prouve que la solution u (ε) de (1), problème variationnelnormalisé 3D associé où figure le terme de couplage, converge fortement dans un espace obtenu par complétion, et quesa valeur moyenne dans l’épaisseur, u (ε), converge fortement également, dans un espace complété qui prend en compte1le 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.2 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(ω; 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 volume occupiedby 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 clamped along Γ±0,ε in a prestressedequilibrium state, that the inflating pressure is Π, the prestress Σ±0 .Figure 1: an example of prestressed systemRemark 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 geometryof the shells and thereby the inner pressure of ∆Π and the stresses which become Σ±. We let n denote the outer unitnormal vector to the current surface and A denote the elasticity tensor in an orthotropy orthonormal base (this is thenatural base to underline the symetry properties of the tensor and to cancel most of the coordinates). Its contravariantcomponents areAijkl = 0 apart from Aiijj = λij and Aijij = Aijji = µij when i 6= jwhere λij = λji and µij = µji are positive constants which only depend on the material, (see [Coirier, 2001] for thecalculus). The equilibrium equations in the reference configuration (prestressed state) are :−div(Σ± − Σ±0 ) = 0 in Ω±ε(Σ± − Σ±0 )n = h on S±ε(Σ± − Σ±0 )n = −∆Πn on S±−ε(Σ± − Σ±0 )n = 0 on Γ±1,ε.2Assumption 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,Σ± − Σ±0 = A : e±ε (u±ε ) where e±ε (u±ε ) =12(∇u±ε + ∇u±εT).The inflating gas is perfect so the inner volume variation ∆V and the inner pressure variation ∆Π are linked by∆Π = −∆VV Π. The volume variation and the displacement field satisfy ∆V = −∫S±−εu±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 depend onε through the bijection (y, x3) 7−→ (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}, Θ : ω × [−ε, ε] −→ Ω±ε is the canonical extension of θ and thus verifiesΘ (y, x3) := θ (y)+x3a3 and det (g1 ,g2 ,g3) > 0 ( (gi)i := (∂iΘ)i is the covariant base of the set Θ(ω× [−ε, ε]) whereas(g1,g2,g3) is the contravariant base such that gj ·gi = δji ). Then, g(ε) denotes the scaled function of gε := det (gi · gj)ijand Aijkl (ε) are the contravariant components of the scaled 3D elasticity tensor A(ε). They satisfy the change of basesand coordinates formulaAijkl(ε)(y, x3) = (|gi| |gj | |gk| |gl|Aijkl)(y, εx3),the symetry conditions Aijkl(ε) = Ajikl(ε) = Aklij(ε) and the first order developments Aijkl(ε) = Aijkl(0) + O(ε). Theorder symbol is meant with respect to the norm ‖w‖0,∞,Ω := sup{|w (x)| , x ∈ Ω}. 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) − Γpαβ (ε) vp, eα‖3(ε;v) := 12 ( 1ε∂3vα + ∂αv3) − Γσα3(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 (inequalityof Korn) to the usual norm. The coupling term Cε(u,v) :=1ε∫Γ−u3√g(ε)dy∫Γ−v3√g(ε)dy is bicontinuous andincreases the positivity of the operator. The second member Lε(u) :=∫Γ+hi±vi√g(ε) dy is continuous.In 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∫ 1−1u(ε) dx3)ε as ε approaches zero. The asymptotic 2D model is very close to theorthotropic membrane model obtained without the coupling.3 Main resultsIn a computing perspective, we are interested in the convergence of (u(ε))ε towards the solution of a variational problemposed over a surface. Analytical preliminaries (we let ε approach zero in (1) when v = εw and w is an arbitrary functionin the space V(Ω) ) suggest to consider the 2D elasticity tensor A(ω) which contravariant components areaαβστ := 2|aα| |aβ | |aσ| |aτ |(Aαβστ − Aαβ33A33στA3333)3and to consider, for any vector field (ηi) in V(ω) := {η = (ηi) ∈ H1(ω); η = 0 on γ0}, the 2D linearized change of metrictensor of covariant components γαβ(η) :=12(∂βηα + ∂αηβ) − Γσαβησ − bαβη3 where bαβ := a3 · ∂αaβ are the covariantcomponents of the curvature tensor and Γσαβ := aσ · ∂αaβ are the surface Christoffel symbols. Let us introduce thesemi-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 norm over V(Ω)defined by |v|MΩ = {|∂3v|20,ω + (|v|Mω )2}1/2. The completion of V(Ω) with respect to |·|Ω is V#M (Ω) . Last, we letaαβ := aα · aβ be the covariant components of the metric tensor and a := det(aαβ). The bilinear form BM (ζ, η) :=∫ωaαβστγστ (ζ) γαβ (η)√ady is obviously continuous with respect to |·|Mω so we let B#M denote the unique continuousextension of the bilinear form BM from V (ω) to V#M (ω). When LM (η) :=∫ωhiηi√a dy and CM (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√ady =∫ωhαβγαβ (η)√ady (i.e. the surface density of forces (hi) is admissible in the sense givenin [Luce-Poutous-Thomas, 2007] ).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 → 0.and 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 [Ciarlet-Lods, 1996] and [Luce-Poutous-Thomas, 2007]. We onlydevelop the proof of the parts where there is a difference due to the orthotropy ( part (iii) ) or to the coupling (parts(ii), (v), (vii) and (viii) ).Proof. Part (ii). We need to add new results to part (ii) : there exists a subsequence still denoted (u(ε))ε such thatlimε→0CM (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 [Luce-Poutous-Thomas, 2007] (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)) = 0.4Moreover,limε→0CM (u(ε)) = limε→0∫ωu3(ε)√g(ε) dy = limε→012∫Ωu3(ε)√g(ε)x3=0dx= limε→012∫∂Ω(x3 − 1)u3(ε)√g(ε)x3=0n3 ds − limε→012∫Ω(x3 − 1)∂3u3(ε)√g(ε)x3=0dx.Hence the result holds since n3 = 0 on ∂ω × [−1, 1] and n3 = −1 on Γ−.Part (iii). With the same technique as in [Ciarlet-Lods, 1996] it becomes : eα‖3 = 0, e3‖3 = −λα3λ33aαα eα‖α.Part (v). It is changed into : There exists a constant cc ∈ R such that for all η ∈ V(ω),∫ωaαβστeσ‖τγαβ(η)√a dy =∫ωhαβγαβ(η)√a dy − cc CM (η).Indeed, let v3 be in H1(ω) such that v3 = 0 on γ0 and∫ωv3√a dy 6= 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) −→∫ωaαβστeσ‖τ bαβv3 dy.These limits are in R, hence there exists cc ∈ R such that limε→01ε∫Γ−u3(ε)√g(ε) dΓ = cc and the result holds.Part (vii). To establish the strong convergency of (ei‖j(u(ε)))ε towards ei‖j in L2(Ω) announced in part (vii) we proceedin two steps. First, we prove thatlimε→0∑i,j‖ei‖j(u(ε)) − ei‖j‖20,Ω ≤ c(limε→0Lε(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∫ΩAijkl(ε)ek‖l(u(ε)) ei‖j√g(ε) dx +∫ΩAijkl(ε)ek‖l ei‖j√g(ε) dx.Thereby, as for all (i, j), ei‖j = ei‖j (they are independant of x3 ), we havelimε→0∫ΩAijkl(ε)ek‖l(u(ε)) ei‖j√g(ε) dx = limε→0∫ΩAijkl(ε)ek‖l ei‖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→0Cε(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→0Bε(u(ε),u(ε)) = limε→0Lε(u(ε)) and letting ε approach zero we obtain the announced inequality.Second, letting η = u(ε) and ε approach zero in part (v) we prove thatlimε→0Lε(u(ε)) = 2∫ωAijkl(0)ek‖l ei‖j√a dy.5Indeed, as limε→0CM (u(ε)) = 0 and Lε(u(ε)) = Lε(u(ε)), we obtainlimε→0Lε(u(ε)) = limε→0∫ωaαβστeσ‖τγαβ(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 =∫ωaαβστeσ‖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 [Ciarlet-Lods, 1996], we prove that ζ, which existence has already been proved, satisfiesthe ill-posed problemζ ∈ 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.4 ConclusionFrom architecture to aeronautics industry, from anatomy to pneumatics, the scope of application of inflated structures isvery wide. This Note is the generalization to any kind of linearly elastic generalized membrane shells of the first kind ofresults obtained in [Poutous, 2006] 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 the coupling termwas proved whereas it is assumed in this Note.References[Chapelle-Bathe, 2003] Chapelle, D. & Bathe, K.J. The Finite Element Analysis of Shells Fundamentals, Springer.[Ciarlet, 2000] Ciarlet, P.G. Mathematical Elasticity, Volume III, Theory of shells, North Holland, Amsterdam.[Ciarlet-Lods, 1996] Ciarlet, P.G. & Lods, V. Asymptotic analysis of linearly elastic shells: ”Generalized membraneshells”, J.Elasticity 43, 147-188.[Coirier, 2001] Coirier, J. Mécanique des milieux continus, Dunod, Paris.[Luce-Poutous-Thomas, 2007] Luce, R. & Poutous, C. & Thomas, J.M. Weakened conditions of admissibility of surfaceforces applied to linearly elastic membrane shells, C. R. Acad. Sci. Paris, ser I (344) 721-726.[Poutous, 2006] Poutous, C. Modélisation asymptotique et analyse numérique d’un problème de couplage fluide-structure, Thèse de doctorat, Université de Pau et des Pays de l’Adour.[Sanchez-Hubert, Sanchez-Palencia, 1997] Sanchez-Hubert, J. & Sanchez-Palencia, E. Coques élastiques minces. Pro-priétés asymptotiques. Masson, Paris.6

Asymptotic modelling of a fluid-structure coupling in the case of a prestressed inflated orthotropic mebrane shell

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Asymptotic modelling of a fluid-structure coupling in the case of a prestressed inflated orthotropic mebrane shell

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