D.A. Chacha

Ismail Merabet

Serge Nicaise

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 338 (2010) 277–282Contents lists available at ScienceDirectComptes Rendus Mecaniquewww.sciencedirect.comSingular layers for transmission problems in thin shallow shell theory:Elastic junction caseIsmail Merabet a,∗, D.A. Chacha a, Serge Nicaise ba Département de mathématiques, Université de Ouargla, BP 511, Algeriab LAMAV, FR CNRS 2956, université Lille nord de France, UVHC, 59313 Valenciennes cedex 9, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 28 March 2010Accepted after revision 13 April 2010Available online 11 May 2010Keywords:Solids and structuresThin shallow shellsInternal layersTransmission conditionsElastic junctionMots-clés :Solides et structuresCoques peu profondes mincesCouches internesConditions de transmissionJonction élastiqueIn this Note we study two-dimensional transmission problems for the linear Koiter’s modelof an elastic multi-structure composed of two thin shallow shells with the same thicknessε  1, in the elastic junction case. We suppose that the loading is singular, that the elasticcoefficients are of different order on each part (O(ε−1) and O(1) respectively) and thatthe elastic stiffness coefficient of the hinge is kε = O(ε). The formal limit problem fails togive a solution satisfying all boundary and transmission conditions; it gives only the outersolution. We derive the inner limit problem which allows us to describe the transmissionlayer.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éCette Note s’intéresse à des problèmes de transmission bidimensionnels pour le modèlelinéaire de Koiter d’une multi-structure élastique mince composée de deux coques peuprofondes de même épaisseur ε  1, pour le cas de jonction élastique avec chargementssinguliers. On suppose que les coefficients d’élasticité sont d’ordre très différent sur chaquepartie (respectivement O(ε−1) et O(1)) et que le coefficient de rigidité de la charnièrekε = O(ε). Le problème limite formel ne donne pas une solution vérifiant toutes lesconditions au bord et de transmission; il donne seulement la solution « extérieure ». Nousidentifions le problème limite « intérieur » qui nous permet de déterminer la couche limitede transmission.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeDans cette Note, on étudie un problème de transmission bidimensionnel, il s’agit d’un problème simplifié du modèlelinéaire de Koiter avec les conditions de jonction élastique (voir [1, III.3]). On considère deux surfaces elliptiques S etS̃ représentant les surfaces moyennes de deux coques peu profondes minces C et C̃ de même épaisseur ε. La surfacemoyenne est elliptique totalement encastrée présentant une arête Γ . Les coefficients d’élasticité dans chaque partie de lamulti-structure sont très différents, c’est-à-dire O(ε−1), O(ε) et O(1) respectivement.Soient ω et ω̃ deux domaines plans, φ et φ̃ deux bijections telles queω = (0,π)x1 × (0,1)x2 , ω̃ = (0,π)x1 × (0,1)x̃2 , Σ ={(x1,0,0)∣∣ 0 < x1 < π} (1)* Corresponding author.E-mail addresses: merabetsmail@gmail.com (I. Merabet), chacha@hotmail.com (D.A. Chacha), snicaise@univ-valenciennes.fr (S. Nicaise).1631-0721/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2010.04.007278 I. Merabet et al. / C. R. Mecanique 338 (2010) 277–282Ω = ω ∪ Σ ∪ ω̃, S = φ(ω), S̃ = φ̃(ω̃)Γ = φ(Σ) = φ̃(Σ), Γ0 = ∂Ω ∩ ∂ω et Γ̃0 = ∂Ω ∩ ∂ω̃ (2)La formulation variationnelle du problème étudié est donnée par (4). Si les chargements sont dans l’espace H−1 × H−1 × H−2et ε est fixé, l’existence et l’unicité de la solution du problème (4) est une conséquence du lemme de Lax–Milgram. Danscette Note, on suppose que la donnée sur ω est de la forme f (x1, x2) = (0,0, F (x1)δ(x2)) et que la donnée (sur ω̃) f̃ estsuffisament régulière, ici δ désigne la masse de Dirac et F ∈ L2(0,π). Pour ε > 0 fixé le problème est bien posé, mais leproblème limite formel, i.e., ε = 0 est mal posé. Les conditions au bord (6) et de transmissions (7) ne peuvent être satisfaites.Une couche limite ordinaire sur le bord et une couche limite de transmission le long de l’interface se développent. Eneffectuant le changement de variables (11) et d’inconnues (12), on obtient un nouveau problè me limite, dont la solutionest une approximation valable au voisinage de la zone de transmission.1. IntroductionThis Note is devoted to a transmission problem for the linear Koiter’s shell model set on an elastic multi-structurecomposed of two thin shallow shells for the elastic junction conditions (see [1, III.3]). The common middle surface ispiecewise elliptic with a simply fixed edge (in the terminology of [2]). We restrict ourselves to the case when the coefficientof the elastic stiffness of the hinge becomes very small. Nevertheless, in the shell theory, it appears a very interesting featurewhen the loading is singular (for example δ-like singularity along a curve C , see [3]), the limit problem becomes ill-posedin the original energy space, leading to the outbreak of internal layers. In this paper we restrict our attention to the limitproblem, when the shell is linked to an other shell with very different elastic coefficients and singular loading along theinterface between the two shells. Hence an appropriate dilation of the normal coordinate leads to an equivalent problemwhose solution is a correct approximation of the original solution in a neighborhood of the transmission zone.Throughout we use the convention of summation of repeated indices, which run from 1 to 2 when they are Greek. Thenotation (.){c} means that the property holds true for both ω, ω̃, u, ũ, x, x̃, . . . , i.e., on both shells.2. The model problem and its formal limitLet us consider two thin elliptic shallow shells C and C̃ with the same thickness ε, whose middle surfaces S and S̃intersect along a common boundary Γ . Namely we assume that the domains are given by (1)–(2), i.e., S{c} is the imagethrough the bijection φ{c} of the bounded open subset ω{c} of the plane of variables x{c}1 , x{c}2 which represent the localcoordinates in the basis:a{c}α = ∂{c}α φ{c} = ∂φ{c}∂x{c}α, α = 1,2 and a{c}3 =∣∣a{c}1 × a{c}2 ∣∣−1(a{c}1 × a{c}2 )We suppose that the common middle surface is clamped along the whole boundary and that the angle between the twoshells along the edge Σ is equal to π/2.We consider a simplified problem, when the coefficients of the second fundamental form of the middle surface S{c}satisfy b{c}11 = b{c}22 = 1, b{c}12 = 0 and the elasticity coefficients Aαβγ λ,{c} satisfyAαβγ λ,{c}γγ λ(v{c}) = T αβ(v{c}) = γαβ (v{c}) = 12(∂{c}α v{c}β + ∂{c}β v{c}α) − b{c}αβ v{c}3Aαβγ λ,{c}12γ λ(v{c}) = Mαβ,{c}(v{c}) = ∂{c}αβ v{c}3 , Ãαβγ λ = ε−1 Aαβγ λThe rigidity coefficient of the hinge is kε = ε.At any point P of the interface, the elastic behavior only insures the continuity of the displacements. The tangentialcomponent Mn of the moment M is proportional to the jump of the tangential component of the rotations, i.e.,⎧⎨⎩u(P ) = ũ(P )Mn = ε[∂uε3∂x2− ε−1 ∂ ũε3∂ x̃2](3)According to [1, III.3], the variational formulation of the problem is{Find(uε, ũε) ∈ H s.t. ∀(v, ṽ) ∈ Ha(uε, v) + ε2b(uε, v) + ε−1(ã(ũε, ṽ) + ε2b̃(ũε, ṽ)) + εcε((uε; ũε), (v; ṽ)) = 〈 f , v〉 + 〈 f̃ , ṽ〉 (4)whereI. Merabet et al. / C. R. Mecanique 338 (2010) 277–282 279H = {(v, ṽ) ∈ V × Ṽ ∣∣ v1 = ṽ1, v2 = ṽ3, v3 = −ṽ2 on Σ}V {c} ={v{c} ∈ H1(ω{c})2 × H2(ω{c}) ∣∣∣ v{c}|Γ {c}0 =(∂v{c}3∂n{c})|Γ {c}0= 0}a{c}(u, v) =∫ω{c}γαβ(u)γαβ(v)dx, b{c}(u, v) =∫ω{c}αβ(u)αβ(v)dxcε((u; ũ), (v; ṽ)) = ∫Σ(∂u3∂x2− ε−1 ∂ ũ3∂ x̃2)(∂v3∂x2− ε−1 ∂ ṽ3∂ x̃2)dΣFor a fixed ε, the left-hand side of (4) is strongly coercive on H and its right-hand side defines a linear form on H , hencethe existence and the uniqueness of a solution (uε, ũε) ∈ H are ensured by the Lax–Milgram lemma. At least formally, it isalso a solution of the transmission system (which is a system of total order 16):{−∂αT αβ(uε) = f β−bαβ T αβ(uε) + ε2∂4αβλμuε3 = f 3 in ω and{−∂αT αβ(ũε) = ε f̃ β−bαβ T αβ(ũε) + ε2∂4αβλμũε3 = ε f̃ 3 in ω̃ (5)with the boundary conditionsuεα = uε3 =∂uε3∂n= ũεα = ũε3 =∂ ũε3∂ñ= 0 on ∂Ω (6)the elastic junction conditions and the action–reaction principle giving (eight) transmission conditions:⎧⎨⎩ũε2 = −uε3ũε3 = uε2uε1 = ũε1on Σ and⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ε[∂uε3∂x2− ε−1 ∂ ũε3∂ x̃2]= MnεT 12 = T̃ 12T 22 = 0 and T̃ 22 = 0εM22 = M̃22on Σ (7)The formal limit problem by taking ε = 0 is the “membrane problem” (which is a system of total order 8) given by⎧⎪⎨⎪⎩−∂αT αβ(u) = f β−bαβ T αβ(u) = f 3uα|Γ0 = 0in ω and⎧⎪⎨⎪⎩−∂αT αβ(ũ) = 0−bαβ T αβ(ũ) = 0ũα|Γ̃0 = 0in ω̃ (8)with four transmission conditions on Σ :u1 = ũ1, T̃ 12 = 0, T 22 = T̃ 22 = 0 on Σ (9)Proposition 2.1. The formal limit problem is not elliptic in the Agmon, Douglis and Nirenberg sense, as the transmission conditions donot satisfy the Shapiro–Lopatinski condition.Proof. We can see that from the system (8) the T̃ αβ vanish but this does not mean that the ũi are nulls (see [4, p. 238])(we take the symmetry in order to obtain u1 = ũ1). Remark 1. The previous proposition asserts that even if f i,{c} are smooth, the formal limit problem is ill-posed. Any solution(u, ũ) does not satisfy the transmission conditions (7) at Σ and the boundary conditions on ∂Ω . Therefore, we may expectthat (uε, ũε) will develop boundary layers (transmission layers) along Σ and (standard) boundary layers along ∂Ω .Remark 2. If the loading f 3 /∈ L2(ω), the limit problem fails to be variational as the loading is out of the dual of the energyspace of the limit problem.3. The case of singular loadingsSuppose now that the loading f̃ on ω̃ is sufficiently smooth and that the loading on ω is of the form:f (x1, x2) =(0,0, F (x1)δ(x2))(10)where δ is the Dirac mass and F ∈ L2(0,π). Note that they are not in the dual of the energy space of the limit problem.Hence to study the limit procedure we perform a dilatation of the coordinate normal to the layer280 I. Merabet et al. / C. R. Mecanique 338 (2010) 277–282y1 = x1, y{c}2 =x{c}2η(ε)for(x{c}1 , x{c}2) ∈ ω{c} (11)The domain ω{c} then becomes B{c}η = ]0,π [y1 × ]0, 1η [y{c}2 .For the consistency of forthcoming developments we shall choose η = √ε as is usual in elliptic shell layers. We shallfurther make the change of unknowns{uεα(x) = uηα(y)uε3(x) = η−1uη3(y)and{ũεα(x̃) = η2ũηα( ỹ)ũε3(x̃) = ηũη3( ỹ)(12)Hence for the test functions the continuity conditions on Σ becomesv1(y1,0) = η2 ṽ1(y1,0), v2(y1,0) = −η ṽ3(y1,0), v3(y1,0) = η3 ṽ2(y1,0) (13)As in [3] we shall introduce the expressions{c}0(v{c}) = [10, 20, 30, 40]{c} =[−v3, 12∂2 v1, ∂2 v2 − v3, ∂22 v3]{c}{c}1 (u) =[11, 21, 31]{c} = [∂1u1, 12∂1u2, ∂1∂2u3]{c}With this choice of loadings (10) and the changes of unknowns and variables, the variational formulation (4) becomes thenew problem⎧⎪⎪⎪⎨⎪⎪⎪⎩Find(uη, ũη) ∈ Hη s.t. ∀(v, ṽ) ∈ Hηaη(uη, v) + ãη(ũη, ṽ) + c(uη; ũη, v; ṽ) =π∫0F (y1)v3(y1,0)dy1 + η2∫B̃0f̃3 ṽ3(y1, ỹ2)d ỹ(14)whereHη ={(v, ṽ) ∈ Vη × Ṽη fulfilling (13)}V {c}η ={v{c} ∈ H1(B{c}η )2 × H2(B{c}η ) ∣∣∣ v{c}|Γ η,c0 =(∂v{c}3∂n)|Γ η,c0= 0}Γη0 = ∂ Bη{y2 = 0} and Γ̃ η0 = ∂ B̃η{ ỹ2 = 0}aη(uη, v) = a00(uη, v) + η(a01(uη, v) + a10(uη, v)) + η2a11(uη, v) + η4a22(uη, v)a00(u, v) =∫B0[10(u)10(v) + 20(u)20(v) + 30(u)30(u) + 40(u)40(v)]dya01(u, v) =∫B0[10(u)11(v) + 220(u)21(v)]dy, a10(u, v) =∫B0[11(u)10(v) + 221(u)20(v)]dya11(u, v) =∫B0[11(u)11(v) + 221(u)21(v) + 31(u)31(v)]dy, a22(u, v) =∫B0∂21 u3∂21 v3 dyc((u; ũ), (v; ṽ)) = ∫Σ(∂u3∂ y2− ∂ ũ3∂ ỹ2)(∂v3∂ y2− ∂ ṽ3∂ ỹ2)dΣ (15)Analogous expressions hold for ã00, ã11, ã10, ã01, ã22 with integration over B̃0.Since we may extend the elements of V η,{c} by zero for y{c}2 1η , we have the inclusion Hη ⊂ Hη′ for η > η′ . Thisextension by zero is also used in the definition of a{c}00 , a{c}01 , a{c}10 , a{c}11 and a{c}22 .3.1. A priori estimatesThe equivalence between (4) and (14) allows us to deduce the next estimates for the solution (uη, ũη) ∈ Hη of (14).I. Merabet et al. / C. R. Mecanique 338 (2010) 277–282 281Lemma 3.1. For a fixed η, if F is in L2(0,π) and (uη, ũη) ∈ Hη is solution of (14) then∥∥uη3∥∥L2((0, π)y1 ;H2(0,∞)) +∥∥ũη3∥∥L2((0,π)y1 ;H2(0,∞)) + η2∥∥∂21 uη3∥∥L2(B0) + η2∥∥∂21 ũη3∥∥L2(B̃0)  C∥∥0(uη)∥∥(L2(B0))4 + ∥∥̃0(ũη)∥∥(L2(B̃0))4 + η∥∥1(uη)∥∥(L2(B0))3 + η∥∥̃1(ũη)∥∥(L2(B̃0))3  Cfor some C independent of η.3.2. Limit problem and convergence theoremWe are now in position to construct the energy space for the limit problem. LetE0 =(L2(B0))4, Ẽ0 =(L2(B̃0))4, E1 =(L2(B0))3and Ẽ1 =(L2(B̃0))3(16)We may consider the space⋃η↘0 Hη with the norm:∥∥(v, ṽ)∥∥V 0×Ṽ 0 = (∥∥0(v)∥∥2E0 + ∥∥1(v)∥∥2E1 + ∥∥0(ṽ)∥∥2Ẽ0 + ∥∥1(ṽ)∥∥2Ẽ1) 12 (17)H0 is the completion of⋃η↘0 Hη with this norm.Lemma 3.2. The quantity ‖(v, ṽ)‖H̄0 := (‖0(v)‖12E0+ ‖̃0(ṽ)‖12Ẽ0)2 is a norm on H0.Hence we can define the larger space H̄0 as the completion of H0 with respect to the norm ‖ · ‖H̄0 .Theorem 3.3. Let (uη, ũη) ∈ Hη be the solution of (14), then (uη, ũη) → (u, ũ) weakly in H̄0 , where (u, ũ) ∈ H̄0 is the solution of :a00(u, v) + ã00(ũ, ṽ) + c((u; ũ), (v; ṽ)) =π∫0F (y1)v3(y1,0)dy1 ∀(v, ṽ) ∈ H̄0 (18)The corresponding limit problem in the inner variables y takes the following form. Choosing appropriate test functionsin (18), we obtain the partial differential equations⎧⎪⎨⎪⎩−∂2T 120 (u) − ∂2T 220 (u) = 0−∂2T 120 (u) = 0−∂2u2 + 2u3 + ∂42 u3 = Fand⎧⎪⎨⎪⎩−∂̃2T 120 (ũ) − ∂̃2T 220 (ũ) = 0−∂̃2 T̃ 120 (ũ) = 0−∂̃2ũ2 + 2ũ3 + ∂̃42 ũ3 = 0(19)with the transmission conditions on Σ :⎧⎨⎩u3 = 0ũ3 = 0∂32 u3 = ∂̃32 ũ3 = 0on Σ⎧⎪⎨⎪⎩∂22 u3 = ∂̃22 ũ3 = ∂2u3 − ∂̃2ũ3∂2u1 = ∂̃2ũ1 = 0∂2u2 − u3 = ∂̃2ũ2 − ũ3 = 0on Σ (20)Remark 3. The conditions (20) satisfied by the solution of the limit problem are the formal limit of the conditions (7) (inthe inner variables y) as ε → 0.4. Concluding remarksWe have shown that for the case of transmission problems of two thin shells the formal limit problem is ill-posed as thetransmission conditions do not satisfy the Shapiro–Lopatinski condition. An appropriate dilation of the normal coordinateleads to an equivalent problem whose solution is a correct approximation of the original solution in a neighborhood ofthe transmission zone. The scaling (12) is consistent with the fact that the layer appears in the part where the elasticitycoefficients are small. The immediate extension apart from more general data (the rigidity coefficient of the hinge andthe elasticity coefficients) is to consider arbitrary kε and Ãαβλμ(ε) = O(εd), d ∈ R, then the behavior of the limit problemdepends on the limit of kε as ε → 0. Using a naive formal asymptotic expansion of the inner solution, we can show that ifkε = O(εl), l < 1, then the inner solution of the elastic case converges to the inner solution of the rigid one, i.e., the solutionsatisfies in addition to (20) the relation ∂2u3 = ∂2ũ3 on Σ.282 I. Merabet et al. / C. R. Mecanique 338 (2010) 277–282AcknowledgementThe authors are indebted to Professor E. Sanchez-Palencia for remarks and comments which allowed the improvementsof the present note.References[1] M. Bernadou, Méthodes d’éléments finis pour les problèmes de coques minces, Masson, Paris, 1994.[2] G. Geymonat, E. Sanchez-Palencia, On the rigidity of certain surfaces with folds and applications to shell theory, Arch. Ration. Mech. Anal. 129 (1995)11–45.[3] E. Sanchez-Palencia, On internal and boundary layers with unbounded energy in shell theory: Elliptic case, Asymptot. Anal. 36 (2003) 169–185.[4] J. Sanchez-Hubert, E. Sanchez-Palencia, Coques é lastiques minces, Masson, 1997.

Singular layers for transmission problems in thin shallow shell theory: Elastic junction case

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Singular layers for transmission problems in thin shallow shell theory: Elastic junction case

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