Evariste Sanchez-Palencia

Jesús Ildefonso Díaz

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 337 (2009) 1–7On a problem of slender, slightly hyperbolic, shells suggestedby Torroja’s structuresJesús Ildefonso Díaz a, Evariste Sanchez-Palencia b,∗a Universidad Complutense de Madrid, Departamento de Matemática Aplicada, plaza de las Ciencias 3, 28040 Madrid, Spainb UPMC Université Paris 06, UMR 7190, Institut Jean-Le-Rond-d’Alembert, tour 65-55, 4, place Jussieu, 75005 Paris, FranceReceived 20 November 2008; accepted after revision 4 February 2009Presented by Evariste Sanchez-PalenciaAbstractA rigidification phenomenon for certain thin slender parabolic shells, with curvature in the transversal direction to the mainlength was addressed in a previous work. The main results are generalized here to the case when the shape of the middle surface isno longer cylindrical, but incorporates a very small curvature in the longitudinal direction (it is “slightly hyperbolic”). The structureof the basic equations changes, but (due to a coupling with the flection rigidity) the main results hold true independently of thelocation of the characteristics. This problem constitutes a tentative modelling of certain remarkably rigid structures built by Torrojain the 1930s. To cite this article: J.I. Díaz, E. Sanchez-Palencia, C. R. Mecanique 337 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméSur un problème de coques élancées, légèrement hyperboliques, suggéré par les structures de Torroja. Un phénomènede rigidification pour un type de coques élancées cylindriques avec courbure dans la direction transversale à celle de la longueurprincipale a été considéré dans un travail précédent. Nous généralisons ici les principaux résultats au cas où la surface moyennen’est plus cylindrique, mais a une petite courbure dans la direction longitudinale (elle est « légèrement hyperbolique »). Le type deséquations change, mais (grâce à un couplage avec la rigidité de flexion) les principaux résultats restent valables indépendammentde la disposition des caractéristiques. Ce problème constitue une tentative de modélisation de certaines structures remarquablementrigides construites par Torroja dans les années 1930. Pour citer cet article : J.I. Díaz, E. Sanchez-Palencia, C. R. Mecanique 337(2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Computational solid mechanics; Slender thin shells; Asymptotic behavior; Hyperbolic shells; V-shaped structuresMots-clés : Mécanique des solides numérique ; Coques minces élancées ; Comportement asymptotique ; Coques hyperboliques ; Structures en V* Corresponding author.E-mail addresses: ildefonso.diaz@mat.ucm.es (J.I. Díaz), sanchez@lmm.jussieu.fr (E. Sanchez-Palencia).1631-0721/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2009.02.0022 J.I. Díaz, E. Sanchez-Palencia / C. R. Mecanique 337 (2009) 1–7Fig. 1. Geometry under consideration.1. IntroductionWe consider the same problem of [1], Sections 1 and 2 with, in addition, a small curvature in the longitudinaldirection. This we proceed to describe, but certain details will only be given in [1]. Let the small parameter ε denotethe thickness of the shell; the shape of the middle surface is nearly a slender rectangle, Ωε = (0, l) × (0, η) in the“plane of the parameters” x = (x1, x2) which is assimilated to the middle surface (see Fig. 1).We take η = ε1/4. The curvature in the x2 direction is equal to the constant b, whereas there is also a small curvatureof order η2 and sign opposite to that of b in the x1 direction. This small curvature constitutes the difference with [1];when it is not present, the shell is cylindrical. When it does, the shell is hyperbolic, with asymptotic curves forminga small angle of order O(η). The shell is fixed by the boundary x1 = 0. In the cylindrical case, the fixation ensuresthe geometric rigidity of the middle surface, which enters in the class of “inhibited shells” (see [2] or [3]). On theother hand, when the small longitudinal curvature is present, geometrical rigidity does no longer hold true as in thehyperbolic case it is only ensured in the “determinacy” triangle defined by the characteristics (i.e. the asymptoticcurves) issued from the fixed boundary; this is obviously a part of Ω , which may not arrive to x1 = l. The aim ofthis Note is to show that some coupling of the asymptotic (for small ε) expressions of the membrane and flectionrigidities allows one to prove that the asymptotic scheme and estimates hold true in the slightly hyperbolic case (atleast concerning magnitude orders).Denoting by ũ = (ũ1, ũ2, ũ3) the displacement (with components in the directions of x1, x2 and the normal to thesurface, the kinematic boundary conditions are0 = ũ1 = ũ2 = ũ3 = ∂̃1ũ3 on {0} × (0, η) (1)and the components of the strain tensor areγ̃11(ṽ) = ∂̃1ṽ1 − η2ṽ3γ̃22(ṽ) = ∂̃2ṽ2 + bṽ3γ̃12(ṽ) = γ̃21(ṽ) = 12(∂̃2ṽ1 + ∂̃1ṽ2) (2)Here we used the notation ∂̃α = ∂∂xα . As the shell is shallow, the tensor of variations of curvature has the componentsρ̃αβ(ṽ) = ∂̃αβ ṽ3 (3)The variational formulation of the shell problem can be writtenεa(uε,v) + ε3b(uε,v) = 〈fε,v〉 (4)The two bilinear forms on the space V = H 1(Ωε) × H 1(Ωε) × H 2(Ωε) are then defined by:a(ũ, ṽ) =∫ΩεAαβλμγ̃αβ(ũ)γ̃λμ(ṽ)dx (5)b(ũ, ṽ) =∫Bαβλμρ̃αβ(ũ)ρ̃αβ(ṽ)dx (6)ΩεJ.I. Díaz, E. Sanchez-Palencia / C. R. Mecanique 337 (2009) 1–7 3where A and B denote the membrane and flexion coefficients. Moreover〈f,v〉 = ε3∫ΩεF3(x1, x2/η)ṽ3(x1, x2)dx (7)Following [1] or [4], we perform the change of variables:{x = (x1, x2) ⇒ y = (y1, y2)y1 = x1, y2 = η−1x2 (8)so, the domain Ωε is transformed into Ω = (0, l) × (0,1) and∂1 = ∂̃1, ∂2 = η∂̃2; ∂α = ∂∂yα(9)The change of unknowns⎧⎨⎩ũ1(x) = η6u1(y)ũ2(x) = η5u2(y)ũ3(x) = η4b−1u3(y)(10)allows one to write the variational problem under the form:Problem Πε . Find uε ∈V satisfyingaε(uε,v) =∫ΩF3(y1, y2)v3(y1, y2)dy (11)∀v ∈ V, whereaε(uε,v) def=∫ΩAαβλμγ εαβ(uε)γ ελμ(v)dy +∫ΩBαβλμρεαβ(uε)ρελμ(v)dy (12)andγ ε11(v) = ∂1v1 − v3 (13)γ ε12(v) = γ ε21(v) = η−112(∂2v1 + ∂1v2) (14)γ ε22(v) = η−2(∂2v2 + v3) (15)ρε11(v) = η2∂21 v3 (16)ρε12(v) = ρ21(v) = η∂1∂2v3 (17)ρε22(ṽ) = ∂22 v3 (18)Defining asymptotic expansions in powers of η for uε , γ ε and the corresponding membrane stresses T ε (see [1]),we see that the terms tending to infinity should vanish. This lead to∂2u1 + ∂1u2 = 0 (19)for the limit (leading order term) and∂2u2 + u3 = 0 (20)for the two leading order terms. Then, proceeding as in [1] (end of Section 2), the limit problem isa0(u,v) def=∫Ω1C1111(∂1u1 − u3)(∂1v1 − v3)dy +∫ΩB2222∂22 u3∂22 v3 dy (21)in a space of functions including the constraints (19) and (20). Here, and in the sequel, irrelevant constants will betaken equal to 1.4 J.I. Díaz, E. Sanchez-Palencia / C. R. Mecanique 337 (2009) 1–72. Main resultBecause of the constraints (19) and (20), the limit may be described in terms of a potential ϕ:v1 = ∂1ψ, v2 = −∂2ψ, v3 = −∂22 ψ (22)withψ(0, y2) = ∂1ψ(0, y2) = 0 (23)We denote by G the completion of the set of the smooth functions defined on Ω satisfying (23) with the norm‖ψ‖2G =∫Ω|ψ |2 dy +∫Ω∣∣∂42 ψ∣∣2 dy (24)where  denotes the Dalembertian = ∂21 − ∂22 (25)Obviously, because of the completion process, the exact definition of the set is irrelevant; “smooth” may be understoodin the sense “of class C2”, for instance. On the other hand, the fact that ‖ψ‖G is a norm is not so evident; the proofwill be given later, at the same time as the proof of Theorem 1 below. The limit problem (21) then becomesProblem Π0. Find ϕ ∈ G such that, ∀ψ ∈ G∫Ωϕψ dy +∫Ω∂42ϕ∂42 ψ dy = −∫ΩF∂22 ψ dy. (26)Our main result is the following:Theorem 2.1. There exist c > 0 such that‖ψ‖G  c‖ψ‖L2(0,l;H 4(0,1)) for any ψ ∈ G (27)This ensures the existence and uniqueness of the solution with (for instance) F ∈ L2(Ω). Because of the presenceof ∂42ψ in the norm of G, obvious equivalence of norms in H4(0,1) show that it is sufficient to prove:Lemma 2.2. There exist c > 0 such that‖ψ‖G  c‖ψ‖L2(0,l;L2(0,1)) (28)for any ψ ∈ G.Remark 2.3. The above lemma shows a remarkable fact: if ψ = 0 in the rectangle Ω we can extend the uniquenessof solution beyond the usual cone of dependence once we know that in the “spatial variable” we have that ∂42 ψ = 0.We also point out that, as in [1], the solution ϕ ∈ G of problem Π0 satisfies a variational principle which is veryimplicit for the formulation Πε.3. First decomposition of L2(Ω)Let us denote by K the subspace of L2(Ω) formed by the functions of the form:ψ(y1, y2) = A(y1) + B(y1)y2 + C(y1)y222+ D(y1)y326(29)with A, B , C, D in L2(0, l), which is clearly closed. By identifying L2(Ω) with L2(0, l;L2(0,1)), the decompositionof L2(Ω) as K × K⊥ is associated with the decomposition of L2(0,1) into the subspace of polynomials of degree 3and its orthogonal.J.I. Díaz, E. Sanchez-Palencia / C. R. Mecanique 337 (2009) 1–7 5Classically, ‖∂42 ψ‖L2(0,1) is the norm of ψ in H 4(0,1)/R4, i.e. in H 4(0,1) up to polynomials of degree 4; fromthe expression of the norm in G we have (where c denotes various constants)‖ψ‖G  c‖ψ‖L2(0,l;H 4(0,1)/R4)  c‖ψ‖L2(0,l;L2(0,1)/R4) (30)But this last expression is precisely ‖ψK⊥‖L2(Ω), so that‖ψ‖G  c∥∥ψK⊥∥∥L2(0,l;L2(0,1)) (31)for any ψ ∈ G. This is the desired inequality “up to the component in K”.4. Second decomposition of L2(Ω)Let us now consider another decomposition of ψ ∈ G (or rather an smooth function of the set which the completionis G).We shall only develop the proof in the case when 1 < l < 2. This amounts to saying that in the forthcomingexplicit construction of the solution of the wave equation with given boundary values, there is only one reflection. Thecase l > 2 with several reflections is analogous but cumbersome, whereas the case l < 1 is simpler by the absence ofreflections. Letψ = ψ1 + ψ2 (32)where ψ1 and ψ2 are defined by:⎧⎪⎪⎨⎪⎪⎩ψ1 = 0 in Ω = (0, l) × (0,1)ψ1(0, y2) = ∂1ψ1(0, y2) = 0ψ1(y1,0) = ψ(y1,0)ψ1(y1,1) = ψ(y1,1)(33)and ⎧⎪⎪⎨⎪⎪⎩ψ2 = ψ in Ω = (0, l) × (0,1)ψ2(0, y2) = ∂1ψ2(0, y2) = 0ψ2(y1,0) = 0ψ2(y1,1) = 0(34)We note that, since ψ ∈ G, functions ψ1 and ψ2 are both well defined as solutions of mixed problems for the waveequation (with initial conditions in y1 = 0 and boundary conditions in y2 = 0 and y2 = 1). From classical estimatesfor mixed problems for the wave equation it follows that‖ψ‖G  c‖ψ‖L2(Ω)  c‖ψ2‖L2(Ω) (35)We are now giving an explicit expression for ψ1. Let us extend both ψ and ψ1 with value 0 for y1 < 0; they are thendefined on (−∞, l) × (0,1); see Fig. 2.Fig. 2. Extension of ψ and ψ1.It is then easily seen that ψ1 can be written:ψ1(y1, y2) = ψ(y1 − y2,0) + ψ(y1 + y2 − 1,1) − ψ(y1 + y2 − 2,0) − ψ(y1 − y2 − 1,1) (36)6 J.I. Díaz, E. Sanchez-Palencia / C. R. Mecanique 337 (2009) 1–7In order to check this expression, we recall that it was supposed that 1 < l < 2, so that the first line amounts tothe “propagation of the boundary data”, and the second one to their reflections (there are not successive reflections asl < 2).Let us now consider the set of functions U(y1, y2) defined on Ω of the form:U(y1, y2) = f (y1 − y2) + g(y1 + y2 − 1) − f (y1 + y2 − 2) − g(y1 − y2 − 1) (37)with f and g smooth functions defined on (−∞, l) and vanishing for y1 < 0. Let us denote by K ′ its completion withthe norm of L2(Ω). It is easily checked that locally, convergence in L2(Ω) of elements of the form (37) implies L2convergence of both f and g in L2, but there is some trouble concerning the points (l,0) and (l,1). This is concernedwith square-integrability in the vicinity of an extremity of a segment versus in a corner of the plane. Anyway, the nextlemma is easily checked, and sufficient for our purpose:Lemma 4.1. The elements of K ′ are of the form (33) with f and g of class L2(−∞, α) (vanishing for negative y1),for any α < l.As ψ1 ∈ K ′, the previous estimate (35) is the desired inequality “up to the component in K ′”.5. Proof of Theorem 2.1According to the two previous sections, the desired estimate (28) was proved “either up to the component in Kor K ′”. Thus it suffices to prove that the intersection of K and K ′ reduces to the zero element of L2(Ω). This is easilydone. Let us consider ψ as an element of K ′; for fixed 0 < y1 < 1/2, as the corresponding f and g vanish for negativevalues of the argument, ψ vanishes in the vicinity of y2 = 1/2 and is a polynomial for 0 < y2 < 1. So, ψ vanishesfor 0 < y1 < 1/2, and f and g vanish for their arguments less than 1/2. Repeating the reasoning, they also vanish for0 < y1 < 1, and so on. Theorem 1 is proved.6. Case when the shell has an edge with slight foldingThere is an interesting variant of the previous problem, when the shell has an slight folding along a curve x2 = c.This is an attempt to modelling elements of shell which, when pasted one after other in the x2 direction, constitutethe very elemental piece of the “Torroja’s structure” [5,6]. It should be noted that the first essays by Torroja before ofconstructing his roofs (reported in [5]) were precisely concerned with a unique element of this kind.Once more, the case when the elements of shell are cylindrical was addressed in [1], Section 3. The modificationswith respect to the case without folding apply exactly to the present case of small curvature in the longitudinal direc-tion. The angle of the folding is small of order η. When passing to the variables and unknowns y and u, the kinematicconditions along the folding write:u+1 = u−1 , u+3 = u−3 , u+2 − u−2 = u3 (38)where the symbols + and − denote the traces on both sides of the folding. In terms of the potential ψ , they become⎧⎪⎨⎪⎩ψ+ = ψ−∂22 ψ+ = ∂22 ψ−2∂22 ψ = ∂2ψ+ − ∂2ψ−(39)It then appears that the space for ψ in the case of the folding is a subspace of the product of the spaces for the twoparts of the shell. As the inequality of Theorem 2.1 holds true for each part, it clearly holds for the product and thenfor a part of it.AcknowledgementResearch partially supported by the project MTM2008-06208 of the Ministerio de Ciencia e Innovacion, Spain.J.I. Díaz, E. Sanchez-Palencia / C. R. Mecanique 337 (2009) 1–7 7References[1] J.I. Díaz, E. Sanchez-Palencia, On slender shells and related problems suggested by Torroja’s structures, Asymptotic Anal. 52 (2007) 259–297.[2] J. Sanchez-Hubert, E. Sanchez-Palencia, Coques élastiques minces. Propriétés asymptotiques, Masson, Paris, 1997.[3] P.G. Ciarlet, Mathematical Elasticity, vol. III. Theory of Shells, Elsevier, Amsterdam, 2000.[4] D. Caillerie, A. Raoult, E. Sanchez-Palencia, On internal and boundary layers with unbounded energy in thin shell theory. Parabolic character-istic and non-characteristic cases, Asymptotic Anal. 46 (2006) 221–249.[5] E. Torroja, New developments in shell structures, in: II Symposium on Concrete Shell Roof Construction, Oslo, 1957.[6] E. Torroja, The structures of Eduardo Torroja, F.W. Dodge Corporation, New York, 1957.

On a problem of slender, slightly hyperbolic, shells suggested by Torroja's structures

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On a problem of slender, slightly hyperbolic, shells suggested by Torroja's structures

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