Jean-Paul Zolésio

John Cagnol

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166Problèmes mathématiques de la mécanique/Mathematical Problems in MechanicsVibration of a pre-constrained elastic thin shellI: Modeling and regularity of the solutionsJohn Cagnola, Jean-Paul Zolésioba Pôle Universitaire Léonard de Vinci, ESILV, DER-CS, 92916 Paris La Défense cedex, Franceb CNRS research director at INRIA, OPALE project, 2004 route des Lucioles, BP 93,06902 Sophia Antipolis cedex, FranceReceived 8 September 2000; accepted after revision 22 October 2001Note presented by Philippe G. Ciarlet.Abstract We study the vibration of an elastic thin shell which is pre-constrained by a largedisplacement with a small deformation. In this first Note we prove the solutions exist andwe investigate both the interior regularity and the boundary regularity which is known to beimportant in the shape differentiation of hyperbolic equations. To cite this article: J. Cagnol,J.-P. Zolésio, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166.  2002 Académie dessciences/Éditions scientifiques et médicales Elsevier SASVibration d’une coque élastique mince pré-contrainte IRésumé On étudie la vibration d’une coque élastique pré-contrainte par grand déplacement enpetites déformations. Dans cette première Note on modélise la vibration et on démontrel’existence des solutions et la régularité intérieure. On termine par une étude de la régularitésur le bord, laquelle est connue pour intervenir dans la dérivée par rapport au domaine dansles équations hyperboliques. Pour citer cet article : J. Cagnol, J.-P. Zolésio, C. R. Acad. Sci.Paris, Ser. I 334 (2002) 161–166.  2002 Académie des sciences/Éditions scientifiques etmédicales Elsevier SASVersion française abrégéeSoit 0 ⊂ R3 une coque mince dans sa position au repos et  ⊂ R3 cette coque dans son état contraintqui est un équilibre statique. Soit T0 une application de R3 dans R3 telle que T0(0)=. On suppose queT0 vérifie que la matrice jacobienne E définie par DT0 ◦ T0−1 a ses coefficients dans W1,∞(R3).La coque est soumise a une vibration. Soit τ le temps final et t < τ , on note (t) la coque au temps t etT (t) l’application telle que T (t)()=(t). On suppose que 0 et (t) sont homogènes et isotropes. Onsuppose également que T appartient à L2([0, τ ],H1(,R3)) ∩ H1([0, τ ],L2(,R3)) et satisfait T|D = Ioù D est une partie du bord. Cette application est une perturbation de l’identité, on note T = I + u avecu petit dans H1. Notons H1D(;R3) l’ensemble des fonctions de H1(,R3) qui s’annulent sur D . Leparamètre u appartient à l’ensemble H qui est défini par (1).On note T = T ◦ T0. Soit ũ= u ◦ T0, on a T(t)= T0 + ũ(t). On note C le tenseur élastique d’ordre 4 eton définit ε(u) et (u) par (2). Soit φ and ψ les valeurs de u et ∂tu à t = 0. On suppose φ ∈ H1(,R3) etψ ∈ L2(,R3). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-073X(02)02182-9/FLA 161J. Cagnol, J.-P. Zolésio / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166En utilisant le principe d’extrémisation de l’action on montre que l’équation satisfaite par E est (6).On note τ < +∞ le temps final et σD = [0; τ ] × D . L’équation de la vibration autour de la forme pré-contrainte  est hyperbolique d’ordre de 2, nous établissons qu’elle est donnée par (3).THÉORÈME 1. –(i) L’équation(3) a une solution uniqueu dansC([0; τ ]; H1D())∩ C1([0; τ ]; L2()) ;(ii) Pour toutν ∈ C1(D) avecν  0 surD et suppν ∈ D , on aν|ε(u)| ∈ L2(σD).L’existence des solutions et la régularité intérieure se démontrent en utilisant les propriétés de l’opérateurspatial et les résultats de semi-groupes de [1], la coercivité nécessaire s’obtenant par l’inégalité de Korn etl’estimation (7).Pour la régularité sur le bord on considère la dérivée, par rapport à s, de :12∫Qs1| det(E)|( ∗Eε(ϕ ◦ T −1s )E)..C..( ∗Eε(ϕ ◦ T −1s )E),où A..B = AijBij représente la double contraction des tenseurs A et B . On note E cette dérivée en 0.On effectue la dérivation sur le domaine mobile en utilisant [4, lemma 12], il vient (8). On dérive ensuiteen se ramenant au domaine fixe grâce à un changement de variables ce qui donne (9). On montre queles expressions obtenues donnent une égalité entre intégrale de volume et intégrale de bord ce qui permetd’établir la régularité du bord, au moyen d’un argument de densité.L’intérêt de cette régularité sur le bord réside dans sa supériorité par rapport à celle que l’on peut obtenirpar les théorèmes de trace. Cette différence de régularité joue un rôle important dans la dérivabilité parrapport au domaine des équations hyperboliques (cf. [3,2,4]).Cette note sera suivie d’une seconde portant sur le modèle p(d.∞) en géométrie intrinsique.1. IntroductionThe vibration of a shell constrained to be in a specific shape and the vibration of a shell with that shapeat its natural reference position are known to be different by physicists.Let 0 ⊂ R3 be a shell in its unconstrained state and  ⊂ R3 be that shell in a constrained state whichis a static equilibrium. Let T0 be the static displacement, that is the mapping from R3 onto R3 suchthat T0(0) = . We assume T0 is such that the coefficients of the matrix E = DT0 ◦ T0−1 belong toW1,∞(R3), which is the case, for instance, if T0 ∈ W2,∞(0,R3).We suppose  is under a vibration. Let τ be the final time and t < τ , we note (t) the shell at thetime t and T (t) the mapping such that T (t)() = (t). We suppose 0 and (t) are homogeneous.Moreover, for the sake of simplicity, we will suppose they are isotropic. We assume T belongs toL2([0, τ ],H1(,R3)) ∩ H1([0, τ ],L2(,R3)) and satisfies the embedding condition of the shell to D ,that is T|D = I . The mapping T is a perturbation of I , we have T = I + u where u is small in H1. Fromnow, u will be the parameter to be considered. We note H1D(;R3) the set of functions of H1(,R3) thatvanish on D . The parameter u belongs to:H = L2([0, τ ],H1D(,R3)) ∩ H1([0, τ ],L2(,R3)). (1)We note T = T ◦ T0. Let ũ= u ◦ T0, we have T(t)= T0 + ũ(t). We note C the 4-order elastic tensor:ε(u)= 12( ∗Du+Du) and (u)= 2| det(E)|E(C..( ∗Eε(u)E)) ∗E, (2)162To cite this article: J. Cagnol, J.-P. Zolésio, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166where A..B =AijBij and E is the matrix (DT0) ◦ T −10 .Let φ and ψ be the value of u and ∂tu at t = 0. We suppose φ ∈ H1(,R3) and ψ ∈ L2(,R3). Theequation of the vibration around the natural shape of the joint  is given by the hyperbolic equation:ρ∂2t u− div((u))= 0 on [0; τ [×,(u) · n= 0 on σ = [0; τ [×,u(0)= φ, ∂tu(0)=ψ on .(3)We will note P(u)= ρ∂2t u− div((u)). We first prove the system above is the equation of the vibration inSection 2. Then, in Section 3, we prove:THEOREM 1. – LetσD = [0; τ [×D.(i) System(3) has a unique solutionu in C([0; τ ]; H1D())∩ C1([0; τ ]; L2());(ii) ∀ν ∈ C1(D) with ν  0 onD andsuppν ∈ D we haveν|ε(u)| ∈ L2(σD).This note will be followed by another one on the p(d.∞) model with intrinsic geometry.2. The modeling2.1. Equation of the vibrationLet C be the 4-order elastic tensor. It satisfies ∀(i, j, k, l) ∈ {1,2,3}4, Cijkl = Cjikl = Cklij and it isconstant since the shell is homogeneous. Moreover we assume ∃α > 0, ∀ξ, Cijklξij ξkl > αξ2i,j this allowsthe definition of norm | · | such that |(|2 =(..C..(. We consider:ε̄(u)= 12( ∗DTDT − I);the elements of this matrix belong to H1([0, τ ],L2(,R3))∩ L2([0, τ ],H−1(,R3)). When no confusionis possible we shall note ε̄ instead of ε̄(u). A similar notation will be used for all subsequent functionsdepending on u and will not be pointed out again. The elastic energy is given by Ep =∫0 ε̄..C..ε̄. At thefirst order 2ε̄ = ∗DT0((I + 2ε) ◦ T0)DT0 − I . The elastic energy of  is given byEp = 14∫( ∗E(I + 2ε(u))E − I)..C..( ∗E(I + 2ε(u))E − I) 1| det(E)| .Let ρ be the density and v be the speed vector field of the vibrating body. It is defined on (t) byv(t, x)= ∂tT ◦ T−1. The kinetic energy of (t) is given byEk(t)= 12ρ∫(∂tu)2.Let τ <+∞ be the final time and the action A be defined by A= ∫ τ0 (Ep −Ek(t))dt . We have:A(u)= 14∫ τ0∫( ∗E(I + 2ε(u))E − I)..C..( ∗E(I + 2ε(u))E − I) 1| det(E)| − 2ρ(∂tu)2.Let θ be a real and w ∈ R3; we note A′(u;w)= ∂∂θA(u+θw)|θ=0. The physical evolutions of the structuresare characterized by the functions u, defined on [0; τ [×, such that A′(u;w)= 0 for all test w such thatw(0)= 0 and w(τ)= 0. Using the linearity of ε we get:A′(u;w)=∫ τ0∫(u)..ε(w)+ ( ∗EE − I)..C..( ∗Eε(w)E) 1| det(E)| − ρ∂tu∂tw.163J. Cagnol, J.-P. Zolésio / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166When u= 0 we have T = T0 which gives a minimum for the energy, thus we get A′(0;w)= 0. Hence∫ τ0∫( ∗EE − I)..C..( ∗Eε(w)E) 1| det(E)| = 0. (4)Using this identity in the general expression of A′(u;w)= 0, we get∫ τ0∫(u)..ε(w)− ρ∂tu∂tw = 0 (5)from Green’s formula∫ τ0∫ − div((u))w + ρ∂2t uw +∫ τ0∫ (u)nw −∫ ∂tu(0)w(0)+ ∂tu(τ )w(τ) = 0that equality holds for all tests w therefore the vibration is governed by (3).Remark1. – The operator  depends on E.2.2. Equation satisfied byEThe matrix E is DT0 ◦ T0−1 where T0 is a local minimum for the elastic energy. We have obtained (4)which strong formulation leads to:div(1detEE(( ∗EE − I)..C) ∗E) = 0, (6)that is div(( 1det(DT0)DT0(ε̄(0)..C)∗DT0) ◦ T0−1)= 0; moreover T0 is assumed to be a small deformationhence ‖ε̄(0)‖L∞ is small as compared to ‖I‖L∞ . On the other hand from that last assertion comes theexistence of a real c > −1 such that∀v ∈ R3, ∥∥ ∗EEv∥∥L2  (1 + c)‖v‖L2 ; (7)the same inequality holds for E ∗E.3. Existence and regularity of the solutions3.1. Interior regularityThe existence and regularity of the solutions will be derived from the theory of semi-groups. Let / bethe linear operator in L2() defined by D(/)= H2(;R3)∩ H1D(;R3) and /= − 1ρ div().LEMMA 2. – The operator/ is self-adjoint.LEMMA 3. – The operator/ is coercive inH1D().Proof. –We use the Hooke’s law, [6] and λ 0 to derive:∫/(u)u 2µρ∫1| det(E)|( ∗Eε(u)E)..( ∗Eε(u)E);then using (7), the Korn’s inequality and u|D = 0 we obtain the the existence of a nonnegative real c suchthat ∫/(u)u(2kcµρ1max | det(E)|)‖u‖H1(). ✷164Pour citer cet article : J. Cagnol, J.-P. Zolésio, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166We noteU =(u∂tu), U0 =(φψ)and A=(0 1−/ 0)then (3) is equivalent to:∂tU =AU, U(0)=U0.Following [1], Proposition 2.12, we havePROPOSITION 4. – The operatorA is the infinitesimal generator of a strongly continuous semi-groupof contractionS on H. Moreover∗A= −A.The first part of Theorem 1 derives from [1], Proposition 3.3 and Proposition 4.3.2. Boundary regularityLet ϕ be a solution to (3). We consider a flow mapping Ts and the associated vector field V (cf. [7] or [4]).Let us compute the derivative with respect to s of12∫Qs1| det(E)|( ∗Eε(ϕ ◦ T −1s )E)..C..( ∗Eε(ϕ ◦ T −1s )E)at s = 0, via two different ways. That derivative will be denoted E and is call the extractor. We want thedistributed integral to be defined for ϕ ∈ C([0; τ ]; H1D())∩ C1([0; τ ]; L2()) such that P(ϕ) ∈ L2().Using [4] we have:LEMMA 5. – One has:E =∫Q1| det(E)|(div(V (0))|∂tϕ|2 + 〈Dϕ.∂tV (0), ∂tϕ〉)dx dt−∫σ1| det(E)|(〈Dϕ.V (0),(ϕ)n〉 + 12( ∗Eε(ϕ)E)..C..( ∗Eε(ϕ)E − |∂tϕ|2)〈V (0), n〉). (8)From the change of variable Ts in the left-hand side and the computation of the derivatives with respectto s we derive the:LEMMA 6. – One hasE =∫Q1| det(E)|(12tr(E−1.DE.V (0))( ∗Eε(ϕ)E)..C..( ∗Eε(ϕ)E)+ ( ∗DE.V (0)ε(ϕ))..C..( ∗Eε(ϕ)E) + 12∗Eε(Dϕ.DV (0))E)..C..( ∗Eε(ϕ)E)+ ( ∗Eε(ϕ)DE.V (0))..C..( ∗Eε(ϕ)E) + 12div(V (0))( ∗Eε(ϕ)E)..C..( ∗Eε(ϕ)E))dx dt . (9)LEMMA 7. – OnD we have:〈(ϕ).n,Dϕ.V (0)〉 = ( ∗Eε(ϕ)E)..C..( ∗Eε(ϕ)E)〈V (0), n〉 = ∣∣ ∗E(Dϕ.n) ∗nE∣∣〈V (0), n〉.PROPOSITION 8. – One has:∫σD1| det(E)|∣∣ ∗E(Dϕ.n) ∗nE∣∣2〈V (0), n〉 = B(ϕ),165J. Cagnol, J.-P. Zolésio / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 161–166whereB is defined byB(ϕ)=∫Q2| det(E)|(div(V (0))|∂tϕ|2 + 〈Dϕ.∂tV (0), ∂tϕ〉 + 〈Dϕ.V (0),P (ϕ)〉− 12( ∗Eε(Dϕ.DV (0))E)..C..( ∗Eε(ϕ)E) − 12div(V (0))( ∗Eε(ϕ)E)..C..( ∗Eε(ϕ)E)− 12tr(E−1.DE.V (0))( ∗Eε(ϕ)E)..C..( ∗Eε(ϕ)E)− ( ∗DE.V (0)ε(ϕ)E + ∗Eε(ϕ)DE.V (0))..C..( ∗Eε(ϕ)E)).Proof. –We use Lemmas 5 and 6 to compute E via two different ways. We have 〈V (0), n〉 = 0 on Nmoreover ∂tϕ = 0 on D . We use Lemma 7 to prove the proposition. ✷Remark9. – The real B(ϕ) is defined when ϕ ∈ C([0; τ ]; H1D()) ∩ C1([0; τ ]; L2()) and P(ϕ) ∈L1([0, τ [,L2()).Let (ϕm)m be a sequence of functions of C∞(Q) that vanish on D and such that ϕm → u in H1(Q)and Pϕm → Pu in L2(Q). The proof of the density is similar to the one given in [5]. Since B(u) exists,‖|ε(ϕm)|√〈V (0), n〉‖L2(σD) is bounded, hence there exists a function ξ in L2(D) and a subsequence suchthatε(ϕm)√〈V (0), n〉⇀ξ weakly in L2(D) as mk → +∞.On σD Green’s theorem proves ξ =(u)..ε(u)√〈V (0), n〉.Therefore for all ν ∈ C1(D) with ν  0 on D and suppν ∈ D we have ν|ε(u)| ∈ L2(σD). The secondpart of theorem 1 derives.References[1] Bensoussan A., Da Prato G., Delfour M.C., Mitter S.K., Representation and Control of Infinite DimensionalSystems, Vol. 1, Birkhäuser, 1993.[2] Cagnol J., Zolésio J.-P., Hidden shape derivative in the wave equation, in: Khan P., Lasiecka I., Polis M. (Eds.),System Modelling and Optimization, Addison-Wesley–Longman, 1998, pp. 42–52.[3] Cagnol J., Zolésio J.-P., Hidden shape derivative in the wave equation with Dirichlet boundary condition, C. R.Acad. Sci. Paris, Série I 326 (9) (1998) 1079–1084.[4] Cagnol J., Zolésio J.-P., Shape derivative in the wave equation with Dirichlet boundary conditions, J. DifferentialEquations 158 (2) (1999) 175–210.[5] Delfour M., Zolésio J.-P., Hidden boundary smoothness in hyperbolic tangential problems of nonsmooth domains,in: Khan P., Lasiecka I., Polis M. (Eds.), System Modelling and Optimization, Addison-Wesley–Longman, 1998.[6] Germain P., Mecanique, Vol. I. Ellipses, École Polytechnique, 1986.[7] Sokolowski J., Zolésio J.-P., Introduction to Shape Optimization, SCM, Vol. 16, Springer-Verlag, 1991.166

Vibration of a pre-constrained elastic thin shell I: Modeling and regularity of the solutions

Cagnol, John

Zolésio, Jean-Paul

Numérisation de Documents Anciens Mathématiques

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Vibration of a pre-constrained elastic thin shell I: Modeling and regularity of the solutions

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