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A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

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Page 1: A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

C. R. Acad. Sci. Paris, t. 329, Série II b, p. 131–135, 2001Acoustique, ondes, vibrations/Acoustics, waves, vibrations(Mécanique des solides et des structures/Mechanics of solids ans structures)

A model problem for vibration of thin elastic shells.Propagation and reflection of singularitiesAlain CAMPBELL

Laboratoire de mécanique, modélisation mathématique et numérique, BP 5186, Université de Caen,14032 Caen cedex, FranceE-mail: [email protected]

Abstract. We consider an elementary model problem to represent some properties of vibration ofthin elastic shells. As classical properties of compactness are not satisfied, there existsan essential spectrum Σess . We study the propagation of singularities when the spectralparameter λ is an interior point of the segment formed by Σess, exhibiting a deteriorationof the regularity of the solutions with respect to the case λ out of Σess (phenomena ofresonance). We also give the reflection law of the singularities at the boundary of thedomain. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

essential spectrum / singularities / propagation / reflection

Étude d’un problème modèle pour représenter la propagation et laréflexion des singularités dans les vibrations de coques élastiques minces

Résumé. On considère un problème modèle élémentaire pour représenter les vibrations de coquesminces. Les propriétés classiques de compacité n’étant pas satisfaites, il existe un spectreessentielΣess. On étudie alors la propagation des singularités lorsque le paramètre spectralλ est dans le segment qui constitueΣess et on montre une dégradation de la régularitédes solutions par rapport au cas oùλ est extérieur àΣess. On étudie ensuite la loi deréflexion des singularités sur le bord du domaine. 2001 Académie des sciences/Éditionsscientifiques et médicales Elsevier SAS

spectre essentiel / singularités / propagation / réflexion

Version française abrégée

Dans les problèmes de vibrations des coques minces, les phénomènes concernant le système membra-naire sont d’autant plus représentatifs que l’épaisseur ε est petite. Le spectre de l’opérateur auto-adjointassocié contient des valeurs propres de multiplicités finies constituant une suite qui tend vers l’infini. Cesvaleurs propres dépendent du domaine et des conditions aux limites.

Comme le problème spectral contient des dérivées d’ordres différents pour les déplacements tangents àla surface ou normal à celle-ci, les propriétés classiques de compacité ne sont pas satisfaites et le spectrecontient aussi un spectre essentiel. Celui-ci est une réunion de segments, chacun d’entre eux étant associéà un point donné de la coque.

À un élément donné de ce spectre essentiel, on peut associer des directions de propagation dessingularités.

Les équations de vibrations des coques minces étant de forme assez compliquée, nous nous intéressonsici au problème modèle (1), à conditions de Dirichlet, u1 = 0 sur le bord du domaine Ω. Le problème étantà coefficients constants, le spectre essentiel se réduit au segment [c− b2, c].

Pour un λ donné dans ]c− b2, c[, il existe deux directions de propagation (cosθ, sin θ) vérifiant (4).

S1620-7742(01)01301-0/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 131

Page 2: A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

A. Campbell

On considère le problème de vibrations forcées (1) avec le second membre :

f1 = 0, f2 = δ(y1)δ(y2)

Si λ est à l’extérieur du spectre essentiel, le problème est elliptique et la solution u1 si elle existe, est doncdans Hs+2(Ω), s <−2. Il n’y a pas de propagation possible.

Si λ est donné dans ]c− b2, c[, le problème (1) s’écrit sous la forme (6) et admet des solutions singulières.Celles-ci peuvent d’écrire sous le forme du développement asymptotique (7) dont les coefficients satisfont à(8) et (9) et donc à (10). Dans ce cas, on peut donc trouver une solution u1 de (1) qui appartient à Hs+3/2(Ω)(s < −2), et donc u2 appartenant à Hs+1/2(Ω), c’est à dire des solutions de singularités beaucoup plusimportantes que dans le cas où λ est en dehors du spectre essentiel.

Dans le cas où λ appartient à ]c− b2, c[, la fonction (11) est définie à une constante d’intégration près.Cependant, la droite caractéristique coupe le bord du domaine en deux points et il y a deux équations pourla détermination d’une seule constante. Il y a donc au moins une réflexion de la singularité sur une autredroite caractéristique. En écrivant que la superposition des termes prépondérants des ondes incidente u1 etréfléchie u1 doit s’annuler sur le bord, on obtient (12).

On remarque que si l’onde incidente est presque tangentielle au contour (cos(φ − θ) petit), alorsl’intensité de l’onde réfléchie est très importante. Ceci montre que dans certains cas, la propagation etla réflexion d’une singularité peuvent en augmenter fortement l’intensité.

On étudie l’exemple d’un contour circulaire dans le cas de caractéristiques formant un rectangle(θ = π/4). Les constantes d’intégration vérifient alors le système (14) dont le déterminant est nul et lesecond membre compactible. Ceci indique que la valeur correspondante de λ peut-être une valeur propre,ce qui se vérifie pour certains choix des paramètres.

On espère que la plupart de ces résultats obtenus sur le problème modèle peuvent aussi s’appliquer auxcoques hyperboliques pour lesquelles, en chaque point, le spectre essentiel est un segment [0,Λ]. On saitque dans le cas de la statique (λ = 0), il y a propagation des singularités mais non réflexion sur les bordsdu domaine.

1. Introduction

In vibration problems of thin shells, the membrane approximation causes phenomena all the morerepresentative as the thickness ε is small. The spectrum of the associated selfadjoint operator containseigenvalues with finite multiplicity constituting a sequence which tends to infinity. These eigenvaluesdepend on the domain Ω and on the boundary conditions.

On the other side, the spectral problem is an elliptic system with mixed order [1]. The classical propertiesof compactness are not satisfied and the spectrum also contains an essential spectrum [2]. This spectrum isa set of segments, each segment corresponding to one point of the shell.

In certain cases, the eigenvalues which could be included in the essential spectrum, may accumulate andeven form a dense subset.

If the spectral parameter is given in this essential spectrum, we can associate some directions of‘resonance’ which are weakness directions of the behaviour. The orthogonal directions are the directions ofpropagation of singularities.

The equations of vibrating elastic thin shells in membrane approximation are rather complicated. So weshall present a model problem, analogous to another one previously considered in [3] for studying otherspectral properties. This simplified problem will give a clear idea about the properties of propagation andreflection of singularities in the problems of vibrations, forced or not.

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Page 3: A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

A model problem for vibration of thin elastic shells

2. The model problem and its essential spectrum

We consider the following spectral problem:−∆u1 + bu2,2 − λu1 = f1

−bu1,2 + cu2 − λu2 = f2(1)

where the unknown (u1, u2) are functions of the two variables y = (y1, y2) in a domain Ω of the plane.In this study, we suppose that b and c are some constant numbers (but it is possible to extend to regularfunctions of y) and f = (f1, f2) is the right-hand side which will be defined later.

The boundary conditions are u1 = 0 on the edge ∂Ω (Dirichlet).We define the spaces H = L2(Ω) × L2(Ω) and V = H1

0 (Ω) × L2(Ω) and we can write the problem inthe form:

(A− λ)u = f (2)

where A is a selfadjoint operator in H .We obtain the essential spectrum by writing that this problem is not elliptic in the sense of Douglis and

Niremberg; so there exists a couple of nonzero real numbers (ξ1, ξ2) which are solutions of:

det

(ξ21 + ξ2

2 ibξ2−ibξ2 c− λ

)= 0 (3)

Therefore, the essential spectrum is [c− b2, c].For λ given into ]c − b2, c[, there exist two different directions (ξ1, ξ2) and then two directions of

propagation (cosθ, sin θ) with:

tan2 θ =λ− (c− b2)

c− λ(4)

By substituting u2,2 into the first equation, we obtain the wave equation:

−u1,11 +λ− (c− b2)

c− λu1,22 − λu1 = f1 −

b

c− λf2,2 (5)

whose characteristic directions correspond to the directions of propagation which have been definedby θ [4].

The set of eigenvalues of this selfadjoint problem is denumerable. Then the measure of its subset alsoincluded in the essential spectrum is zero. For example, if Ω is a square [−a, a]2, it is possible to determinethe eigenvalues and to show that this subset is dense in the essentiel spectrum.

If λ is not an eigenvalue of A, the range of A − λ is dense in V ′ = H−1(Ω) × L2(Ω) and A − λis injective. For some right-hand sides f , there exist solutions which are unique but the resolvent is notcontinuous in these spaces.

At the extremities of the essential spectrum (λ = c− b2, resp. λ = c), the two directions of propagationbecome coincident (θ = 0, resp. θ = π/2).

3. Propagation of the singularities

Let us consider problem (1) with the right-hand side f1 = 0; f2 = δ(y1)δ(y2). This right-hand side doesnot belong to V ′ but we can consider this problem as the research of a fundamental solution [5]. As λ isnot an eigenvalue, if there exists a solution, it is unique.

When λ is exterior to the essential spectrum, equation (5) is elliptic and its right-hand side,−bc−λδ(y

1)δ′(y2), belongs to Hs(Ω) with s <−2 (cf. [6]). If the solution u1 exists, it belongs to Hs+2(Ω).In this case there is no propagation.

133

Page 4: A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

A. Campbell

When λ is given into ]c− b2, c[ and is not an eigenvalue, problem (1) can be written in the form:−∆u1 + bu2,2 − λu1 = 0−bu1,2 + cu2 − λu2 = δ(y1)δ(y2 − y1 tanθ)

(6)

where tanθ is the slope of one of the characteristics (Dθ). Then we seek for solutions in the form of anasymptotic expansion of the singularities:

u1 = U11

(y1)δ(y2 − y1 tan θ

)+U2

1

(y1)Y(y2 − y1 tanθ

)+ · · ·

u2 = U12

(y1)δ′(y2 − y1 tan θ

)+U2

2

(y1)δ(y2 − y1 tan θ

)+ · · ·

(7)

where Y is the Heavyside function. By substituting and identifying the leading terms in (6), we obtain:(1 + tan2 θ

)U1

1 − bU12 = 0

−bU11 + (c− λ)U1

2 = 0(8)

which admits nonzero solutions because of (4). The identification of the next terms gives the system:(1 + tan2 θ

)U2

1 − bU22 = 2tanθU1′

1

(y1)

−bU21 + (c− λ)U2

2 = δ(y1) (9)

the determinant of which is zero. The right-hand sides have to satisfy the compatibility condition:

2(c− λ) tan θU1′1

(y1)+ bδ

(y1)= 0 (10)

and then we can find function U11 which defines the propagation of the singularity along the characteristic

(Dθ):

U11

(y1)=

−b

2(c− λ) tan θ

(Y(y1)+C

)(11)

In this case, we can find a solution u1 of (1) belonging to Hs+3/2(Ω) (s < −2). Then the solution u2

belongs to Hs+1/2(Ω). The singularities of these solutions are much more important than in the case whenλ is out of the essential spectrum. We saw that this singularity can propagate.

4. Reflections of the singularities

When λ belongs to ]c− b2, c[, function U11 is defined by (11) with an arbitrary constant C . This constant

can be obtained by the Dirichlet conditions on the edge of the domain. However, the characteristic straightline cuts the edges in two points. So we have too many equations to determine only one constant C . Itshows that the singularity does not disappear by reaching a point P on ∂Ω. We have at least a reflection onanother characteristic the slope of which is − tanθ.

Let (P,t,n) be a local referential. The vector t is tangent to ∂Ω and n is normal and exterior. We denoteby φ the polar angle of n. We denote by (n, t) the coordinates of a point in this referential. We have on ∂Ω,y1 = t sinφ and y2 = −t cosφ by shifting the origin of y. By writing that the displacement which is thesuperposition of two singularities, incident one u1 and reflected one u1, must vanish at the boundary, weobtain at the leading orders:

U11 (y1)

cos(φ− θ)+

U11 (y1)

cos(φ+ θ)= 0 (12)

which gives a relation between the two constants C and C .If the incident characteristic is almost tangential to the contour of the domain, then cos(φ− θ) is small

and the intensity of the reflected singularity is very strong. This shows that the mechanism of propagationand reflection may in certain cases improve the intensity of the singularities.

134

Page 5: A model problem for vibration of thin elastic shells. Propagation and reflection of singularities

A model problem for vibration of thin elastic shells

By a simple calculus we obtain:

C cos(φ+ θ)− C cos(φ− θ) = 2 sinφ sinθ (13)

Of course the reflected singularity on the characteristic (D−θ) will be reflected again when this straight linewill reach the edge in another point.

As an example, let us suppose that Ω is a disk and that θ = π/4. By following the characteristics froma point inside Ω, we see that a singularity will be propagated and reflected along the sides of a rectanglewhich is inscribed in the circle ∂Ω. We denote by C1, C2, C3 and C4, the constants corresponding to thefour sides and we have the following system,

C1 cos(φ+

π

4

)−C2 cos

(φ− π

4

)=√

2 sinφ

C2 cos(φ− π

4

)+C3 cos

(φ+

π

4

)= −

√2cosφ

C3 cos(φ+

π

4

)−C4 cos

(φ− π

4

)=√

2 sinφ

C4 cos(φ− π

4

)−C1 cos

(φ+

π

4

)= −

√2cosφ

(14)

The determinant is zero and the compatibility condition is satisfied.This shows that the asymptotic structure (or at least its leading terms) exists and is not unique. This

suggests that the corresponding value λ = c − b2/2 is an eigenvalue. It is certainly the case for b2 = 2cbecause λ is equal to zero and the homogeneous equation which is associated to (5) admits some nonzerosolutions vanishing on ∂Ω.

For other Ω and other value of λ the corresponding trajectory of the reflected singularities may besomewhat entangled, so implying complicated phenomena of resonance.

For an arbitrary contour, at the extremities of the essential spectrum (λ = c − b2 and λ = c), the twodirections of propagation are parallel to the axes of coordinates. The incident and reflected directions wouldbe the same so that reflection does not make sense.

5. A remark concerning hyperbolic thin shells

It is clearly hoped that most of the qualitative properties of the model problem will also hold true forshell problems. For a hyperbolic shell, it can be proved that in any point, the essentiel spectrum is asegment [0,Λ].

For λ = 0, corresponding to a static problem, there are two characteristic directions (double) and it isknown that there is no possibility of reflection of the singularities due to the boundary conditions on theedges of the domain [7].

References

[1] J. Sanchez-Hubert, Sanchez-Palencia E., Coques élastiques minces, propriétés asymptotiques, Masson, Paris, 1997.[2] J. Sanchez-Hubert, Sanchez-Palencia E., Vibration and Coupling of Continuous Systems, Springer, Berlin, 1989.[3] E. Sanchez-Palencia, Vassiliev D., Remarks on vibration of thin elastic shells and their numerical computation,

C. R. Acad. Sci. Paris Sér. II 314 (1992) 445–452.[4] Yu. Egorov, Shubin M., Linear Partial Differential Equations. Foundations of the Classical Theory, Encyclopaedia

of Mathematical Sciences, Vol. 30, Springer, 1991.[5] L. Hörmander, The Analysis of Linear Partial Differential Operators, Spinger, Grundlehren, Vols. 256, 257, 1983.[6] J.L. Lions, Magenes E., Problèmes aux limites non homogènes et applications, Vol. 1, Dunod, Paris, 1968.[7] P. Karamian, Réflexion des singularités dans les coques hyperboliques inhibées, C. R. Acad. Sci. Paris. Sér. IIb 326

(1998) 609–614.

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