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Sensitivity phenomena for certain thin elastic shells with edges. (English) Zbl 0989.74047
Summary: We consider two kinds of shells which are sensitive, i.e. they are geometrically rigid and, as the thickness $$\varepsilon$$ tends to zero, the limit problem is unstable in the sense that there are very smooth loadings (belonging to the space $${\mathcal D}$$ of test functions of distributions) such that the corresponding solutions go out of the energy space. The first situation occurs when there is an edge and the middle surface is elliptic on both sides of it. The second situation occurs when there is an edge $$\Gamma_0$$, the surface is respectively elliptic and hyperbolic on both sides of it, and the ‘determination domain’ in the hyperbolic region issued from $$\Gamma_0$$ intersects another edge $$\Gamma_1$$.

##### MSC:
 74K25 Shells 74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics 35B40 Asymptotic behavior of solutions to PDEs 35Q72 Other PDE from mechanics (MSC2000)
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##### References:
 [1] Méthodes d’éléments finis pour les problèmes de coques minces, Masson, Paris, 1994. [2] Analyse fonctionnelle, Masson, Paris, 1983. [3] Caillerie, C. R. Acad. Sci. Paris, Sér. I 323 pp 835– (1996) [4] Mathematical Elasticity, Vol. 3: Theory of Shells, North-Holland, Amsterdam, to appear. [5] Linear Partial Differential Operators, Springer, Berlin, 1963. · Zbl 0108.09301 · doi:10.1007/978-3-642-46175-0 [6] Leguillon, C. R. Acad. Sci. Paris, Sér. IIb 327 pp 485– (1999) [7] and ?Expériences de non-calculabilité dans les problèmes sensitifs. Coques minces chancelantes?, Actes du 2éme Colloque National de Calcul des Structures, pp. 275-280, 1995 Hermès, Paris. [8] Lewy, Ann. Math. 66 pp 155– (1957) · Zbl 0078.08104 · doi:10.2307/1970121 [9] Lions, C. R. Acad. Sci. Paris, Sér. I 319 pp 1021– (1994) [10] and ?Sur quelques espaces de la théorie des coques et la sensitivité?, in: Homogenization and Applications to Material Sciences, ( and eds), pp. 271-278, Gakkotosho, Tokyo, 1995. [11] and ?Problèmes sensitifs et coques élastiques minces?, in: Partial Differential Equations and Functional Analysis in Memory of P. Grisvard, ( and eds), pp. 207-220, Birkhauser, Boston, 1996. · doi:10.1007/978-1-4612-2436-5_14 [12] and ?Examples of sensitivity in shells with edges?, in: Shells, Mathematical Modelling and Scientific Computing, ( and eds), pp. 151-154, University of Santiago de Compostela Press, 1997. · Zbl 1052.74548 [13] and ?Instabilities produced by edges in thin shells?, in: Variation of Domains and Free-Boundary Problems in Solid Mechanics, ( and eds) pp. 277-284, Kluwer, Dordrecht. [14] Functional Analysis, Vol. 2, McGraw-Hill, New York, 1973. [15] and Coques élastiques minces, propriétés asymptotiques, Masson, Paris, 1997.
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