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On the rigidity of certains surfaces with folds and applications to shell theory. (English) Zbl 0830.73040
The subject of this important article is the mathematical study of the properties of shells with folds. An isotropic surface \(S\) is considered under linear Kirchhoff-Love assumptions. Preliminary definitions precise the ideas of pure bending, perfect stiffness or quasi-stiffness. Thus \(S\) admitting pure bending gives inextensional displacements (zero first fundamental form), or \(S\) stiff. It is clear that all that definitions are inherent to elastic surfaces where thickness tends to 0, and useless for three-dimensional bodies. Spectral properties are also strongly connected to these definitions. Hence, a surface with folds is never perfectly stiff. It is precised that such surfaces are considered in their natural configuration, i.e. the folds are not generated by deformations. One considers the subspace \(G\) of pure bendings, i.e. \(S\) is quasi-stiff if \(G\) is finite-dimensional (\(S\) is stiff for \(G= \varnothing\)). A fold is fixed or clamped depending on whether the angle between adjacent surfaces is allowed to vary or not. Points of \(S\) are classified in elliptic, parabolic or hyperbolic. Surfaces parabolic at every point are developable.
Many results are proved with the definitions of the so-called Shapiro- Lopatinskij boundary conditions. In particular, if an elliptic surface with one of their first boundary conditions is quasi-stiff, then conditions are considerably weaker than for fixed or clamped boundaries. Piecewise elliptic surfaces with folds are also examined. The preceding considerations lead to certain rigidity properties of the folds. Some examples are carefully studied, depending on whether locally elliptic of hyperbolic properties on one side of the fold are considered. It is proved, in particular, that shells with folds are not perfectly stiff. Other examples are given of perfectly or imperfectly stiff surfaces.
This study extends considerably some other useful researches which considered only piecewise developable surfaces, and which were based on the consideration of geometric and elastic properties separately.
Reviewer: R.Valid (Paris)

74K15 Membranes
53A05 Surfaces in Euclidean and related spaces
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[1] S. Agmon, A. Douglis & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions – Part II, Comm. Pure Appl. Math., 17 (1965) 35–92. · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[2] J. L. Akian & E. Sanchez-Palencia, Approximation de coques élastiques minces par facettes planes. Phénomènes de blocage membranaire, Comp. Rend. Acad. Sci. Paris. Sér. I, 315 (1992) 363–369. · Zbl 0761.73068
[3] M. Bernadou & P. G. Ciarlet, Sur l’ellipticité du modèle linéaire des coques de W. T. Koiter in Computing Methods in Sciences and Engineering, R. Glowinski, J. L. Lions, editors, 89–136, Lecture Notes in Economies and Math. Systems, Springer, 134 (1976). · Zbl 0356.73066
[4] P. G. Ciarlet & B. Miara, Une démonstration simple de l’ellipticité des modéles de coques de W. T. Koiter et de P. M. Naghdi, Comp. Rend. Acad. Sci. Paris, Sér, I, 312 (1991) 411–415. · Zbl 0762.73050
[5] R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol. 2, Interscience, New York (1962). · Zbl 0099.29504
[6] G. Daraboux, Théorie générale des surfaces, Vol. 4, Gauthier-Villars, Paris (1896).
[7] A. Douglis & L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8 (1958) 503–538. · Zbl 0066.08002 · doi:10.1002/cpa.3160080406
[8] G. Duvaut & J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris(1972). · Zbl 0298.73001
[9] A. L. Goldenveizer, Theory of elastic thin shells, Pergamon, New York (1962). · Zbl 0145.45504
[10] G. Grubb & G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann., 227 (1977) 247–276. · Zbl 0361.35050 · doi:10.1007/BF01361859
[11] A. L. Goldenveizer, V. B. Lidski & P. E. Tovstik, Free oscillations of thin elastic shells (in Russian), Nauka, Moscow (1979).
[12] G. Geymonat & E. Sanchez-Palencia, Remarques sur la rigidité infinitésimale de certaines surfaces elliptiques non régulières, non convexes et applications, Comp. Rend. Acad. Sci. Paris, Sér, I, 313 (1991) 645–651. · Zbl 0734.73051
[13] L. Hörmander, Linear partial differential operators, Springer, Berlin (1963). · Zbl 0108.09301
[14] M. Janet, Leçons sur les systèmes d’équations aux dérivées partielles, Gauthier-Villars, Paris (1929). · JFM 55.0276.01
[15] W. T. Koiter, On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch, B73 (1970) 169–195. · Zbl 0213.27002
[16] H. Le Dret, Folded plates revisited, Comput. Mech., 5 (1989) 345–365. · Zbl 0741.73025 · doi:10.1007/BF01047051
[17] J. L. Lions & E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I. Dunod, Paris (1968). · Zbl 0165.10801
[18] C. B. Morrey & L. Nirenberg, On the analyticity of the solution of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math., 10 (1957) 271–280. · Zbl 0082.09402 · doi:10.1002/cpa.3160100204
[19] F. Niordson, Shell theory, North-Holland, Amsterdam (1985).
[20] A. V. Pogorelov, Extrinsic geometry of convex surfaces, Amer. Math. Soc., Providence (1973). · Zbl 0311.53067
[21] V. I. Smirnov, Course of higher mathematics, Vol. 5, Pergamon, Oxford (1964). · Zbl 0121.25904
[22] J. J. Stoker, Differential geometry, Wiley, New York (1969). · Zbl 0182.54601
[23] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Houston (1975). · Zbl 0306.53002
[24] E. Sanchez-Palencia, Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée, Comp. Rend. Acad. Sci. Paris, Sér. I, 309 (1989) 411–417. · Zbl 0697.73051
[25] E. Sanchez-Palencia, Statique et dynamique des coques minces. II. Cas de flexion pure inhibée, approximation membranaire, Comp. Rend. Acad. Sci. Paris, Sér. I, 309 (1989) 531–537. · Zbl 0712.73056
[26] E. Sanchez-Palencia, Passage à la limite de l’élasticité tridimensionnelle à la théorie asymptotique des coques minces, Comp. Rend. Acad. Sci. Paris, Sér. II, 311 (1990) 909–916. · Zbl 0701.73080
[27] E. Sanchez-Palencia, Asymptotic and spectral properties of a class of singular-stiff problems, Jour. Math. Pure Appl., 71 (1992) 379–406. · Zbl 0833.47011
[28] E. Sanchez-Palencia & D. G. Vassiliev, Remarks on vibration of thin elastic shells and their numerical computation, Comp. Rend. Acad. Sci. Paris, Sér. II, 314 (1992) 445–452. · Zbl 0739.73024
[29] M. E. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton (1981). · Zbl 0453.47026
[30] I. N. Vekua, Generalized analytic functions, Pergamon Press, Oxford (1962). · Zbl 0127.03505
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