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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