On geometrical rigidity of surfaces. Application to the theory of thin linear elastic shells

In this paper we first present a panorama about geometrical rigidity and inextensional displacements (also called infinitesimal bendings) for surfaces with kinematic boundary conditions and for surfaces with edges (in the sense of folds or junctions). This theory is fundamental for thin linear elastic shells, as it rules their asymptotic behavior when the thickness tends to aero. This behavior enlights some difficulties encountered in numerical studies of very thin elastic shells. Our approach is based on the introduction of a nonclassical space denoted by R(S) and related to inextensional displacements. It permits us to obtain new results concerning developable surfaces and hyperbolic surfaces, with one or two edges (most of them assumed to keep constant angle), including a theorem of rigid edge when the edge is an asymptotic line of the surface. By applying these results, we are able to exhibit a new example of sensitive problem for a shell with hyperbolic mean surface and with two edges keeping constant angle. In the Appendix, we give a nonclassical variant of Goursat problem for hyperbolic linear partial differential equations system, used in the proof of a rigidity result.