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Representations of hypersurfaces and minimal smoothness of the midsurface in the theory of shells

Delfour M. C.

Control and Cybernetics

2008

Vol. 37, no 4

879-911

Many hypersurfaces ω in R^N can be viewed as a subset of the boundary Γ of an open subset Ω of R^N. In such cases, the gradient and Hessian matrix of the associated oriented distance function ba to the underlying set Ω completely describe the normal and the N fundamental forms of ω, and a fairly complete intrinsic theory of Sobolev spaces on C1'1-hypersurfaces is available in Delfour (2000). In the theory of thin shells, the asymptotic model only depends on the choice of the constitutive law, the midsurface, and the space of solutions that properly handles the loading applied to the shell and the boundary conditions. A central issue is the minimal smoothness of the midsurface to still make sense of asymptotic membrane shell and bending equations without ad hoc mechanical or mathematical assumptions. This is possible for a C1'1-midsurface with or without boundary and without local maps, local bases, and Christoffel symbols via the purely intrinsic methods developed by Delfour and Zolesio (1995a) in 1992. Anicic, LeDret and Raoult (2004) introduced in 2004 a family of surfaces ω that are the image of a connected bounded open Lipschitzian domain in R² by a bi-Lipschitzian mapping with the assumption that the normal field is globally Lipschizian. >From this, they construct a tubular neighborhood of thickness 2h around the surface and show that for sufficiently small h the associated tubular neighborhood mapping is bi-Lipschitzian. We prove that such surfaces are C1'1-surfaces with a bounded measurable second fundamental form. We show that the tubular neighborhood can be completely described by the algebraic distance function to ω and that it is generally not a Lipschitzian domain in R³ by providing the example of a plate around a flat surface ω verifying all their assumptions. Therefore, the G1-join of K-regular patches in the sense of Le Dret (2004) generates a new K-regular patch that is a C1'1-surface and the join is C1'1. Finally, we generalize everything to hypersurfaces generated by a bi-Lipschitzian mapping defined on a domain with facets (e.g. for sphere, torus). We also give conditions for the decomposition of a C1'1-hypersurface into C1'1-patches.