On the justification of Koiter's model for elliptic membranes subjected to an interior normal compliance contact condition

Abstract

The purpose of this paper is twofold. First, we rigorously justify Koiter's model for linearly elastic elliptic membrane shells in the case where the shell is subject to a geometrical constraint modelled via a normal compliance contact condition defined in the interior of the shell. To achieve this, we establish a novel density result for non-empty, closed, and convex subsets of Lebesgue spaces, which are applicable to cases not covered by the ``density property'' established in [Ciarlet, Mardare \& Piersanti, \emph{Math. Mech. Solids}, 2019]. Second, we demonstrate that the solution to the two-dimensional obstacle problem for linearly elastic elliptic membrane shells, subjected to the interior normal compliance contact condition, exhibits higher regularity throughout its entire definition domain. A key feature of this result is that, while the transverse component of the solution is, in general, only of class $L^2$ and its trace is \emph{a priori} undefined, the methodology proposed here, partially based on [Ciarlet \& Sanchez-Palencia, \emph{J. Math. Pures Appl.}, 1996], enables us to rigorously establish the well-posedness of the trace for the transverse component of the solution by means of an \emph{ad hoc} formula.