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

Paolo Piersanti

TL;DR

This work rigorously justifies Koiter's two-dimensional theory for elliptic membrane shells under an interior normal compliance obstacle by bridging a three-dimensional obstacle problem to a two-dimensional limit through a thorough asymptotic analysis as the thickness $2\varepsilon$ vanishes. It introduces a novel density result for convex subsets of Lebesgue spaces to extend the applicability beyond previous density properties and proves that the 2D limit solution gains enhanced regularity up to the boundary, including a well-defined trace for the transverse displacement. The main results show that the Koiter-type reduced model and the original 3D obstacle problem share the same asymptotic limit, ensuring the reliability of the reduced model for interior-contact constraints and informing numerical approaches. Together, these findings provide a rigorous foundation for convergence and boundary regularity in obstacle problems for elliptic membrane shells and clarify the role of interior contact in Koiter-type theories.

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.

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

TL;DR

This work rigorously justifies Koiter's two-dimensional theory for elliptic membrane shells under an interior normal compliance obstacle by bridging a three-dimensional obstacle problem to a two-dimensional limit through a thorough asymptotic analysis as the thickness vanishes. It introduces a novel density result for convex subsets of Lebesgue spaces to extend the applicability beyond previous density properties and proves that the 2D limit solution gains enhanced regularity up to the boundary, including a well-defined trace for the transverse displacement. The main results show that the Koiter-type reduced model and the original 3D obstacle problem share the same asymptotic limit, ensuring the reliability of the reduced model for interior-contact constraints and informing numerical approaches. Together, these findings provide a rigorous foundation for convergence and boundary regularity in obstacle problems for elliptic membrane shells and clarify the role of interior contact in Koiter-type theories.

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 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.
Paper Structure (7 sections, 13 theorems, 165 equations)

This paper contains 7 sections, 13 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. The three-dimensional obstacle problem for a "general" linearly elastic shell
  4. The scaled three-dimensional problem for a family of linearly elastic elliptic membrane shells
  5. Asymptotic analysis as $\varepsilon\to 0$ departing from Problem \ref{['problem1']}
  6. Justification of Koiter's model for elliptic membrane shells subjected to the interior normal compliance contact condition
  7. Augmentation of the regularity of the solution of Problem \ref{['problem2']} up to the boundary

Key Result

Theorem 2.1

Let $\omega$ be a domain in $\mathbb{R}^2$ and let an immersion $\bm{\theta} \in \mathcal{C}^3(\overline{\omega}; \mathbb{E}^3)$ be given such that the surface $\bm{\theta}(\overline{\omega})$ is elliptic. Define the space Then, there exists a constant $c_0=c_0(\omega,\bm{\theta})>0$ such that for all $\bm{\eta}=(\eta_i)\in\bm{V}_M(\omega)$. ∎

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 3.1
  • Lemma 4.1
  • Theorem 4.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 12 more