Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic behavior of a nonlinear shallow shell model when the shell becomes a plate

Trung Hieu Giang, Ngoc Quynh Nguyen

TL;DR

The paper analyzes the asymptotic behavior of a nonlinear shallow shell model in the Donnell–Vlasov–Mushtari–Galimov–Koiter framework as the shell approaches a plate. It formulates a variational problem with the energy $J_{θ}$ on the admissible space $V(ω)$ and proves a plate-rigidity–based coercivity result, establishing weak lower semicontinuity and compactness. The main result shows that for shells sufficiently close to the flat plate, minimizers exist and any sequence of shells converging to the plate yields minimizers that converge strongly to the plate solution, provided the plate problem has a unique solution. This extends prior work to general applied forces and provides a rigorous link between shallow-shell and plate theories, with implications for both analysis and numerical approximation of thin elastic structures.

Abstract

This paper studies a nonlinear shallow shell model proposed by Donnell, Vlasov, Mushtari, Galimov, and Koiter. More specifically, we address the question concerning the asymptotic behavior of minimizing solutions. Our result can be applied to general applied forces. Thus, it substantially extends the one given in \cite{oana2} whereby the tangential components of the applied forces are assumed to vanish.

Asymptotic behavior of a nonlinear shallow shell model when the shell becomes a plate

TL;DR

The paper analyzes the asymptotic behavior of a nonlinear shallow shell model in the Donnell–Vlasov–Mushtari–Galimov–Koiter framework as the shell approaches a plate. It formulates a variational problem with the energy on the admissible space and proves a plate-rigidity–based coercivity result, establishing weak lower semicontinuity and compactness. The main result shows that for shells sufficiently close to the flat plate, minimizers exist and any sequence of shells converging to the plate yields minimizers that converge strongly to the plate solution, provided the plate problem has a unique solution. This extends prior work to general applied forces and provides a rigorous link between shallow-shell and plate theories, with implications for both analysis and numerical approximation of thin elastic structures.

Abstract

This paper studies a nonlinear shallow shell model proposed by Donnell, Vlasov, Mushtari, Galimov, and Koiter. More specifically, we address the question concerning the asymptotic behavior of minimizing solutions. Our result can be applied to general applied forces. Thus, it substantially extends the one given in \cite{oana2} whereby the tangential components of the applied forces are assumed to vanish.
Paper Structure (3 sections, 3 theorems, 94 equations)

This paper contains 3 sections, 3 theorems, 94 equations.

Table of Contents

  1. Notation
  2. The nonlinear shallow shell model
  3. Main result

Key Result

Lemma 1

Let $\omega\subset {\mathbb{R}}^2$ be a bounded and connected open set with Lipschitz-continuous boundary. Then the set $\boldsymbol{V}(\omega)$ satisfies the following rigidity property:

Theorems & Definitions (7)

  • Lemma 1: The rigidity property of plates
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof