Rigidity of a thin domain depends on the curvature, width, and boundary conditions

TL;DR

This paper investigates how the rigidity of shallow thin domains varies with curvature, width, and boundary conditions, revealing two distinct regimes with different scaling laws and a surprising independence from curvature in the smaller regime.

Contribution

It identifies two scaling regimes for the rigidity of shallow thin domains and shows that in the smaller regime, rigidity is independent of the surface curvature.

Findings

Two regimes:   ( ext{h,}) and (, ext{1})

Rigidity formulas depend on  and h in each regime

Rigidity in the small  regime does not depend on surface curvature

Abstract

This paper is concerned with the study of linear geometric rigidity of shallow thin domains under zero Dirichlet boundary conditions on the displacement field on the thin edge of the domain. A shallow thin domain is a thin domain that has in-plane dimensions of order $O(1)$ and $\epsilon,$ where $\epsilon\in (h,1)$ is a parameter (here $h$ is the thickness of the shell). The problem has been solved in [8,10] for the case $\epsilon=1,$ with the outcome of the optimal constant $C\sim h^{-3/2},$ $C\sim h^{-4/3},$ and $C\sim h^{-1}$ for parabolic, hyperbolic and elliptic thin domains respectively. We prove in the present work that in fact there are two distinctive scaling regimes $\epsilon\in (h,\sqrt h]$ and $\epsilon\in (\sqrt h,1),$ such that in each of which the thin domain rigidity is given by a certain formula in $h$ and $\epsilon.$ An interesting new phenomenon is that in the first (small parameter) regime $\epsilon\in (h,\sqrt h]$, the rigidity does not depend on the curvature of the thin domain mid-surface.