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

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.