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 ribbon 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 domains). The problem has been solved by Grabovsky and the second author in (Ann de l’Inst Henri Poincare (C) An Non Lin 2018 35(1):267–282, 2018) and by the second author in (Arch Ration Mech Anal 226(2):743–766, 2017) 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.

本文关注的是薄薄域在零Dirichlet边界条件下在薄域边缘上的位移场上的线性几何刚度的研究。带状结构是一个薄结构域，其面内尺寸为O（1）和\（\ epsilon，\），其中\（\ epsilon \ in（h，1）\）是参数（此处h是厚度的域）。Grabovsky和第二作者（Ann de l'Inst Henri Poincare（C）An Non Lin 2018 35（1）：267–282，2018）和第二作者（Arch Ration Mech Anal 226）已解决了该问题（2）：743–766，2017 ）具有最优常数\（C \ sim h ^ {-3/2}，\） \（C \ sim的结果的情况下的\（\ epsilon = 1，\）h ^ {-4/3}，\）和\（C \ sim h ^ {-1} \）分别用于抛物线形，双曲形和椭圆形薄域。我们在目前的工作中证明，实际上有两种不同的缩放方式\（\ epsilon \ in（h，\ sqrt {h}] \）和\（\ epsilon \ in（\ sqrt {h}，1），\ ），使得每个薄域的刚度都由h和\（\ epsilon。\）中的某个公式给出。一个有趣的新现象是在第一个（小参数）态\（\ epsilon \ in（h， \ sqrt {h}] \），刚度不取决于薄域中表面的曲率。