论文标题
薄域的刚度取决于曲率，宽度和边界条件
Rigidity of a thin domain depends on the curvature, width, and boundary conditions
论文作者
论文摘要
本文涉及研究域薄边缘的位移场上零dirichlet边界条件下浅薄域的线性几何刚度的研究。一个浅薄域是一个薄域，具有订单$ o（1）$和$ε的平面内尺寸，$，其中$ε\ in（h，1）$是一个参数（这是$ h $是外壳的厚度）。对于最佳常数$ c \ sim h^{ - 3/2}，$ $ $ C \ sim h^{ - 4/3}，$ c \ sim h^{ - 4/3}，$ c \ sim h^{ - $ c \ sim h^{ - 1} $，对于寄生虫，超质子和椭圆形的薄域而言，该问题已在[8,10]中解决了[8,10]，$ $ε= 1，$。在目前的工作中，我们证明实际上有两个独特的缩放状态$ε\ in（h，\ sqrt h] $和$ε\ in（\ sqrt h，1），因此，在每种情况下，薄域刚性在$ h $ and $ h $ and $ε中给出了一个有趣的新现象。 （h，\ sqrt h] $，刚性不取决于薄域中表面的曲率。
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 $ε,$ where $ε\in (h,1)$ is a parameter (here $h$ is the thickness of the shell). The problem has been solved in [8,10] for the case $ε=1,$ 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 $ε\in (h,\sqrt h]$ and $ε\in (\sqrt h,1),$ such that in each of which the thin domain rigidity is given by a certain formula in $h$ and $ε.$ An interesting new phenomenon is that in the first (small parameter) regime $ε\in (h,\sqrt h]$, the rigidity does not depend on the curvature of the thin domain mid-surface.
