Defects and boundary layers in non-Euclidean plates
Defects and boundary layers in non-Euclidean plates
复制标题
非欧几里得板中的缺陷和边界层
DOI：
10.1088/0951-7715/25/12/3553
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发表时间：
2012
期刊：
影响因子：
1.7
通讯作者：
Shankar Venkataramani
中科院分区：
文献类型：
--
作者：
John A Gemmer;Shankar Venkataramani
我们使用 Föppl-von Kármán 简化弹性理论研究具有恒定负高斯曲率的非欧几里得板的行为。受最近实验结果的启发，我们关注具有周期性轮廓的环。我们证明了与厚度平方成比例的弹性能的严格上限和下限。特别是，我们表明只有两种类型的全局最小化：保持平坦的变形和在环带边缘附近具有孤立拉伸区域的鞍形变形。我们还表明，存在具有周期性轮廓的局部最小化器，其在拐点线附近具有额外的边界层。这些额外的边界层是薄弹性片中的一种新现象，对于规范跨拐点的方位曲率的跳跃不连续性是必要的。我们严格推导出这些边界层宽度与板材厚度的函数关系的比例定律。
We investigate the behaviour of non-Euclidean plates with constant negative Gaussian curvature using the Föppl–von Kármán reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers—deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet.