等轴测弯曲需要对自由边进行局部约束
Isometric bending requires local constraints on free edges
摘要：
虽然描述自由弯曲的不可拉伸薄片平衡的形状方程是已知的，但在自由边缘补充这些方程的边界条件仍然难以捉摸。直观地说，不可拉伸性是通过对主体内的度量的约束来捕获的。人们会天真地猜测，这个约束足以确保确定这些边缘上的边界条件的变形遵守等距约束。不幸的是，如果事情是如此简单，它将意味着沿着边缘的不平衡扭矩（以及力），除非边界几何形状满足明显的非物理约束。在本文中，我们确定的问题的来源：不仅是当地弧长，但也测地曲率需要明确限制在所有的自由边缘。我们推导出下列边界条件。与传统观点相反，不需要引入边界层。这个框架适用于孤立的圆锥形缺陷，无论是赤字以及，但更简单地说，作为盈余的角度。利用这些边界条件，我们表明，在一个固定半径的圆锥体内的横向张力是相等的，但相反的径向压缩，和赤字角本身无关。我们继续检查这个锥上的一个倾斜的外边缘的效果扰动证明，无论是校正的几何形状，以及在锥踢在第二阶的边缘的偏心率的应力分布。
英文摘要：
While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability is captured by a constraint on the metric within the bulk. Naively one would then guess that this constraint is enough to ensure that the deformations determining the boundary conditions on these edges respect the isometry constraint. If matters were this simple, unfortunately, it would imply unbalanced torques (as well as forces) along the edge unless manifestly unphysical constraints are met by the boundary geometry. In this paper we identify the source of the problem: not only the local arc-length but also the geodesic curvature need to be constrained explicitly on all free edges. We derive the boundary conditions which follow. Contrary to conventional wisdom, there is no need to introduce boundary layers. This framework is applied to isolated conical defects, both with deficit as well, but more briefly, as surplus angles. Using these boundary conditions, we show that the lateral tension within a circular cone of fixed radius is equal but opposite to the radial compression, and independent of the deficit angle itself. We proceed to examine the effect of an oblique outer edge on this cone perturbatively demonstrating that both the correction to the geometry as well as the stress distribution in the cone kicks in at second order in the eccentricity of the edge.
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