A general framework is developed for finding the equations describing the equilibrium of an inextensional material surface with arbitrary flat reference shape that is deformed by applying tractions or moments to its edge. This is facilitated by using a representation of all isometric deformations of the material surface to convert the bending energy of the material surface to a line integral over the edge of the material surface. Euler–Lagrange equations are derived, leading to a complete and definitive set of equilibrium equations, which are a system of ordinary differential equations for the spatial directrix. Jump conditions that apply at points where the tangent and/or curvature of the edge may be discontinuous are also derived. As a simple but illustrative example, the deformation of a rectangular strip subject to various edge conditions is studied.

开发了一个通用框架，用于找到描述具有任意平面参考形状的非拉伸材料表面的平衡的方程，该平面参考形状通过向其边缘施加牵引力或力矩而变形。这通过使用材料表面的所有等距变形的表示来促进，以将材料表面的弯曲能量转换为材料表面边缘上的积分线。欧拉-拉格朗日方程被推导出来，产生了一套完整且确定的平衡方程，它们是空间准线的常微分方程系统。还导出了应用于边缘的切线和/或曲率可能不连续的点处的跳跃条件。作为一个简单但说明性的例子，