A novel dimensional reduction for the equilibrium study of inextensional material surfaces

Abstract

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.