Inverted cones and their elastic creases.

We study the elastic inversion of a right circular cone, in particular, the uniform shape of the narrow crease that divides its upright and inverted parts. Our methodology considers a cylindrical shell analogy for simplicity where the crease is the boundary layer deformation. Solution of its governing equation of deformation requires careful crafting of the underlying assumptions and boundary conditions in order to reveal an expression for the crease shape in closed form. We can then define the characteristic width of crease exactly, which is compared to a geometrically nonlinear, large displacement finite element analysis. This width is shown to be accurately predicted for shallow and steep cones, which imparts confidence to our original assumptions. Using the shape of crease, we compute the strain energy stored in the inverted cone, in order to derive an expression for the applied force of inversion by a simple energy method. Again, our predictions match finite element data very well. This study may complement other studies of creases traditionally formed in a less controlled manner, for example, during crumpling of lightweight sheets.