Energy Scaling Laws for Conically Constrained Thin Elastic Sheets

We investigate low-energy deformations of a thin elastic sheet subject to a displacement boundary condition consistent with a conical deformation. Under the assumption that the displacement near the sheet’s center is of order h|logh|, where h≪1 is the thickness of the sheet, we establish matching upper and lower bounds of order h2|logh| for the minimum elastic energy per unit thickness, with a prefactor determined by the geometry of the associated conical deformation. These results are established first for a 2D model problem and then extended to 3D elasticity.