Spontaneous free-boundary structure in crumpled membranes

We investigate the strong curvature that appears at the boundaries of a thin crumpled elastic membrane. We account for these high-curvature regions in terms of the stretching-ridge singularity believed to dominate the structure of strongly deformed elastic membranes. Using a membrane fastened to itself to form a bag shape with a single stretching ridge, we show that the creation of points of high boundary curvature lowers the interior ridge's energy. In the limit of small thickness, the induced curvature becomes arbitrarily strong on the scale of the object size and results in sharp edges connecting interior vertices to the boundary. We analyze these edges as conical sectors with no stretching. As the membrane size diverges, the edge energy grows as the square root of the central ridge energy. For comparison, we discuss the effect of truncating a stretching ridge at its ends. The effect of truncation becomes appreciable when the truncation length is comparable to the width of the untruncated ridge.