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Geometric stress focusing, e.g. in a crumpled sheet, creates point-likevertices that terminate in a characteristic local crescent shape. The observedscaling of the size of this crescent is an open question in the stress focusingof elastic thin sheets. According to experiments and simulations, this sizedepends on the outer dimension of the sheet, but intuition and rudimentaryenergy balance indicate it should only depend on the sheet thickness. Weaddress this discrepancy by modeling the observed crescent with a moregeometric approach, where we treat the crescent as a curved crease in anisometric sheet. Although curved creases have already been studied extensively,the crescent in a crumpled sheet has its own unique features: the materialcrescent terminates within the material, and the material extent isindefinitely larger than the extent of the crescent. These features togetherwith the general constraints of isometry lead to constraints linking thesurface profile to the crease-line geometry. We construct several examplesobeying these constraints, showing finite curved creases are fully realizable.This approach has some particular advantages over previous analyses, as we areable to describe the entire material without having to exclude the regionaround the sharp crescent. Finally, we deduce testable relations between thecrease and the surrounding sheet, and discuss some of the implications of ourapproach with regards to the scaling of the crescent size.

Finite curved creases in infinite isometric sheets