Finite curved creases in infinite isometric sheets

Area of Science:

• Physics
• Materials Science
• Geometry

Background:

• Geometric stress focusing in elastic thin sheets leads to point-like vertices with characteristic crescent shapes.
• The scaling of these crescent shapes is an open question, with experimental results conflicting with basic energy balance predictions.

Purpose of the Study:

• To resolve the discrepancy in crescent size scaling in crumpled elastic sheets.
• To model the crescent shape using a geometric approach based on curved creases in isometric sheets.

Main Methods:

• Modeling the crescent as a curved crease in an isometric sheet.
• Developing constraints linking the surface profile to crease-line geometry for crumpled sheet crescents.
• Constructing examples of finite curved creases that satisfy these constraints.

Main Results:

• Demonstrated that finite curved creases are realizable within the constraints of isometric sheets.
• Developed a geometric model that describes the entire material, including the region around the crescent.
• Derived testable relations between the crescent and the surrounding sheet.

Conclusions:

• The geometric approach provides a new framework for understanding crescent formation in crumpled sheets.
• This model offers insights into the scaling of crescent size, potentially resolving previous discrepancies.
• The method allows for a comprehensive description of the material's geometry and stress distribution.