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Confining thin elastic sheets and folding paper. (English) Zbl 1127.74005
Summary: Crumpling a sheet of paper leads to the formation of complex folding patterns over several length scales. This can be understood on the basis of the interplay of a nonconvex elastic energy, which favors locally isometric deformations, and a small singular perturbation, which penalizes high curvature. Based on three-dimensional nonlinear elasticity and by using a combination of explicit constructions and general results from differential geometry, we prove that, in agreement with previous heuristic results in the physics literature, the total energy per unit thickness of such folding patterns scales at most as the thickness of the sheet to the power \(5/3\). For the case of a single fold we also obtain the corresponding lower bound.

MSC:
74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics
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