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Properties of Ridges in Elastic Membranes
When a thin elastic sheet is confined to a region much smaller than its size
the morphology of the resulting crumpled membrane is a network of straight
ridges or folds that meet at sharp vertices. A virial theorem predicts the
ratio of the total bending and stretching energies of a ridge. Small strains
and curvatures persist far away from the ridge. We discuss several kinds of
perturbations that distinguish a ridge in a crumpled sheet from an isolated
ridge studied earlier (A. E. Lobkovsky, Phys. Rev. E. 53 3750 (1996)). Linear
response as well as buckling properties are investigated. We find that quite
generally, the energy of a ridge can change by no more than a finite fraction
before it buckles.Comment: 13 pages, RevTeX, acknowledgement adde
The von Karman equations, the stress function, and elastic ridges in high dimensions
The elastic energy functional of a thin elastic rod or sheet is generalized
to the case of an M-dimensional manifold in N-dimensional space. We derive
potentials for the stress field and curvatures and find the generalized von
Karman equations for a manifold in elastic equilibrium. We perform a scaling
analysis of an M-1 dimensional ridge in an M = N-1 dimensional manifold. A
ridge of linear size X in a manifold with thickness h << X has a width w ~
h^{1/3}X^{2/3} and a total energy E ~ h^{M} (X/h)^{M-5/3}. We also prove that
the total bending energy of the ridge is exactly five times the total
stretching energy. These results match those of A. Lobkovsky [Phys. Rev. E 53,
3750 (1996)] for the case of a bent plate in three dimensions.Comment: corrected references, 27 pages, RevTeX + epsf, 2 figures, Submitted
to J. Math. Phy
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