The University of Arizona

The role of gravity in the self-similar buckling of non-Euclidean elastic sheets

The role of gravity in the self-similar buckling of non-Euclidean elastic sheets

Series: Program in Applied Mathematics Brown Bag Seminar
Location: Math 402
Presenter: Ken Yamamoto, Program in Applied Mathematics, University of Arizona

The intricate wrinkling shapes observed along edges of torn plastic sheets and growing leaves are striking examples of periodic, self-similar patterns. While a generalization of linear elasticity theory posits that such patterns arise from a non-Euclidean sheet buckling to relieve growth-induced residual strains, a full understanding of the shaping mechanism is lacking. I will present investigations suggesting that these complex morphologies result from selecting a potentially non-smooth configuration with vanishing in-plane strain (i.e., stretching) that minimizes the sum of bending and gravity energies. Periodic, self-similar patterns naturally arise from gluing together local isometries along “lines of inflection” that intersect at “branch points.” I propose that the number and locations of such branch-point defects (which generate wrinkles in the surface) are driven by a balance of gravity and bending energies for sheets with constant negative Gauss curvature.

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