Geometry and Elasticity of Strips and Flowers |
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Authors: | M Marder N Papanicolaou |
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Institution: | (1) Center for Nonlinear Dynamics and Department of Physics The University of Texas at Austin, Austin, TX 78712, USA;(2) Department of Physics, University of Crete, and Research Center of Crete, Heraklion, Greece |
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Abstract: | We solve several problems that involve imposing metrics on surfaces. The problem of a strip with a linear metric gradient
is formulated in terms of a Lagrangian similar to those used for spin systems. We are able to show that the low energy state
of long strips is a twisted helical state like a telephone cord. We then extend the techniques used in this solution to two–dimensional
sheets with more general metrics. We find evolution equations and show that when they are not singular, a surface is determined
by knowledge of its metric, and the shape of the surface along one line. Finally, we provide numerical evidence by minimizing
a suitable energy functional that once these evolution equations become singular, either the surface is not differentiable,
or else the metric deviates from the target metric. |
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Keywords: | pattern formation elasticity metric surfaces differential geometry |
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