27 research outputs found

    Shape selection in non-Euclidean plates

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    We investigate isometric immersions of disks with constant negative curvature into R3\mathbb{R}^3, and the minimizers for the bending energy, i.e. the L2L^2 norm of the principal curvatures over the class of W2,2W^{2,2} isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H2\mathbb{H}^2 into R3\mathbb{R}^3. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the LL^\infty norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W2,2W^{2,2}, and have a lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates

    Über die Kräfte, die Kegelschnitte als Bahnen hervorrufen.

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