Abstract
We investigate low-energy deformations of a thin elastic sheet subject to a displacement boundary condition consistent with a conical deformation. Under the assumption that the displacement near the sheet’s center is of order h|logh|, where h≪1 is the thickness of the sheet, we establish matching upper and lower bounds of order h 2|logh| for the minimum elastic energy per unit thickness, with a prefactor determined by the geometry of the associated conical deformation. These results are established first for a 2D model problem and then extended to 3D elasticity.
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Notes
We thank Heiner Olbermann for pointing this out.
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Acknowledgements
We would like to thank Heiner Olbermann for valuable input on an earlier version of this paper, and especially for pointing out a modification of our earlier argument that led to matching upper and lower bounds. Brandman and Nguyen would like to thank the Courant Institute for its warm support. Most of this work was done while they were Courant Instructors.
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J. Brandman’s research is supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship.
R.V. Kohn’s research is supported by NSF grants DMS-0807347 and OISE-0967140.
H.-M. Nguyen’s research is supported by NSF grant DMS-1201370.
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Brandman, J., Kohn, R.V. & Nguyen, HM. Energy Scaling Laws for Conically Constrained Thin Elastic Sheets. J Elast 113, 251–264 (2013). https://doi.org/10.1007/s10659-012-9420-3
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DOI: https://doi.org/10.1007/s10659-012-9420-3