Abstract
In this article we demonstrate the isoperimetric inequality arising in exactly solvable structural optimization problem of stability under torque load. The Greenhill problem describes the forming of a loop in an elastic bar under torsion. The inequality for infinite rod with periodical cross-section with two types of supports is rigorously verified. The optimal shape of the twisted rod is constant along its length and the optimal shape of cross-section is the equilateral triangle. The technique to demonstrate of isoperimetric inequalities exploits the variational method and the Hölder inequality.
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Kobelev, V. Isoperimetric inequality in the periodic Greenhill Problem of twisted elastic rod. Struct Multidisc Optim 54, 133–136 (2016). https://doi.org/10.1007/s00158-016-1404-7
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DOI: https://doi.org/10.1007/s00158-016-1404-7