Polarized Hessian Covariant: Contribution to Pattern Formation in the Föppl-von Kármán Shell Equations
Partha Guha and Patrick Shipman
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Submission date: 06. Feb. 2008
published in: Chaos, solitons and fractals, 41 (2009) 5, p. 2828-2837
DOI number (of the published article): 10.1016/j.chaos.2008.10.025
MSC-Numbers: 58D05, 35Q53
Keywords and phrases: surface geometry, Elastic sheet, transvectant, minimal surfaces, buckling, Whitham method
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We analyze the structure of the Föppl-von Kármán shell equations of linear elastic shell theory using surface geometry and classical invariant theory. This equation describes the buckling of a thin shell subjected to a compressive load. In particular, we analyze the role of polarized Hessian covariant, also known as second transvectant, in linear elastic shell theory and its connection to minimal surfaces. We show how the terms of the Föppl-von Kármán equations related to in-plane stretching can be linearized using the hodograph transform and relate this result to the integrability of the classical membrane equations. Finally, we study the effect of the nonlinear second transvectant term in the Föppl-von Kármán equations on the buckling configurations of cylinders.