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MiS Preprint
11/2008

Polarized Hessian Covariant: Contribution to Pattern Formation in the Föppl-von Kármán Shell Equations

Partha Guha and Patrick Shipman

Abstract

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.

Received:
Feb 6, 2008
Published:
Feb 6, 2008
MSC Codes:
58D05, 35Q53
Keywords:
surface geometry, Elastic sheet, transvectant, minimal surfaces, buckling, Whitham method

Related publications

inJournal
2009 Repository Open Access
Partha Guha and Patrick Shipman

Polarized Hessian covariant : contribution to pattern formation in the Föppl-von Kármán shell equations

In: Chaos, solitons and fractals, 41 (2009) 5, pp. 2828-2837