Polarized Hessian covariant: Contribution to pattern formation in the Foppl-von Karman shell equations

We analyze the structure of the Foppl-von Karman 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 Foppl-von Karman 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 Foppl-von Karman equations on the buckling configurations of cylinders. (C) 2008 Elsevier Ltd. All rights reserved.