MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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

Partha Guha and Patrick Shipman


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.

MSC Codes:
58D05, 35Q53
surface geometry, Elastic sheet, transvectant, minimal surfaces, buckling, Whitham method

Related publications

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