On the governing equations of membrane O surfaces

TL;DR

This paper formulates the governing equations for membrane O surfaces, revealing their relation to Demoulin's Ω surfaces and demonstrating that Bäcklund transformations preserve their structure.

Contribution

It introduces the governing equations for membrane O surfaces of the 1st and 2nd kind, linking them to Guichard and Demoulin's surfaces, and explores their transformation properties.

Findings

Membrane O surfaces are a subclass of Demoulin's Ω surfaces.

Governing equations for membrane O surfaces of both kinds are formulated.

Bäcklund transformations preserve membrane O surface structures.

Abstract

It is known that a shell membrane in equilibrium where a constant purely normal load $q_n$ acts on the membrane, and where the principal curvature lines coincide with the principal stress lines, forms an integrable system called a membrane O surface. This paper formulates the governing equations for membrane O surfaces of the 1st and 2nd kind, which are analogues to Guichard surfaces of the 1st and 2nd kind introduced by Calapso. Furthermore, under this formulation, we show that membrane O surfaces are a subclass of Demoulin's $\Omega$ surfaces, and that the B\"acklund transformation for membrane O surfaces preserves membrane O surfaces of the 1st and 2nd kind, respectively.