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On the governing equations of membrane O surfaces

Yoshiki Jikumaru

TL;DR

This work derives the governing equations for membrane O surfaces, treating them as integrable analogues of Guichard surfaces under a constant purely normal load $q_n$ and with curvature lines aligned to stress lines. It develops a Combescure-based O-surface framework, introduces first-integral constraints, and formulates two reductions (1st and 2nd kind) yielding explicit representations of the first and third fundamental forms and associated PDE systems, including sinh-Gordon and sine-Gordon reductions. The authors establish Bäcklund transformations that map membrane O surfaces within the same kind, supporting a solitonic interpretation and connecting to a Lax/Ribaucour structure. They also place membrane O surfaces inside Demoulin's $\Omega$-surfaces, unifying membrane elasticity with classical integrable geometry and enabling constructive solution-generation for design applications in shells and architectural membranes.

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 $Ω$ surfaces, and that the Bäcklund transformation for membrane O surfaces preserves membrane O surfaces of the 1st and 2nd kind, respectively.

On the governing equations of membrane O surfaces

TL;DR

This work derives the governing equations for membrane O surfaces, treating them as integrable analogues of Guichard surfaces under a constant purely normal load and with curvature lines aligned to stress lines. It develops a Combescure-based O-surface framework, introduces first-integral constraints, and formulates two reductions (1st and 2nd kind) yielding explicit representations of the first and third fundamental forms and associated PDE systems, including sinh-Gordon and sine-Gordon reductions. The authors establish Bäcklund transformations that map membrane O surfaces within the same kind, supporting a solitonic interpretation and connecting to a Lax/Ribaucour structure. They also place membrane O surfaces inside Demoulin's -surfaces, unifying membrane elasticity with classical integrable geometry and enabling constructive solution-generation for design applications in shells and architectural membranes.

Abstract

It is known that a shell membrane in equilibrium where a constant purely normal load 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 surfaces, and that the Bäcklund transformation for membrane O surfaces preserves membrane O surfaces of the 1st and 2nd kind, respectively.
Paper Structure (6 sections, 6 theorems, 86 equations)

This paper contains 6 sections, 6 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equilibrium equations for a shell membrane and an O surface connection
  4. Membrane O surfaces of the 1st and 2nd kind
  5. Bäcklund transformations
  6. Appendix: Relation with Demoulin's $\Omega$ surfaces

Key Result

Theorem 3.1

Assume that a constant purely normal load $q_n$ acts on a membrane and the principal curvature lines coincide with the principal stress lines. Then the membrane is in equilibrium if and only if there exist functions $f(x)$ and $g(y)$ satisfying the constraint eq:constraint under the Gauss-Mainardi-C

Theorems & Definitions (15)

  • Theorem 3.1: RogersSchief, Theorem 4.2
  • Remark 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3
  • Remark 4.4
  • Example 4.5
  • Example 4.6
  • Example 4.7
  • Corollary 4.8
  • ...and 5 more