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A derivation of first variation formulas from the strain-displacement relations in thin shell theory

Yoshiki Jikumaru

Abstract

In this paper, we derive the first variation formulas for surfaces in 3-dimensional Euclidean space by using the ``strain-displacement relations'' known in thin shell theory. For applications to architectural surface design, we focus on the objective function which has linear Weingarten surfaces as stationary points. This article aims to provide an elementary and cross-disciplinary exposition for applications without any tensor calculus, and thus, we do not give any new mathematical results.

A derivation of first variation formulas from the strain-displacement relations in thin shell theory

Abstract

In this paper, we derive the first variation formulas for surfaces in 3-dimensional Euclidean space by using the ``strain-displacement relations'' known in thin shell theory. For applications to architectural surface design, we focus on the objective function which has linear Weingarten surfaces as stationary points. This article aims to provide an elementary and cross-disciplinary exposition for applications without any tensor calculus, and thus, we do not give any new mathematical results.
Paper Structure (6 sections, 10 theorems, 61 equations)

This paper contains 6 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The strain-displacement relations
  4. The first variation of the area
  5. The first variation of the volume
  6. The first variation of the mean curvature

Key Result

Lemma 4.1

The first variation of the area element can be written as the summation of the infinitesimal normal strains:

Theorems & Definitions (20)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 10 more