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The Plateau Problem of Michell Trusses and Orthogonality in Springs

Chengcheng Yang

TL;DR

The paper tackles the Plateau problem for Michell Trusses by casting minimal elastic networks balancing prescribed boundary forces into two GMT frameworks: flat chains and currents. It develops a constructive decomposition theory (via rank-1 stress directions, tangent-space alignment, and cone arguments) to show how boundary forces can be realized by a structurally stressed $k$-chain, and proves existence of minimizers under boundary data using flat-chain compactness and Riesz representation for vector-valued currents. A key geometric insight is that, away from the boundary, compressed and stretched springs meet at right angles. The work advances the mathematical understanding of Michell Trusses in arbitrary dimensions, linking mechanical intuition with rigorous geometric measure theory, while highlighting open questions about algorithmic construction and the precise topology of minimizers.

Abstract

Given finitely many pointed forces in the plane. Suppose that these forces sum up to zero and their net torques also sum up to zero. One can show that there exists a system of springs whose boundary forces exactly counter-balance these pointed forces. We will generalize to higher dimensions using the Cauchy stress tensor for elastic materials. Given a system of springs, we can multiply the length of each spring with its corresponding spring constant and then sum these products up. The result is called the total mass of the system. We are interested in the Plateau problem of the existence of the minimal spring system given a boundary condition. This minimization problem was first introduced in 1904 by A. Michell. He showed that a minimizer could smear out. The Michell Truss became known in mechanical engineering. It raised attention in optimal design, such as minimizing costs in building bridges. In 1960s and 1970s, the problem was developed using PDE and convex analysis by introducing an equivalent dual maximization problem. In 2008, Bouchitté, Gangbo, and Sppecher introduced lines of principal actions to generalize Hencky-Prandtle net to higher dimensional duality and proved that the minimizer can be found provided that it exists. In the unpublished notes of Gangbo, he also showed that if springs of the same kind are optimal. In this paper, we are going to solve the Plateau problem using two different tools in GMT: first, a minimizer can be viewed as a flat chain complex; second, a minimizer can also be viewed as a current. At the end, we are going to show one progress in discovering the topological properties of minimizers: compressed and stretched springs must be perpendicular to each other at non-boundary points. I appreciate my advisor Prof. Robert Hardt for communicating with me regularly on this problem.

The Plateau Problem of Michell Trusses and Orthogonality in Springs

TL;DR

The paper tackles the Plateau problem for Michell Trusses by casting minimal elastic networks balancing prescribed boundary forces into two GMT frameworks: flat chains and currents. It develops a constructive decomposition theory (via rank-1 stress directions, tangent-space alignment, and cone arguments) to show how boundary forces can be realized by a structurally stressed -chain, and proves existence of minimizers under boundary data using flat-chain compactness and Riesz representation for vector-valued currents. A key geometric insight is that, away from the boundary, compressed and stretched springs meet at right angles. The work advances the mathematical understanding of Michell Trusses in arbitrary dimensions, linking mechanical intuition with rigorous geometric measure theory, while highlighting open questions about algorithmic construction and the precise topology of minimizers.

Abstract

Given finitely many pointed forces in the plane. Suppose that these forces sum up to zero and their net torques also sum up to zero. One can show that there exists a system of springs whose boundary forces exactly counter-balance these pointed forces. We will generalize to higher dimensions using the Cauchy stress tensor for elastic materials. Given a system of springs, we can multiply the length of each spring with its corresponding spring constant and then sum these products up. The result is called the total mass of the system. We are interested in the Plateau problem of the existence of the minimal spring system given a boundary condition. This minimization problem was first introduced in 1904 by A. Michell. He showed that a minimizer could smear out. The Michell Truss became known in mechanical engineering. It raised attention in optimal design, such as minimizing costs in building bridges. In 1960s and 1970s, the problem was developed using PDE and convex analysis by introducing an equivalent dual maximization problem. In 2008, Bouchitté, Gangbo, and Sppecher introduced lines of principal actions to generalize Hencky-Prandtle net to higher dimensional duality and proved that the minimizer can be found provided that it exists. In the unpublished notes of Gangbo, he also showed that if springs of the same kind are optimal. In this paper, we are going to solve the Plateau problem using two different tools in GMT: first, a minimizer can be viewed as a flat chain complex; second, a minimizer can also be viewed as a current. At the end, we are going to show one progress in discovering the topological properties of minimizers: compressed and stretched springs must be perpendicular to each other at non-boundary points. I appreciate my advisor Prof. Robert Hardt for communicating with me regularly on this problem.

Paper Structure

This paper contains 27 sections, 16 theorems, 187 equations, 9 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. Rank 1 Diagonal Matrices
  4. Polyhedral $k$-chains in $\mathbb{R}^n$
  5. Stressed $k$-chains in $\mathbb{R}^n$
  6. One Dimensional Example: Stressed Springs.
  7. Higher Dimensional Definitions.
  8. A Uni-directional Stretch of a Rectangle.
  9. A Uni-directional Stretch of a Triangle
  10. Multi-directional Stretches of a Rectangle.
  11. Decomposition Theorem
  12. Generalized Cauchy Stress Tensor
  13. The Converse to Decomposition Theorem
  14. Balancing External Forces
  15. Set Up
  16. ...and 12 more sections

Key Result

Theorem 2.1

For any structurally stressed polyhedral $k$-chain P in $\mathbb{R}^n$, the $(k-1)$-stressed chain $\partial P$ admits a unique decomposition: where $S$ consists of simplices in $M_n \otimes C_{k-1}$ with only normal and shear stresses and $F$ consists of simplices in $M_n \otimes C_{k-1}$ with only externally orthogonal and parallel forces.

Figures (9)

  • Figure 1: The decomposition of the matrix.
  • Figure 2: Make cones for radial polyhedral chains.
  • Figure 3: Case 1 of Three Springs
  • Figure 4: Notation for Case 1 of Three Springs
  • Figure 5: Case 3 of Three Springs
  • ...and 4 more figures

Theorems & Definitions (31)

  • Theorem 2.1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 21 more