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On a generalized principle of fractal stiffness self-similarity

Marcelo Epstein

TL;DR

The paper addresses obtaining an exact in-plane stiffness matrix for a self-similar fractal Sierpiński gasket, incorporating drilling modes, by leveraging symmetry, equilibrium, and geometric self-similarity. It extends stiffness self-similarity to a multi-scale setting, showing that distinct mechanisms (axial vs bending) contribute separate scaling factors, and demonstrates this via a three-copy enlargement and static condensation to relate stiffness across scales. Two numerical solutions identify axial and bending scalings, leading to a relation $\kappa(\rho)=\kappa^{\log_2 \rho}$ that ties stiffness scaling to geometric growth. The approach provides a direct stiffness-based framework for fractal shells and tilings, avoiding the need for smooth displacement fields and enabling shell-like fractal design with exact nodal responses.

Abstract

The principle of fractal stiffness self-similarity is expanded to encompass structures with several differently-scaled contributors to the total stiffness matrix. The generalized principle is applied to solve the problem of a fractal triangular gasket that incorporates drilling modes, with a view to further applications to the modelling of fractal shells.

On a generalized principle of fractal stiffness self-similarity

TL;DR

The paper addresses obtaining an exact in-plane stiffness matrix for a self-similar fractal Sierpiński gasket, incorporating drilling modes, by leveraging symmetry, equilibrium, and geometric self-similarity. It extends stiffness self-similarity to a multi-scale setting, showing that distinct mechanisms (axial vs bending) contribute separate scaling factors, and demonstrates this via a three-copy enlargement and static condensation to relate stiffness across scales. Two numerical solutions identify axial and bending scalings, leading to a relation that ties stiffness scaling to geometric growth. The approach provides a direct stiffness-based framework for fractal shells and tilings, avoiding the need for smooth displacement fields and enabling shell-like fractal design with exact nodal responses.

Abstract

The principle of fractal stiffness self-similarity is expanded to encompass structures with several differently-scaled contributors to the total stiffness matrix. The generalized principle is applied to solve the problem of a fractal triangular gasket that incorporates drilling modes, with a view to further applications to the modelling of fractal shells.

Paper Structure

This paper contains 18 sections, 44 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The simplest example
  3. The classical beam
  4. Symmetry and statics
  5. Geometric self-similarity
  6. A leap of faith
  7. Remarks on stiffness self-similarity
  8. Multi-scaling in a non-fractal structure
  9. A frame
  10. Symmetry considerations
  11. Equilibrium
  12. The constitutive law
  13. The total stiffness matrix
  14. Multi-scaled stiffness of a self-similar fractal
  15. Geometrical self-similarity and its consequences
  16. ...and 3 more sections

Figures (7)

  • Figure 1: An Euler-Bernoulli beam
  • Figure 2: Self-similarity
  • Figure 3: The equilateral frame with its 9 degrees of freedom
  • Figure 4: The fundamental blocks
  • Figure 5: A standard plane beam element with its 6 degrees of freedom
  • ...and 2 more figures

Theorems & Definitions (2)

  • Remark 4.1
  • Remark 4.2