Fractal elements

Adeeb, Samer; Epstein, Marcelo

doi:10.2140/jomms.2009.4.781

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Fractal elements

Samer Adeeb and Marcelo Epstein

Vol. 4 (2009), No. 5, 781–797

DOI: 10.2140/jomms.2009.4.781

Abstract

Self-similar fractals are geometrically stable in the sense that, when generated by a recursive copying process that starts from a basic building block, their final image depends only on the recursive generation process rather than on the shape of the original building block. In this article we show that an analogous stability property can also be applied to fractals as elastic structural elements and used in practice to obtain the stiffnesses of these fractals by means of a rapidly converging numerical procedure. The relative stiffness coefficients in the limit depend on the generation process rather than on their counterparts in the starting unit. The stiffness matrices of the Koch curve, the Sierpiński triangle, and a two-dimensional generalization of the Cantor set are derived and shown to abide by the aforementioned principle.

Keywords

fractals, finite element analysis, stiffness matrix, Koch curve, Sierpiński triangle, Cantor set

Milestones

Received: 28 December 2008

Revised: 15 May 2009

Accepted: 17 May 2009

Published: 5 September 2009

Authors

Samer Adeeb
Department of Civil and Environmental Engineering University of Alberta Edmonton, AB T6G 2W2 Canada

Marcelo Epstein
Department of Mechanical and Manufacturing Engineering University of Calgary 2500 University Drive NW Calgary, AB T2N 1N4 Canada

Vol. 4, No. 5, 2009

Samer Adeeb and Marcelo Epstein

Abstract

Keywords

Milestones

Authors