Vol. 4, No. 5, 2009

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Fractal elements

Samer Adeeb and Marcelo Epstein

Vol. 4 (2009), No. 5, 781–797

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.

fractals, finite element analysis, stiffness matrix, Koch curve, Sierpiński triangle, Cantor set
Received: 28 December 2008
Revised: 15 May 2009
Accepted: 17 May 2009
Published: 5 September 2009
Samer Adeeb
Department of Civil and Environmental Engineering
University of Alberta
Edmonton, AB  T6G 2W2
Marcelo Epstein
Department of Mechanical and Manufacturing Engineering
University of Calgary
2500 University Drive NW
Calgary, AB  T2N 1N4