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
