Fractal elements

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.

The average roughness and fractal dimension of articular cartilage during drying

“…Another important feature yet to be determined is if there is a finite range of drying over which the cartilage is self‐similar. The drying time in this study was up to 5 h. Self‐similarity may be lost with a longer interval (Adeeb and Epstein, ). The fractal information suggests that it is possible for the mechanism that causes the change in roughness on the surface to be generated by lubricant leaving the collagen matrix, and not from structural changes in the collagen–aggrecan complex.…”

The nonlinear elasticity of hyperelastic models for stretch-dominated cellular structures

“…Advancements in manufacturing techniques is also enabling the creation of new types of materials with several nested hierarchical levels [2,17,55,56]. Such structures promise to explore uncharted territory in materials research [23,27,50], while the recursive nature of their hierarchies brings up questions about self-similar and fractal behaviours [1,2,20,31,43].…”