The stiffness of self-similar fractals

A method to derive the stiffness of self-similar elastic fractals is presented based on structural mechanics principles and a physically motivated similarity criterion, which is assumed as a postulate. Using this method, the stiffnesses of both the Von Koch curve and the Sierpiński gasket in the small-deformation regime are derived. For these fractal structures, it is shown that the stiffness matrix is completely determined by a single elastic constant. The procedure to tile a planar domain with Sierpiński gaskets is explored and shown to require the consideration of hexagonal-shaped combinations of gaskets joined continuously along their edges. This continuity leads to a phenomenon of geometrically induced inextensibility along the common edges. After deriving the stiffness matrix for the basic hexagon, the analog of the Boussinesq–Flamant problem for a tiled half-plane is solved numerically to demonstrate the potential of the method in modeling of solid mechanics applications.