Abstract The ubiquity of fractals in nature and manmade structures motivates the investigation of the intricate relationship between self-similar architecture and mechanical behaviour. Accordingly, we examine the static stability of a class of triangular fractals evolving across fractal levels. The focus is on unveiling the progression of stiffness of two-dimensional (2D) linear-elastic fractal media under in-plane translational deformations at their vertices. For the first time, we show that the pertinent stiffness matrices behave as a discrete-time dynamical system, with fractal levels acting as the pseudo-time of the system. To this end, we discover a transformation matrix, which captures the effects of holes in the fractals, to attain analytical closed-form expressions. We further introduce original generalized eigenvalue–eigenvector solutions of matrix pairs formed by the examined stiffness matrices and their block diagonals. The analysis indicates that the fractals’ static stability depends on the asymptotic eigenvalues of these matrix pairs. The results reveal that the stiffness matrices follow a universal scaling law, which approximately halves their stiffness coefficients over two consecutive fractal levels, regardless of their geometrical and mechanical properties. Finally, we generalize the proposed solutions to other triangular fractal families and reveal the topological patterns of their transformation matrices.
Loong et al. (Fri,) studied this question.