Wave Propagation in Quasi-continuous Linear Chains with Self-similar Harmonic Interactions: Towards a Fractal Mechanics

Abstract

Many systems in nature have arborescent and bifurcated structures such as trees, fern, snails, lungs, the blood vessel system, but also porous materials etc. look self-similar over a wide range of scales. Which are the mechanical and dynamic properties that evolution has optimized by choosing self-similarity so often as an inherent material symmetry? How can we describe the mechanics of self-similar structures in the static and dynamic framework? In order to analyze such material systems we construct self-similar functions and linear operators such as a self-similar variant of the Laplacian and of the D’Alembertian wave operator. The obtained self-similar linear wave equation describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The dispersion relation of this system is obtained by the negative eigenvalues of the self-similar Laplacian and has the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We deduce a continuum approximation that links the self-similar Laplacian to fractional integrals which also yields in the low-frequency regime a power law scaling for the oscillator density with strictly positive exponent leading to a vanishing oscillator density at frequency zero. We suggest that this power law scaling is a characteristic and universal feature of self-similar systems with complexity well beyond of our present model. For more details we refer to our recent paper [7].