Structural Analysis of Phononic Crystals and Propagation of Elastic Waves in Cubic Solids in Fractal Dimensions

Abstract

We generalize the classical lattice dynamics model to a fractal setting by writing the basic equations of crystal dynamics according to a so-called LOSA model based on the “product-like fractal measure” concept. The fractal model is used to investigate the solutions to the dynamics of a Bravais lattice. The key role is played by so-called fractal elastic waves, where the mathematical solutions to wave equations have been obtained and analyzed, accounting for the underlying fractal geometry. By comparing the theoretical results with available experimental data graphically, we obtain a good match, say, for a diamond. Our numerical analysis estimates the fractal dimension of the order of \(D \approx 2.52\) which is in agreement with research studies on fractal phononic crystals. Attenuation of waves in cubic solids for small fractal dimensions has been observed. Such results justify the naming of this study as an estimation of the fractal dimension of phononic crystals.