On the dilatational wave motion in anisotropic fractal solids

Abstract

This paper reports a study of wave motion in a generally anisotropic fractal medium (i.e. with different fractal dimensions in different directions), whose constitutive response is represented by an isotropic Hooke's law. First, the governing elastodynamic laws are formulated on the basis of dimensional regularization. It is discovered that the satisfaction of the angular momentum equation precludes the implementation of the classical elasticity theory which results in symmetry of the Cauchy stress tensor. Nevertheless, the classical elastic constitutive model can still be applied to explore dilatational wave propagation; in such a case, the angular momentum balance is "trivially" satisfied. The resulting problem, of eigenvalue type, is solved analytically.A computational finite element method solver is also developed to simulate the problem in its 1d form, it is validated by the reference solutions generated through the modal analysis. A 3d finite-difference-based solver is also developed and the obtained results match those of the 1d simulation.

Continuum Homogenization of Fractal Media

“…More work was done on waves in linear elastic fractal solids under small motions. Several cases of isotropic (Joumaa and Ostoja-Starzewski, 2011) or anisotropic (with micropolar effects) Joumaa and Ostoja-Starzewski, 2016) media have been considered through analytical and computational methods. It was found on the mathematical side that fractal versions of harmonic, Bessel, and Hankel functions.…”

Fractal Logistic Equation

“…Fractal geometry includes shapes which are scale invariant and have fractional dimensions and self-similar properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Analysis on fractals was formulated using different methods such as harmonic analysis, probabilistic methods, measure theory, fractional calculus, fractional spaces, and time-scale calculus [18][19][20][21][22][23][24][25][26][27][28][29].…”