On the wave propagation in unbounded space-fractional elastic solids and identification of its intrinsic microstructure based on Lagrange lattice

This paper addresses the wave dispersion behavior and microstructure identification of space-fractional elastic bodies. Starting from the equation of motion, the exact analytical dispersion relation of an unbounded 1D space-fractional elastic body is found by seeking plane wave harmonic solutions. Building on the exact analytical dispersion relation of the unbounded 1D space-fractional elastic body, the fractional parameters (i.e., length scale and order of fractional derivative) that reproduce the wave dispersion behavior, in terms of frequency and group velocity, of a monoatomic lattice (Lagrange lattice) in the first Irreducible Brillouin Zone are found by enforcing appropriate matching conditions at the end of the first Irreducible Brillouin Zone. The distances between the dispersion curves and group velocity of the space-fractional elastic body and Lagrange lattice are investigated with respect to several metrics, demonstrating that the non-local space-fractional model homogenizes the Lagrange lattice microstructure.