Stress wave in the mesoscopic discontinuous medium by fractional approach

Abstract

A fractional approach based on the generalized Taylor's formula is proposed to describe the dynamic responses in the meso-discontinuous medium (e.g., oyster shell). The governing equations with the Caputo-defined fractional derivatives in space are derived. In order to describe the dynamic effect of geometric structures in the meso- discontinuous medium, the finite element method (FEM) is employed together with the fractional governing equations to establish the relation of the equivalent fractional order (α¯) with the fractal dimension (D), and the linear relation is obtained as: α¯=0.99D−0.97. The results also show that the equivalent effects of porous structure on wave propagation could be determined by fractal dimension and fractional order. The CO2 pulse laser together with VISAR measurement is then employed to investigate the dynamic responses in the oyster shell specimens. The evaluated equivalent fractional order decreases with increasing of the density, and the equivalent fractional orders (α¯) of pure nacre and pure chalk are estimated to be respectively 0.61 and 0.69, while the estimated α¯ of pure nacre and pure chalk by FEM are 0.63 and 0.79. The difference of estimated α¯(0.79) with actual α¯(0.69) of chalk is due to the irregular interface, which would cause greater wave attenuation. Due to the stronger and much more meso-discontinuous interfaces (i.e., the brick-mud structures) in the nacre, wave amplitude decays more seriously in the nacre (denser with higher density) than those in the chalk (looser with lower density), and measured velocities decrease with increasing of the density. This work provided a new approach to investigate the dynamic mechanical properties of meso-discontinuous mediums.