Dynamic responses of 2-D fractional medium subjected to impact

Abstract

It is of great significance to establish a new approach to investigate the 2-D dynamic responses of two-dimensional metamaterials with mesoscopic discontinuous structure due to their excellent anti-impact performance. The spatial fractional governing equations for the 2-D orthotropic medium subjected to impact are deduced, and their finite difference forms are given accordingly. The dynamic responses of the 2-D fractional medium are related to two fractional parameters, i.e., the fractional orders (α₁ and α₂) and the characteristic length (Δl₁ and Δl₂). The 2-D fractional medium shows greater flexibility in response simulation than the 1-D fractional medium. The plane-wave velocities of the 2-D fractional medium are obtained from the characteristics line method, and agree well with the numerical results. The oyster shell samples are impacted by the CO₂ pulse laser, and their 2-D dynamic responses are measured by the two-point VISAR system. As the sample density increases, the velocity amplitudes decrease, and the amplitudes of the two signals become closer. The nacre-rich samples with higher density show obvious orthotropic properties, which is more suitable for the 2-D orthotropic fractional model, while the 2-D isotropic fractional model is more suitable for the chalk-rich samples with lower density. The 2-D orthotropic fractional model shows greater flexibility in fitting dynamic responses. The statistical relations of the fractional orders with the fractal dimension of the oyster shell samples are obtained by the FEM and by the fractional models. The fractional orders evaluated by the 2-D fractional model are much better than those evaluated by the 1-D fractional model. It provides a new approach to understanding the dynamic responses of 2-D discontinuous medium.