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Both auxetic structures and hierarchical honeycombs are marked with lightweight and excellent mechanical properties. Here, we combine the characteristics of auxetic structures and hierarchical honeycombs, and propose two re-entrant hierarchical honeycombs constructed by replacing the cell walls of re-entrant honeycombs with regular hexagon substructure (RHH) and equilateral triangle substructure (RHT). The honeycombs are subjected to in-plane impact in order to investigate the crashworthiness by using the commercial software LS-DYNA. The plateau stress of RHH and RHT in x and y directions are derived by a two-scale method. The results from numerical simulation indicate that the specific energy absorption of RHT and RHH is improved by up to 292% and 105%. RHT and RHH improve the mean crushing force value by 298%, 108% respectively compared with the classic re-entrant honeycomb (RH) under quasi-static loading at stress plateau region. The RHT and RHH still have the characteristic of negative Poisson's ratio. Additionally, the parametric studies are further carried out to investigate the effects of impact velocities and relative densities on crashworthiness. All the findings of this study indicate that the proposed two hierarchical honeycombs exhibit an improved crushing performance, and RHT provides the highest energy absorption capacity among all specimens.

SUMMARYThe paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge-based smoothed finite element method (ES-FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the 'overly stiff' feature of the discrete model. In such an 'overly stiff' FEM model, the wave propagates with an artificially higher 'numerical' speed, and hence the numerical wave-number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge-based gradient smoothing operations, the ES-FEM model, however, behaves much softer than the standard FEM model, leading to the so-called very 'close-to-exact' stiffness. Therefore the ES-FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES-FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model.

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