Control of acoustic properties of network materials by finite pre-deformations: Applications to 3D auxetic network materials

We investigate the impact of large pre-deformations applied to repetitive 3D network materials on their acoustic properties, and especially the tuning of band gaps based on the applied state of deformation in the large strains regime. Based on an incremental scheme for the computation of the nonlinear mechanical response of periodic network materials, we develop a perturbation method of Linstedt-Poincaré type to predict the evolution of the dispersion relation and phase velocity versus the applied deformation and effective density. We focus in this contribution on the transition to auxetic like microstructures showing negative Poisson's ratio, based on a deformation control. When Poisson's ratio becomes negative due to the varying imposed kinematic load, we show that some of the analyzed networks become unstable, since the strong ellipticity condition is not more satisfied. Such instabilities are shown to trigger band gaps associated to discontinuities in the phase velocity plots, the range of which increases with the slenderness ratio of the beams network.