Multistable two-dimensional spring-mass lattices with tunable band gaps and wave directionality

Abstract

In this paper, we consider elastic wave propagation in two dimensional periodic lattices that include an alternating pattern of linear springs and nonlinear bistable springs. Because of the bistable springs, these lattices have multiple stable equilibrium configurations. An analytical model that takes into account both nonlinear geometrical effects and nonlinearity of the springs is developed for the total potential energy of the system. After finding the stable equilibrium configurations of the unit cell by minimizing its potential energy, the propagation of waves of small amplitude is analyzed in each of these stable configurations using Bloch theorem. Examination of the band diagrams demonstrates that, depending on the model parameters, directional or complete band gaps can be observed in some of the deformed configurations, particularly for deformed configurations that are close to a critical point. Furthermore, analysis of the iso-frequency contours of the dispersion surfaces and of polar plots of the phase and group velocities indicates that the low frequency wave directionality of the lattice is tunable. For some designs, switching from one configuration to another configuration makes it possible to dramatically alter the preferred direction of wave propagation. Simulations of the response of lattices of finite size confirm that these lattices can be used as reconfigurable phononic crystals with tunable directivity, both in the low frequency range and within the directional band gaps.