Minimizing finite viscosity enhances relative kinetic energy absorption in bistable mechanical metamaterials but only with sufficiently fine discretization: A nonlinear dynamical size effect

Abstract

Bistable mechanical metamaterials have shown promise for mitigating the harmful consequences of impact by converting kinetic energy into stored strain energy, offering an alternative and potentially synergistic approach to conventional methods of attenuating energy transmission. In this work, we numerically study the dynamic response of a one-dimensional bistable metamaterial struck by a high speed impactor where the impactor velocity is commensurate with the sound speed, using the peak kinetic energy experienced at midpoint of the metamaterial compared to that in an otherwise identical linear system as our performance metric. We make five key findings: (1) The bistable material can counter-intuitively perform better (to nearly $10^3\times$ better than the linear system) as the viscosity decreases but remains finite, however this only occurs when sufficiently fine discretization has been reached (i.e. the system approaches sufficiently close to the continuum limit); (2) This discretization threshold is sharp, and depends on the viscosity present; (3) The bistable materials can also perform significantly worse than linear systems (for low discretization and viscosity or zero viscosity); (4) The dependence on discretization stems from the partition of energy into trains of solitary waves that have pulse lengths proportional to the unit cell size, where, with intersite viscosity, the solitary wave trains induce high velocity gradients and thus enhanced damping compared to linear, and low-unit-cell-number bistable, materials; and (5) When sufficiently fine discretization has been reached at low viscosities, the bistable system consistently outperforms the linear one for a wide range of impactor conditions, without impact condition regions of underperformance. The first point is particularly important, as it shows the existence of a nonlinear dynamical “size effect”, where, given a protective layer of some thickness and otherwise identical quasi-static mechanical properties and total mass, e.g., a $1\mathrm{mm}$ thick layer having 200 unit cells of 5 $\mu m$ thickness is predicted to perform significantly better than one having 20 unit cells of 50 $\mu m$ thickness. The complex dynamics revealed herein could help guide the future design and application of bistable, and perhaps more generally nonlinear, metamaterials in various domains including signal processing, shape changing devices, and shock and impact protection, with particular benefits in the latter case predicted for scenarios where constituent materials with low intrinsic viscosity are needed (e.g., wherein metals or ceramics would be used).