Solitary waves in bistable lattices with stiffness grading: Augmenting propagation control

Abstract

In this study we introduce small perturbations in the forms of graded intersite stiffnesses and graded on-site potentials to a lattice composed of bistable unit cells under elastic interactions. Based on a known soliton solution in the $\ensuremath{\phi}$-4 model, we use a perturbation approach to approximate the effects of the perturbations on the propagation speeds of transition waves. Numerical validations follow on the exact discrete equations of motion, from which we observe eventual stoppage of transition waves in the periodic lattice under physical damping, unidirectional propagation of the waves in the direction of softening properties, and boomerang-like reflection of the waves in the stiffening direction. Finally, we present three-dimensional-printed experimental lattices, confirming the theoretical and numerical results. The observed behaviors imply the extreme controllability of solitary waves through slight engineering manipulations in material-level structures. We further find that both kink (rarefaction) and antikink (compression) waves are allowed at any site in the lattice, extending the functionality of the lattice in engineering applications such as energy harvesting.