Transition waves in a beam coupled to a bistable foundation with a symmetric energy landscape

Multistable metamaterials, capable of adopting multiple stable configurations, have been used to control the shape and mechanical properties of mechanical systems. Transition waves (TWs), which enable the spatial propagation of state transitions, offer a potential strategy for reconfiguring these materials. In systems with asymmetric energy landscapes, TWs have been shown to propagate stably, with the local transition from a higher energy state to a lower energy state releasing sufficient energy to continue propagation despite dissipation in the system. However, due to the asymmetry, this strategy requires additional energy to be inserted into the system to reset the structure before a subsequent TW can propagate. In this work, we consider a multistable system with a reconfigurable surface shape, comprising an elastic slender beam coupled to a foundation of continuously distributed bistable elements with a symmetric energy landscape. Through theoretical and numerical analysis, we demonstrate that the symmetric energy landscape supports TWs with varying speeds, energy, and propagation distances (eventually halted by dissipation), and these parameters can be controlled via boundary impact speed. The final position of the TW forms a stable domain wall, separating regions of the beam in different stable states. Based on the quantitative relationship between impact speed and propagation distance, we propose a dynamic strategy for reversibly reconfiguring the surface shape simply by applying multiple impulses at one end of the system. This strategy generates sequential TWs with controlled speeds, stopping at desired locations to achieve targeted configurations. Experimental validation, using buckled beams to realize bistable elements with symmetric energy landscapes, demonstrates the feasibility of our dynamic reconfiguration strategy. Furthermore, we numerically explore the interaction of two transition waves, which provides further control of the surface configuration.