The Role of Symmetry and Patterning in Hierarchical Multistable Metasheets*

Soft metamaterials have introduced new opportunities for design, programmability, and mechanical computation, by adapting to different input and external stimuli.  Recently, dome-patterned metamaterials consisting of patterned array reconfigurable bistable units had gained interest in the scientific community due to their capability of exhibiting different energy minima, hierarchical multistability, and a high dependency in unit cell interactions.   As the unit cell architecture can be reversibly inverted at the local scale, programmable multistable shapes and tunable mechanical responses at the global scale are generated due to the local prestress and interaction between each unit. This allows the structure to strongly depend on the unit inversion order and interaction between unit cells, enabling it to exhibit multiple global stable states, including high-energy frustrated states. This work investigates the interrelations between unit cells and spatial arrangement in the hierarchical multistable behavior of patterned metasheets. We examine the possible number of global states by utilizing group theory and isolating the unique pattern configuration to count the coexisting states.

Moreover, we explore the spectral properties of the connectivity matrix to unveil the possible coexisting states, strain alignment, orientation, and its connection to the global shapes. We show how the order of prestressing controls the ability to reach these global states and that, in many cases, it does not commute. This sheds further light on the mechanics of hierarchical multistable metasheets, thereby enabling their utilization in applications such as mechanical computation.