  In this Letter, we introduce a paradigm to realize magneto-mechanical
metastructures inspired by multi-layer 2D materials, such as graphene bilayers.
The metastructures are intended to capture two aspects of their nanoscale
counterparts. One is the multi-layer geometry, which is implemented by stacking
hexagonal lattice sheets. The other is the landscape of weak inter-layer
forces, which is mimicked by the interactions between pairs of magnets located
at corresponding lattice sites on adjacent layers. We illustrate the potential
of this paradigm through a three-layer prototype. The two rigid outer lattices
serve as control layers, while the thin inner layer is free to experience
flexural motion under the confining action of the magnetic forces exchanged
with the outer ones, thus behaving as a lattice on elastic foundation. The
inner layer is free to rotate relatively to the others, giving rise to a rich
spectrum of inter-layer interaction patterns. Our objective is to determine how
the dynamical response can be tuned by changing the twist angle between the
layers. Specifically, we demonstrate experimentally that switching between
different stacking patterns has profound consequences on the phonon landscape,
opening and closing bandgaps in different frequency regimes.
