  Competing ground states may lead to topologically constrained excitations
such as domain walls or quasiparticles, which govern metastable states and
their dynamics. Domain walls and more exotic topological excitations are well
studied in magnetic systems such as artificial spin ice, in which nanoscale
magnetic dipoles are placed on geometrically frustrated lattices, giving rise
to highly degenerate ground states. We propose a mechanical spin-ice
constructed from a lattice of floppy, bistable square unit cells. We compare
the domain wall excitations that arise in this metamaterial to their magnetic
counterparts, finding that new behaviors emerge in this overdamped mechanical
system. By tuning the ratios of the internal elements of the unit cell, we
control the curvature and propagation speed of internal domain walls. We change
the domain wall morphology from a binary, strictly spin-like regime, to a more
continuous, elastic regime. In the elastic regime, we inject, manipulate, and
expel domain walls via textured forcing at the boundaries. The system exhibits
dynamical hysteresis, and we find a first-order dynamical transition as a
function of the driving frequency. We demonstrate a forcing protocol that
produces multiple, topologically-distinct steady cycles, which are protected by
the differences in their internal domain wall arrangements. These distinct
steady cycles rapidly proliferate as the complexity of the applied forcing
texture is increased, thus suggesting that such mechanical systems could serve
as useful model systems to study multistability, glassiness, and memory in
materials.
