  Periodic networks on the verge of mechanical instability, called Maxwell
lattices, are known to exhibit zero-frequency modes localized to their
boundaries. Topologically polarized Maxwell lattices, in particular, focus
these zero modes to one of their boundaries in a manner that is protected
against disorder by the reciprocal-space topology of the lattice's band
structure. Here, we introduce a class of mechanical bilayers as a model system
for designing topologically protected edge modes that couple in-plane
dilational and shearing modes to out-of-plane flexural modes, a paradigm that
we refer to as omnimodal polarization. While these structures exhibit a
high-dimensional design space that makes it difficult to predict the
topological polarization of generic geometries, we are able to identify a
family of mirror-symmetric bilayers that inherit the in-plane modal
localization of their constitutive monolayers whose topological polarization
can be determined analytically. Importantly, the coupling between the layers
results in the emergence of omnimodal polarization, whereby in-plane and
out-of-plane edge modes localize on the same edge. We demonstrate these
theoretical results by fabricating a mirror-symmetric, topologically polarized
kagome bilayer consisting of a network of elastic beams via additive
manufacturing and confirm this finite-frequency polarization via finite element
analysis and laser-vibrometry experiments.
