  Flexible mechanical metamaterials possess repeating structural motifs that
imbue them with novel, exciting properties including programmability, anomalous
elastic moduli and nonlinear and robust response. We address such structures
via micromorphic continuum elasticity, which allows highly nonuniform
deformations (missed in conventional elasticity) within unit cells that
nevertheless vary smoothly between cells. We show that the bulk microstructure
gives rise to boundary elastic terms. Discrete lattice theories have shown that
critically coordinated structures possess a topological invariant which
determines the placement of low-energy modes on edges of such a system. We show
that in continuum systems a new topological invariant emerges which relates the
difference in the number of such modes between two opposing edges. Guided by
the continuum limit of the lattice structures, we identify macroscopic
experimental observables for these topological properties that may be observed
independently on a new length scale above that of the microstructure.
