Topology mechanics in axially periodic trusses

Axially periodic trusses are quasi-one-dimensional networks whose zero energy modes (ZMs) correspond to complex flexural and twisting deformations within their three-dimensional embedding space that can be utilized for the actuation of mechanical structures and soft robotics. In this talk, we discuss how such trusses can be programmed to exhibit topologically protected boundary and interface ZMs and states of self stress (SSSs), which might be used to amplify mechanical response or localize stress, when the number of constraints matches the number of degrees of freedom. We distinguish between two topological classes of trusses: (1) When the coordination number is the same on every vertex, the truss generically possesses a self-duality that pairs ZMs localized to opposite boundaries, but permits the existence of novel interface ZMs between different topological phases. (2) When the coordination number varies between vertices, the aforementioned self-duality is broken and the truss can be polarized so that one end possesses an excess of ZMs. We discuss the interplay between this topological polarization with gapless translational and rotational ZMs at zero wavenumber in the one-dimensional Brillouin zone.