Localizing softness and stress along loops in three-dimensional topological metamaterials

Topological states can be used to control the mechanical properties of a material along an edge or around a localized defect. The rigidity of elastic networks is characterized by a topological invariant called the polarization; materials with a uniform polarization display a dramatic range of edge softnesses depending on the orientation of the polarization relative to the terminating surface. However, in all three-dimensional mechanical metamaterials proposed to date, topological modes are mixed with bulk soft modes that organize themselves in Weyl loops. Here, we report the design of a 3D topological metamaterial without Weyl lines and with a uniform polarization that leads to an asymmetry between the number of soft modes on opposing surfaces. We use this construction to localize topological soft modes in interior regions via dislocation lines, which are unique to three dimensions. We derive a general formula that relates the difference in the number of soft modes and states of self-stress localized along the dislocation to the handedness of the vector triad formed by lattice polarization, Burgers vector, and dislocation-line direction. Our findings suggest a novel strategy for programming failure and softness localized along lines in 3D, while avoiding extended soft Weyl modes.