Defect Positioning in Combinatorial Metamaterials

Combinatorial mechanical metamaterials are made of anisotropic, flexible blocks, such that multiple metamaterials may be constructed using a single block type, and the system's response depends on the frustration (or its absence) due to the mutual orientations of the blocks within the lattice. Specifically, any minimal loop of blocks that may not simultaneously deform in their softest mode defines a mechanical defect at the vertex (in two dimensions) or edge (in three dimensions) that the loop encircles. Defects stiffen the metamaterial, and allow to design the spatial patterns of stress and deformation as the system is externally loaded. We study the ability to place defects at arbitrary positions in metamaterials made of a family of block types that we recently introduced for the square, honeycomb, and cubic lattices. Alongside blocks for which we show that any defect configuration is possible, we identify situations in which not all sets are realizable as defects. One of the restrictions is that in three dimensions, defected edges form closed curves. Even in cases when not all geometries of defect lines are possible, we show how to produce defect lines of arbitrary knottedness.