Geometric Frustration in Non-Periodic Mechanical Metamaterials

We study geometric frustration in two-dimensional lattice-based mechanical metamaterials comprised of anisotropic triangular building blocks T, where each one possesses a nontrivial floppy mode of deformation. When each triangle is oriented randomly neighboring triangles typically cannot deform self-consistently. On one hand, we analyze the conditions under which a non-periodic packing of these blocks form compatible, frustration-free large-scale structures, i.e., structures that exhibit a global floppy mode that is compatible with the local deformations of each T. By mapping to an antiferromagnetic Ising model, we find an extensive number of possibilities to construct a compatible structure: Ω₀~exp(T). On the other hand, we study incompatible metamaterials in detail and we reveal two distinct types of source of frustration (defects) which either highly localize the frustrated region to a small and finite domain (local defects) or cause delocalized and long-ranged multi-stable conflicts (topological defects) whose multi-stability scales as Ω~exp(T^1/2). We further investigate the mechanical consequences of topological defects by identifying the corresponding states of self stresses and relating these to the mechanical response of the metamaterial to an external boundary load.