Architectured Floppy Modes in Combinatorial Metamaterials

Much of the functionality of mechanical metamaterials has to do with soft or floppy modes, which allow deformations with very small or even zero energetic cost. We combine two different anisotropic triangular building blocks – one with a single floppy mode and another with two floppy modes. The spatial distribution and mutual alignment of these blocks enable us to design the multiplicity and precise spatial structure of the cooperative floppy modes in the resulting metamaterial. We develop a theoretical framework for exactly constructing all the floppy modes of any such metamaterial. Our strategy allows us to design metamaterials with any number of floppy modes, ranging from zero, one or a few, up to an extensive number that scales linearly with system size. Most interestingly, we can design metamaterials that have an intermediate number of floppy modes that scales with the system surface, namely that their degeneracy scales sub-extensively with system size. Such modes are useful for instance for designing metamaterials with multiple textured functionalities or for concentrating stresses and deformations in multiple chosen regions within the metamaterial.