Doing Math with Layered Combinatorial Mechanical Metamaterials

We construct metamaterials made of two types of anisotropic building blocks. The discrete degrees of freedom of selecting which of the two is positioned at each location in the lattice and how it is orientated enable us to design the spatial form of the system’s floppy modes. These modes may be specifically actuated, or alternatively, sequentially excited by plastic deformations under uniform compression. In two dimensions, we utilize the frustration in odd loops within the lattice to perform simple arithmetic operations, such that the magnitudes of output displacements are the sum or difference of two input displacements. More complicated operations, involving multiple inputs and outputs, require chains of motion in the system to cross without touching each other. We disconnect such chains by going to the third dimension, and we consider layered metamaterials, in which we selectively control the coupling between the in-plane motion in neighboring layers. We demonstrate how this allows, for instance, to multiply a vectorial input by a 3x3 matrix.