Combinatorial Design of Floppy Modes and Frustrated Loops in Metamaterials

Metamaterials are a promising platform for a range of applications, from shock absorption to mechanical computing. These functionalities typically rely on floppy modes or mechanically frustrated loops, both of which are difficult to design. In particular, how to design multiple modes or loops with target deformations remains an open problem. We introduce a combinatorial approach that allows to create an arbitrarily large number of floppy modes and frustrated loops. The design freedom of the mode shapes enables to easily introduce kinematic incompatibility to turn them into frustrated loops. We demonstrate that floppy modes can be sequentially buckled by using a specific instance of elastoplastic buckling. We utilize our combinatorial floppy chains and frustrated loops to achieve matrix-vector multiplication in materia. Our findings bring about new principles for the design and use of floppiness and frustration in soft matter and metamaterials.