Rigidity percolation and geometric information in floppy origami

Significance

Origami structures are a particularly interesting class of thin-sheet–based mechanical metamaterials that rely on folds for their morphology and mechanical properties. Here, we study how excess folds in a simple origami pattern control the rigidity of the structure. Furthermore, we show that the onset of geometrical cooperativity in the system allows for information storage in a scale-free manner. Understanding how mechanical rigidity and geometric information can be simultaneously controlled in folded sheets has implications for structures on a range of scales, from graphene to architecture.

Abstract

Origami structures with a large number of excess folds are capable of storing distinguishable geometric states that are energetically equivalent. As the number of excess folds is reduced, the system has fewer equivalent states and can eventually become rigid. We quantify this transition from a floppy to a rigid state as a function of the presence of folding constraints in a classic origami tessellation, Miura-ori. We show that in a fully triangulated Miura-ori that is maximally floppy, adding constraints via the elimination of diagonal folds in the quads decreases the number of degrees of freedom in the system, first linearly and then nonlinearly. In the nonlinear regime, mechanical cooperativity sets in via a redundancy in the assignment of constraints, and the degrees of freedom depend on constraint density in a scale-invariant manner. A percolation transition in the redundancy in the constraints as a function of constraint density suggests how excess folds in an origami structure can be used to store geometric information in a scale-invariant way.