Rigidity percolation and geometric information in floppy origami

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.

Heterogeneous origami-architected materials with variable stiffness

“…Therefore, it is crucial that we identify how the tessellation changes its shape, regardless of which mechanical property we would tune. To quantify this unique morphological transformability of the origami tessellation, we characterize these various transformed configurations as geometrically stored information 28 . Here, we employ the Shannon information entropy S to quantify the geometric information capacity (i.e., number of configurations).…”

Deterministic and stochastic control of kirigami topology

“…For example, according to Fig. 3J, if we remove redundant links (brown) and add links that reduce the internal DoF by 1 (red), we reduce the total DoF by 1 while keeping the total number of links the same, thereby changing the DoF information using the same amount of materials; indeed this might allow for click kirigami (i.e., with reversible links) to be a substrate for mechanical information storage, similar to origami (20).…”

“…These approaches to controlling kirigami provide guidelines for the design of this class of mechanical metamaterials. of both connectivity and mechanical properties in an exquisitely sensitive way, similar, in some aspects, to rigidity percolation in both planar networks and origami (16)(17)(18)(19)(20). (Here we note that the "rigidity" represents the DoF [or number of floppy modes] introduced above, which is different from the "rigidity percolation" transition that describes a global change in the network; e.g., see Fig.…”

“…We have seen that, when links are added to the system, they do one of three things: change the total DoF, change the internal DoF, or are simply redundant. This notion is generic to a number of different systems made of discrete components with constraints; indeed, recently, we showed that this is also true for origami (20). To address the question here, instead of randomly generating link patterns, we start with no links and add links one by one.…”

Non-Euclidean origami

“…Origami provides a vast space to design novel mechanical metamaterials and folding devices [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The exceptional geometrical, shape-shifting, and mechanical functionalities of these systems ultimately spring from the nonlinear folding motions of the building blocks of origami [1][2][3][4][5][6][7][8][9][10][11][12][13][14][20][21][22][23][24][25][26][27][28][29][30]. These building blocks are n-vertices-units where n straight folds connected to n rigid plates meet at a point [1,2,4,5,…”