Origami-based cellular metamaterial with auxetic, bistable, and self-locking properties

We present a novel cellular metamaterial constructed from Origami building blocks based on Miura-ori fold. The proposed cellular metamaterial exhibits unusual properties some of which stemming from the inherent properties of its Origami building blocks, and others manifesting due to its unique geometrical construction and architecture. These properties include foldability with two fully-folded configurations, auxeticity (i.e., negative Poisson’s ratio), bistability, and self-locking of Origami building blocks to construct load-bearing cellular metamaterials. The kinematics and force response of the cellular metamaterial during folding were studied to investigate the underlying mechanisms resulting in its unique properties using analytical modeling and experiments.

where is the edge length of the four identical parallelograms forming the Miura-ori unit, and , , and ⃗ are unit vectors along the x, y, and z directions, respectively. Now, the following expression defines the angle between the vectors ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ : Now, considering the isosceles triangles, ABF and AED [see Figure S1(b)], the following relations can be obtained for the angles and : where ̅̅̅̅ is the length of the edge (similarly for other edges). We should note that ̅̅̅̅ = ̅̅̅̅ , and ̅̅̅̅ = ̅̅̅̅ , which by substituting into supplementary Equation 4 will result in the following: Finally, Figure S1(a) shows that = 180 − 2 , which by using supplementary Equations 3 and 5 will result in the following equation for :

Miura-ori unit: angle
In order to obtain a closed-form expression for the angle, , as a function of , we first translate (with no rotations) the coordinate system of Figure S1(a) from point A to point M, see Figure S1(c). Note that the angle is basically the angle between vectors ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ . We now begin the analysis by obtaining the which can further be simplified (by using supplementary Equations 5 and 6) into the following: = cos −1 ( 1 − (1 + cos 2 ) sin 2 ( 2 ⁄ ) 1 − sin 2 sin 2 ( 2 ⁄ ) ) (10)

Closed-loop elements foldability vs. rigidity
As we mentioned in the manuscript, the only possible closed-loop element (i.e., polygon) with rigidfoldability property, formed by different types of connection introduced in Figure 1(b), is the one highlighted in green in Figure 1(c)right image. For instance, here, we will prove that the quadrilateral element shown in Figure S2 (a), is completely rigid, though it does not violate the geometrical constraint on internal angles, presented by Equation (2) of the manuscript.
We begin the analysis by noting that the length of the edges of this closed-loop element, ̅̅̅̅ , ̅̅̅̅ , ̅̅̅̅ , and ̅̅̅̅ , can be obtained as the following [see Figure where and are defined in Figure 4(a). Furthermore, the following relation must hold for the edges and internal angles of the quadrangle, :  Figure 2 are rigid, however, they can be used as building blocks to construct rigid tessellations such as the well-known 'Kagome' structure made from triangular and hexagonal elements, which is shown in Figure S2 (b).

Closed-Loop element cross-sectional area and volume
Cross Finally, the expressions derived for cross-sectional area and volume of the closed-loop element are normalized with respect to 2 and 3 , respectively.

Periodic unit cells
The foldable unit cell remains periodic in different folding levels. Figure S3 shows the tessellation of four unit cells, while they are still periodic, in folding ratios 100%, 35%, 14%, and 9%.

Force-folding relations
Here, we calculate both out-of-plane and in-plane folding forces required to attain a desired level of folding.
We first assume that the structure is made of rigid plates, held together at the crease lines which are modeled as linear torsional springs with spring constant per unit length of ( ). We then idealize our tubular On studying the self-locking behavior of closed-loop element (and tubular elements), here we study the critical force at which self-locking first occur in the structure. To this end, we first define the self-locking critical force as the force (in-plane or out-of-plane) required to initiate self-locking behavior (i.e., = 90°) in the structure. First of all, supplementary Equation 6 requires that for self-locking behavior to happen in the structure, the angle, , must be greater than 45 o (this can be obtained by substituting = 90° in supplementary Equation 6). Figure S4 shows the contour plots of the self-locking critical force as a function of 0 and for out-of-plane (left) and in-plane (right) loadings. Note that for loadings greater than the critical force, the structure will go into the locked configuration; otherwise it will remain unlocked.

Experiments
We first fabricated the prototypes of closed-loop and tubular elements using papers (thickness ~0.01in). All cuts and perforations were made using a Silhouette CAMEO cutting machine (Silhouette America, Inc., Lindon, UT).
In order to demonstrate the self-locking behavior of the Origami-based cellular metamaterial we proposed in this paper, two tubular units (each unit is made of four closed-loop elements, stacked on top of each other) were used to perform uniaxial tensile tests for unlocked and locked configurations. For the case of locked configuration, we subjected the prototype under out-of-plane compression using an Instron 5582 testing machine with a 1kN load cell, while for the case of unlocked configuration, no loading was imposed to the structure in the vertical direction. Then, the structure was fixed at one end, while the other end was pulled uniaxially using a digital force gauge (which was implemented to measure the required in-plane force to detach the tubular elements), Figure S5. The results indicate that the required in-plane force to detach the elements is almost 0 N in the case of unlocked configuration, while it is around 35 N for the case of locked configuration (at an arbitrary level of folding). This remarkable difference in detaching force demonstrates the importance of self-locking feature of our proposed Origami-based cellular metamaterial.
See the Supplementary Movie titled "Self-locking feature of the proposed Origami-based cellular metamaterial".

Supplementary Movie legend
Self-locking is achieved by applying an out-of-plane compression which keeps the Origami-based cellular metamaterial at certain folding ratios in which the self-locking behavior is guaranteed. In this Movie, we compare unlocked and locked configurations of the Origami-based cellular metamaterial in term of resistance to separating in-plane force. Figure S1. Geometrical characteristics of a Mira-ori fold at an arbitrary level of folding.