Abstract
We present a novel cellular metamaterial constructed from Origami building blocks based on Miuraori 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 fullyfolded configurations, auxeticity (i.e., negative Poisson’s ratio), bistability, and selflocking of Origami building blocks to construct loadbearing 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.
Introduction
Origami, the ancient Japanese art of paper folding, relies on seemingly straightforward operations of concerted folding of a flat sheet of paper to produce incredibly complicated geometrical objects. This relatively simple control of topology makes Origami an important conceptual paradigm for deployable structures across a wide spectrum of applications. This includes several recent demonstrations in areas as diverse as deployable solar panels^{1,2}, foldcore sandwich panels^{3,4}, threedimensional (3D) cellladen microstructures^{5}, flexible medical stents^{6}, flexible electronics^{7}, soft pneumatic actuators^{8}, and selffolding robots and structures^{9,10,11}. Furthermore, periodic cellular metamaterials have been recently designed by assembling foldable Origami units (i.e., sheets or tubes) which tessellate to fill the 3D space^{12,13,14,15,16,17}. In addition, Origami has found applications in designing mechanical metamaterials with tunable stiffness, auxeticity, bistability, load bearing capacity and selffolding features^{14,15,18,19,20,21,22}.
Although an Origami construction relies on a mechanically simple folding operation, discovering the exact sequence of folds for a desired behavior is a combinatorically intractable problem^{23,24,25}. In this context, simplification is possible through an intricate coupling of topology and mechanical compatibility to design periodic fold sequence that can be repeated to create such Origami^{26,27}. An example is the pioneering work of Tachi and Miura^{13}, who introduced a type of rigid Origami based on the previouslyproposed Miuraori fold^{28}. Miuraori is a single degree of freedom (DOF) rigidfoldable Origami shown in Fig. 1(a) – left image. The four crease lines of Miuraori which result in one mountain and three valley folds define four identical parallelograms with adjacent sides defining an acute angle, α [shown in Fig. 1(a) – left image]. As the flat sheet deforms, these parallelograms become inclined to each other which can be quantified in terms of dihedral angles, , , or the angle between the mountain and front valley folding lines, . Due to the geometrical constraints, only one of these angles (θ, ξ, or β) is independent and can then be used to represent the single DOF of the system in analysis. For example, β and ξ can be expressed in terms of θ, and the constant angle, α, using the following relationships [see Supporting Information for details]:
Putting Miuraori units next to each other results in a Miuraori sheet construction while retaining its single DOF properties and rigidfoldability. Stacking and bonding Miuraori sheets along fold lines are shown to form cellular metamaterials with a single DOF that can be machined into any desired shape while preserving its folding motion^{14,29}.
In this work, we propose a new class of Origamibased cellular metamaterials with a wide range of interesting properties such as auxeticity, bistability, foldability, and selflocking. We start our design with putting together four Miuraori folds as shown in Fig. 1(a) – middle image. First, two Miuraori units were positioned in a zigzag pattern, then mirrored to form a symmetric structure, preserving the single DOF, inherent to the original Miuraori fold. Based on this design, we fold a single sheet of paper to construct a ‘firstorder element’ that will be used in developing the Origamibased cellular metamaterial, Fig. 1(a) – right image. It is noteworthy that folding of the first order element, for example by changing θ, results in change in its overall length; however, the left and right parts of the element stay aligned, independent of the folding level.
Firstorder elements can be attached together in three different ways, shown in Fig. 1(b), to make a ‘secondorder element’. From these three configurations, only the configuration shown on the right can be made by folding a single sheet of paper, and the other two configurations can be constructed by attaching the two firstorder elements. The angle between the two segments in each secondorder element is denoted by γ_{1}, γ_{2}, and γ_{3}, which can be calculated as 180° − β, 180° − β, and β, respectively (recall from Fig. 1(a) that β is an angle varying between 180° − 2α and 180°). Considering γ_{1}, γ_{2}, and γ_{3} as internal angles, these secondorder elements can be connected to generate contiguous geometrically closedloop elements with many different topologies with the following geometrical constraints: 1. Secondorder elements with γ_{1} and γ_{2} cannot be adjacent, 2. The two sides of the secondorder element with γ_{3} cannot be connected to two identical elements with γ_{1} or γ_{2}. Note that ignoring these geometrical constraints will result in closedloop elements with at least one external angle with γ_{1} or γ_{2} or γ_{3} value (i.e., closedloop elements with at least one internal angle not equal to γ_{1} or γ_{2} or γ_{3}). Figure 1(c) shows three possible quadrangular configurations that satisfy above constraints.
We now prove that from all possible closeloop elements only one arrangement leads to a rigidfoldable geometry. For each closedloop element with n sides, the summation of all internal angles must be equal to 180° × (n − 2), where n is the number of firstorder elements used to construct the closedloop element. Denoting m_{i} (i = 1, 2, 3) as the number of γ_{i} (i = 1, 2, 3) angles (i.e., n = m_{1} + m_{2} + m_{3}) yields the following geometrical relationship:
To achieve a foldable configuration, the left hand side of Equation (2) must be independent of the folding variable, β (note that the right hand side of the equation is a constant and independent of β). This yields m_{1} + m_{2} = 2 and m_{3} = 2, meaning that the only possible foldable configuration is a ‘quadrangle’ (n = 4). The examples provided in Fig. 1(c) are the only configurations that satisfy the Equation (2). The left and middle configurations can only built for β = 90°, while the right configuration can be built for any value of . This means that the left and middle configurations are rigid and the only possible foldable polygon is the jigsawpuzzlelike unit cell highlighted in green (see Supporting Information for further discussions on the rigidity of unit cells). All other possible configurations of triangular, quadrilateral, and hexagonal closedloop elements (i.e., the only 2D shapes which can individually tessellate the 2D space to form periodic geometries), formed by different types of secondorder elements introduced in Fig. 1(b), are given in Fig. 2. Note that all these elements are rigid (i.e., nonfoldable), since they don’t satisfy Equation (2), however, they can be used as building blocks to construct rigid tessellations such as the wellknown ‘Kagome’ structure made from triangular and hexagonal elements (see Supplementary Fig. S2 for an illustration of the structure).
It is essential to employ a connecting mechanism to link the adjacent unit cells of a lattice structure together, to form the final configuration of the system. An example of this mechanism is using an adhesive material to connect the unit cells together, however, this may affect the foldability of the structure by restricting degrees of freedom of the system, which will definitely alter the geometrical and mechanical properties of the final assembly. Here, we introduce an embedded selflocking mechanism into the proposed foldable unit, bonding the adjacent units together, which originates from the locking of firstorder elements as shown in Fig. 3(a). To ensure fitting of one firstorder element into another, each element must have a folding level corresponding to β > 90°. Once a contact is established between the two elements, selflocking can manifest by decreasing the folding angle to β < 90°, as for example is achieved in Fig. 3(a) – right image, by applying an outofplane compression.
The foldable closedloop element (i.e., Fig. 1(c) – right image) can be stacked in the outofplane direction to create a foldable tubular topology, which then can be used as building blocks to construct a cellular metamaterial, Fig. 3(b). The selflocking feature of the firstorder elements described above gets transferred to these building blocks and similarly gets activated for folding levels with β < 90°. Note that this locked state would impose effective contact strength between the building blocks in addition to simple frictional assembly. To this end we subjected a prototype, made of paper, to tension, when in locked and unlocked states, Fig. 3(c) (see Supporting Information for details on the experiments). When in the unlocked state, the structure exhibits no force resistance [i.e., force ~ 0 (N)], while in the locked state the structure shows noticeable resisting force [i.e., force ~ 35 (N)] before locking fails (see Supporting Information and Movie). Note that the resisting force strongly depends on folding level as well as the mechanical properties (i.e., elasticity) of the parent material which the plates are made of. However, the main goal of these experiments was to demonstrate the effect of the embedded selflocking mechanism on the structural resistance against the applied inplane tensile load by comparing their resisting force in unlocked versus locked configurations. In theory, since the plates are assumed to be rigid, the resisting force will be infinite in the locked configuration.
The behavior and properties of the cellular metamaterial, which exhibits periodicity in both inplane as well as outofplane directions can be analytically evaluated by assuming an infinite repetition of a representative volume element (i.e., RVE; same as the closedloop element) of the cellular metamaterial, Fig. 4(a) – left and middle images. Thus, we investigate the kinematics and kinetics of the cellular metamaterial by analyzing the closedloop element during folding. Figure 4(a) shows top and side views of the closedloop element as well as the geometrical characteristics of the constituting firstorder element introduced earlier. The inplane diagonals, D_{1} and D_{2}, and outofplane height, H, of the closedloop element at an arbitrary level of folding, illustrated in Fig. 4(a), are given in terms of the geometry of the underlying Miuraori unit as (see Supporting Information for details):
Note that D_{1} and D_{2} are diagonals of a diamond (i.e., the closedloop element) and therefore always perpendicular to each other. In order to quantify the folding process, we define a nondimensional parameter called ‘folding ratio’ as, , which varies from 0% (i.e., θ = 180°) to 100% (i.e., θ = 0°). In other words, 0% and 100% folding ratios correspond to two fullyfolded configurations of the proposed construction.
The crosssectional area of the closedloop element, S, defined as the area of the polygon formed by intersecting the closedloop element with a plane normal to its height, is constant through the height of the closedloop element. The volume of the closedloop element, V, is the volume bounded by the constituting firstorder elements. Figure 4(b) depicts the variation of the crosssectional area and volume of the closedloop element (respectively normalized by a^{2} and a^{3}) as functions of the folding ratio, respectively, presented for four different values of α ranging from 30° to 75°. The results are plotted using the analytical expressions of area and volume derived in the Supporting Information. As the folding ratio increases, the normalized area rises from zero (i.e., fullyfolded configuration) up to a turning point, and then decreases due to the auxetic behavior of closedloop element in both diagonal directions (will be discussed later). This is then followed by a plateau regime as the closedloop element reaches the other fullyfolded configuration. The critical folding ratio associated with the turning point decreases significantly for higher values of α. Similar behavior is observed for the variations of the normalized volume, except the fact that at 100% folding ratio, the volume becomes zero due to the fullyfolded configuration of the closedloop element.
Next, for an uniaxial outofplane load, we calculate the Poisson’s ratio of the closedloop element in D_{1} and D_{2} directions (since they are always perpendicular to each other), defined as , where i = 1 or 2. Differentiating Equation (3) with respect to the folding angles and plugging the results into the above equations yield the following closedform expressions for Poisson’s ratios:
It is noteworthy that although these formulations were derived for a single closedloop unit, they still hold true for the infinite periodic metamaterial. This is due to the fact that the calculations were performed on an RVE, which can be tessellated in diagonal (i.e., D_{1} and D_{2}) and outofplane directions [as the “lattice vectors”]^{30} to form the final configuration of the metamaterial.
Figure 4(c) shows the dependence of Poisson’s ratio on the folding ratio in two orthogonal inplane directions (i.e., D_{1} and D_{2}), for four different values of α ranging from 30° to 75°. is negative for the entire range of folding ratio and α, with a significantly pronounced auxetic response at greater values of α. In contrast, has a positive infinity value at 0% folding ratio [theoretically, the denominator of becomes zero at 0% folding ratio, see Equation (4)], which then reduces to 0 at 100% folding ratio. For , this involves exhibiting a negative Poisson’s ratio after a certain folding ratio. Insets in Fig. 4(b,c) illustarete the effect of changing α in the geometry and folding procedure of unitcell. Figure 4(d) shows folding of a sample closedloop element demonstrated under loading in outofplane compression and inplane stretching along the direction of D_{1} (see Supporting Information for details). For this sample, α = 60° and the fullyfolded states are achieved at β = 180° − 2α = 60° (or θ = 0°) and β = 180° (θ = 180°), as shown under outofplane compression and inplane stretching experiments, respectively. Note that the closedloop element, shown in Fig. 4(d) tessellates the 3D space regardless of folding level – see Supplementary Fig. S3.
Next, we investigated the force required to attain a desired level of folding for each building block of the cellular metamaterial under two loading directions (i.e., outofplane and inplane). We assumed that each building block is made of rigid plates, connected together at straight creases modeled as linear torsional springs^{15} with spring constant per unit crease length of k(N). Also, as mentioned earlier, we idealized a building block of the cellular metamaterial as an infinite array of closedloop elements stacked on top of each other, and analyzed the RVE. In the Supporting Information, we derived the following analytical expressions for the folding force on the RVE under outofplane and inplane loadings using the principle of minimum total potential energy:
where F_{out–of–plane} and F_{in–plane} denote the folding forces for outofplane and inplane loading directions, respectively, θ_{0} and ξ_{0} are the free angles of horizontal and inclined torsional springs, respectively (i.e., the angles at which no potential energy is stored in the springs), and dξ/dθ and dβ/dθ can be calculated using Equation (1).
Figure 5(a) shows the plots of normalized outofplane and inplane folding forces, versus the folding ratio for different values of α, while the free angle of the torsional springs is kept constant as θ_{0} = 90° (i.e., 50% folding ratio; ξ_{0} can be calculated from Equation (1) by plugging θ_{0} instead of θ). In addition, for α = 60°, we plotted the normalized outofplane and inplane folding forces versus the folding ratio for a set of θ_{0} varying between the extreme cases, θ_{0} = 0° and θ_{0} = 180°, Fig. 5(b). The results show a socalled “bistable“ behavior for in outofplane loading, and for under inplane loading. For example, the sample with θ_{0} = 170° exhibits local extremum points at 20% (local maximum) and 66% (local minimum) folding ratios when subjected to outofplane loading. This reveals the two stable configurations – one at the initial state (i.e., F/k = 0) where the folding ratio is 5.5%, and – the other one at the local minimum point at 66% folding ratio. We should note that the structure will go to the “local minimum” point (i.e., 66% folding ratio) only if the load is still there (i.e., a preload), otherwise, if we remove the load, the structure will always go back to its stable state at zero force (i.e., 5.5% folding ratio) after going through a “snapthrough”^{29}. This bistability in the response highlights the potential of the proposed cellular metamaterials for energy absorption, energy harvesting, and impact mitigation applications^{31,32,33}. Next, we compare outofplane and inplane loading responses for an RVE with α = 60° and θ_{0} = 90°, see Fig. 5(c). These calculations show that except for folding ratios greater than 78%, the inplane force associated for achieving a specific folding ratio is lower than the outofplane force for the same value of folding ratio. This means that for folding ratios smaller than 78%, it is easier to fold the structure under inplane loading (compared to an outofplane loading), while the opposite is true for folding ratios greater than 78%. Additionally, the inset of the figure shows that the folding ratio corresponding to the point at which the two curves meet [shown by a hollow circle in Fig. 5(c)], decreases with increasing α, making the outofplane force smaller than the inplane force for a wider span of the folding ratio.
In summary, in this paper we propose an Origamibased paradigm of constructing cellular materials which are capable of undergoing large reversible deformation while exhibiting highly nonlinear auxeticity, bistability and topological locking. Particularly, the locking phenomena is used as a platform for scaling up these structures in a systematic modular fashion into larger cellular structures with single force activation without taking recourse to any special structural or surface modifications. The selflocking is achieved using an applied force on the structure. In the Supporting Information we discussed the force required for achieving the initial selflocking under different loading types and geometrical parameters. Thus, in summary, this present work sets forth an important avenue of novel cellular metamaterial design based on both selfsimilar and selflocking assembly.
Methods
Fabrication of the Origamibased elements and structures
All the elements and structures were fabricated out of paper (thickness ~ 0.01 in), where the cuts and crease lines were made using a Silhouette CAMEO cutting machine (Silhouette America, Inc., Lindon, UT).
Tensile tests
We first subjected the prototype under outofplane compression using an Instron 5582 testing machine with a 1 kN load cell. Next, we manually applied inplane tension using a forcegauge to directly measure the tensile force. The experiments were videotaped in order to qualitatively compare the results between the unlocked and locked states of the structure (see Supporting Information video).
Additional Information
How to cite this article: Kamrava, S. et al. Origamibased cellular metamaterial with auxetic, bistable, and selflocking properties. Sci. Rep. 7, 46046; doi: 10.1038/srep46046 (2017).
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Acknowledgements
The authors thank Prof. Hamid NayebHashemi for fruitful discussion. This report was made possible by a NPRP award [NPRP 78822326] from the Qatar National Research Fund (a member of the Qatar Foundation). The statements herein are solely the responsibility of the authors.
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S.K., D.M., and H.E. performed the analytical works and experiments. All authors analyzed the results and contributed in writing the manuscript.
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Kamrava, S., Mousanezhad, D., Ebrahimi, H. et al. Origamibased cellular metamaterial with auxetic, bistable, and selflocking properties. Sci Rep 7, 46046 (2017). https://doi.org/10.1038/srep46046
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DOI: https://doi.org/10.1038/srep46046
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