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Reentrant Origami-Based Metamaterials with Negative Poissons Ratio and Bistability

H. Yasuda1 and J. Yang2,*1Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, USA

2Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195, USA(Received 23 December 2014; published 5 May 2015)

We investigate the unique mechanical properties of reentrant 3D origami structures based on the Tachi-Miura polyhedron (TMP). We explore the potential usage as mechanical metamaterials that exhibit tunablenegative Poissons ratio and structural bistability simultaneously. We show analytically and experimentallythat the Poissons ratio changes from positive to negative and vice versa during its folding motion. Inaddition, we verify the bistable mechanism of the reentrant 3D TMP under rigid origami configurationswithout relying on the buckling motions of planar origami surfaces. This study forms a foundation indesigning and constructing TMP-based metamaterials in the form of bellowslike structures for engineeringapplications.

DOI: 10.1103/PhysRevLett.114.185502 PACS numbers: 81.05.Xj, 46.70.De

Origami is defined as the handcrafted art of paperfolding. In recent decades, origami has attracted significantinterest by mathematicians and engineers, not only becauseit stimulates intellectual curiosity, but also because it haslarge potential for engineering applications. One goodexample is the usage of origami patterns for the enhance-ment of structural bending rigidity for thin-walled cylin-drical structures [1]. By leveraging their compactness,origami structures are also employed for space applica-tions, such as space solar sails [2,3] and deployable solararrays [4]. It is not surprising that biological systemsexhibit origami patterns, e.g., tree leaves [5].Mechanical metamaterials are another topic of active

research in the scientific community nowadays. As acounterpart of electromagnetic metamaterials, mechanicalones are constructed in an ordered pattern of unit-cellelements to achieve unusual mechanical properties [6,7].Among them are negative Poissons ratio and controllableinstability of structures. For example, negative Poissonsratio has been exploited to manipulate wave propagationin reentrant cellular structures [8]. Previous studiesalso reported that structural instability can be used toachieve tailored damping characteristics in mechanicalmetamaterials [9,10].In this study, we adopt origami structures as a building

block of mechanical metamaterials to achieve simulta-neous negative Poissons effect and structural bistability.Specifically, we employ the Tachi-Miura polyhedron(TMP), which is a bellowslike 3D origami structure basedon Miura-ori cells [11,12] (Fig. 1). Lateral assembly ofMiura-ori cells in the form of 2D Miura-ori sheets has beenpreviously explored for the construction of metamaterials[13]. However, there have been limited efforts to study thecylindrical derivative of Miura-ori cells in the form of TMP[14]. In contrast to 2D origami structures such as Miura-orisheets and the waterbomb [15], the TMP holds a volume

that changes continuously from zero to a certain value, andthen returns to zero again at the ends of the folding motion[see Fig. 1(a)]. This implies that we can obtain a very largestroke from its folding motion, which is useful in designingactuators and impact absorbers. Also, compared to otherorigami-based cylindrical structures [1621], the TMP hasa unique feature of rigid foldability. That is, the deforma-tion takes place only along crease lines instead of relyingon the elasticity of materials. Therefore, the structure can

FIG. 1. (a) Folding motion of the Tachi-Miura polyhedron.(b) Folded TMP cell. (c) Top view of the TMP. (d) Flat front andrear sheets of TMP with the crease pattern consisting of mountainand valley folds. (e) Folded configuration of the front sheetcorresponding to the shaded areas in (c) and (d).

PRL 114, 185502 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending8 MAY 2015

0031-9007=15=114(18)=185502(5) 185502-1 2015 American Physical Society

consist of only rigid panels and hinges without incurringbending of planar origami surfaces.In this Letter, we first examine the kinematics of the

TMP to show the tunable characteristics of its Poissonsratio. We verify analytically and experimentally the auxeticeffect of volumetric 3D TMP prototypes in bilateraldirections, which is an improvement over the conventional2D origami structures with a single-directional negativePoissons effect [13,22,23]. Second, we investigate theforce-displacement relationship of the TMP structure toshow that it exhibits a bistable nature under the reentrantconfigurations. While previous studies investigatedstructural stability in other types of origami structures[15,2426], bistableyet foldable3D origami structureswithout introducing defects have not been reported yet.Last, a cellular structure consisting of the TMP cells isexplored to form origami-based metamaterials with a viewtoward potential engineering applications.We begin with characterizing the geometry of the TMP.

Figures 1(b) and 1(c) show the folded TMP cell in slantedand top views, respectively. This unit cell consists of twoflat sheets, whose geometry can be characterized by lengthparameters (l; m; d) and an inner angle of parallelogram ()[Fig. 1(d)]. The point Q is defined by the two crossingedges of the front and rear surfaces as shown in Fig. 1(c),which passes through the quarter line of the sheets [dashed-dotted line in Fig. 1(d)]. Accordingly, the half breadth(B=2) of the TMP cell corresponds to the distance betweenpoints O and Q along the y axis, and the half-width (W=2)is the distance between points O and R along the x axis[Figs. 1(c)]. The half height (H=2) of the TMP cell isalso illustrated in Fig. 1(b). Note that the TMP cell exhibitsa reentrant shape when the given geometrical angle isabove 45 [e.g., see the insets of Fig. 2(a)].To calculate W, B, and H under various folding

configurations, we consider a quarter model of the TMP[Fig. 1(e)], which corresponds to the dark colored area inFigs. 1(b)1(d). Here, the folding angles M and S arefunctions of (determined by the given geometry) and G(varies by the degree of folding). The mathematicalexpressions for these folding angles are described in theSupplemental Material [27]. It should be noted that whileM 0; 90 and S 0; 90, the range of is limitedto satisfy 2l d cot 2m cos 2 > 0. This is to avoidthe collision between points P and P0 during folding.Accordingly, G 0; 2. Based on the geometrydescribed in Fig. 1, W, B, and H are obtained as follows:

B 2m sin G d cos M;W 2l d

tan 2m cos G;

H 2d sin M:1

We investigate the Poissons ratios of the TMP bydefining them as

HB dB=BdH=H and HW

dW=WdH=H : 2

Differentiating Eq. (1) with respect to the folding anglesand plugging them into Eq. (2), we obtain the Poissonsratios as follows:

HB 4m tan cos Gcos2G=2 d

2m sin G d cos Msin M tan M;

HW 4m tan sin Gcos2G=22l d= tan 2m cos G

sin M tan M:


To verify this analytical expression, we fabricate threeprototypes of the TMP ( 30, 45, and 75) by usingpaper (see the Supplemental Material [27] for details). Thenumber of Miura-ori layers used in each configuration isN 7, and the characteristic lengths of the prototypes areidentical (l m 50 and d 30 mm). We conduct threemeasurements of B and W at each H, as we graduallychange the folding angle. We compare the measuredPoissons ratios with the analytical results from Eq. (3).Figure 2 shows the Poissons ratios as a function of a

folding ratio defined as 90 M=90. The Poissonsratio HW in the cases of 45, 30, and 70 are plotted inFig. 2(a), while the insets show the folded configurationsunder 70. We find HW is always negative regardlessof the folding ratio and . On the other hand, the Poissonsratio HB related to width B is positive in the initial foldingstage, and it approaches zero. It is notable that in thereentrant case (e.g., 70) HB becomes negative asshown in Fig. 2(b). As seen in the inset, we evidentlyobserve that B increases and then decreases as the foldingratio of the reentrant TMP increases. We find excellentagreement between the experimental and analytical results.The areal change of the TMP is also measured andcompared with the analytical predictions in theSupplemental Material [27].The analytical contour plot of HB as a function of

continuous and the folding ratio is shown in Fig. 2(c).If is above approximately 55, HB becomes negativeduring the folding motion. Figure 2(d) also shows thecontour plot of HB but d 60 mm. By choosing a certain angle [e.g., 70 as shown in the inset of Fig. 2(d)],we observe HB changes from positive to negative in theinitial folding stage, and then it becomes positive againaround 62%. The sign of HB changes multiple times inone folding motion. This is a unique feature of the TMPcompared to the conventional 2D Miura-ori cells, inwhich negative Poissons effect has been reported, butmultiple sign flips of the Poissons ratio have not beendiscovered [13,22,23]. Solving dB 0, we obtain theanalytical expression for the transition between positiveand negative HB:

PRL 114, 185502 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending8 MAY 2015


cos2M2m tand2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim tanm tan2dp

dtan2: 4

This boundary is plotted in a dashed curve in Figs. 2(c)a


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