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April 3rd, 2015

SUPPLEMENTAL MATERIAL

Re-entrant origami-based metamaterials with negative Poisson’s ratio and bistability

H. Yasuda1 and J. Yang*2 1Department of Mechanical Engineering, University of Washington, Seattle, WA 98195

2Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195

1. Folding angles relationships

The Tachi-Miura Polyhedron consists of two sheets (one of the sheets is shown in Fig. S1(a)), and a building block for the TMP is the Miura-ori unit (Fig. S1(b,c)). In Fig. S1(c), we define the three different folding angles; θM, θS, and θG.

FIG. S1 (a) Tachi-Miura Polyhedron. (b) Miura-ori unit cell which is folded into (c).

There are relationships among these three angles as

0

0 0 0

tan tan cos2G

MA CCD CD

O C O C A Cθ

α θ= ⇔ = (S1)

1 1 1

1 2 1 2 1

sin sin cos2G

SB E B F B EB B B B B F

θα θ= ⇔ = (S2)

Taking a derivative of above two equations, we obtain

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22 tan cos sin2G

G M Md dθθ α θ θ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(S3)

1 cos sin sin2 2

GG S Sd dθθ α θ θ= − (S4)

Therefore

3cos cos sin1 2 2

2 sin sin cos sin

G GM

S G MS S

d d d

θ θθ

θ θ θα θ α θ

= − = (S5)

Also, the breadth (B), width (W), and height (H) of the TMP are described as follows:

B = 2msinθG + d cosθM (S6)

W = 2l + d

tanα+ 2mcosθG (S7)

sin MH Nd θ= (S8)

Taking a derivative of these equations, we obtain

( ) ( )2 cos sinG G M MdB m d d dθ θ θ θ= − (S9)

( )2 sin G GdW m dθ θ= − (S10)

( )cos M MdH Nd dθ θ= (S11)

Here, we defne the Poisson’s ratio of the TMP (νHB and νHW) as  

( )( )HB

dB BdH H

ν = − (S12)

( )( )HW

dW WdH H

ν = − (S13)

Substituting Eqs.(S5-S11) into Eqs.(S12) and (S13), we obtain

νHB =

4m tanα cosθG cos2 θG

2+ d

2msinθG + d cosθM

sinθM tanθM (S14)

νHW = −4m tanα sinθG cos2 θG

2

2l + dtanα

+ 2mcosθG

⎛⎝⎜

⎞⎠⎟

sinθM tanθM (S15)

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2. Experimental setup  

In order to verify analytical expression of the Poisson’s ratio, we fabricate three prototypes of the TMP (α = 30°, 45°, and 75°) by using paper (Strathmore 500 Bristol, 2-Ply, Plate Surface, 235–72) as shown in Fig. S2. Length parameters (l, m, d) = (50 mm, 50 mm, 30 mm) are chosen.

FIG. S2 Prototype of the TMP made of paper

Figure S3 shows the experimental setup to measure the cross-sectional area of the TMP. A glass plate is placed on the top surface of the TMP to control the height of the TMP, and a camera captures a digital image of the cross-sectional area of the TMP from above. Based on digital images from a camera, we measure the breadth (B), width (W), and cross-sectional area of the TMP with different height (H). To measure B and W, we use Image J software [1]. The measurement was conducted three times on each TMP prototype. In order to obtain the Poisson’s ratio from the experiment, we modify Eqs. (S12) and (S13). Substituting Eqs. (S5) and (S9-A11) into Eqs. (S12) and (S13), we obtain

( )( )

24 cos2 tan cos sin sin

cosG G M M

HBM

H m dB Nd

θ α θ θ θν

θ

+= (S16)

( )

24 sin 2 tan cos sincos

G G MHW

M

mHW Ndθ α θ θ

νθ

= − (S17)

Where θM is calculated from Eq.(S8), and θS and θG are calculated from Eqs. (S1) and (S2).

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FIG. S3 (a) Experimental setup. (b) Digital image taken from a camera mounted on top of a TMP prototype.

3. Cross-sectional area change results.

In addition to the measurement on B and H, we also measure the cross-sectional area of the TMP. Figure S4 shows the cross-sectional area change as a function of a folding ratio defined as (90°–θM)/90°. We have an excellent agreement between the analytical and experimental data. Note that the low folding ratios are difficult to achieve in experiments, since the TMP prototypes made of paper are initially folded at certain degrees.

FIG. S4 Cross-sectional area change of the TMP. Error bars indicate standard deviations.

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4. Force-displacement response

We model the crease lines of the TMP by using a torsional spring as shown in Fig. S5. Focusing on one main crease line (red line in Fig. S5(a)).

FIG. S5 Modeling of a crease line with a torsional spring. (a) TMP cell. Red line corresponds to (b).

Let MM and MS be bending moment along horizontal (related to θM) and inclined (related to θS) crease lines respectively, we assume that relationship between bending moment and angle is linear as follows:

M M = 2kθ θM −θM

(0)( ) (S18)

MS = 2kθ θS −θS

(0)( ) (S19)

where kθ is a spring constant, and θ(0)M and θ(0)

S are the initial folding angles for horizontal and inclined crease lines respectively.

By using the principle of virtual work, we the following equation:

Fδu = 2nM M MδθM + 2nS MSδθS (S20)

where nM = 8(N-1) and nS = 8N are the number of the main crease lines and sub-crease lines respectively. By applying variation to Eqs.(S1) and (S2) with respect to the folding angles, we obtain

22 tan cos sin2G

G M Mθ

δθ α θ δθ= − (S21)

1 cos sin sin2 2

GG S S

θδθ α θ δθ= − (S22)

Therefore

3cos cos sin1 2 22 sin sin cos sin

G GM

S G MS S

θ θθ

δθ δθ δθα θ α θ

= − = (S23)

Also, the height of the TMP is

0 sin MH u Nd θ− = (S24)

(a)xyz

(b)

kθθM

u

HH0

F

yz

Torsional spring

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Then

cos M Mu Ndδ θ δθ= − (S25)

Substituting Eqs.(S18), (S19), (S23) and (S25) into Eq.(S20), we obtain

( ) ( ){ } ( ){ }(0) (0)cos 2 2 2 2M M M M M M S S S SF Nd n k n kθ θθ δθ θ θ δθ θ θ δθ− = − + −

( ) ( )3

(0) (0)cos sin4 2

cos cos sin

GM

M M M S S SM S

kF n nNd

θ

θθ

θ θ θ θθ α θ

⎧ ⎫⎪ ⎪

= − − + −⎨ ⎬⎪ ⎪⎩ ⎭

( )( ) ( )3

(0) (0)cos sin4 28 1 8

cos cos sin

GM

M M S SM S

kF N NNd

θ

θθ

θ θ θ θθ α θ

⎧ ⎫⎪ ⎪

= − − − + −⎨ ⎬⎪ ⎪⎩ ⎭

Hence,

( ) ( ) ( )

3

(0) (0)cos sin32 1 2

cos cos sin

GM

M M S SM S

F Nk d Nθ

θθ

θ θ θ θθ α θ

⎧ ⎫⎪ ⎪−

= − − + −⎨ ⎬⎪ ⎪⎩ ⎭

(S26)

5. Multiple equilibrium states

Figure S6 shows the force-folding ratio relationship based on Eq.(S26), when (0) 80Mθ = °, 70α = ° , and N = 7. If the normalized force reaches the local minima and passes this point (e.g., F/(kθ/d) = 45), it is possible for this TMP to have three different configurations as shown in Fig. S6. The insets show the geometrical configurations of the TMP structure under these three states.

FIG. S6 Three different configurations of the TMP under the same normalized force.

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6. Negative stiffness, snap-through, and hysteresis

Force-folding ratio relationship (Eq.(S26)) is derived based on an equilibrium state at each folding ratio. Therefore, the loading and unloading curves are identical. However, if we consider dynamical circumstances (i.e., the compressive force is applied in an incremental manner), the TMP can exhibit snap-through response. Figure S7 shows the force-folding ratio relationship when (0) 80 , 53 , and 7M Nθ α= ° = ° = . In this figure, there is a region where the curve exhibits negative stiffness (see red solid line in Fig. S7), which indicates that the structure is unstable. For example, Fig. S8 shows that the force-folding ratio curve evolves in the stable regime from zero to P1, and if we further increase the force at P1, this will cause the structure to snap through from P1 to P2. Similarly, under the unloading condition, the force decreases passing through P2 to P3, and further reduction of the force will cause a snap-through to P4. This implies that we can achieve hysteresis effect in origami-based metamaterials under dynamical circumstances, which can be exploited for building an efficient structure with high damping [2].

FIG. S7 Force-folding ratio relationship and snap-through response ( θM(0) = 80°, α = 53°, and N = 7 )

Reference

[1] Rasband, W.S., ImageJ, U. S. National Institutes of Health, Bethesda, Maryland, USA, http://imagej.nih.gov/ij/, 1997-2014.

[2] Dong, L., and Lakes, R. S., “Advanced damper with negative structural stiffness elements,” Smart Mater. Struct. 21:075026, 2012.