UPPLEENTAL ATERIAL Re-entrant origami-based metamaterials with negative Poisson s ratio and bistability April 3 rd, 5 H. Yasuda and J. Yang * Department of echanical Engineering, University of Washington, eattle, WA 9895 Department of Aeronautics and Astronautics, University of Washington, eattle, WA 9895. Folding angles relationships The Tachi-iura Polyhedron consists of two sheets (one of the sheets is shown in Fig. (a, and a building block for the TP is the iura-ori unit (Fig. (b,c. In Fig. (c, we define the three different folding angles;,, and. FI. (a Tachi-iura Polyhedron. (b iura-ori unit cell which is folded into (c. There are relationships among these three angles as CD OC AC CD = tan = tanαcos ( OC AC BE BF BE = sin = sinαcos ( BB BB BF Taking a derivative of above two equations, we obtain
d α = tan cos sin d (3 cos d = sin α sin d (4 Therefore 3 cos cos sin d = d = d (5 sinαsin cosαsin Also, the breadth (B, width (W, and height (H of the TP are described as follows: B = msin + d cos W = l + d tanα + mcos (6 (7 H = Ndsin (8 Taking a derivative of these equations, we obtain ( cos ( sin db = m d d d (9 ( dw = msin d ( ( dh = Nd cos d ( Here, we defne the Poisson s ratio of the TP (ν HB and ν HW as HB ( db B ( dh H ν = ( HW ( dw W ( dh H ν = (3 ubstituting Eqs.(5- into Eqs.( and (3, we obtain 4mtanα cos cos ν HB = + d sin msin + d cos tan (4 4mtanα sin cos ν HW = l + d sin tanα + mcos tan (5
. Experimental setup In order to verify analytical expression of the Poisson s ratio, we fabricate three prototypes of the TP (α = 3, 45, and 75 by using paper (trathmore 5 Bristol, -Ply, Plate urface, 35 7 as shown in Fig.. Length parameters (l, m, d = (5 mm, 5 mm, 3 mm are chosen. FI. Prototype of the TP made of paper Figure 3 shows the experimental setup to measure the cross-sectional area of the TP. A glass plate is placed on the top surface of the TP to control the height of the TP, and a camera captures a digital image of the crosssectional area of the TP from above. Based on digital images from a camera, we measure the breadth (B, width (W, and cross-sectional area of the TP with different height (H. To measure B and W, we use Image J software []. The measurement was conducted three times on each TP prototype. In order to obtain the Poisson s ratio from the experiment, we modify Eqs. ( and (3. ubstituting Eqs. (5 and (9-A into Eqs. ( and (3, we obtain ν HB ( 4 costanαcos sin + sin H m d = (6 B Nd ( cos ν HW 4mH sin tanα cos sin = (7 W Nd ( cos Where is calculated from Eq.(8, and and are calculated from Eqs. ( and (. 3
FI. 3 (a Experimental setup. (b Digital image taken from a camera mounted on top of a TP 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 TP. Figure 4 shows the cross-sectional area change as a function of a folding ratio defined as (9 /9. 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 TP prototypes made of paper are initially folded at certain degrees. FI. 4 Cross-sectional area change of the TP. Error bars indicate standard deviations. 4
4. Force-displacement response We model the crease lines of the TP by using a torsional spring as shown in Fig. 5. Focusing on one main crease line (red line in Fig. 5(a. Torsional spring F u z y x (a z y (b k H H FI. 5 odeling of a crease line with a torsional spring. (a TP cell. Red line corresponds to (b. Let and be bending moment along horizontal (related to and inclined (related to crease lines respectively, we assume that relationship between bending moment and angle is linear as follows: ( = k ( (8 ( = k ( (9 where k is a spring constant, and ( and ( are the initial folding angles for horizontal and inclined crease lines respectively. By using the principle of virtual work, we the following equation: Fδu = n δ + n δ ( where n = 8(N- and n = 8N are the number of the main crease lines and sub-crease lines respectively. By applying variation to Eqs.( and ( with respect to the folding angles, we obtain δ = tanα cos sinδ ( cos δ = sinα sin δ ( Therefore 3 cos cos sin δ = δ = δ sinαsin cosαsin (3 Also, the height of the TP is H u= Ndsin (4 5
Then δu= Ndcos δ (5 ubstituting Eqs.(8, (9, (3 and (5 into Eq.(, we obtain ( ( ( ( { } { ( } F Ndcos δ = n k δ + n k δ 3 cos sin 4k ( ( F = n ( + n( Nd cos cosαsin 3 cos sin 4k ( ( 8( F = N ( + 8N( Nd cos cosαsin Hence, 3 cos sin F 3 N ( ( = ( + ( k d cos N cosαsin ( (6 5. ultiple equilibrium states ( Figure 6 shows the force-folding ratio relationship based on Eq.(6, when = 8, α = 7, and N = 7. If the normalized force reaches the local minima and passes this point (e.g., F/(k /d = 45, it is possible for this TP to have three different configurations as shown in Fig. 6. The insets show the geometrical configurations of the TP structure under these three states. FI. 6 Three different configurations of the TP under the same normalized force. 6
6. Negative stiffness, snap-through, and hysteresis Force-folding ratio relationship (Eq.(6 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 TP can exhibit snap-through response. Figure ( 7 shows the force-folding ratio relationship when = 8, α = 53, and N = 7. In this figure, there is a region where the curve exhibits negative stiffness (see red solid line in Fig. 7, which indicates that the structure is unstable. For example, Fig. 8 shows that the force-folding ratio curve evolves in the stable regime from zero to P, and if we further increase the force at P, this will cause the structure to snap through from P to P. imilarly, under the unloading condition, the force decreases passing through P to P 3, and further reduction of the force will cause a snap-through to P 4. 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 []. FI. 7 Force-folding ratio relationship and snap-through response ( ( = 8, α = 53, and N = 7 Reference [] Rasband, W.., ImageJ, U.. National Institutes of Health, Bethesda, aryland, UA, http://imagej.nih.gov/ij/, 997-4. [] Dong, L., and Lakes, R.., Advanced damper with negative structural stiffness elements, mart ater. truct. :756,. 7