Research ArticleMATERIALS SCIENCE

# Unraveling metamaterial properties in zigzag-base folded sheets

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Science Advances  18 Sep 2015:
Vol. 1, no. 8, e1500224
DOI: 10.1126/sciadv.1500224

### Figures

• Fig. 1 From Miura-ori to zigzag-base foldable metamaterials with different scales of zigzag strips.(A) A Miura-ori unit cell containing two V-shapes aligned side by side, forming one concave valley and three convex mountain folds (or vice versa if the unit cell is viewed from the opposite side). (B) Top view of a V-shape fold including two identical parallelogram facets connected along the ridges with length a. Its geometry can be defined by the facet parameters a, b, and α, and by the angle ϕ ∈ [0, α]. (C) Two different scales of V-shapes, with the same angle ϕ, connected along joining fold lines. The length b of parallelogram facets in the left zigzag strip of V-shapes is half that of the strip on the right in the unit cell shown.
• Fig. 2 Geometry of BCHn pattern.(A) Geometry of the unit cell. The geometry of a BCHn sheet can be parameterized by the geometry of parallelogram facets (a, b, and α), the half number of small parallelogram facets (n), and the fold angle ϕ ∈ [0, α], which is the angle between fold lines b and the x axis. Other important angles in the figure include the fold angle between the facets and the xy plane (that is, θ ∈ [0, π/2]), the angle between the fold lines a and the x axis (that is, ψ ∈ [0, α]), and the dihedral fold angles between parallelograms β1 ∈ [0, π] and β2 ∈ [0, π] joined along fold lines a and b, respectively. (B) A BCH2 sheet with m1 = 2, m2 = 3, and outer dimensions L and W.
• Fig. 3 Sample patterns of BCHn and cellular folded metamaterials.(A) A BCH2 sheet. (B) A BCH3 sheet-adding one layer of small parallelograms to the first row reduces the DOF of the system to 1 for rigid origami behavior. (C) Combination of BCH2 and layers of large and small parallelograms with the same geometries as the ones used in BCH2. (D) Combination of BCH3 and layers of large and small parallelograms with the same geometries as the ones used in BCH3. (E) A BCH3 sheet and layers of small parallelograms with the same geometries as the ones used in BCH3. (F) A sheet composed of various BCHn and Miura-ori cells with the same angle ϕ. (G) A stacked cellular metamaterial made from seven layers of folded sheets of BCH2 with two different geometries. (H) Cellular metamaterial made from two layers of 3 × 3 sheets of BCH2 of different heights tailored for stacking and bonded along the joining fold lines. The resulting configuration is flat-foldable in one direction.
• Fig. 4 In-plane Poisson’s ratios of finite configurations of metamaterials.(A) A 5 × 4 (m1 = 5 and m2 = 4) BCH2 sheet (left) and its corresponding Miura-ori sheet (right) with the same basic geometry and same amount of material. Projected lengths of zigzag strips along the x′–x′ line parallel to the x axis are used to obtain υz, and L is used to obtain υe-e. Both sheets have identical υz but have different υe-e. (B) In-plane kinematics (υz) of the class of metamaterials. (C) In-plane Poisson’s ratio considering the end-to-end dimensions (υe-e) of a single unit cell of Miura-ori and BCH2 patterns with a = b. (D) In-plane Poisson’s ratio considering the end-to-end dimensions (υe-e) of sheets of Miura-ori and BCH2 with m1 = 5 and a = b.
• Fig. 5 In-plane Poisson’s ratio of a BCH2 sheet with infinite configuration.Poisson’s ratio obtained by considering the projected length of zigzag strips υz versus Poisson’s ratio considering the end-to-end dimensions of the sheet when the sheet size approaches infinity, υe-e (a = b and m1 → ∞). The latter is equivalent to the Poisson’s ratio of a repeating unit cell of BCH2 in an infinite tessellation. Contrary to Miura-ori, the transition to a positive Poisson’s ratio is present with an infinite configuration of the BCH2 sheet.
• Fig. 6 Ratio of the in-plane stiffness of a Miura-ori cell to the in-plane stiffness of BCH2 in the x and y directions.The results show that, depending on the geometry and considering the same amount of material, BCH2 can be stiffer or more flexible than its corresponding Miura-ori cell in the x and y directions. (A) a/b = 2. (B) a/b = 1. (C) a/b = 1/2.
• Fig. 7 Behavior of a BCH2 sheet upon bending and results of the eigenvalue analysis of a 3 × 3 BCH2 pattern.(A) A BCH2 sheet deforms into a saddle shape upon bending (that is, a typical behavior seen in materials with a positive out-of-plane Poisson’s ratio). (B) Twisting deformation, (C) saddle-shaped deformation, and (D) rigid origami behavior (planar mechanism) of a 3 × 3 pattern of BCH2 (a = 1, b = 2, and α = 60°). Twisting and saddle-shaped deformations are the softest modes observed for a wide range of material properties and geometries. For large values of Kfacet/Kfold, rigid origami behavior (planar mechanism) is simulated.
• Fig. 8 Outcomes of the current study.Inspired by Miura-ori to create BCHn zigzag-base patterns with a broad range of applications.

### Supplementary Materials

• Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/8/e1500224/DC1

Supplementary text

Fig. S1. BCH2, BCH3, and their combinations with rows of small and/or large parallelograms.

Fig. S2. Constrained DOF by implicit formation of the structure of the Miura-ori unit cell between adjoining unit cells of BCH2 and BCH3 in the pattern.

Fig. S3. Concept of Poisson’s ratio considering end-to-end dimensions.

Fig. S4. Geometry of a Miura-ori cell.

Fig. S5. In-plane stiffness for BCH2 with a = b = 1.

Fig. S6. Behavior of a BCH3 sheet upon bending and results of the eigenvalue analysis of a 3 × 3 pattern of BCH3.

Fig. S7. Behavior of a sheet of the pattern shown in Fig. 3C upon bending and results of the eigenvalue analysis of a 2 × 3 sheet of the pattern.

Fig. S8. Behavior of a sheet of the pattern shown in Fig. 3D upon bending and results of the eigenvalue analysis of a 2 × 3 sheet of the pattern.

Table S1. Main points of the in-plane Poisson’s ratio of the class of zigzag-base folded metamaterials.

Movie S1. In-plane behavior of the patterns.

Movie S2. A cellular folded metamaterial made by stacking seven layers of a 3 × 3 sheet of BCH2 pattern with two different geometries.

Movie S3. Out-of-plane behavior of the patterns.

Reference (38)

• ## Supplementary Materials

This PDF file includes:

• Supplementary text
• Fig. S1. BCH2, BCH3, and their combinations with rows of small and/or large parallelograms.
• Fig. S2. Constrained DOF by implicit formation of the structure of the Miura-ori unit cell between adjoining unit cells of BCH2 and BCH3 in the pattern.
• Fig. S3. Concept of Poisson’s ratio considering end-to-end dimensions.
• Fig. S4. Geometry of a Miura-ori cell.
• Fig. S5. In-plane stiffness for BCH2 with a = b = 1.
• Fig. S6. Behavior of a BCH3 sheet upon bending and results of the eigenvalue analysis of a 3 × 3 pattern of BCH3.
• Fig. S7. Behavior of a sheet of the pattern shown in Fig. 3C upon bending and results of the eigenvalue analysis of a 2 × 3 sheet of the pattern.
• Fig. S8. Behavior of a sheet of the pattern shown in Fig. 3D upon bending and results of the eigenvalue analysis of a 2 × 3 sheet of the pattern.
• Table S1. Main points of the in-plane Poisson’s ratio of the class of zigzag-base folded metamaterials.
• Legends for movies S1 to S3
• Reference (38)

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Other Supplementary Material for this manuscript includes the following:

• Movie S1 (.mov format). In-plane behavior of the patterns.
• Movie S2 (.mov format). A cellular folded metamaterial made by stacking seven layers of a 3 × 3 sheet of BCH2 pattern with two different geometries.
• Movie S3 (.mov format). Out-of-plane behavior of the patterns.

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