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Unraveling metamaterial properties in zigzag-base folded sheets.

Eidini M, Paulino GH - Sci Adv (2015)

Bottom Line: We show that our class of patterns, by expanding on the design space of Miura-ori, is appropriate for a wide range of applications from mechanical metamaterials to deployable structures at small and large scales.We further show that, depending on the geometry, these materials exhibit either negative or positive in-plane Poisson's ratios.By introducing a class of zigzag-base materials in the current study, we unify the concept of in-plane Poisson's ratio for similar materials in the literature and extend it to the class of zigzag-base folded sheet materials.

View Article: PubMed Central - PubMed

Affiliation: Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801, USA.

ABSTRACT
Creating complex spatial objects from a flat sheet of material using origami folding techniques has attracted attention in science and engineering. In the present work, we use the geometric properties of partially folded zigzag strips to better describe the kinematics of known zigzag/herringbone-base folded sheet metamaterials such as Miura-ori. Inspired by the kinematics of a one-degree of freedom zigzag strip, we introduce a class of cellular folded mechanical metamaterials comprising different scales of zigzag strips. This class of patterns combines origami folding techniques with kirigami. Using analytical and numerical models, we study the key mechanical properties of the folded materials. We show that our class of patterns, by expanding on the design space of Miura-ori, is appropriate for a wide range of applications from mechanical metamaterials to deployable structures at small and large scales. We further show that, depending on the geometry, these materials exhibit either negative or positive in-plane Poisson's ratios. By introducing a class of zigzag-base materials in the current study, we unify the concept of in-plane Poisson's ratio for similar materials in the literature and extend it to the class of zigzag-base folded sheet materials.

No MeSH data available.


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.
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Figure 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.

Mentions: In this section, we look closely at the kinematics of Miura-ori as a zigzag-base folding pattern, which provides inspiration to create a class of mechanical metamaterials. A regular Miura-ori sheet contains zigzag strips of parallelogram facets in which each unit cell can be decomposed into two V-shapes (Fig. 1A). Each V-shape includes two rigid parallelogram facets connected via a hinge along joining ridges, as shown in Fig. 1B. The Poisson’s ratio considering the in-plane kinematics of a one-DOF V-shape (for more details, see the Supplementary Materials) is given by(υwℓ)V=−εℓvεwv=−dℓv/ℓvdwv/wv=−tan2 ϕ,(1)where is the projected length of the edges a in the xy plane and in the x direction, wv is the width of the semifolded V-shape in the xy plane and along the y direction, and ϕ is the angle in the xy plane between the edge b and the x axis. The abovementioned expression shows that the Poisson’s ratio of a V-shape is only a function of the angle ϕ. In particular, it shows that, in a unit cell containing two V-shapes arranged side by side in a crease pattern, we can scale down the length b of parallelogram facets to 1/n that of the other joining V-shape (where n is a positive integer) while preserving the capability of folding and unfolding. Using this insight in our current research, we create a class of zigzag-base metamaterials in which the unit cell includes two different scales of V-shapes with equivalent ϕ angles (Fig. 1C). For instance, n is equal to 2 for the unit cell shown in Fig. 1C. In the case of n = 2 from numerical models and constructed geometry, the ideal unit cell has only one planar mechanism (see Section 7-1 in the Supplementary Materials); that is, the geometry of the unit cell properly constrains the V-shapes to ideally yield a single-DOF planar mechanism. Therefore, the condition for which we have studied the kinematics of the V-shape is met.


Unraveling metamaterial properties in zigzag-base folded sheets.

Eidini M, Paulino GH - Sci Adv (2015)

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.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4643767&req=5

Figure 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.
Mentions: In this section, we look closely at the kinematics of Miura-ori as a zigzag-base folding pattern, which provides inspiration to create a class of mechanical metamaterials. A regular Miura-ori sheet contains zigzag strips of parallelogram facets in which each unit cell can be decomposed into two V-shapes (Fig. 1A). Each V-shape includes two rigid parallelogram facets connected via a hinge along joining ridges, as shown in Fig. 1B. The Poisson’s ratio considering the in-plane kinematics of a one-DOF V-shape (for more details, see the Supplementary Materials) is given by(υwℓ)V=−εℓvεwv=−dℓv/ℓvdwv/wv=−tan2 ϕ,(1)where is the projected length of the edges a in the xy plane and in the x direction, wv is the width of the semifolded V-shape in the xy plane and along the y direction, and ϕ is the angle in the xy plane between the edge b and the x axis. The abovementioned expression shows that the Poisson’s ratio of a V-shape is only a function of the angle ϕ. In particular, it shows that, in a unit cell containing two V-shapes arranged side by side in a crease pattern, we can scale down the length b of parallelogram facets to 1/n that of the other joining V-shape (where n is a positive integer) while preserving the capability of folding and unfolding. Using this insight in our current research, we create a class of zigzag-base metamaterials in which the unit cell includes two different scales of V-shapes with equivalent ϕ angles (Fig. 1C). For instance, n is equal to 2 for the unit cell shown in Fig. 1C. In the case of n = 2 from numerical models and constructed geometry, the ideal unit cell has only one planar mechanism (see Section 7-1 in the Supplementary Materials); that is, the geometry of the unit cell properly constrains the V-shapes to ideally yield a single-DOF planar mechanism. Therefore, the condition for which we have studied the kinematics of the V-shape is met.

Bottom Line: We show that our class of patterns, by expanding on the design space of Miura-ori, is appropriate for a wide range of applications from mechanical metamaterials to deployable structures at small and large scales.We further show that, depending on the geometry, these materials exhibit either negative or positive in-plane Poisson's ratios.By introducing a class of zigzag-base materials in the current study, we unify the concept of in-plane Poisson's ratio for similar materials in the literature and extend it to the class of zigzag-base folded sheet materials.

View Article: PubMed Central - PubMed

Affiliation: Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801, USA.

ABSTRACT
Creating complex spatial objects from a flat sheet of material using origami folding techniques has attracted attention in science and engineering. In the present work, we use the geometric properties of partially folded zigzag strips to better describe the kinematics of known zigzag/herringbone-base folded sheet metamaterials such as Miura-ori. Inspired by the kinematics of a one-degree of freedom zigzag strip, we introduce a class of cellular folded mechanical metamaterials comprising different scales of zigzag strips. This class of patterns combines origami folding techniques with kirigami. Using analytical and numerical models, we study the key mechanical properties of the folded materials. We show that our class of patterns, by expanding on the design space of Miura-ori, is appropriate for a wide range of applications from mechanical metamaterials to deployable structures at small and large scales. We further show that, depending on the geometry, these materials exhibit either negative or positive in-plane Poisson's ratios. By introducing a class of zigzag-base materials in the current study, we unify the concept of in-plane Poisson's ratio for similar materials in the literature and extend it to the class of zigzag-base folded sheet materials.

No MeSH data available.