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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.


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.
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Figure 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.

Mentions: For sheets made by tessellations of the same BCHn (for example, Fig. 3A), υe-e is given by(υWL)e−e=−εLεW=−dL/LdW/W=−tan2 ϕκλcos α−cos2 ϕκλcos α+cos2 ϕ,(6)withκ=2n⋅m1m1(n−1)+1 and λ=a/b,(7)in which n = 2 (n = 1 reduces to the relation for the Miura-ori sheet). Considering end-to-end dimensions, for a unit cell of BCH2 (m1 = 1), υe-e is identical to that of a Miura-ori unit cell (Fig. 4C) and is given by:(υWL)e−e=−tan2 ϕ2λcos α−cos2 ϕ2λcos α+cos2 ϕ.(8)Therefore, unlike υz, which is always negative (Fig. 4B), υe-e can be positive for some geometric ranges (Fig. 4, C and D). Moreover, υz is only a function of the angle ϕ, but υe-e can be dependent on other geometric parameters [that is, the geometry of parallelogram facets (a, b, and α), tessellations (n and m1), and angle ϕ]. The Poisson’s ratio considering end-to-end dimensions can be positive even for a Miura-ori unit cell (Fig. 4C). Furthermore, the shift from negative Poisson’s ratio to positive Poisson’s ratio in Miura-ori is only an effect of the tail (32), and the difference between two Poisson’s ratios (that is, υz and υe-e) vanishes as the length of the Miura-ori sheet approaches infinity. However, for BCH patterns, the transition to positive Poisson’s ratio is primarily a result of the effect of holes in the sheets; unlike Miura-ori, the difference between these two approaches (that is, υz and υe-e) does not disappear even for a BCH sheet with an infinite configuration (Fig. 5). Figure 5 presents the Poisson’s ratio of a repeating unit cell of BCH2 pattern (in an infinite tessellation) that corresponds to the following expression:(υ∞)e−e=−tan2 ϕ4λcos α−cos2 ϕ4λcos α+cos2 ϕ.(9)From Eq. 9, (υ∞)e-e for the BCH2 sheet is positive if 4λ cos α < cos2 ϕ and negative if 4λcos α > cos2 ϕ.


Unraveling metamaterial properties in zigzag-base folded sheets.

Eidini M, Paulino GH - Sci Adv (2015)

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

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Show All Figures
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Figure 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.
Mentions: For sheets made by tessellations of the same BCHn (for example, Fig. 3A), υe-e is given by(υWL)e−e=−εLεW=−dL/LdW/W=−tan2 ϕκλcos α−cos2 ϕκλcos α+cos2 ϕ,(6)withκ=2n⋅m1m1(n−1)+1 and λ=a/b,(7)in which n = 2 (n = 1 reduces to the relation for the Miura-ori sheet). Considering end-to-end dimensions, for a unit cell of BCH2 (m1 = 1), υe-e is identical to that of a Miura-ori unit cell (Fig. 4C) and is given by:(υWL)e−e=−tan2 ϕ2λcos α−cos2 ϕ2λcos α+cos2 ϕ.(8)Therefore, unlike υz, which is always negative (Fig. 4B), υe-e can be positive for some geometric ranges (Fig. 4, C and D). Moreover, υz is only a function of the angle ϕ, but υe-e can be dependent on other geometric parameters [that is, the geometry of parallelogram facets (a, b, and α), tessellations (n and m1), and angle ϕ]. The Poisson’s ratio considering end-to-end dimensions can be positive even for a Miura-ori unit cell (Fig. 4C). Furthermore, the shift from negative Poisson’s ratio to positive Poisson’s ratio in Miura-ori is only an effect of the tail (32), and the difference between two Poisson’s ratios (that is, υz and υe-e) vanishes as the length of the Miura-ori sheet approaches infinity. However, for BCH patterns, the transition to positive Poisson’s ratio is primarily a result of the effect of holes in the sheets; unlike Miura-ori, the difference between these two approaches (that is, υz and υe-e) does not disappear even for a BCH sheet with an infinite configuration (Fig. 5). Figure 5 presents the Poisson’s ratio of a repeating unit cell of BCH2 pattern (in an infinite tessellation) that corresponds to the following expression:(υ∞)e−e=−tan2 ϕ4λcos α−cos2 ϕ4λcos α+cos2 ϕ.(9)From Eq. 9, (υ∞)e-e for the BCH2 sheet is positive if 4λ cos α < cos2 ϕ and negative if 4λcos α > cos2 ϕ.

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.