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Zigzag-base folded sheet cellular mechanical metamaterials
Extreme Mechanics Letters 6 (2016) 96–102
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Extreme Mechanics Letters journal homepage: www.elsevier.com/locate/eml
Zigzag-base folded sheet cellular mechanical metamaterials Maryam Eidini Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, 205 North Mathews Ave., Urbana, IL 61801, USA
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Article history: Received 26 September 2015 Received in revised form 18 December 2015 Accepted 20 December 2015 Available online 29 December 2015 Keywords: Miura-ori Auxetic Metamaterial Deployable structure Origami Kirigami Zigzag Negative Poisson’s ratio
abstract The Japanese art of turning flat sheets into three dimensional intricate structures, origami, has inspired design of mechanical metamaterials. Mechanical metamaterials are artificially engineered materials with uncommon properties. Miura-ori is a remarkable origami folding pattern with metamaterial properties and a wide range of applications. In this study, by dislocating the zigzag strips of a Miura-ori pattern along the joining ridges, we create a class of one-degree of freedom (DOF) cellular mechanical metamaterials. The unit cell of the patterns comprises two zigzag strips surrounding a hole with a parallelogram shape. We show that dislocating zigzag strips of the Miura-ori along the joining ridges preserves and/or tunes the outstanding properties of the Miura-ori. The introduced materials are lighter than their corresponding Miura-ori sheets due to the presence of holes in the patterns. Moreover, they are amenable to similar modifications available for Miura-ori which make them appropriate for a wide range of applications across the length scales. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Miura-ori, a zigzag/herringbone-base origami folding pattern, has attracted substantial attention in science and
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.eml.2015.12.006 2352-4316/© 2015 Elsevier Ltd. All rights reserved.
engineering for its remarkable properties [1–11]. The exceptional mechanical properties of the Miura-ori [4,6], the ability to produce its morphology as a self-organized buckling pattern [1,2] and its geometric adaptability [7, 8] has made the pattern suited for applications spanning from metamaterials [4] to fold-core sandwich panels [9]. Moreover, Miura-ori is a mechanical metamaterial with
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Fig. 1. Creation of zigzag-base folded materials by dislocating the zigzag strips of a Miura-ori crease pattern. (A) A sample Miura-ori crease pattern. (B) Changing the direction of the offset from a zigzag strip to the next adjoining one results in a pattern with holes oriented in different directions—the direction of the offsets are shown with blue arrows. (C) Arranging the offsets all to one side, results in zigzag strips with holes all oriented in the same direction. (D) Crease pattern of the unit cell. In the figures, the blue and red lines show mountain and valley folds, respectively, and hatched black areas represent the places of the cuts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
negative Poisson’s ratio for a wide range of its geometric parameters [10,11]. Mechanical metamaterials are artificially engineered materials with unusual material properties arising from their geometry and structural layout. In-plane Poisson’s ratio is defined as the negative ratio of the transverse to axial strains. Poisson’s ratios of many common isotropic elastic materials are positive, i.e., they expand transversely when compressed in a given direction. Conversely, when compressed, materials with negative Poisson’s ratio or auxetics contract in the directions perpendicular to the applied load. Discovery and creating of auxetic materials has been of interest due to improving the material properties, for instance, increasing the indentation resistance or hardness and increasing the shear modulus in isotropic auxetic materials among others [12–15]. Auxetic behavior may be exploited through rotating rigid and semi-rigid units [16,17], chiral structures [18,19], reentrant structures [20–22], elastic instabilities in switchable auxetics [23,24], creating cuts in materials [25], and in folded sheet materials [4,10]. The latter is the concentration of the current research and the materials introduced in this work can be categorized under the designer matter [26]. Research studies have shown that the herringbone geometry leads to auxetic properties in folded sheet materials [4,10] and textiles [27,28], and its morphology arises in biological systems [29–31]. Due to possessing unprecedented deformability (i.e., the ability to change the configuration of the material by folding and unfolding of the corrugated surface and without deforming the base material), the herringbone structure fabricated by bi-axial compression, has been also used in deformable batteries and electronics [32,5,33]. Kirigami, the art of paper cutting, has been applied in the construction of three dimensional (3D) core cellular structures and solar cells among others [34,35]. The current research expands on a recent study by Eidini and Paulino [10] where origami folding has been combined with cutting patterns to create a class of cellular metamaterials. Specifically, Ref. [10] introduces zigzag-base
metamaterials comprising various scales of zigzag strips. A regular BCHn sheet [10] contains tessellations of the identical BCHn (Basic unit Cell with Hole) in which the unit cell includes 2 large and 2n small parallelogram facets. In the present study, we use the concept of the Poisson’s ratio of a one-DOF zigzag strip (i.e., υz = − tan2 φ ) [10] which provides inspiration to tune and/or preserve the mechanical properties of the Miura-ori. In this regard, by dislocating the zigzag strips of the Miura-ori pattern along the joining fold lines, we create a novel class of metamaterials. 2. Geometry of the patterns As shown in Fig. 1, the arrangement of zigzag strips with offsets creates the parallelogram holes in the patterns. The crease pattern of the Zigzag unit Cell with Hole (ZCH) is shown in Fig. 1(D). Changing the values and the directions of the offsets result in infinite configurations of the patterns. Sample patterns formed by dislocating the zigzag strips of a regular Miura-ori sheet are shown in Fig. 2 and can be generally categorized as follows:
• Regular ZCH sheet: in this sheet, both the values of the offsets are identical and the directions of the offsets can either alternate (e.g., the patterns shown in Fig. 2(A) and Supplemental Video 1, Appendix A) or remain the same (e.g., the pattern shown in Fig. 2(B)). Accordingly, a regular ZCH sheet comprises identical unit cells. • Irregular ZCH sheet: in this sheet, the values of the offsets are not identical and/or the directions of the offsets do not alternate regularly (e.g., the pattern shown in Fig. 2(C)). • Combined ZCH sheets: the values of the offsets are zero for some instances. Hence, a combined ZCH sheet includes both ZCH and Miura-ori unit cells (e.g., patterns shown in Fig. 2(D) and (E)). The unit cell of ZCH contains two zigzag strips surrounding a hole with a parallelogram shape. For a regular ZCH sheet, the unit cell is parameterized in Fig. 3(A). The
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Fig. 2. Sample patterns of ZCH. (A)–(E) Sample ZCH sheets created by changing the directions and/or the values of the offsets in the patterns. Note that by changing the height h, the length and width of the parallelogram facets, the hole width (e.g., pattern C) and other changes (e.g., similarly to the Miura-ori, changing the geometry of the facets to get the curved versions and others) we can produce numerous graded and/or shape morphing materials/structures.
Fig. 3. Geometry of ZCH pattern. (A) Geometry of the unit cell. The geometry of a regular ZCH sheet can be parameterized by the geometry of a parallelogram facet, hole width bh , fold angle φ ∈ [0, α] which is the angle between the edges b0 (and b) and the x-axis in the xy-plane, and the direction of the offsets. Other important angles in the figure are: the fold angle between the facets and the xy-plane, i.e., θ ∈ [0, π/2]; the angle between the fold lines a and the x-axis, i.e., ψ ∈ [0, α]; dihedral fold angles between parallelogram facets β1 ∈ [0, π] and β2 ∈ [0, π], joining along fold lines a and b0 , respectively. (B) A ZCH sheet with m1 = 2 and m2 = 3 and outer dimensions L and W .
equations defining the geometry of a regular ZCH sheet are given by
w = 2 b sin φ
l = 2a
h = a sin α sin θ
cos α cos φ
(1)
b0 = (b − bh ) /2.
have υz equal to − tan2 φ [10]. Using the outer dimensions, the Poisson’s ratio of a regular ZCH sheet (for example, sample patterns shown in Figs. 2(A) and (B) and 3(B)) is given by
(υWL )e−e = −
The expression relating the angle φ and the fold angle θ is as follows tan φ = cos θ tan α.
(2)
The outer dimensions of a regular sheet of ZCH (Fig. 3(B)) are given by W = m2 (2 b sin φ)
LZCH = m1
2a
cos α cos φ
(3)
+ 2bh cos φ + (b − bh ) cos φ. (4)
If considered in the context of rigid origami, the ZCH unit cell is a one-DOF mechanism system and its tessellation creates a one-DOF sheet as well. Sample patterns containing ZCH unit cells are presented in Fig. 2. We can obtain the DOF of the patterns in this work using the approach mentioned in [10]. 3. Basic mechanical behaviors of the patterns Being in the class of zigzag-base patterns with one-DOF planar mechanism, the patterns of ZCH shown in Fig. 2 all
dL/L εL =− εW dW /W
= − tan2 φ in which m1 a
η=
m1 bh + b0
η cos α − cos2 φ η cos α + cos2 φ
.
(5)
(6)
The in-plane Poisson’s ratio of regular ZCH sheets are shown in Figs. 4–6. The effects of the tail (i.e., the last term in Eq. (4)) and holes (i.e., the second term in parentheses given in Eq. (4)) in the sheets cause the transition of the Poisson’s ratio towards positive values. By increasing the number of rows in ZCH sheets, Poisson’s ratio shifts towards negative values as presented in Fig. 4. This phenomenon happens because the effect of the tail, which causes the transition towards positive Poisson’s ratios, cancels out for the ZCH sheets when m1 → ∞. In Eqs. (3) and (4), the dimensions of a repeating unit cell of the sheet are as follows
w = 2 b sin φ cos α Lr = 2a + 2bh cos φ. cos φ
(7) (8)
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Fig. 4. In-plane Poisson’s ratio of a regular ZCH sheet. Figure shows the effect of the number of rows (m1 ) on the Poisson’s ratio of sample sheets.
Fig. 5. In-plane Poisson’s ratio of regular ZCH sheets with infinite tessellations. The effect of the hole width on the Poisson’s ratio of ZCH sheets for two values of α and hole width bh . (A) a = b. (B) b/a → ∞. The values correspond to the Poisson’s ratios of the repeating unit cells of sheets as well. Note that Miura-ori is a special case where bh = 0.
Hence, the Poisson’s ratio of a repeating unit cell (in an infinite tessellation) is given by
(υ∞ )e−e = υwLr repeating unit cell =−
εLr dLr /Lr =− εw dw/w
= − tan2 φ
a cos α − bh cos2 φ a cos α + bh cos2 φ
.
(9)
The value given in the relation above presents the Poisson’s ratio of a regular ZCH sheet for an infinite tessellation as well (see Fig. 5). The value is positive if a cos α < bh cos2 φ . The value is negative if a cos α > bh cos2 φ . If a/bh → ∞, the Poisson’s ratio of a repeating unit cell approaches υz . If b/a → ∞, the Poisson’s ratio of a repeating unit cell of the Miura-ori remains as − tan2 φ , i.e. the υz (Fig. 5(B)), but the Poisson’s ratio of a repeating unit cell of ZCH approaches tan2 φ . This phenomenon happens due to the existence of
the holes in the ZCH patterns. In general, unlike Miura-ori, Poisson’s ratio of a ZCH sheet with an infinite tessellation shifts from negative to positive values by increasing the ratio of b/a (see Fig. 6). Upon bending, a ZCH sheet exhibits a saddle-shaped deformation (see Fig. 7) which is a property for materials with positive Poisson’s ratio [15]. Using the bar-framework numerical approach [4], the results of the eigen-value analysis of sample ZCH patterns reveal similar behavior to those observed in Miura-ori and BCHn [4,10] (see Fig. 8, Fig. S1 and Supplemental Videos 2–4 for more details, Appendix A). For small values of the ratio of the stiffness of the facet to that of the fold line, i.e. small values of Kfacet /Kfold , which cover a wide range of material properties, twisting and saddle-shaped modes are the first and the second softest modes, respectively (see Fig. 8(A) and (B), Fig. S1(A) and (B) and Supplemental Videos 2 and 3, Appendix A). That the second softest bending mode is a saddle-shaped deformation (see Fig. 8(B) and Fig. S1(B),
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Fig. 6. The effect of the facet aspect ratio (b/a) on the in-plane Poisson’s ratio of regular ZCH sheets for a wide range of hole width (bh ).
Fig. 7. Behavior of sample sheets of ZCH patterns under bending. ZCH sheets deform into saddle-shaped curvatures under bending.
Fig. 8. Results of the eigenvalue analysis of a sample regular ZCH sheet with the alternating directions of offsets. (A) Twisting, (B) saddle-shaped and (C) rigid origami behavior (planar mechanism) of a 4 × 4 regular ZCH sheet with the holes oriented in various directions (a = 1; b = 2; α = 60°). Twisting and saddle-shaped modes are the first and the second predominant bending modes observed in materials with small values of Kfacet /Kfold . Rigid origami behavior is the predominant behavior for large values of Kfacet /Kfold .
Appendix A) further shows that the material, constructed from a ZCH sheet, has a positive out-of-plane Poisson’s ratio. Moreover, the patterns exhibit rigid origami behavior (Fig. 8(C) and Fig. S1(C), Appendix A) for large values of Kfacet /Kfold which is in accordance with our expectation. 4. Other variations of ZCH and their assemblages The variation of the ZCH pattern shown in Fig. 9(A) provides additional bonding areas on the crests of a corrugation for applications as folded-core sandwich panels. Furthermore, by changing the geometry of the facets similarly to the curved version of the Miura-ori [36], we can create
a curved version shown in Fig. 9(B). In addition to the variations available for the Miura-ori (e.g., Ref. [7,36]) which are applicable to these patterns, changing the hole width bh (Fig. 2(C)) and the width of the bonding areas at the crests (shown in Fig. 9(A)) combined with other changes can provide extensive versatility to create various variations/shapes from the patterns. Sample one-DOF cellular materials designed based on the ZCH patterns are shown in Fig. 10. The stacked materials shown in Fig. 10(A) and (B) are appropriate for applications such as impact absorbing devices [37]. The interleaved ZCH tubular material shown in Fig. 10(C) is similar to the one made from the Miura-ori pattern [38], which
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Fig. 9. (A) A developable ZCH pattern with augmented bonding areas. (B) A curved ZCH pattern.
Fig. 10. Sample cellular foldable metamaterials. (A), (B) Stacked cellular metamaterials made from 7 layers of folded ZCH sheets. Each material includes two different geometries of similar sheets. (C) An interleaved ZCH tubular materials. (D)–(F) Materials made from various assemblages of ZCH tubes. (G) A sample ZCH tube with a parallelogram cross section.
is a bi-directionally flat-foldable material. Its geometry results in a material which is soft in two directions and relatively stiff in the third direction. The samples shown in Fig. 10(D)–(F) are bi-directionally flat-foldable materials made from different assemblages of the ZCH tubes. 5. Concluding remarks In this study, we have presented a method to tune and/or keep the basic mechanical properties of the Miuraori, i.e., an origami folding pattern with outstanding properties which has attracted considerable attention in science and engineering. The resulting configurations are zigzag-base patterns which are flat-foldable and developable and have one DOF for rigid origami behavior. The main advantages of the patterns are highlighted as follows: (i) the patterns are amenable to similar modifications and/or applications available for the Miura-ori (for example, see Figs. 9 and 10 and Refs. [4,38,8,36,37]). (ii) Due to possessing holes in their configurations, they are less dense than their corresponding Miura-ori patterns (see Fig.
S2–S4, Appendix A). (iii) They extend geometrical and mechanical design space of the prior known zigzag-base patterns such as Miura-ori and BCH2 (for example, see Figs. 4– 6). (iv) Despite the existence of holes, they are all singledegree of freedom (SDOF) systems for the rigid origami behavior. SDOF rigid mechanisms are appropriate for low energy, efficient and controllable deployable structures. (v) Compared with BCHn patterns [10] whose unit cell includes two large and 2n small parallelogram facets, the unit cell of the patterns introduced in this work has identical number of facets on each side of the hole. Hence, they can be more appropriate than their corresponding BCHn patterns when considering the thickness of the facets (e.g., when separate thick panels are connected with frictionless hinges [39]). (vi) For applications such as sandwich folded-cores, unlike Zeta core [40], the patterns remain developable by adding surfaces at the top and bottom of the patterns to increase the bonding areas (see Fig. 9(A))— developable sheets are well-suited for continuous manufacturing techniques available for folded core structures. (vii) Dislocating the zigzag strips along the joining ridges makes the ZCH patterns appropriate for the construction of
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programmable materials and structures. Accordingly, the programmable ZCH-base materials are fabricated so that one zigzag can move with respect to the neighboring one along the joining ridges in which the directions and values of the offsets (bh ) can be adjusted depending on the external excitations to achieve a desirable objective (e.g., Supplemental Video 1 shows the transformation from Miuraori to a ZCH pattern by dislocating and moving the zigzag strips along the joining ridges, Appendix A). In summary, the characteristics of the introduced patterns make them suitable for a broad range of applications from folded-core sandwich panels and morphing structures to metamaterials at various length scales. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.eml.2015.12. 006. References [1] L. Mahadevan, S. Rica, Self-organized origami, Science 307 (2005) 1740. [2] K. Miura, The science of miura-ori, in: R.J. Lang (Ed.), 4th International Meeting of Origami Science, Mathematics, and Education, AK Peters, 2009. [3] K. Tanizawa, K. Miura, Large displacement configurations of biaxially compressed infinite plate, Japan Soc. Aeronaut. Space Sci. Trans. 20 (1978) 177–187. [4] M. Schenk, S.D. Guest, Geometry of miura-folded metamaterials, Proc. Natl. Acad. Sci. 110 (9) (2013) 3276–3281. [5] Z. Song, T. Ma, R. Tang, Q. Cheng, X. Wang, D. Krishnaraju, R. Panat, C.K. Chan, H. Yu, H. Jiang, Origami lithium-ion batteries, Nature Commun. 5 (2014) 3140. [6] Z.Y. Wei, Z.V. Guo, L. Dudte, H.Y. Liang, L. Mahadevan, Geometric mechanics of periodic pleated origami, Phys. Rev. Lett. 110 (21) (2013) 215501–215505. [7] T. Tachi, Generalization of rigid foldable quadrilateral mesh origami, in: Proc. of the Int. Association for Shell and Spatial Structures, IASS, Symp., Valencia, Spain, 2009. [8] F. Gioia, D. Dureisseix, R. Motro, B. Maurin, Design and analysis of a foldable/unfoldable corrugated architectural curved envelop, J. Mech. Des. 134 (3) (2012) 031003. [9] S. Heimbs, Foldcore sandwich structures and their impact behaviour: An overview, in: Dynamic Failure of Composite and Sandwich Structures, Springer, Netherlands, 2013, pp. 491–544. [10] M. Eidini, G.H. Paulino, Unraveling metamaterial properties in zigzag-base folded sheets, Sci. Adv. 1 (8) (2015) e1500224. http: //dx.doi.org/10.1126/sciadv.1500224. Avilable online pdf: http:// advances.sciencemag.org/content/1/8/e1500224. [11] C. Lv, D. Krishnaraju, G. Konjevod, H. Yu, H. Jiang, Origami based mechanical metamaterials, Sci. Rep. 4 (2014) 5979. [12] R. Lakes, Foam structures with a negative Poisson’s ratio, Science 235 (4792) (1987) 1038–1040. [13] K.E. Evans, A. Alderson, Auxetic materials: functional materials and structures from lateral thinking!, Adv. Mater. 12 (9) (2000) 617–628. [14] W. Yang, Z.-M. Li, W. Shi, B.-H. Xie, M.-B. Yang, Review on auxetic materials, J. Mater. Sci. 39 (10) (2004) 3269–3279. [15] A. Alderson, K.L. Alderson, Auxetic materials, Proc. Inst. Mech. Eng. G: J. Aerosp. Eng. 221 (4) (2007) 565–575.
[16] J.N. Grima, A. Alderson, K.E. Evans, Auxetic behaviour from rotating rigid units, Phys. Status Solidi (b) 242 (3) (2005) 561–575. [17] J.N. Grima, V. Zammit, R. Gatt, A. Alderson, K.E. Evans, Auxetic behaviour from rotating semi-rigid units, Phys. Status Solidi (b) 244 (3) (2007) 866–882. [18] D. Prall, R.S. Lakes, Properties of a chiral honeycomb with a Poisson’s ratio of −1, Int. J. Mech. Sci. 39 (3) (1997) 305–314. [19] A. Spadoni, M. Ruzzene, Elasto-static micropolar behavior of a chiral auxetic lattice, J. Mech. Phys. Solids 60 (1) (2012) 156–171. [20] R.F. Almgren, An isotropic three-dimensional structure with Poisson’s ratio −1, J. Elasticity 15 (4) (1985) 427–430. [21] K.E. Evans, A. Alderson, F.R. Christian, Auxetic two-dimensional polymer networks. An example of tailoring geometry for specific mechanical properties, J. Chem. Soc. Faraday Trans. 91 (16) (1995) 2671–2680. [22] H. Yasuda, J. Yang, Reentrant Origami-based metamaterials with negative Poisson’s ratio and bistability, Phys. Rev. Lett. 114 (18) (2015) 185502. [23] K. Bertoldi, P.M. Reis, S. Willshaw, T. Mullin, Negative Poisson’s ratio behavior induced by an elastic instability, Adv. Mater. 22 (3) (2010) 361–366. [24] S. Babaee, J. Shim, J.C. Weaver, E.R. Chen, N. Patel, K. Bertoldi, 3D soft metamaterials with negative Poisson’s ratio, Adv. Mater. 25 (36) (2013) 5044–5049. [25] Y. Cho, J.-H. Shin, A. Costa, T.A. Kim, V. Kunin, J. Li, S.Y. Lee, S. Yange, H. Nam Hanf, I.-S. Choi, D.J. Srolovitze, Engineering the shape and structure of materials by fractal cut, Proc. Natl. Acad. Sci. 111 (49) (2014) 17390–17395. [26] P.M. Reis, H.M. Jaeger, M. van Hecke, Designer Matter: A perspective, Extreme Mech. Lett. 5 (2015) 25–29. [27] Z. Wang, H. Hu, 3D auxetic warp-knitted spacer fabrics, Phys. Status Solidi (b) 251 (2) (2014) 281–288. [28] M. Glazzard, P. Breedon, Weft-knitted auxetic textile design, Phys. Status Solidi (b) 251 (2) (2014) 267–272. [29] M.B. Amar, F. Jia, Anisotropic growth shapes intestinal tissues during embryogenesis, Proc. Natl. Acad. Sci. 110 (26) (2013) 10525–10530. [30] A.E. Shyer, T. Tallinen, N.L. Nerurkar, Z. Wei, E.S. Gil, D.L. Kaplan, C.J. Tabin, L. Mahadevan, Villification: how the gut gets its villi, Science 342 (6155) (2013) 212–218. [31] H. Kobayashi, B. Kresling, J.F.V. Vincent, The geometry of unfolding tree leaves, Proc. R. Soc. Lond. B Biol. Sci. 265 (1998) 147–154. [32] J.A. Rogers, T. Someya, Y. Huang, Materials and mechanics for stretchable electronics, Science 327 (5973) (2010) 1603–1607. [33] Y. Zhang, Y. Huang, J.A. Rogers, Mechanics of stretchable batteries and supercapacitors, Curr. Opin. Solid State Mater. Sci. (2015). [34] T. Nojima, K. Saito, Development of newly designed ultra-light core structures, JSME Int. J. Ser. A 49 (1) (2006) 38–42. [35] A. Lamoureux, K. Lee, M. Shlian, S.R. Forrest, M. Shtein, Dynamic kirigami structures for integrated solar tracking, Nature Commun. 6 (2015) 8092. [36] J.M. Gattas, W. Wu, Z. You, Miura-base rigid origami: Parameterizations of first-level derivative and piecewise geometries, J. Mech. Des. 135 (11) (2013) 111011. [37] M. Schenk, S.D. Guest, G.J. McShane, Novel stacked folded cores for blast-resistant sandwich beams, Internat. J. Solids Structures 51 (25) (2014) 4196–4214. [38] K.C. Cheung, T. Tachi, S. Calisch, K. Miura, Origami interleaved tube cellular materials, Smart Mater. Struct. 23 (9) (2014) 094012. [39] S.A. Zirbel, R.J. Lang, M.W. Thomson, D.A. Sigel, P.E. Walkemeyer, B.P. Trease, S.P. Magleby, L.L. Howell, Accommodating thickness in origami-based deployable arrays, J. Mech. Des. 135 (11) (2013) 111005. [40] K. Miura, Zeta-core sandwich — Its concept and realization, Inst. of Space and Aeronautical Science, University of Tokyo, vol. 480, 1972, pp. 137–164.
Contents lists available at ScienceDirect
Extreme Mechanics Letters journal homepage: www.elsevier.com/locate/eml
Zigzag-base folded sheet cellular mechanical metamaterials Maryam Eidini Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, 205 North Mathews Ave., Urbana, IL 61801, USA
graphical
article
abstract
info
Article history: Received 26 September 2015 Received in revised form 18 December 2015 Accepted 20 December 2015 Available online 29 December 2015 Keywords: Miura-ori Auxetic Metamaterial Deployable structure Origami Kirigami Zigzag Negative Poisson’s ratio
abstract The Japanese art of turning flat sheets into three dimensional intricate structures, origami, has inspired design of mechanical metamaterials. Mechanical metamaterials are artificially engineered materials with uncommon properties. Miura-ori is a remarkable origami folding pattern with metamaterial properties and a wide range of applications. In this study, by dislocating the zigzag strips of a Miura-ori pattern along the joining ridges, we create a class of one-degree of freedom (DOF) cellular mechanical metamaterials. The unit cell of the patterns comprises two zigzag strips surrounding a hole with a parallelogram shape. We show that dislocating zigzag strips of the Miura-ori along the joining ridges preserves and/or tunes the outstanding properties of the Miura-ori. The introduced materials are lighter than their corresponding Miura-ori sheets due to the presence of holes in the patterns. Moreover, they are amenable to similar modifications available for Miura-ori which make them appropriate for a wide range of applications across the length scales. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Miura-ori, a zigzag/herringbone-base origami folding pattern, has attracted substantial attention in science and
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.eml.2015.12.006 2352-4316/© 2015 Elsevier Ltd. All rights reserved.
engineering for its remarkable properties [1–11]. The exceptional mechanical properties of the Miura-ori [4,6], the ability to produce its morphology as a self-organized buckling pattern [1,2] and its geometric adaptability [7, 8] has made the pattern suited for applications spanning from metamaterials [4] to fold-core sandwich panels [9]. Moreover, Miura-ori is a mechanical metamaterial with
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Fig. 1. Creation of zigzag-base folded materials by dislocating the zigzag strips of a Miura-ori crease pattern. (A) A sample Miura-ori crease pattern. (B) Changing the direction of the offset from a zigzag strip to the next adjoining one results in a pattern with holes oriented in different directions—the direction of the offsets are shown with blue arrows. (C) Arranging the offsets all to one side, results in zigzag strips with holes all oriented in the same direction. (D) Crease pattern of the unit cell. In the figures, the blue and red lines show mountain and valley folds, respectively, and hatched black areas represent the places of the cuts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
negative Poisson’s ratio for a wide range of its geometric parameters [10,11]. Mechanical metamaterials are artificially engineered materials with unusual material properties arising from their geometry and structural layout. In-plane Poisson’s ratio is defined as the negative ratio of the transverse to axial strains. Poisson’s ratios of many common isotropic elastic materials are positive, i.e., they expand transversely when compressed in a given direction. Conversely, when compressed, materials with negative Poisson’s ratio or auxetics contract in the directions perpendicular to the applied load. Discovery and creating of auxetic materials has been of interest due to improving the material properties, for instance, increasing the indentation resistance or hardness and increasing the shear modulus in isotropic auxetic materials among others [12–15]. Auxetic behavior may be exploited through rotating rigid and semi-rigid units [16,17], chiral structures [18,19], reentrant structures [20–22], elastic instabilities in switchable auxetics [23,24], creating cuts in materials [25], and in folded sheet materials [4,10]. The latter is the concentration of the current research and the materials introduced in this work can be categorized under the designer matter [26]. Research studies have shown that the herringbone geometry leads to auxetic properties in folded sheet materials [4,10] and textiles [27,28], and its morphology arises in biological systems [29–31]. Due to possessing unprecedented deformability (i.e., the ability to change the configuration of the material by folding and unfolding of the corrugated surface and without deforming the base material), the herringbone structure fabricated by bi-axial compression, has been also used in deformable batteries and electronics [32,5,33]. Kirigami, the art of paper cutting, has been applied in the construction of three dimensional (3D) core cellular structures and solar cells among others [34,35]. The current research expands on a recent study by Eidini and Paulino [10] where origami folding has been combined with cutting patterns to create a class of cellular metamaterials. Specifically, Ref. [10] introduces zigzag-base
metamaterials comprising various scales of zigzag strips. A regular BCHn sheet [10] contains tessellations of the identical BCHn (Basic unit Cell with Hole) in which the unit cell includes 2 large and 2n small parallelogram facets. In the present study, we use the concept of the Poisson’s ratio of a one-DOF zigzag strip (i.e., υz = − tan2 φ ) [10] which provides inspiration to tune and/or preserve the mechanical properties of the Miura-ori. In this regard, by dislocating the zigzag strips of the Miura-ori pattern along the joining fold lines, we create a novel class of metamaterials. 2. Geometry of the patterns As shown in Fig. 1, the arrangement of zigzag strips with offsets creates the parallelogram holes in the patterns. The crease pattern of the Zigzag unit Cell with Hole (ZCH) is shown in Fig. 1(D). Changing the values and the directions of the offsets result in infinite configurations of the patterns. Sample patterns formed by dislocating the zigzag strips of a regular Miura-ori sheet are shown in Fig. 2 and can be generally categorized as follows:
• Regular ZCH sheet: in this sheet, both the values of the offsets are identical and the directions of the offsets can either alternate (e.g., the patterns shown in Fig. 2(A) and Supplemental Video 1, Appendix A) or remain the same (e.g., the pattern shown in Fig. 2(B)). Accordingly, a regular ZCH sheet comprises identical unit cells. • Irregular ZCH sheet: in this sheet, the values of the offsets are not identical and/or the directions of the offsets do not alternate regularly (e.g., the pattern shown in Fig. 2(C)). • Combined ZCH sheets: the values of the offsets are zero for some instances. Hence, a combined ZCH sheet includes both ZCH and Miura-ori unit cells (e.g., patterns shown in Fig. 2(D) and (E)). The unit cell of ZCH contains two zigzag strips surrounding a hole with a parallelogram shape. For a regular ZCH sheet, the unit cell is parameterized in Fig. 3(A). The
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Fig. 2. Sample patterns of ZCH. (A)–(E) Sample ZCH sheets created by changing the directions and/or the values of the offsets in the patterns. Note that by changing the height h, the length and width of the parallelogram facets, the hole width (e.g., pattern C) and other changes (e.g., similarly to the Miura-ori, changing the geometry of the facets to get the curved versions and others) we can produce numerous graded and/or shape morphing materials/structures.
Fig. 3. Geometry of ZCH pattern. (A) Geometry of the unit cell. The geometry of a regular ZCH sheet can be parameterized by the geometry of a parallelogram facet, hole width bh , fold angle φ ∈ [0, α] which is the angle between the edges b0 (and b) and the x-axis in the xy-plane, and the direction of the offsets. Other important angles in the figure are: the fold angle between the facets and the xy-plane, i.e., θ ∈ [0, π/2]; the angle between the fold lines a and the x-axis, i.e., ψ ∈ [0, α]; dihedral fold angles between parallelogram facets β1 ∈ [0, π] and β2 ∈ [0, π], joining along fold lines a and b0 , respectively. (B) A ZCH sheet with m1 = 2 and m2 = 3 and outer dimensions L and W .
equations defining the geometry of a regular ZCH sheet are given by
w = 2 b sin φ
l = 2a
h = a sin α sin θ
cos α cos φ
(1)
b0 = (b − bh ) /2.
have υz equal to − tan2 φ [10]. Using the outer dimensions, the Poisson’s ratio of a regular ZCH sheet (for example, sample patterns shown in Figs. 2(A) and (B) and 3(B)) is given by
(υWL )e−e = −
The expression relating the angle φ and the fold angle θ is as follows tan φ = cos θ tan α.
(2)
The outer dimensions of a regular sheet of ZCH (Fig. 3(B)) are given by W = m2 (2 b sin φ)
LZCH = m1
2a
cos α cos φ
(3)
+ 2bh cos φ + (b − bh ) cos φ. (4)
If considered in the context of rigid origami, the ZCH unit cell is a one-DOF mechanism system and its tessellation creates a one-DOF sheet as well. Sample patterns containing ZCH unit cells are presented in Fig. 2. We can obtain the DOF of the patterns in this work using the approach mentioned in [10]. 3. Basic mechanical behaviors of the patterns Being in the class of zigzag-base patterns with one-DOF planar mechanism, the patterns of ZCH shown in Fig. 2 all
dL/L εL =− εW dW /W
= − tan2 φ in which m1 a
η=
m1 bh + b0
η cos α − cos2 φ η cos α + cos2 φ
.
(5)
(6)
The in-plane Poisson’s ratio of regular ZCH sheets are shown in Figs. 4–6. The effects of the tail (i.e., the last term in Eq. (4)) and holes (i.e., the second term in parentheses given in Eq. (4)) in the sheets cause the transition of the Poisson’s ratio towards positive values. By increasing the number of rows in ZCH sheets, Poisson’s ratio shifts towards negative values as presented in Fig. 4. This phenomenon happens because the effect of the tail, which causes the transition towards positive Poisson’s ratios, cancels out for the ZCH sheets when m1 → ∞. In Eqs. (3) and (4), the dimensions of a repeating unit cell of the sheet are as follows
w = 2 b sin φ cos α Lr = 2a + 2bh cos φ. cos φ
(7) (8)
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Fig. 4. In-plane Poisson’s ratio of a regular ZCH sheet. Figure shows the effect of the number of rows (m1 ) on the Poisson’s ratio of sample sheets.
Fig. 5. In-plane Poisson’s ratio of regular ZCH sheets with infinite tessellations. The effect of the hole width on the Poisson’s ratio of ZCH sheets for two values of α and hole width bh . (A) a = b. (B) b/a → ∞. The values correspond to the Poisson’s ratios of the repeating unit cells of sheets as well. Note that Miura-ori is a special case where bh = 0.
Hence, the Poisson’s ratio of a repeating unit cell (in an infinite tessellation) is given by
(υ∞ )e−e = υwLr repeating unit cell =−
εLr dLr /Lr =− εw dw/w
= − tan2 φ
a cos α − bh cos2 φ a cos α + bh cos2 φ
.
(9)
The value given in the relation above presents the Poisson’s ratio of a regular ZCH sheet for an infinite tessellation as well (see Fig. 5). The value is positive if a cos α < bh cos2 φ . The value is negative if a cos α > bh cos2 φ . If a/bh → ∞, the Poisson’s ratio of a repeating unit cell approaches υz . If b/a → ∞, the Poisson’s ratio of a repeating unit cell of the Miura-ori remains as − tan2 φ , i.e. the υz (Fig. 5(B)), but the Poisson’s ratio of a repeating unit cell of ZCH approaches tan2 φ . This phenomenon happens due to the existence of
the holes in the ZCH patterns. In general, unlike Miura-ori, Poisson’s ratio of a ZCH sheet with an infinite tessellation shifts from negative to positive values by increasing the ratio of b/a (see Fig. 6). Upon bending, a ZCH sheet exhibits a saddle-shaped deformation (see Fig. 7) which is a property for materials with positive Poisson’s ratio [15]. Using the bar-framework numerical approach [4], the results of the eigen-value analysis of sample ZCH patterns reveal similar behavior to those observed in Miura-ori and BCHn [4,10] (see Fig. 8, Fig. S1 and Supplemental Videos 2–4 for more details, Appendix A). For small values of the ratio of the stiffness of the facet to that of the fold line, i.e. small values of Kfacet /Kfold , which cover a wide range of material properties, twisting and saddle-shaped modes are the first and the second softest modes, respectively (see Fig. 8(A) and (B), Fig. S1(A) and (B) and Supplemental Videos 2 and 3, Appendix A). That the second softest bending mode is a saddle-shaped deformation (see Fig. 8(B) and Fig. S1(B),
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Fig. 6. The effect of the facet aspect ratio (b/a) on the in-plane Poisson’s ratio of regular ZCH sheets for a wide range of hole width (bh ).
Fig. 7. Behavior of sample sheets of ZCH patterns under bending. ZCH sheets deform into saddle-shaped curvatures under bending.
Fig. 8. Results of the eigenvalue analysis of a sample regular ZCH sheet with the alternating directions of offsets. (A) Twisting, (B) saddle-shaped and (C) rigid origami behavior (planar mechanism) of a 4 × 4 regular ZCH sheet with the holes oriented in various directions (a = 1; b = 2; α = 60°). Twisting and saddle-shaped modes are the first and the second predominant bending modes observed in materials with small values of Kfacet /Kfold . Rigid origami behavior is the predominant behavior for large values of Kfacet /Kfold .
Appendix A) further shows that the material, constructed from a ZCH sheet, has a positive out-of-plane Poisson’s ratio. Moreover, the patterns exhibit rigid origami behavior (Fig. 8(C) and Fig. S1(C), Appendix A) for large values of Kfacet /Kfold which is in accordance with our expectation. 4. Other variations of ZCH and their assemblages The variation of the ZCH pattern shown in Fig. 9(A) provides additional bonding areas on the crests of a corrugation for applications as folded-core sandwich panels. Furthermore, by changing the geometry of the facets similarly to the curved version of the Miura-ori [36], we can create
a curved version shown in Fig. 9(B). In addition to the variations available for the Miura-ori (e.g., Ref. [7,36]) which are applicable to these patterns, changing the hole width bh (Fig. 2(C)) and the width of the bonding areas at the crests (shown in Fig. 9(A)) combined with other changes can provide extensive versatility to create various variations/shapes from the patterns. Sample one-DOF cellular materials designed based on the ZCH patterns are shown in Fig. 10. The stacked materials shown in Fig. 10(A) and (B) are appropriate for applications such as impact absorbing devices [37]. The interleaved ZCH tubular material shown in Fig. 10(C) is similar to the one made from the Miura-ori pattern [38], which
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Fig. 9. (A) A developable ZCH pattern with augmented bonding areas. (B) A curved ZCH pattern.
Fig. 10. Sample cellular foldable metamaterials. (A), (B) Stacked cellular metamaterials made from 7 layers of folded ZCH sheets. Each material includes two different geometries of similar sheets. (C) An interleaved ZCH tubular materials. (D)–(F) Materials made from various assemblages of ZCH tubes. (G) A sample ZCH tube with a parallelogram cross section.
is a bi-directionally flat-foldable material. Its geometry results in a material which is soft in two directions and relatively stiff in the third direction. The samples shown in Fig. 10(D)–(F) are bi-directionally flat-foldable materials made from different assemblages of the ZCH tubes. 5. Concluding remarks In this study, we have presented a method to tune and/or keep the basic mechanical properties of the Miuraori, i.e., an origami folding pattern with outstanding properties which has attracted considerable attention in science and engineering. The resulting configurations are zigzag-base patterns which are flat-foldable and developable and have one DOF for rigid origami behavior. The main advantages of the patterns are highlighted as follows: (i) the patterns are amenable to similar modifications and/or applications available for the Miura-ori (for example, see Figs. 9 and 10 and Refs. [4,38,8,36,37]). (ii) Due to possessing holes in their configurations, they are less dense than their corresponding Miura-ori patterns (see Fig.
S2–S4, Appendix A). (iii) They extend geometrical and mechanical design space of the prior known zigzag-base patterns such as Miura-ori and BCH2 (for example, see Figs. 4– 6). (iv) Despite the existence of holes, they are all singledegree of freedom (SDOF) systems for the rigid origami behavior. SDOF rigid mechanisms are appropriate for low energy, efficient and controllable deployable structures. (v) Compared with BCHn patterns [10] whose unit cell includes two large and 2n small parallelogram facets, the unit cell of the patterns introduced in this work has identical number of facets on each side of the hole. Hence, they can be more appropriate than their corresponding BCHn patterns when considering the thickness of the facets (e.g., when separate thick panels are connected with frictionless hinges [39]). (vi) For applications such as sandwich folded-cores, unlike Zeta core [40], the patterns remain developable by adding surfaces at the top and bottom of the patterns to increase the bonding areas (see Fig. 9(A))— developable sheets are well-suited for continuous manufacturing techniques available for folded core structures. (vii) Dislocating the zigzag strips along the joining ridges makes the ZCH patterns appropriate for the construction of
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programmable materials and structures. Accordingly, the programmable ZCH-base materials are fabricated so that one zigzag can move with respect to the neighboring one along the joining ridges in which the directions and values of the offsets (bh ) can be adjusted depending on the external excitations to achieve a desirable objective (e.g., Supplemental Video 1 shows the transformation from Miuraori to a ZCH pattern by dislocating and moving the zigzag strips along the joining ridges, Appendix A). In summary, the characteristics of the introduced patterns make them suitable for a broad range of applications from folded-core sandwich panels and morphing structures to metamaterials at various length scales. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.eml.2015.12. 006. References [1] L. Mahadevan, S. Rica, Self-organized origami, Science 307 (2005) 1740. [2] K. Miura, The science of miura-ori, in: R.J. Lang (Ed.), 4th International Meeting of Origami Science, Mathematics, and Education, AK Peters, 2009. [3] K. Tanizawa, K. Miura, Large displacement configurations of biaxially compressed infinite plate, Japan Soc. Aeronaut. Space Sci. Trans. 20 (1978) 177–187. [4] M. Schenk, S.D. Guest, Geometry of miura-folded metamaterials, Proc. Natl. Acad. Sci. 110 (9) (2013) 3276–3281. [5] Z. Song, T. Ma, R. Tang, Q. Cheng, X. Wang, D. Krishnaraju, R. Panat, C.K. Chan, H. Yu, H. Jiang, Origami lithium-ion batteries, Nature Commun. 5 (2014) 3140. [6] Z.Y. Wei, Z.V. Guo, L. Dudte, H.Y. Liang, L. Mahadevan, Geometric mechanics of periodic pleated origami, Phys. Rev. Lett. 110 (21) (2013) 215501–215505. [7] T. Tachi, Generalization of rigid foldable quadrilateral mesh origami, in: Proc. of the Int. Association for Shell and Spatial Structures, IASS, Symp., Valencia, Spain, 2009. [8] F. Gioia, D. Dureisseix, R. Motro, B. Maurin, Design and analysis of a foldable/unfoldable corrugated architectural curved envelop, J. Mech. Des. 134 (3) (2012) 031003. [9] S. Heimbs, Foldcore sandwich structures and their impact behaviour: An overview, in: Dynamic Failure of Composite and Sandwich Structures, Springer, Netherlands, 2013, pp. 491–544. [10] M. Eidini, G.H. Paulino, Unraveling metamaterial properties in zigzag-base folded sheets, Sci. Adv. 1 (8) (2015) e1500224. http: //dx.doi.org/10.1126/sciadv.1500224. Avilable online pdf: http:// advances.sciencemag.org/content/1/8/e1500224. [11] C. Lv, D. Krishnaraju, G. Konjevod, H. Yu, H. Jiang, Origami based mechanical metamaterials, Sci. Rep. 4 (2014) 5979. [12] R. Lakes, Foam structures with a negative Poisson’s ratio, Science 235 (4792) (1987) 1038–1040. [13] K.E. Evans, A. Alderson, Auxetic materials: functional materials and structures from lateral thinking!, Adv. Mater. 12 (9) (2000) 617–628. [14] W. Yang, Z.-M. Li, W. Shi, B.-H. Xie, M.-B. Yang, Review on auxetic materials, J. Mater. Sci. 39 (10) (2004) 3269–3279. [15] A. Alderson, K.L. Alderson, Auxetic materials, Proc. Inst. Mech. Eng. G: J. Aerosp. Eng. 221 (4) (2007) 565–575.
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