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Research Papers

Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs

[+] Author and Article Information
Fei Wang, Xiaogang Guo, Jingxian Xu, Yihui Zhang

Department of Engineering Mechanics,
CNMM & AML,
Tsinghua University,
Beijing 100084, China

C. Q. Chen

Department of Engineering Mechanics,
CNMM & AML,
Tsinghua University,
Beijing 100084, China
e-mail: chencq@tsinghua.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 17, 2017; final manuscript received April 11, 2017; published online April 24, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(6), 061007 (Apr 24, 2017) (7 pages) Paper No: JAM-17-1155; doi: 10.1115/1.4036476 History: Received March 17, 2017; Revised April 11, 2017

Originated from the art of paper cutting and folding, kirigami and origami have shown promising applications in a broad range of scientific and engineering fields. Developments of kirigami-inspired inverse design methods that map target three-dimensional (3D) geometries into two-dimensional (2D) patterns of cuts and creases are desired to serve as guidelines for practical applications. In this paper, using programed kirigami tessellations, we propose two design methods to approximate the geometries of developable surfaces and nonzero Gauss curvature surfaces with rotational symmetry. In the first method, a periodic array of kirigami pattern with spatially varying geometric parameters is obtained, allowing formation of developable surfaces of desired curvature distribution and thickness, through controlled shrinkage and bending deformations. In the second method, another type of kirigami tessellations, in combination with Miura origami, is proposed to approximate nondevelopable surfaces with rotational symmetry. Both methods are validated by experiments of folding patterned thin copper films into desired 3D structures. The mechanical behaviors of the kirigami designs are investigated using analytical modeling and finite element simulations. The proposed methods extend the design space of mechanical metamaterials and are expected to be useful for kirigami-inspired applications.

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Figures

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Fig. 1

Design of developable surfaces using kirigami tessellations: (a) the target cylindrical surface and (b) the kirigami array approximating the surface (front view). A piecewise-linear curve passes through Pi (i=1, 2, 3...): (c) the kirigami array before bending (front view), which is the last configuration of step 1 and (d)–(g) folding process of the kirigami array. (d) → (e) → (f): step 1, shrinkage of the array. (f) → (g): step 2, bending of the array, (h)–(k) kinematics of eight quadrilaterals surrounding one rhombus cut during folding, corresponding to (d)–(g), rectangular plates labeled by l are kept coplanar during step 1, whereas plates labeled by II and III inclined. The four sides of the rhombus coincide into one line after folding.

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Fig. 2

Examples of programmable kirigami designs for developable surfaces. Each example shows the target surface, the kirigami approximation by the algorithm, the experimental result with copper film, and the corresponding 2D tessellation: (a) unit to make the 3D array bendable in step 2. αi (i=1,2,3,4) alternate between acute and obtuse angle in both x and y directions, that is, α1,α3∈(0, 90 deg), whereas α2,α4∈(90 deg, 180 deg), (b) a semicircle as the planar curve, in which all the cut out rhombuses are made identical and uniformly distributed, (c) a sinusoid as the planar curve with continuously varying curvature, (d) an equiangular spiral as the planar curve with continuously varying curvature, and (e) a helical surface. The scale bars in the experiments are 10 mm.

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Fig. 3

Analytical and FEA predicted elastic responses of a 5×5 kirigami array during shrinking: (a) the geometric model and (b) kirigami array's auxetic feature for α=75 deg, characterized with negative Poisson's ratio νyx. (c) Normalized in-plane force F̃ in the x and y directions versus θ. The adopted parameters are α=60  deg, initial state θ0=45  deg, and a1:b:l=1:1:1. The insets illustrate FEA predicted configurations at θ=0  deg, 45  deg, and 90  deg, respectively, (d) load ratio c=F̃x/F̃y for α=40 deg and 90 deg, and (e) contour plot of the instantaneous stiffness Kx in the θ0−α space. The black-dashed curve shows the optimized design path to obtain the minimum of Kx.

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Fig. 4

Design of nondevelopable surfaces using Miura-ori based kirigami patterns: (a) a Miura-ori 2D pattern and its folded configuration, (b) illustration of the algorithm using a spherical surface as an example, in which a piecewise-linear curve P1P2⋯Pn+1¯ (i=1,2...n+1) is used to approximate the generatrix, and (c) the virtual 2D patterns and real 3D origami strip to approximate a part of the spherical surface

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Fig. 5

Examples of programmable kirigami designs for nonzero Gauss curvature surfaces with rotational symmetry. Each example shows the target surface, the kirigami approximation by the algorithm, the experimental result with copper film, and the corresponding 2D tessellation: (a) arc-pattern of Miura-ori is used to approximate a piece of revolution surface, analogous to the cyclotomic method, (b) spherical surface, (c) hyperboloid, and (d) torus. The scale bars in the experiments are 10 mm.

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Fig. 6

(a) A quadrilateral used to construct the surfaces with rotational symmetry. The upper limit of height h is determined by its angles αi−1 and αi and the length of Pi−1Pi¯ and (b) the origami strip units in the circumferential direction before and after cutting.

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Fig. 7

Variation of the number of quadrilaterals needed to approximate the surface as the number of vertexes changes: (a) α decreases with increasing θ, (b) the number of quadrilaterals needed increases rapidly with increasing number of vertexes, by taking a spherical surface as an example, and (c) comparison of the accuracy and complexity when 8 and 12 vertexes are picked, respectively

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