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Research Papers

Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs OPEN ACCESS

[+] 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.

FIGURES IN THIS ARTICLE
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As an ancient art, origami leverages strategically designed folding to transform planar paper sheets into 3D sculptures with a variety of topologies [14]. Kirigami, cousin of origami, involves additional cutting of planar paper sheets before the folding-induced 2D–3D transformation [5,6]. Recently, the principle of origami and kirigami has received growing attention from the research communities of both science and engineering [711], due to their promising potentials in a wide range of applications ranging from reconfigurable architected materials [1215], deformable batteries [16,17], microscale 3D self-assembly [1821], energy absorption [22], topological mechanics [23], and compact deployable structures [3,2426]. In particular, the intriguing mechanical properties of origami/kirigami structures, such as auxiticity, afford significant advantages in building mechanical metamaterials [2731]. To facilitate the aforementioned applications, a fundamental aspect is the development of inverse design methods of origami/kirigami that map complex target 3D surfaces into their corresponding 2D patterns of cuts and/or creases. Although notable advancements were made in the inverse design of origami [32,33], some inherent limitations and constraints also exist, mainly due to the complexity of interlocking folds [34,35]. The kirigami methods can avoid these issues by strategically designing patterns of cuts in the planar sheets [36,37]. With cuts of “line [38,39]” or “polygon [10,40]” patterns, lightweight and deployable 3D structures can be constructed by mastering relevant folding principles. Meanwhile, the introduction of cut pattern increases the possible modes of folding, thereby leading to difficulties in developing an algorithmic approach for inverse design, especially for those involving complex cut shapes [36]. A recent work proposed an elegant algorithm based on “lattice kirigami [6],” in which stepped structures with hexagonal or triangular cuts were adopted to approximate a certain class of smooth 3D surfaces.

In this paper, we propose two inverse design methods of kirigami tessellations to construct curvy structures that approach the geometries of developable surfaces and nonzero Gauss curvature surfaces with rotational symmetry. First, we show that singly curved surfaces can be approximated using reprogrammable kirigami tessellations. This kirigami design offers a tunable bendability and allows the formation of 3D structures with tunable thicknesses. As for the second method, through combining kirigami techniques and Miura-ori pattern, a particular class of nondevelopable surfaces with rotational symmetry can be approximated. This method provides a solution to program varying curvatures, avoiding the possible folding limitations (e.g., interlocking folds) in origami. It is noted that studies on the mechanics of kirigami structures, especially for those with rigid plates, are limited. In this paper, we present analytical and numerical studies on the mechanics of kirigami arrays. We illustrate how to adjust the mechanical responses of the kirigami structures by redesigning their geometric parameters. In the analysis of both the geometric design and the mechanical behaviors, the results suggest the significant effects of controlling the degrees-of-freedom (DOFs) of rigid arrays on constructing mechanical metamaterials [2,41,42].

Inverse Design Method.

Figure 1 shows a schematic illustration of the first design method by taking a cylindrical surface (Fig. 1(a)) as an example to describe the procedure of deformation with kirigami tessellations. Once a set of nodes Pi (i=1,2,3...) on the target curve are chosen (Fig. 1(b)) and a desired thickness of the cylindrical structure is prescribed, another set of nodes Qi (i=1,2,3...) is obtained (Figs. 1(b) and 1(c)). Accordingly, the patterns of cuts and creases can be determined i=1,2,3... using the algorithm of a modified lattice kirigami, as detailed below. Here, cut patterns of rhombuses, rather than regular polygons (e.g., squares or hexagons) as explored previously [6], are adopted to allow further bending after shrinking the 2D sheet into an array of polyhedrons. The entire process includes two steps, i.e., shrinking and bending, as shown in Figs. 1(d)1(g). In step 1 (i.e., Figs. 1(d)1(e)1(f)), the 2D sheet with lattice kirigami shrinks into a periodic array of 3D structures while maintaining the plate components (labeled by “I”) coplanar (Fig. 1(h)). For ideal creases (i.e., rigid folds), such folding in step 1 corresponds to motions of a system with single degree of freedom (DOF). In step 2 (Figs. 1(f)1(g)), the flat array of polyhedrons is bent until the plates labeled by “II” and “III” contact each other, forming a stable 3D structure that approximates the target curvature. Figures 1(c)1(b) show the bending process in front view, corresponding to angled view in Figs. 1(f)1(g), respectively. Figures 1(h)1(k) detail the spatial kinematics of eight quadrilaterals surrounding a rhombus cut. The folding process, through spatial shrinkage and bending of the array, leads to spatial coincidence of the four sides of the rhombus (see the Figs. 1(h)1(k)).

Experimental Validation.

The procedure illustrated above can be implemented into an inverse design algorithm to allow approximation of a variety of developable surfaces through 2D–3D transformation. Figure 2 shows a couple of representative examples, including semicircular, sinusoidal, equiangular spiral, and helical surfaces. As shown in Fig. 2(a) for the design of a unit, the angles (α1, α3(0,90deg) and α2, α4(90deg,180deg)) are the key parameters used to adjust the bendability to match the curvature of target surfaces. In particular, the sinusoidal and equiangular spiral surfaces in Figs. 2(c) and 2(d) possess continuously varying curvature. Among them, the sinusoidal surface in Fig. 2(c) even involves the change of concavity/convexity at the inflection point. The helical surfaces with desired chirality can also be accessed in a deterministic manner, as shown in Fig. 2(e).

Experiments using patterned copper film with 30 μm in thickness serve to validate the proposed inverse design algorithm. In particular, slits are introduced at the creases to reduce the bending stiffness and localize the folding deformation. As a result, relatively rigid folds can be obtained, as shown in Fig. 2. In all of the examples shown here, the experimental results show good agreement with the predicted 3D configurations and reproduce well the corresponding target 3D surfaces.

Mechanical Modeling of Rigid Folding.

To further investigate the mechanical responses of the kirigami arrays, we provide both analytical and numerical solutions using a simplified model with rigid plates connected by torsional springs, in view of limited available mechanics studies of such structures compared with abundant investigation on the mechanics of Miura-ori [28,4345]. Here, we focus on the shrinkage process (i.e., step 1), aiming at establishing quantitative relations between the external forces and the geometric parameters of the kirigami patterns. Linear elastic torsional springs are adopted to model the folding of the creases. A constant, k, is used to represent the torque required to twist one spring of unit length over one unit radian. In this paper, the dihedral angle (θ) between the inclined plates II and the horizontal plates I is chosen as the actuation angle to characterize the kinematics. Therefore, the height H of a designed kirigami array during folding is H=lsinαsinθ=lsinφ, where φ is the dihedral angle between the inclined plates III and horizontal plates I (see Figs. 1(h)1(k)).

The Poisson's ratio is given by νyx=εx/εy=(dLx/Lx)/(dLy/Ly). We use finite number of plates to build the geometric and physical models, with Lx=(m1)[a1+l(cosα+cosφ)]/2+a1+2lcosα and Ly=(n1)(b+lsinαcosθ)/2+b being the lengths of the array in the x and y directions, and m and n being the corresponding number of plates, respectively. Note that m,n4N++1 (N+=1,2,3...) and the expressions of Lx and Ly differ slightly from those when m,n4N++3). Figure 3(a) illustrates one configuration of the array during folding and Fig. 3(b) shows the analytical predictions of the dependence of νyx on θ, for two typical sets of parameters a1:b:l=1:1:1 and a1:b:l=1/3:3:1. The obtained results show clearly that νyx is negative, indicating an auxetic feature. For the special case of the geometric parameters that give a semicylinder (Fig. 2(b)) through the above folding process, we consider b/l=a1/l=1, m=n=5, and identical rhombuses as the cut shapes. The corresponding strain energy U and external work T associated with the kirigami array are

Display Formula

(1)U=(m1)(n+1)4kb(φφ0)2+(m+1)(n1)4k(a1+lcosα)(θθ0)2T=0sF(s)ds=θ0θF(θ)sθdθ

where θ0 and φ0 represent the initial state of the array, and s represents the displacement under the corresponding force F(s) in one of the three possible directions, i.e., x, y, and z (see Fig. 3(a)). F(θ) can then be analytically obtained according to the principle of minimum potential energy, i.e., δ(UT)=0. Using finite element analyses (FEA), additional numerical simulations on F(θ) versus θ are conducted. Figure 3(c) shows the predicted normalized balanced forces F̃x and F̃y versus θ, respectively. The forces are normalized by the torsional spring constant k. Excellent agreement between analytical solutions and numerical predictions is obtained. Our results show that, to stretch or shrink the array, the required forces differ considerably in different directions, especially in the vicinity of the limit positions θ=0 and 90deg. The load ratio c=F̃x/F̃y characterizes the degree of such anisotropy Display Formula

(2)c=n1m11+cot2αsec2θ

which indicates that c is independent of the length ratio b/l and a1/l. In particular, when α=90deg (i.e., the cut potions are identical squares, as shown in the literature [6,42]), c becomes a constant (see Fig. 3(d)). The normalized instantaneous in-plane stiffness of the kirigami-based metamaterials can be defined as Display Formula

(3)Ks(α,θ0)=F̃sθ|θ=θ0(s=x,y)

which implies that the stiffness is variable and reprogrammable by shrinking or stretching the kirigami array. This is one of the significant features of kirigami-based metamaterials. Optimized in-plane stiffness can be obtained by choosing appropriate geometric parameters. To illustrate this, Fig. 3(e) shows the contours of Kx in the design space in terms of α and θ0, in which the black-dashed curve represents the optimum design to yield the minimum Kx by setting Kx/θ0=0.

Inverse Design Method.

Approximation of nondevelopable surfaces with kirigami or origami arrays needs more complex designs than that of developable surfaces. Note that this study focuses on the development of algorithms to approximate surfaces with rigid plates rather than elastic ones [46,47]. This requires careful selections of varying Gauss curvatures [33] and avoiding possible intersection of vertexes and edges [34,35]. Here, we present an intuitive method to construct a particular class of nondevelopable surfaces with rotational symmetry. This method combines kirigami tessellations with Miura-ori principles. The latter are widely used to approximate planar or cylindrical surfaces in previous literature [32,48]. The 2D pattern of a Miura-ori unit is characterized by an angle α, as shown in Fig. 4(a). During folding, Miura-ori has one single DOF [24], and the four geometric parameters, i.e., the line angles (θ and η) and dihedral angles (ϕ and φ), satisfy the following equations:

Display Formula

(4)η=arccos[cos2αsin2αcosϕ]θ=arccos[14cos2α/(1+cosη)]φ=arccos[(cosθ+cos2α)/sin2α]

Taking a spherical surface with positive Gauss curvatures as an example (see Fig. 4(b)), the design process is shown as follows. First, a piecewise-linear curve P1P2Pn+1¯ is used to approximate the generatrix of the sphere. The line angles θ1, θ2,… θn1 are then determined by the locations of vertexes Pi. To simplify the analyses, the dihedral angles ϕ of all the Miura-ori quadrilaterals are kept the same in this algorithm [48]. Once ϕ is specified, the geometric parameter α can be determined inversely as Display Formula

(5)αi=cos1[(1+cosϕ)(1cosθi)4(1cosϕ)(1cosθi)]

As shown in Fig. 4(c), after figuring out the virtual quadrilaterals in 2D planes by the above equations and deploying them to 3D structures, the Miura-ori units are cut by two planes passing through the axis of the sphere, forming an origami strip. The cutting angle between the two planes is κ, as detailed in Sec. 3.3. By repeating this 3D origami strip circumferentially, a discrete 3D structure can be generated to approximate the desired rotational surface. Note that this algorithm can provide a solution to the 2D kirigami patterns in a deterministic manner.

Experimental Validation.

Experiments were performed to validate the design algorithm described above, as shown in Fig. 5. Here, three representative 3D geometries are considered, including a spherical surface, hyperboloid, and torus. Note that the hyperboloid in Fig. 5(c) has negative Gauss curvatures, in contrast to the spherical surface. Figure 5(d) shows a torus, representing a topology qualitatively different from a spheroid. This algorithm shows favorable applicability to deployment of various surfaces with rotational symmetry. Compared with the previous design methods using origami tessellations [32,33], our algorithm systemically cuts and bends arc-Miura-pattern [49], locating all of the chosen vertices accurately on the target surfaces. By cutting the possible overlapping regions between adjacent parts, this method successfully avoids the intersections of vertexes [34].

Balance Between Complexity of Cuts and Accuracy.

As shown in Figs. 4(b) and 6(a), all the quadrilaterals for spherical surface have the same height h satisfying

Display Formula

(6)h=c1min[li/(cotαi1+cotαi)],(i=2,3...n1),c1(0,1)

where c1 is a dimensionless parameter that can be adjusted to tune the required shape of quadrilaterals. The height of the origami strip before cutting is H=2hsin(ϕ/2), as shown in Fig. 6(b). The initial cutting angle (κ0) should be smaller than 2arctan(H/R), with R being the radius of the target sphere. By choosing the vertexes Pi(i=1,2,...,n+1) on the generatrix, it is clear that θi increases with increasing the number n. According to Eq. (5), αi decreases with increasing θi, as shown in Fig. 7(a). Together with Eq. (6), it can be deducted that κ0 decreases with increasing n. The number (N1) of the origami strip to form the rotational surface is then given by

Display Formula

(7)N1=Int(2π/κ0)+1

where operator “Int” denotes the largest integer that is no larger than 2π/κ0. Importantly, Eq. (7) indicates that κ0 only provides an upper limit of the cutting angle. Using a feedback mechanism, the real cutting angle κ is determined by the integer N1Display Formula

(8)κ=2πN1<κ0

The number of the quadrilaterals needed in the algorithm is N=N1n2. Figure 7(b) shows the variation of N with the number of vertexes picked along the generatrix. Two designs with different numbers of vertexes (i.e., n1+1) to approximate the spherical surface are given in Fig. 7(c), showing clearly that the accuracy of approximation can be improved by increasing n.

In summary, this work provides validated inverse design methods of kirigami tessellations for constructing 3D structures with geometries of developable surfaces (e.g., cylinder and helical surfaces) or surfaces (e.g., spheres and hyperboloids) with rotational symmetry. These methods allow formation of 3D structures with tunable thicknesses by adjusting the geometric parameters in a deterministic manner. They offer new options to create and tune the properties of mechanical metamaterials. The DOFs-controllable, bendable, scale-independent, and lightweight nature of these kirigami structures is believed to hold promising potentials for applications in a broad range of areas, such as architectures, aerospace, and robotics.

This work was supported by the National Natural Science Foundation of China (No. 11472149), the National Basic Research Program of China (No. 2015CB351900), and the Tsinghua University Initiative Scientific Research Program (No. 2014z22074).

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Kim, J. , Hanna, J. A. , Byun, M. , Santangelo, C. D. , and Hayward, R. C. , 2012, “ Designing Responsive Buckled Surfaces by Halftone Gel Lithography,” Science, 335(6073), pp. 1201–1205. [CrossRef] [PubMed]
Guo, X. , Li, H. , Ahn, B. Y. , Duoss, E. B. , Hsia, K. J. , Lewis, J. A. , and Nuzzo, R. G. , 2009, “ Two-and Three-Dimensional Folding of Thin Film Single-Crystalline Silicon for Photovoltaic Power Applications,” Proc. Natl. Acad. Sci., 106(48), pp. 20149–20154. [CrossRef]
Wang, F. , Gong, H. , Chen, X. , and Chen, C. Q. , 2016, “ Folding to Curved Surfaces: A Generalized Design Method and Mechanics of Origami-Based Cylindrical Structures,” Sci. Rep., 6(1), p. 33312. [CrossRef] [PubMed]
Gattas, J. M. , Wu, W. , and You, Z. , 2013, “ Miura-Base Rigid Origami: Parameterizations of First-Level Derivative and Piecewise Geometries,” ASME J. Mech. Des., 135(11), p. 111011. [CrossRef]
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Silverberg, J. L. , Evans, A. A. , McLeod, L. , Hayward, R. C. , Hull, T. , Santangelo, C. D. , and Cohen, I. , 2014, “ Using Origami Design Principles to Fold Reprogrammable Mechanical Metamaterials,” Science, 345(6197), pp. 647–650. [CrossRef] [PubMed]
Kim, J. , Hanna, J. A. , Byun, M. , Santangelo, C. D. , and Hayward, R. C. , 2012, “ Designing Responsive Buckled Surfaces by Halftone Gel Lithography,” Science, 335(6073), pp. 1201–1205. [CrossRef] [PubMed]
Guo, X. , Li, H. , Ahn, B. Y. , Duoss, E. B. , Hsia, K. J. , Lewis, J. A. , and Nuzzo, R. G. , 2009, “ Two-and Three-Dimensional Folding of Thin Film Single-Crystalline Silicon for Photovoltaic Power Applications,” Proc. Natl. Acad. Sci., 106(48), pp. 20149–20154. [CrossRef]
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Figures

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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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