Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: 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.

Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: 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.

Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: 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.

Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: 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

Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: 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.

Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: (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.

Date of download: 11/11/2017 Copyright © ASME. All rights reserved. From: Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476 Figure Legend: 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