Accepted Manuscript Origami and kirigami inspired self-folding for programming three-dimensional shape shifting of polymer sheets with light Qiuting Zhang, Jonathon Wommer, Connor O’Rourke, Joseph Teitelman, Yichao Tang, Joshua Robison, Gaojian Lin, Jie Yin PII: DOI: Reference:
S2352-4316(16)30126-2 http://dx.doi.org/10.1016/j.eml.2016.08.004 EML 207
To appear in:
Extreme Mechanics Letters
Received date: 31 May 2016 Revised date: 10 August 2016 Accepted date: 22 August 2016 Please cite this article as: Q. Zhang, J. Wommer, C. O’Rourke, J. Teitelman, Y. Tang, J. Robison, G. Lin, J. Yin, Origami and kirigami inspired self-folding for programming three-dimensional shape shifting of polymer sheets with light, Extreme Mechanics Letters (2016), http://dx.doi.org/10.1016/j.eml.2016.08.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Origami and kirigami inspired self-folding for programming three-dimensional shape shifting of polymer sheets with light Qiuting Zhang#, Jonathon Wommer#, Connor O’Rourke#, Joseph Teitelman, Yichao Tang, Joshua Robison, Gaojian Lin, Jie Yin* Applied Mechanics of Materials Laboratory, Department of Mechanical Engineering, Temple University, 1947 North 12th Street, Philadelphia, PA 19122, USA
Abstract: Origami and kirigami guided programmable shape shifting is explored via self-folding and spontaneous buckling of a thin sheet of shape memory polymer with light. By patterning the sheet with printed black ink lines as actuating hinges, we show that the folding angle can be well controlled by tuning the ink line width, which is predicted by both a simplified localized bilayer folding model and corresponding finite element method (FEM) simulation. Inspired by the approach of paper origami and kirigami combining folding and cutting, we then explored the design of prescribed patterned creases (i.e. ink lines) and/or cuts in the polymer thin sheet for programming a library of light-responsive three-dimensional (3-D) surfaces in a controlled fashion. Through the design of prescribed straight and curved crease patterns in origami, we demonstrated the generation of light-driven self-folding cylinders, helices, and pyramids with zero Gaussian curvature, as well as spontaneous formation of saddles with negative Gaussian curvature through localized curved folding induced global buckling. The quantitative underlying mechanism governing the geometry of the different self-folded 3-D structures is revealed through simple geometrical modeling and FEM simulations. Lastly, through kirigami combining both folding and cutting in the form of line cuts or cut-outs, we demonstrated the spontaneous formation of light-responsive, more complex pop-up kirigami structures.
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Keywords: origami and kirigami, self-folding, buckling, programmable matter, light-responsive #
These authors contributed equally to this work
*Corresponding Author: J. Y. E-mail:
[email protected]
1. Introduction The beauty of starting from a two-dimensional (2-D) thin sheet to generate a desired nontrivial three-dimensional (3-D) structure on demand has fascinated both artists and scientists for decades[1-7]. Inspired by the ancient art of paper folding and cutting, origami and kirigami (“ori” means “folding” and “kiri” means “cutting”), researchers have applied the concept of folds and cuts beyond paper to generate a variety of compactable or expandable 3-D materials and structures from macro- to nanoscale[8-12], which have enabled broad applications in space exploration[13], biomedical devices[14], energy storage[15, 16], reconfigurable robotics[17, 18], stretchable electronics[19, 20], and reconfigurable mechanical metamaterials[21-23]. Recently, there are growing interests in self-folding origami for producing desired programmable 3-D objects from 2-D polymeric sheets with prescribed crease patterns[21, 24-30], including boxes[27, 31], Miura-ori patterns[25], boat-, airplane-, and bird-like shapes[24, 28]. Normally, creases are localized regions where active materials (e.g. gels, liquid crystals, and shape memory polymers (SMP) ) actuate to fold in response to external stimuli (e.g. temperature, pH, solvent, and light), thus creases play the role of stimuli-responsive hinges. Rather than global bending of the entire sheet, folding in active origami is triggered by localized bending deformation along the hinges, inducing a localized curvature in hinge regions[24, 28]. Despite the simple yet straightforward way to form complicated shapes through folds, most of the studied self-folded 3-D origami structures still fall into the category of developable objects
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with zero global curvature due to the limitation of rigid rotation of facets around hinges. Meanwhile, different from the localized folding in active origami, global buckling instability in thin sheets provides an effective way to form nontrivial 3-D structures with non-zero global curvatures[4], arising from differential in-plane strains induced out-of-plane buckling. Nature provides a number of such examples of harnessing buckling instability for self-assembly formation of varieties of 3-D shapes during growth, including leaves, flowers, fruits, and growing soft tissues[32-34]. By controlling the inhomogeneous swelling/shrinkage in a thin sheet of soft materials, researchers have generated 3-D stimuli-responsive surfaces with constant Gaussian curvature in the shape of spherical caps, saddles, and cones, and other complex shapes through global buckling and local wrinkling and creasing [1, 4, 35-37]. However, the buckling strategy of shaping thin sheets into desired such simple 3-D shapes with non-zero curvature often involves either heterogeneous materials, or complicated multi-step fabrication techniques, or complex control of differential swelling or shrinkage patterning in soft materials. Such simple yet fundamental surface shapes make one wonder whether there exist other relatively simpler way to form the similar 3-D stimuli-responsive surfaces with constant Gaussian curvature (defined as the product of the two principal curvatures), as well as other complex shapes of structured surfaces, in a programmable and controllable manner. The possibility of simple origami and kirigami inspired approach combining folding and/or cutting is explored in this study through either localized self-folding or localized folding induced global buckling. Although there are many ways in actuating the self-folding of creases or hinges as mentioned above, here we employ the similar approach introduced by Liu et al. [38] of harnessing local light absorption to trigger the localized self-folding of creases in our studied origami and
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kirigami structures due to its extreme simplicity. The studied polymer thin sheet in this paper is a thin sheet of commercial shrink paper prescribed with patterned printed black ink lines. The shrink paper is composed of pre-strained polystyrene (“Shrinky-Dinks”), a type of SMP that shrinks in plane upon heating. Upon local absorption of near-infrared (NIR) light, the black ink line shrinks to actuate the localized folding, acting as a mountain or valley fold in self-folding origami and kirigami depending on which side it is printed. To quantitatively guide and control the self-folding of hinges, a simplified, localized bilayerfolding model is first discussed in this paper to correlate the quantitative relationship between the folding angle and the width of ink stripes in experiment, as well as to compare with finite element method (FEM) simulation. Such a simple mechanics model provides a fundamental tool for guiding the controllable self-folding in the following discussed origami and kirigami-inspired 3-D structures and surfaces using prescribed patterned ink lines. Inspired by the folding-based paper origami approach, we then explore the design of prescribed straight and curved crease pattern in a thin sheet for achieving programmable light-responsive 3-D surfaces with controllable zero, positive, and negative Gaussian curvature K. We develop simple geometrical models to predict such different 3-D shapes and compare with corresponding FEM simulations. Lastly, through combined bi-directional folding and cutting in the form of line cuts or cut-outs with a certain shape, the actuation of responsive pop-up kirigami structures is discussed.
2. Experimental Method The experiment was set up as follows guided by Ref. [38]: a commercial Grafix transparent ink jet shrink paper with thickness of 0.3 mm was chosen and used as is. The optical transparency allows for bi-directional folding when printed on both sides. With the design of
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crease patterns in computer aided design software (e.g. AutoCAD) including geometry and line width control as inputs, we used a desktop printer (HP 1536 LaserJet) to introduce different black ink printed patterns to single or both sides of the shrink paper. The self-folding of samples was actuated by one or two clear infrared heat lamps (BR40 250W) hanging in series on a customized holder as the light source. The distance between the lamp and samples was designed to be adjustable through the holder. In this study, a distance of about 10 to 12 cm was set for uniform light intensity distribution. A short exposure time of about 10 to 30 seconds was set to avoid over-heating and over-deformation.
3. Control of Folding Angle through Ink Line Width As schematically illustrated in Fig. 1a, upon exposure to near-infrared (NIR) light, the black ink absorbs light locally and heats the underlying polymer gradiently throughout the thickness to above its glass transition temperature Tg. The resulting stress relaxation induces local gradient shrinkage, and thus triggers the self-folding along the ink line (Fig. 1a). The black ink line acts as an actuating hinge for guiding both the folding direction and folding angle by controlling the width of ink pattern, light intensity, and exposure time. After uniformly heating above Tg, the folded structure can be unfolded into its original planar sheet with a shrinking size [38]. Fig. 1b shows the optical images of folded rectangle samples (40 mm × 20 mm) with the width of black ink line w in the center changing from 0.7 mm to 2 mm under the same exposure time of 10 seconds. The folding angle is defined as the interior angle between two folded planes (Fig. 1a). Fig. 1c shows that decreases approximately linearly with w for the studied range, changing from 180o (no folding) to below 90o (large folding) averaged by 5 measurements for each line width.
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To better understand the quantitative relationship between and w, we used a simplified localized bilayer beam bending model [39] and compared it with FEM simulation. It should be noted that Liu et al. [38] also developed a simple geometrical model of predicting the folding angle for the same studied material system. However, the information on the role of material properties in determining the folding angle is missing in their geometrical model. Fig. 1d schematically shows a composite beam with thickness t and length L. The localized black part represents the shrinking region underneath the ink line with a coefficient of thermal expansion
(), where its length is set to be equal to the ink line width w and thickness is set to be ts = m t with m being the thickness ratio The grey part represents the unshrinking medium. The materials are assumed to be isotropic and linearly elastic. Es and Eu are the Young’s modulus of shrinking and unshrinking parts in the composite, respectively, representing the respective rubbery and glassy modulus. It should be noted that underneath the black ink lines, it undergoes complex photochemical reactions and gradient stress relaxation throughout the thickness, which is induced by thermal gradient via localized light absorption. The employed localized bilayer model is reasonably oversimplified to shed some light on controlling the folding angle through ink line width. Upon the mismatched thermal deformation with a temperature change of ΔT, the angle between the cross-section sides of the bilayer (Fig. 1d) is given by[39]
w r
(1)
1 6T r t 4 2m2 2m m3n 1 m 1 m 3 mn
(2)
where r is the bending radius (Fig. 1d) and n = Es / Eu. Then the folding angle can be given by
w r
(3)
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Eq. (3) shows that the folding angle is negatively proportional to w and the mismatched thermal strain T = ΔT, i.e. the shrinkage strain. As seen from Eq. (3), unlike the geometrical model in Liu et al. [38], it shows that is dependent on the modulus ratio n, implying its non-negligible role in determining the folding angle. To validate the simplified model and compare to the experiment, we use a 3-D FEM modeling with 3-D hexahedral elements (C3D8R element in ABAQUS) to simulate the selffolding of the thin sheet. The increase of temperature is used to simulate the time period of light exposure time. Fine mesh was applied to the actuating hinge regions with the displacement of one bottom edge being fixed. The geometrical size of the 3-D model (length 40 mm width 20 mm thickness 0.3 mm) is set to be the same as that of the experiment. The modulus ratio n and thickness ratio m are estimated to be 0.1 and 0.5 from experiments, respectively. The thermal shrinkage T is estimated to be 30% for the exposure time, i.e. T = 30%. In the simulation, the width w varies from thinner ink line (w = 0.5 mm) to thicker one (w = 2.0 mm) with other parameters fixed to investigate its effect on the folding angle. Figure 1c shows an excellent agreement between FEM simulation (red solid dots) and theoretical model (blue line) in predicting the folding angle as a function of ink line width, which validates the localized bilayer beam bending model. As the ink line becomes thicker, more severe stress concentration is observed in the hinge area as shown in simulations (inset of Fig. 1c). The simulation and theoretical model show a good agreement with the experiments when the line width is larger than 1.4 mm, while the model overestimates the folding angle for thinner ink lines, suggesting the oversimplified model need to be improved by considering the continuous thermal gradient throughout the thickness, as well as the constitutive modeling of stress relaxation in the hinge materials with temperature in the future study. 7
4. Origami-Inspired Self-Folding 3-D Structures with Different Gaussian Curvatures Armed with the knowledge of controlling with w, we apply the controllable self-folding hinge method to generate a library of 3-D origami-inspired structures with zero, positive, and negative Gaussian curvature K from light-responsive shrink paper sheet with prescribed ink patterns. The ink lines are applied to the shrink paper with the same line width of 1.2 mm. Cylinders (K = 0) through parallel folds: To generate a cylindrical shape from a rectangle paper stripe, we can fold along the parallel creases (blue lines) to form a closed cylinder with a polygonal cross-section (left of Figure 2a). Similarly, to generate a self-folding cylinder with light, parallel patterned ink lines were introduced to a rectangle sheet with the parallel distance setting to be 5 mm in Fig. 2a. The lines act as the folding creases in the paper origami. The right of Fig. 2a shows the captured images from videos on the self-rolling process of a shrink paper upon exposure to NIR light (Supplementary video S1). When the ink line absorbs the light, it starts to fold sequentially along the ink line with slight and constant movement of the light source, driving the global folding of sheets into a closed cylinder, where the patterned ink lines are observed to align with the longitudinal axis of the self-folded cylinder. It should be noted that similar self-rolling of a bilayer thin sheet into a cylindrical shape has been extensively studied before[6, 7, 40], however, the deformation mechanism is different. The deformation in a bilayer thin sheet is a global folding due to the mismatched strain between the layers, induced by differential swelling/expansion, shrinkage, or mechanical strain. However, in our work, each segment folds locally into a global cylindrical shape. The simple origami approach allows the accurate control of the localized folding angle in each segment, i.e. the localized curvature, and thus the control of uniform or non-uniform global curvature of the whole structure on demand.
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Helices (K = 0) through oblique parallel folds: To generate a helical structure, we can fold a paper sheet along patterned parallel oblique creases (left of Fig. 2b). Accordingly, in the selffolding helical structure, the parallel ink lines in Fig. 2a were tilted to the left with a certain angle, leading to the self-rolling into a helical shape (right of Fig. 2b and supplementary video S2). The oblique folding lines spontaneously impose a handedness determined by the inclined orientation. When they are inclined to the left, it generates right-handed helices (Fig. 2b), while creases inclined to the right generate left-handed helices. The sequential self-folding through the control of movable light source provides an effective way to generate a long helical structure (Fig. 2b).Similar self-folding helical shapes were reported by using composite hydrogel sheet with inplane periodic structures of oblique soft and rigid stripes along the long axis[3], rather than patterned localized bilayer structure across the sheet thickness in our work. Despite the formation of similar helical shapes, their underlying mechanisms are distinct from each other. One is through global buckling in composite hydrogel, arising from the competition between stretching and bending energy in the hard and soft domains with differential swelling, and thus the orientation of stripes is not necessarily in alignment with the axis of the formed helix[3]. The other is through localized folding in this work. Each hinge folds locally along the helix axis and collectively leading to the global folding into a helical shape without the occurrence of abrupt and uncontrollable buckling, and thus is more controllable. Pyramids (K = 0) and domes (K > 0) through intersecting folds and cut-outs: In addition to the demonstrated parallel patterning of creases and folds, it can also actuate non-parallel creases to self-fold into polyhedral shapes with K = 0 and dome-like shapes with K > 0. In paper origami, it remains challenging to start from a regular hexagonal shape to fold along its diagonals into a pyramid without compressing and bending the triangular facets. To avoid bending deformation
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in non-folded rigid facets, we can cut one triangle out of the hexagon to release the compression, and thus to fold it into a pentahedron manually using hardcopy paper (left of Fig. 2c). Similarly, when applying ink lines to the diagonals of a hexagon of a shrink paper sheet with a triangular cut-out, it shows that it can first self-fold into a pentahedron and then a tetrahedron with one facet overlapped upon a prolonged light exposure time (right of Fig. 2c and Supplementary video S3). Despite the generation of similar pyramid shapes in previous studies, e.g. a tetrahedron starting from closed square-shaped hinges as a bottom base surface[27, 38], the method of selffolding of hinges combined with cut-out here allows for multiple shape transformation from a single layout design. Similarly, through cut-outs to release the stretching or compression energy, it is possible to generate non-developable surfaces and structures such as a dome-like shape with positive Gaussian curvature. Fig. 2d shows such an example of a self-folding dome-like shape from a circular disk with a triangular cut-out. The disk is patterned with radial ink lines as hinges and a small area of ink in the center. Similar to the formation of polyhedron, upon light exposure, the disk folds up and then closes tightly with overlapping (Supplementary video S4), showing positive and zero Gaussian curvature near and away from the apex, respectively. Saddles (KN2, and thus an overlapping feature in terms of Eq. (7). Fig. 3c shows the simulated shape evolution starting from a planar sheet with parallel patterned ink lines to an overlapped cylinder with the increase of temperature (i.e. to simulate the time period of light exposure time), where all the hinges are activated to fold simultaneously. Each hinge is shown to fold the same angle by following the theoretical prediction by Eq. (3). As temperature increases, the occurrence of overlapping feature is consistent with the theory in Eq. (7). 5.2 Self-folding helix via inclined straight creases Fig. 4a schematically illustrates the generation from a flat stripe with patterned oblique ink lines (left) to a helical structure (right). As shown in Fig. 4a, the initial flat stripe has a length of Lh. The distance between ink lines is d and the oblique angle of ink lines is The geometry of the resulting folded helix can be characterized by the radius Rh, pitch p, and helix angle . Fig. 4b shows the simulated evolution of shape change during the self-folding process of helix formation using FEM. As temperature T increases, all the hinges start to fold simultaneously and slightly to form a spiral structure with a large radius and pitch distance. When T further increases, the hinges fold more, rendering the spiral structure become tighter and smaller by reducing both the gap distance between helix and the tubular radius. The bottom row in Fig. 4b shows the corresponding top-view of the helical structure evolution. The simulated self-folding process is consistent with experiments.
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By assuming the same folding angle in all the oblique ink lines, the geometrical connection between the patterning of ink lines and the shape of resulting self-folded helix can be readily established, which gives
2
(8)
Lh 2 w d
n
Rh
p
(9)
d 2cos 2
(10)
2 d w cos
(11)
where n is the number of helical loops. The radius in Eq. (9) is the same as that of cylinders in Eq. (4), and thus follows the same results as those found in the case of cylinders. Eq. (11) shows that the pitch p is solely determined by the geometry of ink patterns and is independent of the original size of flat stripes. To validate the geometrical model, we conducted parametric studies of FEM simulations by changing the geometry of ink patterns. Fig. 4c shows the comparison between the geometrical modeling and the corresponding FEM simulation results on the normalized pitch p by parallel distance d, as a function of and . Due to the symmetry, is set to vary from 0o to 90o. The prediction from Eq. (10) (represented by solid curves) shows an excellent agreement with the corresponding simulation results (represented by solid symbols), which validates the geometrical model. It shows that when is fixed (e.g. = 105o), p/d decreases nonlinearly with , i.e. a larger oblique angle leads to a tighter helical structure with smaller gaps, as can be seen from the simulated helix shape in the insets of Fig. 4c. However, when is fixed (e.g. = 70o), p/d
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increases nonlinearly with . The insets of Fig. 4c show that at fixed when increases from 105o to 131o, it leads to a relatively looser helix with larger gaps. The geometrical model is further qualitatively validated by experiments. To examine the effect of on the shape of self-folded helices, we let increase gradually from the left to right in the helices shown in Fig. 4d while all the other parameters are kept the same, i.e. the size of rectangle stripes, w, d, and light exposure time are set to be identical. Fig. 4d shows that increasing leads to a decreasing pitch p, i.e. the shape changes from a loose to a tight helix with an increasingly closing gap, which is consistent with the prediction from Eq. (11) and Fig. 4c. Meanwhile, as increases, the radius of the helix Rh also increases, which is not consistent with Eq. (10) since Rh is independent of . Such a discrepancy between the idealized geometrical model and experiments originates from the slight optical opacity of the polymer sheet. Compared to the helix with large gaps, the tight helix with small gaps produced from larger blocks the light transmission and absorbs less light, resulting in a smaller folded angle, i.e. a larger . In terms of Eq. (10), an increasing leads to an increasing Rh, which corresponds with the experiments. 5.3 Self-folding saddles via curved creases Based on the curved crease origami approach, we can introduce different curved crease patterns to generate self-folding shapes with negative Gaussian curvature. Fig. 5a shows a selffolded saddle produced from a shrink paper annulus with a central, circular ink line. Upon shrinkage and folding along the circular crease, the annulus buckles out of plane to form a saddle shape, similar to the paper origami saddle in Fig. 2d. Similarly, after applying a spiral ink pattern with equal distance to a circular disk, it can also self-fold into a saddle shape via buckling (left of Fig. 5b). The observed saddle shapes are well reproduced by corresponding FEM simulations by 15
assuming the localized shrinkage only in the crease patterns (right of Fig. 5a and 5b). Simulation results show that the localized shrinkage induces severe stress concentration along the curved creases highlighted by red color. However, it should be noted that a localized shrinking spiral crease pattern does not always lead to a saddle shape upon actuation. Very recently, a printed similar spiral fibril was applied to a circular disk of hydrogel bilayer materials using biomimetic 4D printing[43]. Upon immersed in water, it produced a conical dome-like structure due to the different anisotropic swelling behavior of spiral fibrils from the isotropic shrinking of spiral ink line in this work. Since the mechanics of buckling induced formation of origami saddles has previously been studied [41, 44], here we focus on its geometrical modeling of the self-folded saddle shape controlled by the localized shrinkage of a central ink annulus with width w. As shown in Fig. 5c, the self-folded saddle shape can be described by the equation x2 y2 = a2z, where a is constant that represents the same level of curvature in both the x z and y z planes. A cut plane of y = k x (shaded plane in Fig. 5c) intersecting with the saddle gives a parabolic curve, which can be expressed by the equation z = (1 k2) x2 / a2. Point P (red color point) is the intersection point with coordinates of (x*, y*, z*). Geometrically, the length of the intersected parabolic curve l is equal to the reduced diameter Drof the disk due to folding along the circular creases with a folded angle . Dr is given by Dr = D 2wsin(/2) with D being the original diameter. Hence, l = Dr gives
2
x*
0
1 4 1 k 2 x 2 a 4 dx D 2w sin 2 2
(12)
is a function of w and temperature induced shrinkage strain T = ΔT. From Eq. (12), the coordinates of point P can be expressed as a function of k, a, w, and ΔT, i.e.
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2 2 x* f k , a, w, T , y* kx* g k , a, w, T , z* x* y* a 2 h k , a, w, T
(13)
The geometrical compatibly condition requires that the outer circumferential length of the disk not be changed when it transforms to the outer edge of a saddle, which gives
D f g h dk 4
2
2
2
(14)
0
where “ ' ” means the differentiation with respect to k. After integrating over k, Eq. (14) gives a relationship between a, w, T, and D. However, it remains challenging to numerically solve the above equations and find the relationship. Thus, we rely on FEM simulation to shed some light * on controlling the saddle shape through w and T . Fig. 5d shows the curves of Z max , defined as
the maximum out-of-plane displacement, normalized by D as a function of shrinkage strain T = * ΔT and ink line width w. It shows that Z max increases nonlinearly and monotonically withT .
* A thicker ink line leads to a higher Z max due to its relatively larger folded angle.
5.4 Other self-folding shapes via curved creases When replacing the straight crease (straight ink line) in Fig. 1a with a more generalized case of a sinusoidal crease45 (a sinusoidal ink line) as shown in Fig. 6a, it self-folded into an arc shape by bending both parts of the sheet (Fig. 6b) oppositely when exposure to NIR light. This is in contrast to the localized folding in straight creases without globally bending the sheet (Fig. 1b). The corresponding FEM simulation shows a similar cylindrical inversion shape by bending the two halves in an opposite way when the sinusoidal crease is introduced to the middle (Fig. 6c). Similar to the self-rolling of cylinders induced by parallel straight creases in Fig. 2a, we can apply parallel sinusoidal creases (ink lines) to the thin sheet of shrink paper as schematically illustrated in Fig. 6d. The localized curved folding leaded to the global folding of the thin sheet into a shallow arc (Fig. 6e). We observed that upon a longer exposure time to NIR light, the 17
shallow self-folded arc shape couldn’t be further folded and bended due to the large bending rigidity of the shrink paper. However, a more folded cylindrical arc shape was observed in the corresponding FEM simulation (Fig. 6f). Despite the similar self-folded arc shapes, the curved creases lead to the global folding along the direction perpendicular to the creases, while the straight creases result in the global folding along the direction parallel to the creases (Fig. 3c). Thus, when the straight creases become curved in parallel, the global self-folding direction switches to its orthogonal direction due to the bending of the folded surface.
6. More Complex 3-D Shapes via Self-Folding Kirigami All the 3-D structures discussed so far are generated through uni-directional folding by patterning the shrink paper with single side printing. However, ink line patterns on both sides of shrink paper allow for the control of bi-directional folding, and thus provide the possibility to generate more complex 3-D structures. In addition, compared to the folding-based origami approach, cutting-based kirigami can bring an extra level of design space to open or close materials to achieve more unique compactable and deployable 3-D structures, which are unattainable by origami alone. To demonstrate the uniqueness of the kirigami approach, we use two representative kirigami designs, for example: one is a combination of folds and cut-outs, which renders a fold-to-close structure (Fig. 7a-7b); the other is a combination of folds and line cuts, which generates a fold-to-pop-up structure (Fig. 7c). Fig. 6a shows the one-unit planar design of a kirigami structure with a combined hexagonal cut-out and folds (valley and mountain creases) from Ref.[46]. After folding the paper kirigami along the creases, it can close and hide the cut out (bottom of Fig. 7a), more efficiently using materials to build 3-D structures. The unit structure can be repeated to create multiple-unit
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structures through folding (Fig. 7b). By replacing the mountain and valley creases in the top of Fig. 7a with printed ink lines on both sides of a shrink paper, it shows that the patterned sheet can self-fold into the same designated structure as paper kirigami by closing the pore upon light exposure (Fig. 7d and Supplementary video S6). To generate nearly the same folding angle from both valley and mountain creases, we used a thickness of 1 mm for the ink lines printed on the front (i.e. valley creases), and a thickness of 2 mm for the lines printed on the back (i.e. mountain creases). The reason for the difference is that the ink lines on the back side do not absorb as much heat as the front side lines due to the slight tint of the shrink paper sheet, generating a smaller folding angle along the mountain creases when given the same thickness. Fig. 7e further demonstrates the self-folding kirigami in multiple units with cut-outs (Supplementary video S7). The top of Fig. 7c shows the design of a typical, hierarchical pop-up kirigami structure with line-cuts and folds. Folding the paper along the central folding crease results in the spontaneous folding along all the other creases in the design by opening the pores of line cuts, thus a hierarchical pop-up structure is formed as shown in the bottom of Fig. 7c. Similarly, by replacing the hardcopy paper with a shrink paper sheet patterned with the same layout of cuts and doublesided ink lines for actuating hinges, we demonstrated the self-folding, level 3 step-like pop-up kirigami structure in Fig. 7f and supplementary video S8, where one edge is adhered to the substrate with tapes to guide the folding up. 7. Conclusions In summary, through the prescribed planar design of patterned folds and cuts, we have demonstrated the origami and kirigami-inspired programmable self-folding of 3-D structures with different curvatures from a thin polymer sheet with light. The printed ink line on the polymer sheet acts as a light-responsive actuating hinge for guiding both the folding direction
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and folding angle, depending on the side it’s printed on and line width. In origami-inspired selffolding structures, straight creases fold locally rendering a localized curvature, providing an effective way to construct global 3-D structures with zero Gaussian curvature, whereas curved creases fold locally to buckle the sheet out of plane globally, and thus provides an exceptional ability to construct 3-D structures with non-zero global curvatures. In kirigami-inspired selffolding structures, the uniqueness of cuts enables the formation of self-folding 3-D structures that are unattainable by folding-based origami. The origami and kirigami inspired approach provides a simple and powerful way for generating programmable matter via actuating the self-folding along the designed creases. Although we only demonstrated the self-folding structures made from pre-strained SMPs on the macro-scale via local light absorption, we believe that the approach of active origami and krigami can be scaled down to the micro and even nanoscale, as well as be applied to more types of light or temperature stimuli-responsive materials such as shape memory polymers, hydrogels, and liquid crystals[47]. The results presented in this work will provide useful guidance for design of patterned cuts and folds for achieving more variety of controllable and desired self-folding 3-D shapes with extraordinary properties and functionalities.
Acknowledgements J. W, C. O’R, J. T, and J. R acknowledge the financial support from the Merit Scholar Program at Temple University. J. Y. acknowledges the funding support from the start-up at Temple University.
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Figure Captions:
(a)
Light Pre-strained polystyrene
Folding
w Black ink
Hinge
(b)
1 cm
Width (mm):
w = 0.70
0.90
1.20
(c)
2.00
1.58
(d) 0.5 mm
w
π
t Folding 2.0 mm Shrinking
,
Unshrinking (1- m)t,
1.5 mm
,
r
w
w Figure 1 Controllable folding angle in light-responsive pre-strained polystyrene sheet through printed ink lines (a) Schematic illustration of self-folding of a polymer sheet via localized light absorption in black ink lines, acting as actuating hinges (b) Optical images of self-folded polymer stripes by increasing the ink line width under the same period of light exposure time. (c) Comparison between experiments, FEM simulation, and theoretical modeling on the folding angle as a function of ink line width. (d) Schematic illustration of localized bilayer folding model
24
(a)
Ink print
crease
1 cm
2 cm
0s
3s
8s
10 s
0s
3s
6s
14 s
0s
3s
4s
11 s
K > 0 & K=0
0s
6s
7s
8s
K
