Origami structures are a traditional Japanese art w hich have recently found their way into engineering applications due to their powerful capa bility to transform flat 2D structures into complex 3D structures along their creases. Here, gr adient Miura-ori origamis are introduced as a method to pre-program out-of-plane curvatures. Ni ne types of unit cell distributions in the origami lattice structure including checkered, line ar gradient, concave radial gradient, convex radial gradient, and striped have been considered. These distributions of Miura-ori origami can create twisting, saddling, bending, local inflation , and wavy shapes, as well as their combinations when the origami lattice structure is loaded in compression/tension. Keywords : Origami; Gradient, Miura-Ori; Metamaterials; Curv ature 2 Designer materials, where rationally designed geome try at the small-scale gives rise to unusual materials properties at the macro-scale, ar e often called metamaterials [1, 2]. Origami structures are a traditional Japanese art w hich have recently found their way into engineering applications due to their powerful capability to transform flat 2D structures into complex 3D structures along their c reases. Particularly, Miura-ori origami, also known as herringbone geometry (Figure 1), has been proposed as an origami-based mechanical metamaterial [3]. This des ign was first invented to pack solar panels efficiently, but it is relevant to note that this pattern also occurs in natural structures such as leaves, embryonic intestine [4], and vertebrate guts [5]. The mathematical richness and tunability of Miura-ori g eometry allows using it in different scales from nanometric level to architecture [6]. T he suitability of Miura-ori for engineering applications lies in its four spectacul ar characteristics: being able to be folded rigidly, having only one degree of freedom, having negative Poisson’s ratio, and being flat-foldable [6]. c (a) (b) Figure 1: (a) Geometry and dimensions of unit cells of a Miura-ori origami structure, (b) Miura-ori origami with uniform unit cell distribution before and after compression 3 3D exotic materials with negative Poisson’s ratio ( known as auxetics) offer extreme mechanical properties such as high shear resistance , energy absorption, indentation resistance, and toughness [7]. When used in 2D sett ing, a structure with spatially incompatible auxetic properties can create various 3D shapes through out-of-plane buckling [8]. Given the richness of the Miura-ori origami, the qu estion that arises here is whether or not it is possible to create basic well-known Eucli dean and non-Euclidean 3D curvatures using Miura-ori tessellations upon application of a n external stimulus such as mechanical load. There has been a few studies which have used origami tessellations to create pre-programed (not necessarily 3D) curvature s [3, 6, 9]. Although a recent study presented an optimization-based method procedure to produce Miura-ori tessellations [6] programmed to create 3D geometries with some approx imations, no study has been carried out on how basic gradient distributions of Miura-ori unit cells can be used to create various basic 3D curvatures. Basic Miura-ori distributions (rather than ones created by numerical models) have the advantages of being easy to understand and thus easy to implement for practical problems. In this paper, for the first time, we introduce bas ic gradient tessellations as a tool to create programmable basic 3D curvatures. In our pre vious work [10], we demonstrated how using gradient distributions of re-entrant auxe tics can lead to action-at-a-distance actuators. While such block-shaped actuators are us eful in many applications such as soft robotics, sensors, and controllers [10], curve d actuators seem to offer more exciting functionalities useful in more advanced configurati ons. Geometries Eight types of tessellations with variable mountain height namely checkered, linear gradient (in both the X and Y directions), concave radial gradient, convex radial 4 gradient, striped (in both the X and Y directions), and regular were considered. It must be noted that the XY is the plane parallel to the m id-plane of the origami lattice structure (see Figure 1a). In most of the structure s, regardless of the mountain peak height, the parameters , , and  were set constant and equal to  = 19.64 mm,  = 19.84 mm, and  = 22.68 mm. The overall length of the origami lattice stru ctures in each direction was therefore 392.7 mm. The maximum and minimum heights of the mountains in all the configurations was considered as  = 22.5 mm and  = 4.5 mm, respectively. Some of the above-mentioned geome tries, due to including non-flat surfaces, or due to including particular patterns w ere not possible to be manufactured by folding flat plates. Therefore, to make the orig amis foldable from flat papers, in addition to the noted structures, two additional ge ometries (linear X and striped X) in which instead of mountain height, unit cell width w as varied was also considered. The dimensions of Miura-origami are [11]: =+  (1) =sin  ! (#)% (2) = &' () (! (#) (3) )=1+cos ()tan (#) (4) For the unit cells with maximum height,  = # =. / and  = ) = 3 11 . For the unit cells with minimum height, we set # =. 2. Results In several tessellations, flat-foldability conditio ns based on Kawasaki’s theorem were met which made them manufacturable by folding. Base d on Kawasaki’s theorem, a structure is flat-foldable if the sum of even and o dd sequence of angles, each, equal to 3. The tessellations which did not satisfy the Kawasak i’s conditions were only modelled 5 numerically. Thanks to novel manufacturing technolo gies such as Additive Manufacturing such non-flat-foldable structures ar e now realizable [12]. The flat- foldable origamis namely checkered, linear X gradie nt, linear Y gradient, striped in the X direction, and striped in the Y direction provide d twisting, local inflammation, bending, asp shape, and wavy deformations, respecti vely (Figure 2). 6 Checkered (Twist) Linear X (Local inflammation) Linear Y (Bending) Striped X (Asp) Striped Y (Wavy) Unloaded After deformation (a) (b) (c) (d) (e) Figure 2: Deformation of (a) checkered, (b) linear X gradient, (c) linear Y gradient, (d) striped X, a nd striped Y tessellations. 7 The numerical results show that structures which de part (slightly or highly) from flat- foldability conditions namely concave radial gradie nt, convex radial gradient, linear X gradient (with variable mountain height), and strip ed X (with variable mountain height) distributions of unit cells show respective ly saddle-shape, bending, combined twisting and bending, and combined wavy and bending deformations (Figure 3). Therefore, the results show that different proposed tessellation types are able to transform their shape into different 3D curvatures by means of in-plane incompatibilities. Overall, the achieved 3D curvatu res from the nine proposed tessellations includ twisting, local inflammation, bending, asp-shape, wavy, saddle- shape, combined twisting/bending, and combined wavy /bending deformations (Figure 3). The maximum out-of-plane deformation in the che ckered, linear X gradient (with variable unit cell width), linear Y gradient, strip ed X (with variable unit cell width), striped Y, concave radial gradient, convex radial g radient, linear X gradient (with variable mountain height), and striped X (with vari able mountain height) structures are 19.9 mm, 4.5 mm, 9.4 mm, 4.7 mm, 2.1 mm, 8.8 mm, 11 mm, 14.9 mm, and 7.2 mm, respectively. 8 Front view Isometric view Side view Top view Favourable view (a) Checkered Twist (b) Linear X Local inflation (c) Linear Y Bending (d) Striped X Asp ﺍﻓﻌﯽ (e) Striped Y Wavy (f) Radial concave Saddle (g) Radial Convex Bending (h) Linear X - VH Twist & bending (i) Striped X - VH Wavy & bending 9 Figure 3: The curvatures resulting from Miura-ori origa mis with (a) checkered, (b) linear X gradient (with variable unit cell width), (c) linear Y gradient, ( d) striped X (with variable unit cell width), (e) striped Y, (f) concave radial gradient, (g) convex r adial gradient, (h) linear X gradient (with variable mountain height), (i) striped X (with variabl e mountain height), and (j) regular distributions of unit cells. VH stands for variable (m ountain) height. Discussions It is well known that flat-foldable origamis which have one degree of freedom are easily deployable and energy efficient. One way of making the non-flat foldable structures (Figure 3f-i) flat-foldable is to consider some hol es in their structure and thus to transform the Miura-Ori origami structures into Mir a-Ori hybrid origami-kirigami structures. Kirigami is a variation of origami that includes cutting in the sheet in addition to foldable crease lines. One such combina tion based on zig-zag strips was introduced in [13], and it was shown that such hybr id structures are flat-foldable. Manipulation of Poisson’s ratio, which was achieved by changing Miura-Ori characteristics spatially in this work, can be perf ormed by other means to create similar 3D curvatures. As long as different regions of a flat plate are programmed to demonstrate different lateral behavio rs upon receiving external stimulus, the plate would demonstrate curvatures si milar to what was proposed in this study. Examples include plates with non-uni form/gradient distribution of thermal expansion coefficient under heat stimilu s and plates with non- uniform swelling factor distribution under adjustab le environmental (humidity and/or chemical) conditions. In fact, the mechanica l actuation technique proposed in this study has the advantages of being inexpensive, easy to control, and mechanically durable. As origamis are free from geometrical scales, the p resented approaches can be generalized to other scales all the way from nano-s cale to mega-structures. Therefore, 10 these configurations can have applications in sever al engineering devices such as robotics [14], self-folding assemblies [15], and self-morph ing structures [16]. Such applications are more realizable due to recent advances in manuf acturing technologies such as additive manufacturing [12], stress within thin fil ms [17], nano-lithography [18], and small-scale hinge constructions [19, 20]. The actua tion can be performed either globally or locally. Locally, the actuation can be done by m aking regions at the sides of each unit cell inflatable or stretchable to create respe ctively pressure or contraction at the two ends of each cell. This actuation mechanism can be compared to auto-stress phenomenon in organs [21]. Regardless of the actuat ion technique being local or global, each lattice structure deforms in such a way that i t minimizes the total mechanical energy of the overall system [21]. This can cause d ifferent bifurcations such as local buckling, single curvature, or double curvature dep ending on the unit cell distribution (see Figure 2-3). The other fact to mention is that the proposed desi gns can also be used in hinged configurations. In fact, we tried constructi ng such hinged assemblies (Figure 4). However, as the structures get larger a nd larger, and tessellation pattern numbers increase, the attention required to be paid to precision and accuracy increase exponentially. This is more prono unced in gradient assemblies. In case insufficient attention is paid to precision, very small inaccuracies in the placement of origami faces and attachment of the hinges can lead to very strong unwanted forces in sufficie ntly large structures. This not only decreases the robustness of the structure locally and as a whole, it also departs the structure from rigid-foldability, which , in turn, introduces unwanted energy-consumption through strain energy d issipation. 11 → (a) (b) Figure 4: Hinged Miura-ori origami tessellation (a) befo re and (b) after folding Finally, the basic gradient tessellations could be combined in dual or multiple configurations to give more complex shapes on deman d. We have shown the benefits of such combinations in 2D gradient auxetic configurat ions for programming vase, barrel, and narrow-waisted shapes into the flat rectangular plates [10]. Combinations of the designs proposed in this study can be implemented t o develop deformable surfaces such as flexible yet strong clothing, soft robotics, def ormable batteries [22, 23], deformable electronics [24], and complex actuators [25]. Origa mi-based deformable electronics could demonstrate excellent performance of conventional r igid electrical components as well as high 3D deformability levels. The electronic perfor mance in currently existing highly- deformable thin-film-based solutions are usually di minished to a high degree due to low areal coverage [23]. In summary, gradient Miura-ori origamis were introd uced as a method to create out-of- plane curvatures. Nine types of unit cell distribut ions in Miura-ori origami lattice structure including checkered, linear gradient, con cave radial gradient, convex radial gradient, and striped were considered. These distri butions of Miura-ori origami led to several out-of-plane 3D curvatures with single- or double-curvatures such as twisting, 12 saddling, bending, local inflation, and wavy shapes , as well as their combinations when the origami lattice structure was loaded in compres sion in the direction parallel to the origami mid-plane. All the Gaussian curvatures (neg ative, positive, and zero) was achieved using the proposed models. Our approach wi ll help tailoring complex pre- programmed surface geometries by employing basic gr adient distributions of Miura-ori origami. Methods The specimens were generated from the original Miur a-ori crease pattern. According to 3D models, creases were designed, transferred to paper s, and folded by hand. Thin kraft papers (120 gsm) with thickness of 2.5 mm were used as initial flat sheets. All the specimens were made in one piece. All the fabricated origamis were placed vertically, and their lower edge was constrained using glue. The top side of the origami was compressed vertically manually for ≈ 2% strain (similar to the simulations). ANSYS finite element (FE) modeling package was used to construct the geometry of the origami lattices and to carry out the simulations. An algorithm was developed to vary the unit cell height or width  (Figure 1) depending on the type of tessellation. Each crease line was discretized using 10 elements. Therefore, each latt ice structure consisted of 40,000 planar elements (SHELL 181 in ANSYS library). The material properties of Kraft 120 gsm paper was given to the models ( 56= 1.7 89 and : = 0.4 ). The thickness of the origami was set to 2.5 mm. One side of each origami tessellation was const rained in the Y direction and the other side was compressed in the Y direction ( <== 6 mm). The other degrees of freedom were kept free to allow out-of-plane displacements. 10×10 tessella tions were used for all the computational models. 13