The fabrication of thick origami structures is usually rigid and not flat foldable. This work demonstrates a technique for creating non-rigid, fully flat foldable, and thick origami. By harnessing the material properties of 3D-printed Polylactic Acid (PLA), thick origami panels that selectively exhibit non-rigid be- haviors can be fabricated using living hinges. Constraining the deformation of these living hinges within the elastic regime of the material we can create robust flat foldable thick origami structures. These methods are demonstrated in a para- metric model of a thick origami flat foldable Kresling pattern. The kinematics and bistable mechanical properties of these Kresling structures will be characterized to demonstrate the viability and scalability of this technique. 1 Introduction This study presents a novel design technique for creating flat foldable derivatives of the Kresling pattern using thick, nonrigid origami methods. The Kresling origami pattern arises naturally when a thin sheet of paper is wrapped over the ends of two cylinders and a twist is applied to the cylinders which are held apart, leaving a gap. This twist results in the spontaneous formation of a belt with the Kresling pattern on the piece of paper over the gap, allowing it to fold to a flat diaphragm as illus- trated in Figure 1 (a-c). A pattern similar to the Kresling pattern is also found in nature within the air sac of a hawkmoth. The pattern is able to expand and con- tract similar to a set of bellows [Dalaq and Daqaq 22]. The simple design and flat foldability of the Kresling pattern make it ideal for use in deployable structures as the belt can be easily extended and collapsed [Kresling 08]. However, the induced torque of the helical creases is not always desired and can be eliminated by adding multiple Kresling units with opposing chirality. As the Kresling transitions from the collapsed state to the expanded state, one can feel the presence of an energy barrier resulting in the bistability of the pattern. The presence of this energy barrier indicates that some deformation occurs outside of the predefined creases. Thus the pattern is inherently non-rigid. This non-rigid deformation poses a challenge for current thick origami methods that rely on the use of thick rigid materials [Lang et al. 18]. Another challenge posed by the Kresling pattern regarding thick origami BERSHADSKY , PAULINO (a) (b) (c) Figure 1: Spontaneous generation of a Kresling pattern using two cylindrical tubes and a piece of paper. (a) the paper is wrapped around the two tubes coaxially, leaving a gap between them equal to the height of the desired Kresling. (b) shows the paper cylinder collapsing over the gap when the paper cylinder is twisted. The twist draws the two ends of the cylinders together closing the gap. (c) shows the fully collapsed Kresling. is the number of overlapping layers of material present within the flat folded state of the structure. The spontaneously generated Kresling contains many overlapping layers of thin paper that fold nicely into each other, however, if the paper has non- negligible thickness it results in the structure jamming before being able to fully flat fold. The present study seeks to overcome these limitations and create a framework for generating fully flat foldable, non-rigid, thick origami Kresling structures. 2 Thick Origami Methods In this section, we discuss the simple methods developed to design fully flat fold- able thick origami structures that enabled the creation of the thick Kresling pattern. The methods outlined may also be applied to most origami patterns. 2.1 Hinge Design The simplest method of creating a crease in thick origami is to mimic the behavior of a thin origami fold, creating a hinge that pivots around a single point Figure 2 (a). This primitive approach works for creating simple thick origami structures but inherently does not allow for flat foldability as the two adjacent panels will always intersect. A slight modification of the design allows for full 180°rotation of the hinge Figure 2 (b). Moving on from the simplest design, one can create a hinge with two pivot points, one on each panel, with a small third panel between them. This third panel will have a width equal to at least two times the thickness of the material, allowing the two panels to fold flat against each other, linked together through the smaller panel. This very simple double-pivot hinge, shown in Figure 2 (c), allows each panel to rotate a full 360°around the other panel. A third method is that of cutting small slats into the location of the crease, creating thinner pieces of the rigid material and taking advantage of elastic deformations to create a living hinge Figure 2 (d). This method is slightly more complicated and depends greatly FRAMEWORK FOR THE FABRICATION OF SCALABLE ,FLAT FOLDABLE ,THICK ORIGAMI KRESLING STRUCTURES VIA NON -RIGID METHODS (a) (b) (c) (d) Figure 2: Hinge mechanisms that can be used in thick origami (a) A simple single pivot hinge has a very limited range of motion due to the intersection highlighted in red. (b) An offset single pivot hinge that can go a full 180°. (c) A dual pivot hinge that allows for a full 360°range of motion as each panel can pivot around the complementary panel to lay flat on either side, enabling flat foldability. (d) A living hinge is made by removing material from the hinge and relying on the elastic properties of the hinge material to enable bending. on the properties of the thick material being used. Living hinges offer a distinct advantage over the other types of folds discussed in Figure 2. Their flexible nature, arising from their reliance on elastic deformation of the material, enables them to flex along multiple axes and not only around a pivot point. This flexibility allows for out-of-plane deformations and even facilitates compression and stretching of the panel. This is discussed in section 2.3. 2.2 Pattern Intersection Reduction An origami pattern is considered flat foldable when there exists some folded state where all the panels of the structure are parallel to each other. In the limit of zero thickness, thin origami allows one to create patterns where multiple layers of material occupy the same volume in the folded state. In reality, however, this is BERSHADSKY , PAULINO (a) (b) Figure 3: (a) Kresling with thick walls in an extended state, where collisions are not present. (b) Three cross-sections of the collapsed state where the intersections of the panels become evident. never the case. In an example like paper origami, the thickness of the material is so thin that, in most cases, this approximation is a helpful one. Even though the faces of the pattern are not truly parallel, the angles between them are so small that, to an outside observer, the structure appears flat. Even with thin materials like paper, this approximation begins to break down as more and more layers of paper are folded into the same space. This can be easily observed when trying to fold a piece of paper in half multiple times. To achieve truly flat foldable structures, it is necessary to modify the geometry of the origami to resolve any conflicts where materials are folded into the same space. In practice, when looking at the cross-section of the Kresling pattern with thick walls, we see that the intersections between panels are minimal in the extended state, as shown in Figure 3 (a). As the structure is flat folded, these intersections become more and more severe as shown in Figure 3 (b). To create a thick, flat foldable Kresling pattern, it is necessary to reduce the number of material overlaps (Ni) at any given location to be less than or equal to the number of creases ( Nf) along the vertical axis of the structure plus one. Ni≤Nf+1 (1) This equation arises from the fact that the maximum number of layers that can be stacked on top of each other is directly related to the number of folds along the vertical direction of the structure. Each fold along the vertical direction allows another layer of material to be staked on the structure in the folded state. Thus the maximum overlaps at any point is Nf+1. For the Kresling pattern, the maximum allowed number of overlapping layers of material in the flat folded state at any point FRAMEWORK FOR THE FABRICATION OF SCALABLE ,FLAT FOLDABLE ,THICK ORIGAMI KRESLING STRUCTURES VIA NON -RIGID METHODS is four, as the structure contains three folds along the vertical structure. To achieve this, the material within the faces of the Kresling pattern was re- duced to enable panels within the same layer to snake around each other while still preserving the location of the creases. This can be seen in Figure 4 (a). This op- timization also enables the fabrication of the Kresling pattern using a single print- in-place assembly if 3D printed in the flat-folded state. This reduces the time and labor required to assemble the completed units. 2.3 Non-Rigid Methods The energy barrier that is felt in the deployment of the Kresling structure is a result of non-rigid deformation in the panels. The presence of a non-rigid deformation, by Gauss’s Theorema Egregium1, implies a change in the Gaussian curvature of the structure during the transition. In thick origami, however, the panels are often made from thick rigid materials that do not permit stretching. It is possible to cre- ate a composite structure using several materials, such as the inclusion of rubber sections in the panels, the inclusion of a spring, or even a telescopic structure, thus enabling the panel to change shape and size [Tachi 11]. In this study, however, we explore the inclusion of living hinges within the panel faces to enable this non- rigid deformation as shown in the last panel of Figure 4 (b). As described above (in the Hinge Design section), the addition of living hinges is done by creating thin sections of material that can deform due to the natural elasticity of the material – see Figure 2 (d). The living hinges enable the panel material to be compressed and stretched and even allow for out-of-plane deformation. In this work, the living hinge is placed in the middle of the Kresling side panel, just above the middle fold. This provides sufficient flexibility to transition between the open and closed states, while also ensuring a stiff structure. Altering the size, placement, and orientation of the living hinges can be used to tune the actuation force of the Kresling pattern. Increasing the length of the living hinge section reduces the required bending force and the magnitude of elastic deformation on each segment is reduced. In our case, the living hinge is positioned near the center of the Kresling structure and oriented orthogonally to the top and bottom of the Kresling, as shown in Figure 4. This is because the orthogonal living hinge experiences significant outward bowing and twisting during actuation as seen in Figure 4 (c). Altering the location and orienta- tion of the living hinge to be in line with the structure and located near the top and bottom plates would presumably result in far less bowing, but more twisting of the flexible hinge. The location and orientation of the living hinges in our design are largely dictated by our efforts to reduce material collisions in the flat folded state as discussed previously in section 2.2. 3 Designing the Kresling The key parameters of a fully flat foldable Kresling structure for any regular N-gon are thickness (T), side length (L), and number of sides (N). If non-rigid deformation 1Any Change in Gaussian curvature must be accompanied by stretching of the material [Pressley 10]. BERSHADSKY , PAULINO (a) (b) (c) Figure 4: (a) Layers of the 6-sided thick Kresling in its collapsed state. (b) The 6-sided Kresling in its extended state. The last panel shows the alignment of the two arms in the fully extended state (c) Six-sided thick Kresling being actuated using an Instron tensile tester (while collecting force and torque data) FRAMEWORK FOR THE FABRICATION OF SCALABLE ,FLAT FOLDABLE ,THICK ORIGAMI KRESLING STRUCTURES VIA NON -RIGID METHODS (a) (b) (c) Figure 5: Sketches for designing a 4-sided Kresling. (a) Bottom polygon with hinge locations marked. (b) Lower arm with hinge locations. (c) Upper arm with hinge locations. is achieved through a living hinge, one also needs to define the thickness, location, and length of the living hinge portion. From the geometric parameters, we can define the angle γand radius ras follows: γ=360/N (2) r=L 2∗sinγ/2(3) One starts by drawing a regular polygon of side length L and then radially subdi- viding it into equal triangles – see Figure 5 (a). The regular polygon will act as both the top and bottom face of the completed structure. Hinges are placed along each of the polygon edges, allowing one to affix another layer to the structure. The constraints on the inner layers of the Kresling structure are defined as follows. Any two arms corresponding to any edge of the structure do not intersect with any of the other arms. The axis through the hinges located at the ends of the arms where they attach to the polygonal faces is parallel to the axis of the hinges on the polygon face. This normally contains the extended arms to a plane orthogonal to the poly- gon face, however, this constraint can be relaxed for cases where the side lengths of the two regular polygons are different, resulting in a slanted Kresling structure such as in [Lu et al. 22], with some non-rigid deformation permitted within the inner layers. In the examples shown here the hinges connecting the two inner layers are located along the radial edges of the subdividing triangles, thus being rotated by an angle of γfrom the X-axis – see Figure 5 (b). This hinge connects the two middle layers. To connect the upper of the two middle layers to the top of the structure, the location of the hinge must be selected such that when rotated around the middle hinge it becomes parallel to the hinge linking the middle arms to the base structure. This is done by placing it at an angle of 180 −γfrom the location of the middle hinge – see Figure 4 (b), Figure 5 (c). Non-rigid deformation is enabled by the BERSHADSKY , PAULINO (a) (b) (c) (d) Figure 6: Various layers of the: (a) 3-sided Kresling (b) 4-sided Kresling (c) 5- sided Kresling (d) 6-sided Kresling inclusion of a living hinge portion in at least one of the two middle arms, enabling the arms to stretch, compress, and otherwise deform as needed to account for the change in Gaussian curvature experienced by the Kresling pattern. In doing this, one ensures that in the extended state of the Kresling, the hinges connecting to the top face and bottom face to the middle sections are parallel, allowing the structure to actuate. Following this method results in N-gon Kreslings where in the flat folded state the top face is rotated relative to the bottom face by an angle of γ. This means that for any even N the two faces will be aligned and for any odd N they will be offset. This, of course, can be modified by changing the angles and locations of the hinges as long as they still satisfy the required constraints outlined above. (a) Figure 7: 3D printed flat foldable Kresling structures FRAMEWORK FOR THE FABRICATION OF SCALABLE ,FLAT FOLDABLE ,THICK ORIGAMI KRESLING STRUCTURES VIA NON -RIGID METHODS 4 Fabrication Using the technique described above, 3, 4, 5, and 6-sided Kresling Structures were generated as depicted in Figure 6 (a-d). The individual components were then 3D printed out of polylactic acid (PLA) on a consumer FMD 3D printer using a 0 .4mm nozzle and 0 .2mm layer height. The serpentine living hinge pattern of the top arm section of each Kresling structure enables a non-rigid deformation by harnessing the elastic deformation of thin-walled PLA extrusions. To improve the longevity and flexibility of these 3D-printed springs, the printer was set to use 3 continuous perimeters, ensuring that no infill structure would be printed in the spring area. Do- ing this greatly increases the spring lifespan because it aligns the extrusion axis with the axis of deformation in the spring. Aligning the extrusion direction with the axis of tensile force enables one to take advantage of the full strength of the extruded polymer and rely on interfacial adhesion forces between nearby extruded filaments. The removal of infill also avoids the presence of misaligned filaments that would restrict the deformation of the spring. A subset of the fabricated samples are de- picted in Figure 7. In this work, each component was printed separately and then assembled. However, due to the removal of intersections in the flat folded state, as discussed in section 2.2, these structures can be printed as a single print-in-place assembly in the flat folded state, (assuming that sufficient air gaps are incorporated into the structure to prevent unwanted boding of the stacked components during the printing process). 5 Preliminary Results The generated Kresling structures, pictured in Figure 6 and Figure 7 were placed into a modified Instron tensile tester that collects tensile force from the freely ro- tating top surface as well as torque data from the constrained bottom surface as depicted in Figure 4 (c) [Zang et al. 24]. Each sample was then tested to its full extension length of approximately 2 times its radius. All 4 structures exhibited bi- stability at an actuation range of approximately 85 −98% extension – see Figure 8 (a). The critical point is a function of many parameters including: N (number of sides), spring length, spring orientation, radius, side length, and the elasticity of the 3D-printed material. As a result of these numerous contributing factors, our samples are not directly comparable as many of these parameters vary between them. We can however gain intuition about the general properties of these struc- tures. Looking at the torque exerted on the constrained bottom face in Figure 8 (b) we can see the axial forces required to actuate the structure. At the critical points, we see a rapid change in the torque as the structure switches to being in compres- sion rather than tension. These Kresling structures exhibit strong bi-stability, easily snapping between stable points once a critical strain is reached. The energy barrier between these states depends on the elastic modulus of the material and various spring parameters. In Figure 6 (a), the triangular structure has quite a low energy barrier as the three arms are short and the required deformation is small. However, as the arms grow longer and the required deformation to actuate increases, so does the required energy as seen with the hexagonal structure of Figure 8 (d). In Figure BERSHADSKY , PAULINO Figure 8: (a) Tensile force vs displacement plots for 3, 4, 5, and 6-sided thick Kresling structures. (b) Torque on the constrained face of the Kresling structure as a function of displacement. (c) A closer look at the forces around the bi-stability point (the X axis has been rescaled to percent actuation). (d) The magnitude of the energy barrier of each Kresling structure. The integral of the force from 0 displacement to the point when the force crosses zero. 6 (b) the spring is in a straight line between the two hinges, in Figure 6 (c, d) the spring is oriented at an angle and then linked with a rigid bar. This kink in the arm increases the required deformation of the structure, the height of the energy barrier between the two states, and therefore the magnitude of the reversal of force observed in Figure 8 (a,c). 6 Discussion and Applications This technique for designing fully flat foldable, thick Kresling structures, scales while preserving the intrinsic properties of the Kresling pattern. The structures exhibit a bistable behavior with a clear energy barrier between the collapsed and extended states. By varying the chosen material and geometry on the nonrigid sec- tion of the structure, one can tune the energy barrier between the bistable states for various energy absorption levels. The twisting motion of the Kresling pattern is preserved and remains additive, as a result of this it can be removed by staking sev- eral Kresling structures with opposing chirality. Potential applications include flat foldable deployable shelters, prefabricated assemblies construction of larger build- ings, and deployable spacecraft [Navarro et al. ]. This design framework enables the creation of larger and more complicated deployable origami structures by en- abling a transition from thin sheet materials to thicker structural materials. Using hinge design optimizations and techniques for the incorporation of non-rigid seg- FRAMEWORK FOR THE FABRICATION OF SCALABLE ,FLAT FOLDABLE ,THICK ORIGAMI KRESLING STRUCTURES VIA NON -RIGID METHODS ments in thick origami structures, we can translate many classical non-rigid thin origami patterns to their thick counterparts. This framework allows us to create complex origami-inspired structures while harnessing the structural properties of thick materials such as various plastics, woods, and metals. 7 Conclusions In this work, we demonstrated and characterized a framework for the creation of fully flat foldable, non-rigid, thick origami through a generalized implementation of the Kresling pattern. The demonstration of non-rigid techniques with thick origami enables the translation of thin non-rigid origami patterns to their thick origami equivalents. This in turn enables the implementation of many deployable origami structures, currently only manufacturable out of thin sheet materials, to be made using thick origami methods. Acknowledgement The work presented in this study originated from independent work by David Ber- shadsky which was then quantified with the help of the Paulino research group which graciously provided access to an Instron tensile tester and custom fixtures designed for testing Kresling structures. We acknowledge Dr. Tuo Zhao (of the Paulino Research Group) for his help with the collection of tensile testing data. We also thank Ms. Biruta Kresling for reviewing this publication and providing valuable feedback and suggestions.