Design and Modeling of a Non-Flat Foldable Tubular Kirigami With Compliant Joints

Tubular origami, such as Kresling, waterbomb, and Yoshimura, has become an emerging topic over the recent decades. Their properties have great potential in a wide range of applications. For example, their tubular shape provides a closed environment for medical devices [1] and pneumatic actuators [2] and benefits the omnidirectional robot arms [3]. Their non-rigid foldability makes them suitable for use in energy absorbers [4]. Metamaterials exploit their tunable mechanical response properties [5]. With their variable radius, their application even extends to valves [6]. However, a significant category of tubular origami cannot be folded rigidly [7,8], which significantly limits its potential applications and fabrication methods. Its folding requires the in-plane deformation or the traveling of hinges [9]. The current common solution is to use flexible materials for the in-plane deformation. Butler et al. tested five kinds of flexible sheet materials for the origami bellows used in Mars rovers [10]. Melancon et al. 3D printed an inflatable Kresling structure with flexible TPU 95A filament [11]. Flexible materials such as paper and plastic sheets are easy to deform, while most engineering materials are too rigid to allow large deformations. Therefore, modifying the structure to achieve rigid foldability can greatly expand its engineering applications.

Most tubular origami structures are built by isometrically embedding folding patterns in 3D space [8]. Non-rigid foldability comes from the over-constrained connections between each folding unit, as vertex positions conflict when defined by the motion of different folding units [12]. This feature indicates that modifying the connected edges between each folding unit alters the structure's non-rigid foldability, which results in a kirigami structure. While kirigami has gained popularity in recent research due to its impressive stretchability and deformability [13,14], its utilization in origami structures has been limited to the best of our knowledge. However, in this study, we propose the design and modeling of a radially closable tubular kirigami (RC-kiri) structure based on a non-rigid foldable origami [6].

Folding along the creases of engineering materials is a significant challenge. A fold is a reduction in stiffness along the creases [15]. For thin materials with high flexibility, folding at the crease can be achieved by plastic deformation or punching [10,16–19]. However, to withstand loads and insulate heat, radiation, and sound, the thickness and stiffness of engineering materials should be considered [20]. High stiffness strongly limits the possible folding motions [21]. In addition, engineering materials always require additional mechanical hinges or membranes [22]. For example, Lang et al. introduced the synchronized-offset rolling-contact elements (SORCE) in the context of thick origami mechanisms, incorporating additional hinges into the design [23]. Similarly, Butler et al. presented the regional sandwiching of compliant sheets (ReCS) technique, which involves the inclusion of additional membrane materials [24].

The compliant joint has been the focus of many studies over the years [25,26]. It allows the material to maintain stiffness on the panels while reducing stiffness at the creases without adding additional hinges or membranes, which is an effective way to surrogate folds [21,22]. With high rotational compliance, the LET joint is well-suited for application in origami among all compliant joints; it requires very little footprint and allows origami-like folding of panels and localized joints [27]. Some studies can provide valuable guidance in the design of LET joints. Delimont et al. summarized and evaluated several commonly used LET joints [15,28]. Jacobson et al. introduced the equations for the stiffness of LET joints [29]. Pehrson et al. proposed the LET array by placing LET joints in series and parallel, which provided the joint with tunable stiffness to facilitate desired behaviors [27]. The tunable stiffness is crucial for our kirigami model, as it ensures precise diameter control during folding and facilitates the intended folding behavior of the RC-kiri structure.

Despite the high rotational compliance, the LET joints remain susceptible to failure when subjected to large deformations. While this largely depends on the flexural strength of the material, reducing stress and strain is still an effective way to improve their performance. This can be achieved by reducing the maximum folding angle. As a result, a non-flat foldable kirigami structure is introduced, which implies that the structure retains a 3D shape when fully folded.

The paper is organized as follows. In Sec. 2, the kirigami technique is applied to Kresling, Yoshimura, and RC-ori structures. The kinematic analysis of the non-flat foldable RC-kiri is performed. Section 3 applies the outside LET joint to the tubular RC-kiri. The configuration design, theoretical modeling, and load–displacement relationship analysis are carried out. In Sec. 4, the settings of the finite element model are described in detail, and the FEA results are analyzed in comparison to theoretical results. Section 5 illustrates the fabrication of the physical model and experimental setup, followed by conclusions in Sec. 6.

To the best of our knowledge, most kirigami structures are planar [30–34], and the spatial kirigami structures are often analyzed based on their tensile movements [13,35,36]. This paper, however, analyzes the motion generated by compressing the tubular kirigami.

Kresling and Yoshimura are two examples of non-rigid foldable tubular origami structures that can be applied with the kirigami technique (Figs. 1(a) and 1(c)). It is necessary to clarify several conditions here. First, to maintain the polygonal shape of the ends during folding, it is essential to have symmetrical stacking of the two layers. Second, all folding units should be folded identically to ensure that there is only one degree-of-freedom.

As shown in Fig. 1, when the connected edges between each folding unit are cut, the structure becomes rigid foldable. Each folding unit consists of four symmetrically arranged triangular facets, which can be folded inward or outward, resulting in a total of four folding configurations, as indicated in Figs. 1(b) and 1(d). Nonetheless, when certain geometric parameters are present, Kresling and Yoshimura kirigami structures may only exhibit three folded configurations due to interference between panels when all of the facets are folded inward. This phenomenon is illustrated in Fig. 1(b). In fact, the Yoshimura and Kresling patterns are constructed from a parallelogram that consists of two identical triangles. Yoshimura patterns use isosceles triangles [37], while Kresling patterns generally use obtuse triangles. This perspective further illustrates that the configuration of the models is significantly influenced by the triangle parameters. Additionally, the number of folding units is another important parameter to consider. Here, we only provide examples of structures with six folding units.

Our design aims to create a radially closable structure, and the aforementioned examples also demonstrate radially closable characteristics. However, the deployed configuration of Yoshimura kirigami does not exhibit a perfect hexagonal prism due to the non-flat state required for the connection between each folding unit. Additionally, the stacking of two layers in Kresling kirigami increases the axial length of the entire structure, leading to manufacturing complexities. To address these issues, we propose the application of an alternative kirigami structure (RC-kiri) in this study, which can circumvent the aforementioned problems.

RC-kiri is a modified version of RC-ori. The latter is an origami structure that can be radially closed and exhibits both non-rigid foldability and flat foldability. The folding unit of RC-ori comprises two right-angled trapezoidal facets and two right triangular facets, as depicted in Fig. 2(a). By applying the kirigami technique on RC-ori, the structure becomes rigid foldable with three different folding configurations [38]. When the trapezoidal facets are folded inward and the triangular facets are folded outward, a radially closable kirigami (RC-kiri) configuration is produced (Figs. 2(b)–2(d)). In simple terms, each folding unit consists of a straight-minor degree-four vertex (α₁ + α₂ = α₃ + α₄ = π) [23].

The authors' earlier paper established that the RC-ori crease pattern can be completely characterized by the number of folding units n and the polygon side length a [6]. The side length a only affects the structure size, and the configuration is fully defined by the number of folding units n. Nonetheless, an additional parameter is applied in this paper for the non-flat foldable structure, the base angle of the trapezoid θ, as shown in Fig. 3(b).

The implementation of non-flat folding results in a notable reduction in the stress and strain experienced by the structure, as well as a decrease in the number of overlapping panels, since the maximum folding angle of each crease is decreased. These greatly improve the performance of the structure. A geometric definition is suggested here: “In the closed state, the structure exhibits multiple identical isosceles triangles with coinciding apexes. If the apexes coincide before the structure reaches its flat-folded state, it is possible to close the structure without folding $180deg$ at each crease. Hence, by adjusting the base angle θ of the isosceles triangle (trapezoid panel), it is feasible to define non-flat folding. To ensure a closed state when fully folded, the value range of θ is maintained between $(π2−πn)$ and $π2$⁠.”

Here, Δγ₅ and Δγ₆ are the folding angles between the trapezoid facet and the ends extension facet. It can be seen from Figs. 3 and 5 that the two ends are lengthened to ensure every folding unit is evenly stressed.

For theoretical modeling in Sec. 3, the geometric parameters are defined as n = 6 and a = 40 mm. The base angle θ will be determined by considering the maximum folding angle (fully folded) of each crease, the original axial length of the structure, and the maximum displacement. Figure 4(a) shows that as the base angle θ increases, the maximum folding angles $Δγi_max$ decrease. When θ exceeds $65deg$⁠, the maximum folding angle decreases less noticeably. Among the creases, $Δγ1/3_max$ experiences the greatest decrease, while $Δγ2/4_max$ decreases only slightly. This indicates that the joints at creases c₂ and c₄ (Fig. 3(b)) may experience the highest levels of stress and strain and should be assigned less stiffness in comparison to other creases to ensure their desired folding direction.

Although increasing the base angle θ leads to smaller deformations, the initial axial length of the structure h₀ is also an important factor to consider. As shown in Fig. 4(b), when $θ>80deg$⁠, the initial axial length increases dramatically, which increases the difficulty of manufacturing. Additionally, as θ increases, the maximum displacement H_max gradually decreases, which can adversely affect the control of the structure's diameter. Therefore, in this study, $θ=65deg$ is applied for the design. Based on this setting, the values of the relevant parameters are compared in Table 1. With an increase of $5deg$ at the base angle θ, $Δγ1/3_max$ and $Δγ5/6_max$ decreased by 40%, and $Δγ2/4_max$ decreased by 19%. The initial axial length increased h₀ by 24%, and the maximum displacement H_max decreased by 49%.

As previously demonstrated, the RC-kiri structure exhibits three folding configurations, and the parasitic motion has a significant effect on the diameter precision. Therefore, to achieve the desired folding behavior, it is necessary to assign varying stiffness values to the creases. The LET array can be easily modified to achieve tunable stiffness, rendering it well-suited for our RC-kiri model.

The dimensions of the joint region should be determined first. The joint region length is generally limited by the crease length, while the region width depends on factors such as model size, required deformation, material yield strength, and manufacturing precision. In order to simplify the model and reduce parasitic motion, this paper specifies that the joint region width of all creases is identical, and only two torsion segments arranged in series are applied to the region width. The standard LET joint (outside LET and inside LET), with four torsion segments (two in series and two in parallel), is favorable for this paper since it has been intensively studied. To modify the joint stiffness, this paper employs a varying number of outside LET joints placed in parallel along the creases (Fig. 5).

The geometric parameters are labeled in Fig. 5(b), and their values are listed in Table 2 (t is the thickness). The material applied in this model is Ultimaker (Utrecht, Netherlands) black CPE, with Young's modulus E = 1863 MPa and Poisson's ratio v = 0.37. To simplify the model, most of the geometric parameters of the joints are identical; the only difference is the length of the joints, L_Ti.

The quantity of outside LET joints at each crease is dictated by the precision of the internal diameter of the structure, which is mainly influenced by the movement of the trapezoidal panels. Therefore, it is necessary to minimize the parasitic motion of the joints located on creases c₁, c₅, and c₆. One effective solution is to enhance the stiffness of these joints by decreasing the length of the torsion segment and increasing the number of parallel joints. As a result, c₁ is composed of two parallel joints, while c_5/6 comprises four parallel joints. Folding the joints on creases c₂ and c₄ poses a challenge as their movement has minimal impact on the trapezoidal panels. Therefore, it is necessary to minimize their stiffness. However, since these creases have considerable length, the presence of parasitic motion at c_2/4 can affect the accuracy of the theoretical model. As a result, two parallel joints are utilized at c_2/4, as depicted in Fig. 5(a). Each folding unit in the system is comprised of six joints that rotate along their x-axis during the folding process. These folding units are connected indirectly through the extension ends, with one end fixed and the other end subjected to either load or displacement.

The theoretical load–displacement relationship consists of two stages, divided by the critical buckling point. Before reaching the critical buckling load, the structure undergoes a small displacement due to the in-plane parasitic motion of the compliant joint. When the load reaches the critical point, the panels lose stability and fold, causing large displacements. The equivalent spring model of the folding unit is shown in Fig. 5(c).

By substituting the length of each joint into Eq. (13), their torsional stiffness K_Ti can be obtained. Their bending stiffnesses K_B are identical. Thus, their rotational stiffnesses are listed in Table 3.

When plugging in the folding angle Δγ_i and the rotational stiffness k_Ri into Eq. (11) and taking the virtual work derivative, the theoretical relationship between force and displacement during folding is obtained. Thus, the critical buckling load can be applied to Eq. (8) for the small displacement Δy, and the theoretical load–displacement curve and virtual work–displacement curve are generated (Fig. 6). After reaching a critical buckling load of 10.2 N and experiencing a displacement of 0.96 mm at that point, the load decreases dramatically and eventually stabilizes at 4.3 N upon folding.

The simulation is performed by abaqus explicit dynamic analysis with the nonlinearity solver due to the large elastic deformations. All of the geometrical and material characteristics remain identical to those of the theoretical model, except for the thickness at panels, which is twofold that of the thickness at the joints, allowing for most of the deformation to occur at the joints rather than throughout the panels. The flexural strength of the material is 72.8 MPa, according to the technical data sheet of Ultimaker CPE. The computational domain is discretized via triangular mesh elements set at 0.5 mm in the joint and 1 mm in the panels.

To reduce computational time, the finite element analysis is performed on only one folding unit, taking advantage of the symmetrical configuration of the structure. The structure is fixed on one end, and a displacement is applied to the other end. To constrain the folding direction of every facet, two small rigid pieces are placed and fixed near the crease c₃ and the trapezoidal facet, as shown in Fig. 7. In this case, the trapezoidal facets are restricted to folding inward, while the triangular facets can only fold outward.

The reaction force of the FEA simulated results is fitted by the curve shown in Fig. 8(a) with the polynomial curve fitting. This fluctuation is caused by dynamic processes and can be reduced by setting damping. The dashed line in Fig. 8(b) shows the external work value, while the solid fitting curve is generated by integrating the fitting curve of the force. They show little difference, which means the fitting curve of the force–displacement relationship has a strong correlation with the original data and can be used to verify the correctness of the theoretical model.

When comparing the fitted curves of FEA with the theoretical ones, as depicted in Fig. 9, the critical buckling loads show minimal differences. The folding process shows a similar pattern in both curves, but there are still variations. In the early stages of folding, the forces observed in the FEA exceed the theoretical results. This variation can be attributed to the presence of parasitic motions in the joints 2/3/4, which are not considered in the theoretical calculations. These parasitic motions cause an additional force requirement for the folding process. On the other hand, as the process moves forward, these parasitic motions influence the folding angle of the joint, leading to a reduction in the required force. This verification can be observed in Fig. 10, where the folding angle of creases c₁, c₃, and c_5/6 is in good agreement, while the folding angle of c_2/4 is smaller than the theoretical value.

Another significant reason is found to be the maximum strain exceeding the flexural strength of the material, leading to plastic deformation and a reduction in stiffness. The relationship between the maximum stress and displacement during the folding process is shown in Fig. 10(b), where the blue curve represents the fitted curve of the max Mises stress of the FEA result, and the red horizontal line represents the flexural strength of the material. It can be speculated that the model undergoes plastic deformation when the displacement reaches around 14 mm. This is consistent with the lower force observed in the FEA results compared to the theoretical values in Fig. 9(a) (after 14 mm).

Metals and plastics are two typical materials for compliant mechanisms [44,45]. Plastic is flexible and easily deformable, which is a cost-effective option for our model. In terms of the manufacturing method, 3D printing is commonly regarded as the most appropriate method for manufacturing plastic materials among techniques such as laser cutting, CNC, etching, and etching. After evaluating different materials for 3D printing, we found that PLA and PC were too rigid and prone to brittleness. On the other hand, TPU 95A and PP were excessively flexible and unable to bear the required load. As a result, we selected CPE material as it offered a suitable balance of hardness and flexibility. CPE demonstrated the ability to withstand the load while providing the necessary level of flexibility for meeting the requirements of joint torsion. (These materials are from the same brand, Ultimaker, and detailed information can be found on their technical data sheet, available on the official website.)

Each folding unit is individually printed using an Ultimaker S3 and then inserted into two hexagonal bases that feature grooves at both ends (Fig. 11(b)). However, some trapezoidal panels tend to fold outward, and the joint at c_2/4 is difficult to fold [38]. Therefore, controlling the folding direction of each panel is a significant challenge in the experiment.

Two innovative designs are introduced into this model. First, small rectangular sheets are placed in proximity to the trapezoidal panels to facilitate inward folding without the need for external intervention. These rectangular sheets are affixed to the end bases and are composed of the same material as the end bases. To prevent interference with the folding of the triangular panel, the width of these rectangular sheets is less than half the side length a, and their height is determined by the maximum displacement H_max. Second, to achieve the desired folding direction for the hinge at c_2/4, the triangular panels are extended (2 mm) beyond the adjacent edge. Thus, the triangular panels are prevented from folding inward by the movement of the neighboring trapezoidal panels. The assembled model is shown in Fig. 11.

The test is conducted by a TA.HDplus Texture Analyser (Stable Micro Systems Ltd, Godalming, UK). The reaction force is obtained when applying a displacement in the axial direction with a speed of 1 mm/s. The environmental temperature is 14 °C, which may affect the plastic properties. To prevent errors caused by inconsistencies in the equipment or model, multiple tests are conducted by rotating the model six times along the axis. The model was initially tested in six rounds with a displacement setting of 14.00 mm (according to the result from Fig. 10(b)) to better compare with the theoretical model and to see if any plastic deformation occurred. The next six rounds of tests were conducted with a displacement set at 34.00 mm, which is smaller than the maximum displacement (35.20 mm). This can reduce the differences between the experimental results and the theoretical results that are caused by the contact force between the panels and also prevent excessive folding that could potentially damage the model. The experiment process is shown in Fig. 12.

The results of the multiple tests are shown in Fig. 13. In each test, the hysteresis effect is noticed, which could be attributed to the friction between the panels. There is no significant plastic deformation that occurs when the displacement is 14.00 mm, as shown in Fig. 13(a). However, with the displacement reaching 34.00 mm (Fig. 13(b)), the critical buckling load point and the reaction force gradually decrease with each round of tests. In addition, the reaction force in Fig. 13(b) demonstrates a lesser magnitude of reaction force compared to Fig. 13(a), also indicating that significant plastic deformation has occurred with large displacement. This observation highlights the need for optimized geometric parameters to prevent plastic deformation or the consideration of a material with higher flexural strength for our design would be more appropriate.

When comparing the first-round test results with the theoretical result, as shown in Fig. 14(a), the results show a consistent trend, but there are differences at the critical buckling point. The delayed critical buckling point may have occurred due to the initial non-folding of the triangular panel, which can be attributed to the parasitic motion of joint 3, as can be seen in Fig. 12(b). The abrupt decrease in reactive force can be attributed to the sudden buckling of joint 3, leading to the rapid folding of the triangular panel. As a result, joints 2/3/4 experience simultaneous torsion. Due to the thickness of the panel, contact force was generated, which resulted in the experimental reactive force being higher than the theoretical value toward the end of the folding process.

As a radially closable structure, the most important parameter of this model is the diameter, as shown in Fig. 15. The diameter is defined as the distance between the central point P of the folding unit on the opposite sides of the structure. The outcomes exhibit satisfactory concurrence, except for some divergence observed at the end of folding, which may be mainly due to the thickness of the panel and the width of joint 1. The experimental results provide new support and evidence for the theoretical model while also providing crucial directions for enhancing the theoretical model further.

This study provides a new method to solve the manufacturing difficulty of non-rigid foldable tubular origami by applying the kirigami technique. The non-flat folding structure design helps decrease the maximum folding angles of every crease in the structure. The tunable stiffness property of LET joints is applied to the RC-kiri model, which facilitates the folding direction and the diameter accuracy. Other methods to control the folding direction are also provided in both the FEA model and the experimental model.

The theoretical model is established via the virtual work principle. Finite element analysis and experimental investigations are carried out, both of which showed good agreement with theoretical predictions. Verification and analysis of simulation and experimental results provide further evidence for the accuracy and reliability of the theoretical model. This study not only confirmed the applicability and effectiveness of the theoretical model but also provided important insights and directions for future research and development. The RC-kiri structure finds application in a variety of fields, including valves, grippers, metamaterials, antennas, and soft continuum robots, among others.

Authors Siyuan Ye, Pengyuan Zhao, and Shiyao Li are grateful for the financial support from the China Scholarship Council (CSC). Gratitude is extended to SEFS for generously supporting the conference travel expenses for Siyuan Ye.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.