Development of a Cubic Core Lightweight Panel Using Origami-Kirigami Engineering

: Honeycomb core panels are commonly used in industry as lightweight structures. However, because the bond between the face plate and the regular hexagonal core is weak, there are strength problems, such as heat and shear loads. In this study


Introduction
The development of lightweight structures against the backdrop of global energy problems is an important research topic in such industries as automobiles, aircraft, and machine tools [1][2][3]. To achieve this goal, the optimal design for structural weight reduction has been studied under conditions that satisfy mechanical performance [4][5][6]. In addition, lightweight structures using aluminum alloys and composite materials instead of ordinary metal materials are being studied [7][8][9][10][11].
Sandwich lightweight panels play an important role among the lightweight structures used in modern industry [12][13][14][15]. Among them, honeycomb core panels are the most widely used owing to their excellent specific strength and specific stiffness per weight [16][17][18][19]. In addition, the mechanical properties of honeycomb core panels under various conditions have been studied in detail. Stiffness and strength problems of honeycomb core panels in the presence of complex loads, such as dynamic and impact, have been investigated [20][21][22]. Because the honeycomb core panel is composed of thin plates, the problem of buckling under the compressive load has also been investigated [23][24][25]. Problems of stress concentration and fracture caused by microdefects in materials and production processes have also been investigated [26][27][28]. Furthermore, the optimization of the shape parameters of the honeycomb core panel was investigated [29,30]. These research results provide basic technical results useful for the practical development of honeycomb core panels. However, the honeycomb core panel has the drawback of being difficult to apply to shear loads and thermal environments because the regular hexagonal core material and the face plate are bonded with glue through only the line-shaped contact area. There is also a processing problem for honeycomb core panels [31,32].
Therefore, to improve the drawbacks of honeycomb core panels, which are used in large quantities in industry, it is important to promote research and development on new lightweight panels that can be processed by simpler methods and have good mechanical properties. Furthermore, such research is expected to have significant economic value.
In this article, a new cubic core panel is proposed instead of a conventional honeycomb core panel. A cubic core can be processed from a single flat metal plate by simple punching of a square hole and bending. In addition, glues and rebates can be used to assemble integral cubic core panels with double bonding through square contact surfaces between the faceplate and the cubic core. For investigation, a cubic core panel was created using an aluminum plate, and the mechanical properties of the cubic core panel were examined in a three-point bending test. The finite-element method (FEM) was used to analyze and compare the bending stiffness of the honeycomb core panel and that of the cubic core panel under the same numerical analysis conditions. Furthermore, for practical development, a processing method for a curved cubic core panel that is applied to the corner part of an actual vehicle structure was devised and investigated. A dedicated cubic core manufacturing system using a four-bar linkage mechanism was developed and examined for future mass production. Figure 1 shows a conventional honeycomb core panel. As shown in Figure 1, the honeycomb core and face plate are bonded with glue. Because the bonding surface is in the form of a thin line, the bonding strength at this portion is relatively weak.

Cubic Core Panel
In this study, the cubic core shown in Figure 2 was employed instead of the conventional honeycomb core by making use of the knowledge obtained from origami and kirigami engineering [33,34]. As shown in Figure 2, first, on a flat plate, square holes are drilled in every other horizontal and vertical direction. The sides of each square are then alternately folded up and down to obtain a cubic core. Finally, the cubic core panel is obtained by gluing the face plates to the upper and lower sides of the cubic core with glue and fixing the face plate and cubic core to the central point with rivets.   Figure 3 shows the simplified I-shaped equivalent cross section of a honeycomb panel and a cubic core panel with the same dimensions. The plate thickness t is assumed to be the same for the face plate and the plate composing the core. As indicated by the red dotted line in the figure, the number n of vertical walls on both sides is the same; therefore, the central width of both I-shaped equivalent sections is the same nt. The thickness of the face plate of the I-shaped equivalent section of the honeycomb panel is t, and the thickness of the face plate of the I-shaped equivalent section of the cubic core can be approximated as 1.5t. Therefore, the equivalent moment of inertia of the honeycomb and cubic core panels and their difference can be calculated by the following equations. (1)

Comparison of Cubic and Honeycomb Core Panels
In Equation (3), 0 and 2 3 , so the following equation is obtained.
(4) Equation (4) indicates that the cubic core panel has a higher bending stiffness than the honeycomb panel. However, as shown in Figure 3, the thickness of the equivalent face plate of the cubic core panel is larger than that of the honeycomb panel, and the weight is likely to be heavier. Therefore, it is necessary to evaluate the bending stiffness per unit weight of the panel.
If the materials of the two panels are the same, the material volume should be used for evaluation. From Figure 3, the material volume of the honeycomb panel and the cubic core panel can be calculated by Using the honeycomb panel as a comparison standard, the ratio of the geometrical moment of inertia and the material volume of the cubic core panel compared with the honeycomb panel can be summarized by using Equations (1), (2), (4), and (5). Figure 4 shows the comparison results. The horizontal axis in Figure 4 represents the width-to-thickness ratio h/b of the lightweight panel. However, the plate thickness is 1.0 mm, and the number of cores in the cubic and honeycomb core panels is the same.  Figure 4 reveals that the increase in the geometric moment of inertia is greater than the increase in the material volume when cubic core panels are used instead of honeycomb panels.

Manufacturing Test of Cubic Core Panel
A prototype cubic core panel was fabricated according to the procedure shown in Figure 5. The plate material used was aluminum A5052 with a thickness of 0.8 mm. First, as shown in Figure 5(a), a 450mm×300mm aluminum flat plate was cut using a laser to cut square holes with a side length of 30 mm at intervals of 30 mm horizontally and vertically. To bend it vertically, as shown in Figure 5 (b), wooden blocks were attached to the upper and lower sides of the flat plate part at equal intervals, and rubber rings were attached to the end faces of the wooden blocks so that they could slide easily. Then, as shown in Figure 5 (c), a pressing machine applied forming pressure to the flat plate part, the forming block slid horizontally while moving vertically, and the flat plate was bent, as shown in Figure 5 (d). The forming work was gradually divided into several steps, and, finally, the cubic core shown in Figure 5 (e) was obtained through hand finishing.

Examination by Three-Point Bending Test
To examine the mechanical properties of the cubic core panel manufactured in the previous section, a three-point bending test was performed, as shown in Figure 6. A Shimadzu AG-300kNG autograph machine was used. The support width of the three-point bending test was 200 mm. The curvature radius of the corner part of the three-point bending jig that contacted the cubic core panel was 10 mm. The loading speed was 3 mm/m, and the maximum displacement was 20 mm.  First, in the initial stage of applying pressure, the jig and test piece slide sideways, and a phenomenon in which a slightly low load continues from the origin to point a can be observed. After passing point a, the graph between a and b becomes a straight line, indicating a linear relationship between the load and displacement. Its elastic modulus is 560 N/mm, and its linear elastic limit load value is approximately 4200 N.
Next, there is a slight curvature between b and c, and a nonlinear relationship exists between the load and displacement. However, the resistance to bending deformation continues to the maximum load point c while changing slowly, and the maximum load value is 5312 N.
Finally, after the maximum load point c has been exceeded, local buckling deformation occurs in the inner core material. Thus, the load continued to deform while it gradually decreased.
Two different cubic core panels were fabricated to investigate the effect of glue strength between the face plate and the core material. One was fixed only with rivets without using glue, as shown in Figure 8 (a). The other sheet was fixed using glue and rivets, as shown in Figure 8 (b). A three-point bending test was performed on each of the two cubic core panels. The measurement results are shown in Figure 9. In Figure 9, the black line represents the measurement results when fixed with rivets and glue, and the blue line represents the measurement results when fixed with rivets only.
First, in the first elastic deformation process, the modulus of elasticity when fixed with rivets and glue was 560 N/mm, but when fixed with rivets alone, it was 525 N/mm, approximately 6.25% smaller. The elastic limit load when fixed with rivets and glue was 4200 N, and when fixed with rivets only, it was 2258 N, which is approximately 42.24% smaller. This shows that the elastic modulus and linear elastic limit load value are increased by using glue.
Next, after the load exceeded the elastic limit load, the cubic core panel fixed only with rivets broke owing to the stress concentration at the fixed part of the rivets, and the load value dropped sharply in the load-displacement graph. However, when the panel was fixed with rivets and glue, there was no partial breakage of the rivets, and the load value did not drop sharply in the load-displacement graph. The maximum load value was 3082 N when fixed with rivets alone and 5312 N when fixed with rivets and glue, which was approximately 41.98% smaller.

Comparison with Honeycomb Core Panel by FEM
Using the FEM analysis model shown in Figure 10, a static analysis of the cubic and honeycomb core panels was performed to compare the bending stiffness results of both.  As shown in Figure 10, a uniformly distributed load was applied to a fixed square panel with a side length of 900 mm and a thickness of 30 mm. The core size of each panel was also 30 mm. The plate material used was aluminum A5052 with a Young's modulus of 72 MPa, Poisson's ratio of 0.33, and plate thickness of 0.8 mm. The average edge length of the analysis mesh was 5 mm. The cubic core panel was connected between the face plate and the core material under the rigid contact conditions of computer-aided engineering (CAE), and rivets were omitted. Figure 11 (a) shows the static analysis results of the cubic core panel, and Figure 11  To evaluate the bending stiffness of the lightweight panel, it is necessary to use the displacement value corresponding to the unit load.
The external load P can be calculated using the uniformly distributed load q and panel area A with the following equation.
Therefore, the stiffness of the cubic core panel and honeycomb panel can be evaluated by the following equations.
Using the material volume value calculated from the FEM analysis model and an aluminum plate density of 2700 kg/m 3 , a cubic core panel weight of m cubic = 9.72 kg and a honeycomb core panel weight of m honey = 7.83 kg were obtained. Therefore, the bending stiffness per unit weight can be evaluated by the following equations.
Comparing Eqs. (10) and (11) reveals that, when the cubic core panel is used instead of the honeycomb core panel, the stiffness index per unit weight increases from 2873.6 to 3333.3 N/kgm (by 16.0%).

Application to Curved Corner Structure
In the actual structure, as shown in Figure 12, there are flat parts, such as the side walls and roof, and curved parts, such as the corner joints. In line with this, the cubic core panel proposed can be manufactured with a curved cubic core by partially applying a kirigami operation, as shown in Figure 13.
The actual manufacturing procedure is as follows. First, as shown in Figure 13 (a), the width of the curved part is determined according to the actual structural dimensions. Then, inside the curved corner, the shaded area in Figure 13 (b) is cut, and the panel is bent inward. As a result, curved cubic cores are obtained, as shown in Figures 13 (c) and (d). Finally, by fixing the bent face plate and cubic core to both sides of the curved cubic core using glue and rivets, a cubic core panel for curved corners, as shown in Figure 12, is obtained. For practical use, it is important to determine the width of the curve and dimensions of the cutout portion appropriately for the cubic core.   Figure 14 shows the design parameters of the curved part of the cubic core. In Figure 14, l is the bending width, R is the radius of curvature of the inner surface of the curved part, and D is the bending angle.
In the actual design, l, R and D are given by the requirements of the curved structure. To manufacture the curved part accurately according to the designed dimensional requirements, it is necessary to determine appropriately the length E of the part to be cut and the width to be cut from the cubic core on the inner surface.
According to Figure 14, the following relationship exists between the design parameters of the curved part.
where n is the number of cubic cores used to cut the material inside the bend. Equation (13) is the design equation for the curved part of the cubic core panel. The width can be calculated by substituting the actual bending dimensions l, R, and D into Equation (13).

Mass Manufacturing System Design
As shown in Figure 4, a prototype cubic core panel was manufactured using a simple pressing method with a wooden block and rubber ring to examine the performance. However, it is necessary to develop a more precise manufacturing method for practical applications.
In the forming process of the cubic core, the flat plate is bent vertically and simultaneously shrinks horizontally. Therefore, a four-bar link mechanism is proposed that connects adjacent bending blocks with parallel links (Figure 15). As shown in Figure 15, to perform bending, the forming plate is sandwiched between the upper and lower forming blocks, and pressure is applied in the vertical direction. An elastic spring attached to one end of the forming block is compressed, and the parallel movement of the four-bar linkage mechanism allows the adjacent forming blocks to shift vertically and move closer to each other in the horizontal direction. According to the order shown in Figure 15, the formed plate is bent up to a right angle. A ball caster is attached to the end face of the bending block so that the bending block slides easily in the horizontal direction. As shown in Figure 16, the forming block is expanded to five pieces, and pressure is applied in the vertical direction. The central bending block moves downward, whereas the four surrounding bending blocks move upward. Five cubic cores can be bent in the order shown in Figure 16.
According to the actual processing dimensions, multiple bending blocks can be connected in the horizontal direction to configure one dedicated processing die set, as shown in Figure  17. When applying pressure in the vertical direction, multiple cubic cores can be bent in one process according to the order shown in Figure 17. To study the bending load, Figure 18 shows the load condition of the bending part of the plate material. Here, P is the bending load, P s is the reaction force of the compression spring, r is the curvature radius of the local bending, b is the side length of the cubic core, t is the plate thickness, and x is the bending displacement. The distance between adjacent blocks during forming and the reaction force of the elastic spring can be calculated using the following equations.
where K is the spring constant. Using the local coordinate O shown in Figure 18, the local bending moment is expressed by the following equation using stress P.
The material properties of a plate are approximated by rigid plastic linear hardening, and the stress-strain relationship is expressed by where P is the yield stress, and F is the linear hardening factor. The strain can be calculated using the following equation.
By substituting Equations (17) and (18) According to Figure 18, the external force P -P s intersecting in the vertical direction becomes a couple, and the bending moment is expressed by Using the equilibrium relationship between Equations (19) and (20), the following equation is obtained.
By substituting Equations (14) and (15), the load required for bending is obtained as The results were verified using the cubic core forming example in the previous section. The parameters of cubic core (side length b = 30 mm, aluminum plate yield stress P = 215 MPa, work hardening factor F = 0, plate thickness t = 0.8 mm, and spring constant K = 0.15 N/mm) are substituted into Equation (22). As a result, the relationship between the load and displacement during molding is obtained ( Figure 19).
As shown in Figure 19, the resistance to bending deformation increases gradually from the initial stage of forming, indicating that bending is relatively easy. However, after 80% bending, the forming load tends to increase as the relative distance between the forming blocks decreases. However, the results of Equation (22) and Figure 19 are the calculation results of the forming load required for bending one vertical wall. If the number of vertical walls of the cubic core panel to be processed is N, then the forming load required for processing is NP.

Conclusions
A cubic core lightweight panel was proposed, its mechanical properties and manufacturing methods were examined, and the following conclusions were obtained.
In the proposed cubic core panel, the square area of the bonding between the core material and the face plate is larger than that of the honeycomb core panel, and double bonding using glue and rivets is used; thus, the cubic core panel has a stronger bonding strength than the honeycomb structure. In addition, the cubic core reinforced the face plate in a relatively thick manner. It was confirmed theoretically and by FEM analysis that the cubic core panel has a higher bending stiffness than the honeycomb panel under the same conditions.
For the investigation, a cubic core panel was manufactured using an aluminum plate, and the cubic core was processed by simple punching and bending. Cubic core panels were also easily assembled using glue and rivets. It was confirmed that the cubic core panel exhibited good manufacturing performance.
A three-point bending test was performed on the manufactured cubic core panel, and it was found that there are three stages in the relationship between the bending load and deformation. After the first stage with an initial linear relationship, the second stage with gradually nonlinear characteristics continues until the selected maximum load is reached. Finally, in the third stage, which begins after passing that load, the load value tends to drop gradually owing to local buckling. When a cubic core panel is used, it seems better to limit it to the first stage, which has a linear relationship.
According to the requirement for a lightweight structure with a curved portion, the curved portion of the cubic core panel can be manufactured by cutting part of the material from one side of the cubic core panel. A design equation for the curved portion of the cubic core panel was derived using the curvature radius and curvature angle of the actual curved structure.
A design for a cubic core forming system was studied for future practical development. By utilizing the parallel movement characteristics of the four-bar link mechanism, adjacent bending blocks can be shifted up and down. Simultaneously, they can gradually approach each other in the horizontal direction. Furthermore, equations for the bending load and displacement were derived, and important design factors for practical use were obtained.