Mechanical properties of two types of cubic cores with different folding methods

Abstract

The cubic core panel is a lightweight metamaterial structure with a core made by three-dimensionally forming a sheet into a shape resembling an array of cubes using origami (folding paper) and kirigami (cutting paper) technology. Even if the initial sheet is of the same cut, by changing the pattern of arrangement of the fold lines, two different types of fabrication processes exist for the cubic core, and although the component arrangement of the finished core is the same, the connections between the components are different, which allows for different mechanical properties to be provided. Hear the most fundamental property, stiffness in compression, is quantified using experimental, analytical, and theoretical methods. We focus on the vertical plane that comprise the cubic core panel and quantitatively predict the compressive stiffness of the two core types based on their buckling strength. In addition, we investigate the post-buckling crushing behavior from mechanical measurements and simulations, and discuss the impact absorption performance. The structural design provides an engineering application that optimizes the two formation methods of cubic core panels.

Introduction

Honeycomb core sandwich panels Bitzer (1997), Gibson and Ashby (1997) is an essential structural material used in vehicles such as aircraft and spacecraft. Recently, their applications have expanded to include architecture, automobiles, and railway vehicles. In the automotive industry, the transition from conventional fuel vehicles to Electric vehicles has led to a rapid increase in vehicle weight. Thus, automakers are experimenting with honeycomb core sandwich panels, which offer low weight along with high rigidity, strength, and impact resistance Hazizan and Cantwell (2003), Sun et al. (2021), making them suitable for such applications. Moreover, paper honeycomb cores Chen and Yan (2012) are widely used in furniture and interior materials. Therefore, producing high-strength panels from environmentally friendly materials such as paper broadens the potential range of applications Smardzewski and Tokarczyk (2024).

Although honeycomb core sandwich panels are excellent structural materials, they have several limitations: They are expensive because of their complex manufacturing process, exhibit poor flexibility by deforming into a saddle shape when subjected to bending stress, and are difficult to machine because core material is thin Bitzer (1997). Research is being conducted on developing honeycomb cores with special cell shapes for improved structural flexibility Kitajima et al. (2023) and manufacturing processes for creating three-dimensional (3D) honeycomb structures similar to origami and kirigami Saito et al. (2014), Wang et al. (2019), Saito et al. (2011, 2023), Xu et al. (2023). Accordingly, new core materials that can replace the honeycomb cores have been developed. Foldcore is a promising technology based on origami and kirigami techniques and utilizes a single sheet to form 3D cell structures Miura (1972), Klett et al. (2011), Heimbs (2012), Schenk and Guest (2013), Wei et al. (2013), Yasuda et al. (2016), Chen et al. (2017), Nojima and Saito (2006), Scarpa et al. (2013), Tian et al. (2023), Niu et al. (2023). Miura proposed the Zeta-core, which is built using metal sheets subjected to plastic deformation and features the Miura fold and its derivative patterns Miura (1972). Zeta-core has widely utilized for designing composite materials Klett et al. (2011), Heimbs (2012), Wang et al. (2024). 3D cell structures fabricated using origami and kirigami techniques have also been extensively researched as models for mechanical metamaterials Schenk and Guest (2013), Wei et al. (2013), Yasuda et al. (2016), Chen et al. (2017).

Nojima and Saito (2006) proposed various cell structures formed by folding thin sheets with periodic slits and crease lines. These structures are based on space-filling systems using regular and semi-regular polyhedra. Among these, the cubic core, which is created from a sheet with rectangular cutouts, has attracted significant research interest as a core material and a metamaterial due to the simplicity of the folding process and the resulting structural stability Scarpa et al. (2013), Tian et al. (2023), Fathers et al. (2015). In addition to the basic pattern, derivative models such as elongated rectangular, cylindrical, angular, and dome-shaped structures were also proposed by Nojima and Saito (2006). Scarpa et al. Scarpa et al. (2013) prototyped a cubic core using aramid fiber prepreg and numerically analyzed its vibrational characteristics. Hagiwara et al. Tian et al. (2023) elucidated the performance of a cubic core as a structural core through numerical analysis, aiming for its application in automotive components.

Nojima et al. Nojima and Saito (2006) proposed two different folding methods (Type A and Type B, as shown in Figure 1) to obtain a cubic core from the same flat pattern, resulting in a similar 3D structure (see Supplementary Videos 1 and 2). Although the folded shapes are very similar, as shown in Figure 1(iv), they exhibit different performance characteristics as core materials due to differences in the bonding conditions of the vertical plane components. Fathers et al. (2015) analyzed the shock absorption properties of four types of cores—Type A, Type B cubic cores, Eggbox, and Diamond Strip—and found that Type B cores exhibit higher stronger stiffness. In contrast, Type A cores demonstrated superior curved surface formability due to their unique mechanical property known as “auxetic” behavior, in which the material expands laterally when stretched Scarpa et al. (2013). More recently, simplification of the deformation process has enabled mass production Tian et al. (2023). Although previous studies have investigated the mechanical properties of these structures through experiments and numerical simulations, the theoretical explanation of compressive strength based on the configuration of constituent planes has not been sufficiently explored. Once the mechanical properties resulting from the two folding methods proposed by Nojima and Saito are thoroughly analyzed, these approaches are expected to find applications in a wide range of research areas.

Figure 1.

Type A and Type B are mentioned as methods for forming a cubic core. ‘A’ and ‘B’ indicate the Type-A and Type-B deformations, respectively. (i) Cut a grid of holes in the sheet material, add different mountain and valley crease patterns (red and blue) for each of Type A and Type B, and (ii)–(iii) 3D the cubic core according to the crease pattern. (iv) The shape of the completed core is very similar to Type A and Type B.

This study aims to develop theoretical equations for compressive strength—the most fundamental mechanical property—for each of the two types of cubic cores derived from the same development pattern. In particular, by focusing on the differences between Types A and B, we seek to elucidate the structural characteristics of each type to better understand their respective applicability.

The remainder of this paper is organized as follows. Section 2 describes the development of the cubic core and the two folding processes and clarifies their characteristics. Section 3 describes a prototype made of kraft paper and its compression test results. Section 4 presents a numerical analysis and discusses the experimental results for the two core types. Section 5 presents the discussion, and lastly, Section 6 concludes this paper.

Structure

The cubic core consists of a three-dimensional sheet with periodic square cutouts, arranged in an $n\times n$ configuration, as shown in Figure 1A-F1(i) or B-(i). In this study, we analyze two types of cubic cores, each defined by a distlinct fold-line pattern, referred to as Types A and Type B.

In Type A, the crease lines—indicated by the red (mountain folds) and blue (valley folds) lines in Figure 1A-F1(i)—are arranged such that the square surfaces, each bounded by the four adjacent cube sides, are alternately pushed forward and backward while remaining attached to the surface panel. This folding pattern results in a cell structure in which unevenly elevated cubes are arranged in a checkerboard-like configuration.

In Type B, a similar three-dimensional structure is obtained by placing the crease lines as shown in Figure 1B-F1(i), corrugating the entire sheet such that the square surfaces adjacent to the holes are attached to the surface panel, and then alternately shifting them to the left and right (Figure 1B-F1(ii) and (iii)). Although the resulting core geometry is identical for both Types A and B (Figure 1A-F1(iv) and B-(iv)), the bonding conditions of the vertical cell walls differ, as indicated by the red and blue crease lines in Figure 1A-F1(i) and B-(i). This difference leads to variation in the load-bearing performance of the core panels under out-of-plane compression.

In Type A core, the vertical plane compornets are connected via their top and bottom edges, which are originally crease lines, as shown in Figure 2A-F2(b). As a result, the planes, fixed at both ends, buckle independently under load, as shown in Figure 2A-F2(c).

Figure 2.

Surface attachment configurations in cubic core Types A and B. (a) Mountain folds (red lines) and valley folds (blue lines) are applied in different patterns to sheets with identical square grids and hole arrangements. To compare the relative positions after forming, surface elements are labeled with numbers. (b) A single cube is shown after forming. The locations of crease lines where surfaces are attached differ between Types A and B. (c) When vertical loads are applied, the boundary conditions of the vertical surfaces vary between Types A and B, resulting in different buckling modes.

On the other hand, in Type B core, the vertical plane components are also connected at a right angle to one of its orthogonal neighbors, as shown in Figure 2B-F2(b)). As a result, it exhibits higher buckling Strength than the Type A core, which consists of several independent planes.

Experiments

Methods

We conducted mechanical experiments in which the cores were vertically compressed to investigate the compressive strength of the cubic core panels. Kraft paper (OJI F-TEX) with a thickness $t=0.2\mathrm{mm}$ was used as the core material, and a test model was constructed using an acrylic plate as the surface material, which was sufficiently stiff compared with the core material. The Young’s modulus of the kraft paper was assumed to be $E=10\mathrm{GPa}$ . The Young’s modulus and density of paper are known to vary with temperature and humidity. In this study, we adopted the approximate values measured for common kraft paper. Each side of the cubic core measured $a=20\mathrm{mm}$ . The core was strongly bonded to the surface material using a double-sided tape. Considering the periodic boundary conditions, a portion of the vertical plane of Type B core was fixed to the surface material using Scotch tape. Moreover, to account for errors in the formation, four test models were prepared for Types A and B. As shown in Figure 3, a compression testing machine (MCT-2150W, A&D) was used to measure the load when the upper surface material was displaced sufficiently slowly while maintaining parallelism between the panels.

Figure 3.

Type A cubic core model collapsing in a compression test. The reaction force is measured when the top end is pushed down. The vertical planes, which is flat when the magnitude of displacement is $z/a=0.0$ , buckles and deflects by $z/a=0.26$ . At this point, the shape of the vertical planes indicates that the top and bottom edges are pin-ended. In some cases, the buckling shape may be of the second-order mode because it may be in contact with adjacent planes. The maximum collapse was about $z/a=0.94$ .

Results

The loads on a cubic core panel with a core height $a$ when it was compressed in the hight direction to obtain displacement $z$ are plotted in Figure 4. As an initial response, the load increased almost linearly with compression; however, buckling occurred in the vertical plane of the cubic core with a peak at the buckling load. In both results, larger buckling loads were observed for Type B cubic core panels than for Type A. The experimentally obtained buckling loads are listed in Table 1. On average, the ratio of the buckling loads for Types A and B in the physical experiments was $P_{\mathrm{crB}}/P_{\mathrm{crA}(\mathrm{Exp}.)}\approx 3.6$ , with Type B exhibiting greater stiffness.

Figure 4.

Measured load (N) versus compressive deformation of cubic core. Measurements were taken on different models, four times for each of Types A and B. Load is the maximum in each model, with the points A1–A4 and B1–B4 showing the buckling load.

Table 1.

The experimental buckling loads $P_{\mathrm{cr}}$ for Type A and Type B. $P_{\mathrm{crA}}$ and $P_{\mathrm{crB}}$ denote the respective average values for Type A and Type B.

Specimen ID	$P_{\mathrm{cr}}[(\mathrm{N}])$	Specimen ID	$P_{\mathrm{cr}}[(\mathrm{N}])$
A1	66.6	B1	242.1
A2	83.1	B2	316.3
A3	83.9	B3	278.5
A4	78.3	B4	301.3
$P_{\mathrm{crA}}$	78	$P_{\mathrm{crB}}$	280

After compression, the test model collapsed, as shown in Figure 5. For Type A, although there was contact between the vertical planes, each vertical plane generally buckled in a single curvature. Whereas, for Type B, because the vertical walls were connected, deformation was observed in a complex folding pattern.

Figure 5.

Cubic core test model after compression. Note the shape of the vertical plane: in Type A, each vertical plane is curved independently, while in Type B, the deformation is not as simple as in Type A, because some planes are connected to another vertical plane.

Simulations

Methods

To quantify the compressive strength of the two cubic core panels formed using different approaches, a finite element analysis (FEA) was performed using the commercial software Abaqus (Dassault Systèmes). A square linear isotropic elastic shell element with geometric nonlinearity was specified with element size in the range of $0.1-1.0\mathrm{mm}$ , Poisson’s ratio $\nu =0.3$ , Young’s modulus $E=10\mathrm{GPa}$ , density $\rho =500\mathrm{kg}/{\mathrm{m}}^3$ , thickness $t=0.2\mathrm{mm}$ and height of the core $a=20\mathrm{mm}$ . Here, the joint was fixed at a right angle, the relative positions of the upper and lower ends of the connected vertical panel were fixed and arranged assuming zero deformation of the surface panel, and the contact between the core materials and the resultant friction were considered negligible.

First, for Type A, the buckling load $F_{\mathrm{crA}}$ of the core was calculated from the buckling eigenvalue analysis when the upper and lower surfaces were approached while maintaining parallelism between them. For Type B, the eigenvalue analysis could not be used to calculate the buckling load because the stiffness was maximized at the point where the vertical panels were connected to each other, and the analysis did not show a buckling mode in which the entire core deformed uniformly.

In addition, the total reaction force $F$ applied to the top edge of the approaching displacement $z$ was calculated for Types A and B using the dynamic explicit method.

The purpose of the simulations in this study is not to fully reproduce the complex behavior observed in the experiments, but rather to complement the theoretical framework. Therefore, to reduce computational cost, contact and plastic deformation were not considered. In future work, for quantitative evaluations tailored to specific materials or applications, it will be necessary to conduct simulations that incorporate these factors as well as additional deformation modes.

Results

The buckling eigenvalue analysis of the Type A cubic core with several mesh size changes showed that the buckling load in the lowest-order buckling mode was $P_{\mathrm{crA}(\mathrm{Sim}.)}=83.2N$ . These findings highly agree with the experimental results.

The results of the explicit dynamic analysis of the compressive loads are presented in Figure 6. Here, the buckling load for Type A was approximately $10$ times larger than the eigenvalue analysis result $P_{\mathrm{crA}(\mathrm{Sim}.)}$ . This is because the analysis results do not indicate a static deformation, suggesting that there is a velocity dependency. Comparing the buckling load with the explicit results of displacement at the same velocity for Types A and B, the peak of the force curve plotted in Figure 6 shows that $F_{\mathrm{crA}}/P_{\mathrm{crA}(\mathrm{Sim}.)}=9.46$ and $F_{\mathrm{crB}}/P_{\mathrm{crB}(\mathrm{Sim}.)}=37.5$ for Types A and B, respectively, yielding a ratio $F_{\mathrm{crB}}/F_{\mathrm{crA}}\approx 3.96$ . Comparing the buckling load with the explicit results of displacement at the same velocity for both types A and B, the peak of the force curve in Figure 6 shows that $F_{\mathrm{crA}}/P_{\mathrm{crA}(\mathrm{Sim}.)}=9.46$ for type A, $F_{\mathrm{crB}}/P_{\mathrm{crB}(\mathrm{Sim}.)}=37.5$ for type B, with a Ratio of $F_{\mathrm{crB}}/F_{\mathrm{crA}}\approx 3.96$ .

Figure 6.

Dynamic explicit analysis of load $F$ and rescaled displacement $z$ for type A buckling load. Here the type A buckling load $P_{\mathrm{crA}(\mathrm{Sim}.)}$ is the value obtained by buckling eigenvalue analysis and $a$ is the cell size. A snapshot of the deformed shape in FEA is shown in the inset. The color map represents the magnitude of von Mises stress.

Discussion

The buckling load resulting from the cubic core compression was theoretically derived. The buckling load of the Type A cubic core was expected to match the buckling loads of multiple parallel vertical planes. Assuming the vertical planes to have a rectangular cross section and Eulerian buckling with pin connections at the top and bottom ends, and under the condition of no collision with adjacent planes, the buckling load per vertical plane was calculated as ${\pi }^2\mathrm{EI}/a^2$ according to Euler’s formula, where $E$ is the Young’s modulus of the component and $I$ denotes its moment of inertia. For a rectangular cross-section, such as a sheet component, $I=a{t}^3/12$ . If the number of vertical planes in the entire core is $m$ , then the buckling load in the entire core is

$$P_{\mathrm{crA}}=m\frac{{\pi }^{2}E{t}^3}{12a}.$$

(1)

Here for Type A, the number of vertical planes is $m=2n(n-1)$ . For the cubic core considered in this study, where $n=4$ , substituting each variable into equation (1) yields $P_{\mathrm{crA}}=79.0\mathrm{N}$ , which agrees well with the experimental results and the buckling load obtained from the FEA buckling eigenvalue analysis. This agreement confirms the validity of equation (1) for the case of $n=4$ .

In the case of Type B, the deformation is not as simple as that of Type A. This is because, in addition to vertical panels with top and bottom edges connected to the surface panel, there are also surfaces connected at right angles to their adjacent surfaces.

Assuming that the deformation is similar to Eulerian buckling in the first-order mode, the effective buckling length $a_e$ is unknown. However, if the effective buckling length coefficient $K=a_e/a$ , then $K=1$ at the pin end and $K=0.5$ at the clamp end. Consequently, the effective buckling length coefficient should be equivalent because Type B cubic core also has the same number of vertical planes as Type A. Considering $K$ to be an unknown constant, we propose the following empirical buckling load predictions:

$$P_{\mathrm{crB}}=m\frac{{\pi }^{2}E{t}^3}{12K^{2}a}.$$

(2)

Considering that $P_{\mathrm{crB}}=280\mathrm{N}$ from the experimental results, $K=0.53$ , and the ratio of the buckling load between Types B and A is obtained as

$$\frac{P_{\mathrm{crB}}}{P_{\mathrm{crA}}}=\frac{1}{K^2}=3.5.$$

(3)

Thus, Type B exhibits $3.5$ times higher compressive stiffness than Type A. This theoretical calculation was in close agreement with the results of the explicit dynamic simulation. Whether the obtained effective buckling length coefficient of vertical planes $K=0.53$ is universally valid for compression of Type B cubic cores with arbitrary geometric parameters remains an open question.

For Type B, the experimental and simulation results shown in Figures 4 and 6 indicate that the compressive strength varied after buckling. This was due to the formation of wrinkles and fractures at the perpendicular connections of the adjacent vertical walls of the cell owing to jump-over deformation. In particular, the simulation results did not consider contact with other members, and there was no relaxation of the deformation, which resulted in a noticeable load reduction jump. In the experimental results shown in Figure 4, the work done by the core material was calculated over the range of $z/a=0$ to $0.7$ . When averaged for each type, Type B exhibited a value $3.7$ times greater than that of Type A, suggesting its higher capacity to absorb kinetic energy. In contrast, Type A maintained a nearly constant load from the initial peak through the plateau region, indicating a more stable crushing behavior. These results suggest that Type A possesses characteristics ideal for applications requiring gradual energy absorption.

Conclusion

By combining model building, mechanical experiments, FEA, and theoretical calculations of buckling, this study investigated the mechanical properties under compression of paper-based cubic cores formed by two different folding processes. Through systematic experiments and explicit numerical analyses, it was quantitatively demonstrated that the Type B core, formed by corrugation, exhibits higher compressive stiffness than the Type A core. In addition, by focusing on the configuration of the constituent planes, we developed an elastic buckling theory that predicts the buckling loads of Type A and Type B cubic cores and successfully explains the results of both experiments and numerical simulations. These findings demonstrate that even when using the same sheet material, the load-bearing performance of folded core structures can be significantly enhanced—by several times—through the careful design of the forming process, particularly by strengthening the connections between vertical planes. This insight provides a valuable basis for strength-oriented design in a wide range of core panels fabricated by three-dimensional forming of sheet materials. The effective buckling length of Type B cores needs further investigation through experiments and geometric theories.