Introduction

At present, environmentally friendly vehicles are in demand. Such demand will increase further as interest in next-generation transportation, such as flying cars, increases. Flying cars are electric automobiles having flying capabilities and are currently being developed worldwide. The characteristic of flying cars over conventional aircraft is short take-off and landing (STOL) or vertical take-off and landing (VTOL), thus they can take off and land in tight spaces, which allows to fly closer to the destination. To commercialize flying cars, it is essential to reduce the weight of the body to increase the cruising range and reduce the manufacturing costs by increasing the rate of production of vehicle body parts.

A few studies have been conducted on methods to reduce the weight of flying cars by using inexpensive materials. For example, Sun et al. analyzed the use of lightweight frames of 6061 aluminum alloy for flying cars using the finite element method (FEM) [1]. However, the weight reduction achieved by changing the structure using conventional metallic materials such as aluminum alloys is limited. The National Aerospace Laboratory of Japan estimated [2] that the weight of the structural frame of a 100-seat class VTOL aircraft modeled by adding equipment to enable flight, such as lift-generating fans to a conventional aircraft, could be reduced from 14,650 to 12,030 kg (18%) by changing most of the airframe structure to carbon fiber reinforced plastic (CFRP). As a result, the weight increase arising from the VTOL equipment could be mitigated. Putnam et al. [3] also showed that the use of composite materials leads to a significant weight reduction. In addition, although changing the material allows an increase in load without protection mechanisms, Federal Aviation Authority (FAA) regulations for passenger loads are met with the addition of protection mechanisms. The need for weight reduction is not limited to flying cars, and reducing the weight of conventional vehicle bodies reduces energy consumption, which is important for improving their environmental credentials.

Currently, the main use of composite materials is in airframe structures. According to Misumi [4], the use of CFRP, which began to be applied in secondary structural members without large load transmission in the 1970s, was also used in primary structural members such as tail wings and cabin girders as a result of the increased reliability of CFRP compared to conventional materials. The application of CFRP to aircraft not only improves fuel efficiency and cruising range but also improves comfort by improving barometric pressure and humidity inside the cabin. According to Shimizu [5], CFRP has also been used in racing cars. Because racing cars are more strongly required to have high mechanical characteristics and light weights than standard vehicles, the use of CFRP in these vehicles is actively adopted. However, CFRP is far more expensive than other widely used materials because of its lower rate of production, especially compared to materials prepared by conventional forming, such as the stamping of thin steel sheets. Owing to the high price and low productivity, to date, the applications of CFRP have been limited.

Various CFRP molding methods that shorten the production cycle to increase the production rate and reduce costs have been developed. Yamasaki et al. [6] developed an ultra-high-speed curing-type molding resin and high-speed resin impregnation molding technology for application in a high-cycle resin-transfer molding (RTM) method. This technique was used to mold the inner panels for automobiles in complex three-dimensional shapes. Terada [7] stated that carbon fiber reinforced thermoplastic (CFRTP), which uses a thermoplastic resin and chopped fibers, is effective in high-cycle injection molding and press molding. In addition, Yanagimoto et al. [8] proposed a new method that enables the high-productivity stamping of long-continuous-fiber CFRP with excellent mechanical properties. Using this method, the stamping of the CFRP sheets is possible even after curing in an autoclave by sandwiching the CFRP between metal dummy sheets.

On the other hand, proposals for the development of alternative lightweight materials having high strength and rigidity via structure engineering have also been investigated. Sandwich structures are one solution, and the design of the core structure has been investigated previously [9]. For example, a sheared dimple core was proposed by Seong et al. [10] as a high-strength structure because of the small gap between the attachment points. In addition, Zhang et al. proposed and analyzed a sandwich sheet with a CFRP core sandwiched between two ductile thin metallic face sheets [11,12,13]. As the core structure composed of CFRP, a truncated dome shape was proposed considering the bending strength and possible production rate. However, this core structure was obtained by comparing some typical structures. To date, various additively manufactured core structures have been proposed for new composite structures, but few investigations have been conducted on thin composite structures having stampable cores.

Therefore, in this paper, a new core–shell-type thin composite structure is proposed. The core–shell structure is designed to have sufficient stiffness and deformability while having a stampable core and sufficient interfacial contact with the thin metallic face sheets. The core structure was proposed based on the moment of the inertia of the area, and a core composed of square domes was obtained. The pure bending deformation of the sandwich sheet using the proposed core and face sheets was modeled numerically by finite element (FE) analysis. The FE analysis and practical three-point bending test were also conducted to evaluate the performance of the newly proposed core–shell-type composite structure. The proposed sheet is capable of bearing sufficient force to enable its use in monocoque structures and could be produced easily in large quantities because both the core and sandwich sheet can be stamped from the sheet into the desired geometry.

Proposed core structure

Features required for the core

The structural material of monocoque bodies used in future vehicles must have high specific strength and specific rigidity and be produced by high-productivity methods. One solution is to prepare sandwiched sheets having a core–shell structure made of ductile thin metallic sheets and a CFRP core.

For the face sheets, if the metallic material is defined, the thickness of the sheets is the only design parameter. Clearly, increasing the thickness of the face sheets increases rigidity. On the other hand, increasing the volume of the metallic face sheets, which have high densities, results in an increase in the density of the sandwich sheet, which reduces the specific strength and rigidity. In this study, the thickness of the metallic sheets was selected as having the standard thickness of a thin sheet (0.5 mm).

Thus, in this study, we concentrated on the core structure, and the following two points were considered as limitations for the design of the core structure. First, to enable future mass production, the core structure was limited to a shape having a constant thickness that can be manufactured by warm stamping proposed in previous research [8]. Secondly, there must be sufficient interfacial surface area between the face sheet and core so that the face sheets and core have sufficient bonding strength. In other words, a core structure having increased bending rigidity while satisfying the conditions of high productivity and interfacial strength between the face sheet and CFRP core was the target of this study.

Design conditions

The bending stiffness was used as the key parameter in the design of the core geometry. A structure that minimizes the displacement with respect to the moment in the X- and Y-axes was obtained. In Fig. 1, only the Y direction is shown, but the same condition applies in the X direction. For symmetry with respect to the X and Y-axes, the periodic structure of the core must have a 90° rotational symmetry, where X denotes the length, Y denotes the width, and Z denotes the thickness.

Fig. 1
figure 1

Loading conditions in the design of the core structure: pure bending

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The structure of the core is periodic, and the width of the periodic structure in the XY plane was set to 12 mm, which is the same size as the truncated dome core proposed in our previous study [11]. The thickness of the core sheet was set to 1 mm, which is standard thickness, and the apparent thickness of the 3D core structure including unevenness was set to 4.5 mm for simplicity.

The proposed periodic structure is shown in Fig. 2. The tilt angle, the periodic width, and the bonding surface denote the parameters indicated in Fig. 2. In designing the core structure, the tilt angle of the core sheet and maximum depth, that is, the difference in height between the upper and lower surfaces, were considered as parameters. Here, maximum depth was fixed to 4.5–1.0 = 3.5 mm. Stamping is easier if the tilt angle is small, so the core was designed with a tilt angle in the range of approximately 30°.

Fig. 2
figure 2

Periodic core structure proposed in this paper: A square domes are formed vertically from the neutral surface. These periodic structures are lined up in bending and orthogonal direction

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The proposed structure was developed considering the moment of inertia of the area, which represents the structural rigidity against bending, which is increased by distributing the material at positions away from the neutral plane. When the cross-section is not uniform, the moment of inertia of the area cannot be calculated using beam theory. However, if the material is distributed away from the neutral plane where the bending stress is large, the rigidity increases. In the obtained structure, the material distribution of the core on its intermediate surface, which is a neutral surface, was very small. Thus, the obtained structure has a higher specific flexural rigidity than fiber–metal laminates (FMLs) [14], for which the intermediate core layer does not have a 3D geometry.

FE analysis of the designed sandwich sheet to evaluate the rigidity

Modification of the designed structure

Reducing the tilt angle of the periodic structure facilitates stamping molding, which shortens the molding time and improves the molding accuracy. In contrast, increasing the tilt angle results in a large amount of material outside of the neutral plane, which would increase the flexural rigidity from the viewpoint of the moment of inertia of the area. Therefore, bending deformation analysis using the FEM was performed to evaluate the effect of the tilt angle on the bending rigidity.

In addition, the change in the tilt angle influences the area of the interface bonding surface between the metallic face sheet and CFRP core. As the adhesive surface becomes smaller, peeling between the core and face sheets is more likely to occur. By changing the periodicity (cycle length) without changing the shape of the irregular and height, the ratio of the adhesive area to the entire area can be changed. The change in flexural rigidity with respect to the change in the cycle length was also investigated. In the subsequent text, each structure is denoted, for example, 12mm_30deg, indicating the cycle length and tilt angle separated by the “_” symbol.

Abaqus CAE was used for FE analysis. For the analysis mesh, tetrahedral elements were used for the core, and hexahedral elements were used for the face sheets. The pure bending deformation was analyzed for each structure to compare the flexural rigidity. For the opposite end faces, a rotational constraint of 0.01 deg and − 0.01 deg was given, respectively. Namely, a bending deformation of 0.02 deg was given, and the bending moment was calculated. The flexural rigidity, EI, was calculated using Eq. (1) based on the bending moment obtained from the FE analysis, for which a bending deformation of θ = 0.02° occurs for a one-period structure.

$$I={\left.\frac{M}{\rho}\right|}_{\theta =0.02\ \deg }$$
(1)

Here, M is the bending moment, and ρ is the curvature. This equation is derived from beam theory.

The material properties used in the analysis and experiment are listed in Table 1.

Table 1 Material properties used in analysis and experiment: (a) ONYX was used for the core and (b) SUS304 stainless-steel was used for the face sheets
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Effect of tilt angle change on flexural rigidity

Figure 3 shows the periodic structure used in the analysis. The tilt angle was changed in intervals of 15° from 30° to 90°. Table 2 shows the relative flexural rigidity and relative specific flexural rigidity of the core and sandwich sheet calculated for each structure. The relative flexural rigidity is given by Eq. (2).

Fig. 3
figure 3

Different core structures having varying tilt angles with a cycle length of 12 mm: (a) 12mm_30deg, (b) 12mm_45deg, (c) 12mm_60deg, (d) 12mm_75deg, and (e) 12mm_90deg

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Table 2 Flexural rigidity, specific flexural rigidity, and ratio of adhesive surface with varying tilt angle
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$$\mathrm{relative}\ \mathrm{flexural}\ \mathrm{rigidity}=\frac{EI}{(EI)_0}$$
(2)

Here, (EI)0 is the flexural rigidity of a 1-mm flat-plate core or that of a sandwich sheet with a solid core of 4.5 mm for the core only and sandwich structures, respectively.

First, when the analysis was performed on the core-only structure, the bending rigidity increased as the tilt angle increased. In addition, the flexural rigidity of 12mm_30deg and 12mm_90deg differed by approximately five times. However, as the tilt angle increased, the volume of the core increased, and the mass increased accordingly. However, the difference in mass is less than the increase in rigidity, so a structure having a larger tilt angle is better, even in terms of the specific flexural rigidity.

Next, the flexural rigidity of the core–shell sandwich sheets was compared. The core structure with a smaller tilt angle exhibited a larger specific bending rigidity. This is because the change in the rigidity of the core has only a very small effect on the flexural rigidity of the sandwich sheet. This point was examined from the viewpoint of the moment of inertia of the area. The face sheets are located on the exterior of the sandwich sheet and are made from stainless steel, whose Young’s modulus is higher than that of CFRTP. Therefore, the contribution of the face sheet to the moment of inertia of the area was much larger than that of the core. As a result, the flexural rigidity was hardly changed by the tilt angle, and a structure with a smaller tilt angle, which has a small mass, has a larger specific flexural rigidity.

In contrast, as shown in Table 2, when the tilt angle decreased, the adhesive surface with the face sheet became small. In this analysis, the peeling of the adhesive surface cannot be modeled, but it can be expected that peeling is more likely to occur when the area of the adhesive surface is small. As Kobayashi pointed out, the peeling of the outer layer before destruction means that the core and face sheet have excessive strength and weight [15], and this situation should be avoided.

Effect of cycle length on flexural rigidity

To cope with the decrease in the adhesive area arising from the change in the tilt angle, structures in which the fraction of contact surface was increased by changing the cycle length without changing the shape and height were considered, and the change in flexural rigidity with respect to the change in cycle length was examined. The tilt angle was set to 30°, which had the highest specific flexural rigidity in the previous section. Five types of cycle lengths (12, 14, 20, 40, and 100 mm) are analyzed in this section, and Fig. 4 shows the examined core structures. Table 3 lists the calculated values of the flexural rigidity and specific flexural rigidity of each structure. As in the previous section, the standard structure for normalization was a 1-mm flat plate and a sandwich sheet having a 4.5-mm solid core.

Fig. 4
figure 4

Core structures having varying cycle lengths and a tilt angle of 30°: (a) 12mm_30deg, (b) 14mm_30deg, (c) 20mm_30deg, (d) 40mm_30deg, and (e) 100mm_30deg

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Table 3 Flexural rigidity, specific flexural rigidity, and ratio of adhesive surface for different cycle lengths
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First, the rigidity of the cores was compared. In this analysis, unlike the previous section, it is necessary to consider that the size of the periodic structure changes. Thus, considering the differences in width, the flexural rigidity per unit width, which is the flexural rigidity divided by the cycle length, was used. The specific flexural rigidity was also obtained by dividing the flexural rigidity per unit width by the surface density instead of the mass. Thus, it was confirmed that increasing the cycle length of the core increases the surface area for adhesion with the face sheets and flexural rigidity, although increasing the cycle length too much reduces the flexural rigidity significantly. This is because the area of the planar portion becomes excessively large when the cycle length is increased too much.

On the other hand, when the specific flexural rigidity of the core–shell structure sheets was compared, we found that the increase in flexural rigidity with increase in cycle length was slight. This is because the face sheets had a large effect on the moment of inertia of the area, as described in the previous section.

In this analysis, only the flexural rigidity obtained by pure bending analysis was considered. However, when a plate material is used as a structural material, it is subject to various forces other than pure bending. For example, the body structural material of an aircraft receives tension in the in-plane direction owing to the difference in barometric pressure between the inside and outside of the aircraft. In addition, depending on the situation, pressure can be applied in the plate thickness direction. As shown in Fig. 5 and Table 4, as the cycle length increases, the cross-sectional area of the core in the neutral plane of the sandwich sheet increases proportionally. However, the load applied to a periodic structure increases in proportion to the square of the cycle length according to the area of the entire structure. Therefore, for the same applied pressure, a larger stress is generated in structures having longer periodic widths. In addition, Iwama et al. [16] showed that high tension rigidity, dent resistance, and surface quality are required for the outer panels of automobiles. Increasing the cycle length of the core and reducing the portion connecting the upper and lower surfaces may result in a decrease in the dent resistance.

Fig. 5
figure 5

Cross-section on the neutral surface of a periodic structure having a tilt angle of 45°

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Table 4 Cross-sectional areas of the neutral plane with respect to the periodic structural area
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Therefore, increasing the cycle length unnecessarily will result in a reduction in the mechanical properties, so the cycle length should be reduced within a range where the adhesive strength is maintained sufficiently high. Compared with the existing structure proposed by Zhang et al. [11], a structure with a cycle length of 14 mm ensured a sufficient bonding area. Therefore, in this study, 14mm_30deg was considered to be the best structure. However, if peeling is observed at the interface in experiments, the adhesive strength can be increased by adjusting the cycle length.

Comparison with existing structures

We performed the same analysis on the existing truncated dome core and confirmed that the new structure has a slightly higher flexural rigidity.

Next, the deformability of the two structures was compared. The difference in the length of a cross-section indicates that the elongation at the time of molding differs with location. When considering the stamping of CFRP using long continuous carbon fibers, if the fiber lengths in various locations are different, the fibers may break or wrinkle during molding, and the strength and reproducibility of the product will decrease. The cross-sectional length and the ratio of the maximum length to the minimum length were calculated for the new structure and the truncated dome structure. Table 5 shows the examined directions and ratios.

Table 5 Lines used to consider the length of the neutral plane and the difference ratios
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For the truncated dome, the difference in the dome arrangement direction was large although the difference in the diagonal direction was small. In contrast, in the new structure, the differences in both the dome arrangement and diagonal directions were very small. To compensate for the anisotropy of CFRP arising from the fiber arrangement, carbon fibers are often stacked in two orthogonal directions or in various directions, as reported by Takahashi et al. [17]. The new structure, in which the fiber lengths are close in various directions, allows various combinations of fiber orientations, which improves the mechanical properties.

Furthermore, if there is a large curvature in the structure, a large bending deformation is required during the molding process, which makes molding difficult and increases the risk of fiber breakage. In the truncated dome core, the curvature is vertical at the boundary between the plane and dome. On the other hand, in 14mm_30deg, the curvature does not exceed 45°. Therefore, the new structure should be easy to stamp without defects.

In summary, the structure proposed in this study is not only superior in rigidity but also excellent in moldability compared to the existing truncated dome structure.

Three-point bending test of the sandwich sheet

Test conditions

The core was created using a Markforged Mark Two 3D printer. Mark Two uses the material extrusion method to manufacture parts additively. ONYX was used as the material for the modeled objects. ONYX is a CFRTP that is reinforced by adding fine carbon to tough nylon. ONYX has twice the strength of standard plastics used in 3D printers and has an excellent surface finish. In addition, it has a high heat resistance so can withstand heat treatment during bonding. For the face sheets, a SUS304 stainless-steel sheet with a thickness of 0.5 mm was used. IW2460 adhesive (3 M) was used to bond the core and face sheets. IW2460 is a one-component epoxy adhesive with features of high adhesiveness, high moisture resistance, and high heat resistance.

The test pieces were prepared by applying the adhesive to the face sheet to a constant thickness, stacking the sheets on the core, and then holding the assembly at 120 °C for 75 min in an autoclave manufactured by Hanyuda Iron Works. To prevent adhesion failure as a result of the warpage of the core, a weight was placed on the test piece, and pressure was applied to the adhesive surface during the heating process.

The sizes of the dies for three-point bending are shown in Fig. 6. The lower die is fixed, and the upper die moves downward. During this process, the displacement of the die and the reaction force applied to the die were measured.

Fig. 6
figure 6

Die used for three-point bending tests (units are in millimeters)

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Comparison of the flexural rigidity of structures having different tilt angles

Figure 7 shows the results of the three-point bending tests. For 12 mm_30deg, two test pieces were prepared for reproducibility. After several experiments, we confirmed that when the die hits a place where the core and the face sheet are separated, only the face sheet in the hit part is significantly deformed. As shown in Fig. 8(a), to reduce the influence of the dies, the test pieces were placed in contact with the upper die at a point with high rigidity, that is, where the core and face sheets were stuck together.

Fig. 7
figure 7

Effect of tilt angle on flexural rigidity: (a) relationship between displacement and force during the three-point bending test and (b) relationship before structural destruction (enlarged view of upper figure)

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Fig. 8
figure 8

Three-point bending experiments: (a) 12mm_30deg before bending, (b) deformation at the center of 12mm_30deg when the displacement of the upper die was 3 mm and (c) deformation at the center of 12mm_45deg when the displacement of the upper die was 3 mm

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As shown in Fig. 7, 12mm_45deg, 12mm_60deg, 12mm_75deg, and 12mm_90deg have the same reaction force for the same displacement. This means that the flexural rigidities of these structures are very similar. On the other hand, the reaction force of 12mm_30deg is smaller than that of the other structures. This is because one of the convex parts in the center of the core was crushed earlier during deformation, as shown in Figs. 8(b) and (c). This resulted in a decrease in the thickness of the sandwich sheet at the center and a decrease in rigidity. Therefore, this structure cannot be compared with other structures under the same conditions.

For structures without local destruction, it was confirmed that changing the tilt angle had almost no effect on the flexural rigidity. This result is consistent with the FE analysis results. However, the flexural rigidity obtained by the analysis is approximately 10 times that calculated using experiment data and Eq. (3).

$$EI=\frac{l^3}{48}\frac{P}{\delta }$$
(3)

Here, l, δ, and P represent the length between the fulcrums, mold displacement, and reaction load, respectively. That is, P/δ represents the slope of the load graph with respect to die displacement. Eq. (3) is derived from beam theory, and its applicability to our structures will be examined using new analyses in the next section.

Comparison of the flexural rigidity of structures having different cycle lengths

Figure 9 shows the results of the three-point bending tests when the cycle length was changed. In the previous experiments, it was found that core fracture occurs locally at a tilt angle of 30°. Therefore, in subsequent experiments, the test pieces were prepared by changing the cycle length with a tilt angle of 45°, at which angle the central convex part was not destroyed. Because of the size of the die, there is a difference in the number of periodic structures in the test pieces depending on the cycle length. Test pieces with cycle lengths of 12 and 14 mm have two periodic structures across the width and eight along the length, and those having cycle lengths of 20 and 30 mm have one periodic structure across the width and four along the length.

Fig. 9
figure 9

Effect of periodic width on flexural rigidity: (a) relationship between displacement and force during the three-point bending test and (b) relationship only before structural destruction (enlarged view of upper figure)

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For 12mm_45deg and 14mm_45deg, the difference in rigidity was small, and it was considered that their rigidities were almost identical. On the other hand, the rigidities of 20mm_45deg and 30mm_45deg are relatively low. This result was not observed in pure bending analysis. Furthermore, the flexural rigidity calculated from Eq. (3) also differs from that obtained in the analysis. The reasons for such differences are discussed in the next section.

FE analysis of three-point bending

Change in the analytical model

In the previous chapter, it was found that the values of the flexural rigidity obtained by analysis and experiment differed. A possible reason for this difference is error in the analytical model. Eq. (3), which was used to calculate the flexural rigidity from the bending moment and is shown in Section “Comparison of structures with different tilt angles”, is based on beam theory. However, in the core–shell sandwich sheets, because the cross-sectional shape is not constant, the Bernoulli–Euler assumption is not valid. Figure 10 shows the reaction force when a three-point bending displacement of 2 mm was applied to the sandwich sheet at the center of the upper face sheet while changing the width of the fulcrum, that is, the length of the specimen. For these tests, a solid beam and the 12 mm_45deg periodic sandwich structure were used. As for the solid beam structure, the rigidity decreased according to the -3rd power of the length. This is consistent with the equation derived from beam theory. In contrast, for the periodic structure, the stiffness decreased in proportion to the −1.5th power of the length. Thus, beam theory cannot be applied to the periodic structures. Therefore, to obtain consistency between experiment and analysis, it is necessary to perform analysis under the same conditions as in the experiment, that is, three-point bending, as shown in Fig. 11.

Fig. 10
figure 10

Reaction force of structures with varying lengths against 2 mm displacement

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Fig. 11
figure 11

Three-point bending FE analysis: (a) before deformation and (b) deformation magnified five times showing Mises stress contours when a mold displacement of 2 mm was applied, and (c) Mises stress contours of the core when a mold displacement of 2 mm was applied

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Therefore, using the modified procedure, the same test pieces were modeled and analyzed using the FE method. As in the previous analysis, the analysis mesh consisted of tetrahedral elements for the core and hexahedral elements for the face sheets. Each die was made of rigid shell elements. An upper die displacement of 2 mm was applied to this model, and the reaction force during that period was obtained. The purpose of this analysis was to determine the rigidity of the pieces. Thus, we determined that it was sufficient to analyze the conditions where the displacement was 2 mm or less because the structures were destroyed at displacements of approximately 2 mm in the experiments. In this analysis, to evaluate the rigidity of the structure, only the parameters related to elastic deformation (Young’s modulus and Poisson’s ratio) were input.

Furthermore, because the dies were assumed to be rigid bodies, it is possible that the deformation around the molds would be larger than in the experiments. Therefore, in this analysis, the width of the mold was adjusted so that the mold hit the portions of the structures where the face sheets were adhered to the core. Then, using the index obtained in Fig. 10, it was converted to the same width as in the experiments.

Comparison of structures with different tilt angles

Figure 12 shows the change in the reaction force when a displacement of up to 2 mm is applied. The same trends as those obtained by experiment were observed in the analysis. The rigidities of the structures having a tilt angle of 45° or more were similar, and that of the structure having a tilt angle of 30° was approximately half those of the other structures. Next, the reason for the decrease in rigidity in the structure with a tilt angle of 30° was considered. Figure 13 shows the deformation at 12mm_30deg. As the tilt angle decreased, the rigidity of the core decreased. If the rigidity is low, the adhesive surface of the core cannot be maintained parallel. In addition, the gap between the attachment points of 12mm_30deg is larger than that of the other structures. The face sheets then underwent wavy buckling. This resulted in a decrease in the rigidity.

Fig. 12
figure 12

Three-point bending FE analysis of structures with varying tilt angles

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Fig. 13
figure 13

Deformation magnified five times: (a) 12mm_30deg and (b) 12mm_45deg

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Comparison of structures with different cycle lengths

Figure 14 shows the reaction force for displacements of up to 2 mm for structures having varying cycle lengths.

Fig. 14
figure 14

Three-point bending FE analysis for structures with varying periodic widths

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Regarding the cycle length, the result was almost the same as that in the experiment. The rigidity decreased as the cycle length increased. Figure 15 shows the deformation of 30mm_45deg. As the cycle length increased, the distance between the tilted portions connecting the upper and lower face sheets increased. Thus, if the width of the face sheets cannot be kept constant, the thickness of the sandwich sheet is partially reduced, and the rigidity is lowered.

Fig. 15
figure 15

Deformation magnified five times: (a) 30mm_45deg and (b) 12mm_45deg

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Summary

In a core–shell sandwich sheet consisting of stainless-steel face sheets and a CFRP core, a core structure with square irregularities is proposed. The structure has high rigidity from the viewpoint of the moment of inertia of the area because the material is located away from the neutral surface. Furthermore, the moldability and adhesive strength of this structure was improved without compromising rigidity by varying the tilt angle and the ratio of the adhesive surface area through the cycle length, respectively.

Test pieces of the proposed sandwich sheets having a CFRTP core were prepared with a 3D printer and SUS304 face sheets, and three-point bending tests were performed. There was no significant change in the rigidity with respect to the change in the tilt angle of the core in the range where local destruction did not occur. In addition, the rigidity of the sandwich sheets decreased as the cycle length increased.

In the experiments, phenomena that were not modeled in the elastic pure bending analysis, such as local deformation and destruction, were observed. Therefore, in addition to the bending moment, the shear stress or load in the plate thickness direction must be considered when determining the structure of the core.

Further, by comparing the theoretical and experiment results, it was found that beam theory cannot be applied to periodic structures whose cross-sectional shape is not constant. However, by performing the analysis under the same conditions as the experiment, the reason for the reduction in rigidity arising from structural change was clarified. Specifically, the rigidity of the core alone decreased or the gap between the adhesive points increased on reducing the tilt angle. In addition, the increase in the cycle length decreased the fractional area connecting the upper and lower face sheets. This also caused a decrease in rigidity because the plate thickness was not maintained constant. In addition, it was found that the effect of the core structure on the flexural rigidity of the sandwich sheet was small because the face sheets have a significant influence. Therefore, a better mechanical property to enhance the mechanical properties of the entire sandwich sheet should be identified.

Finally, when using continuous fiber CFRP for the core, the fiber direction of the CFRP must also be considered. Takahashi et al. [17] used a genetic algorithm and the finite element method to determine the optimum stacking direction for tension and embossing. Therefore, if the fiber direction is included in determining a core structure by combination with genetic algorithms, structure design will be greatly assisted. In summary, the emphasized mechanical property to produce more excellent structural material must be reconsidered in the future.