Transverse energy absorption performance of sandwich tubes with various origami foldcores

Abstract

Nowadays, composite sandwich tubes are extensively utilized in the civil and aerospace industries due to their superior strength-to-weight mechanical properties. Origami-based core offers a large enhancement of the mechanical properties, yet little study research focuses on the effect of various foldcore configurations on the transverse mechanical properties of sandwich tubes, necessitating the design method for applications. This study introduces an innovative approach by incorporating origami into the composite sandwich core to enhance the transverse energy absorption capacity. The quasi-static transverse mechanical properties of carbon fibre-reinforced polymer (CFRP) sandwich tubes with foldcores are studied under three-point bending-like local compression and transverse structural compression. A systematic geometric design framework and numerical modelling technique are provided. By integrating finite element analysis and experiments, the research investigates the effects of various origami foldcore configurations and geometric parameters on transverse energy absorption capacity. The experiment setup is provided by sandwich tubular specimens with a full-diamond configuration as the foldcore. The cylindrical tubes (foldcore) of the sandwich structures were manufactured using four plies [0°]₄ of T700 (T300) woven CFRP with the hot press moulding (vacuum bag using female and male moulds) technique respectively. Then, the parametric study and damage mode analysis of eight different foldcore patterns (axial Miura, circumferential Miura, diamond, Kresling, and their curved-creased counterparts) were studied. The results showed the superior energy absorption performance of the sandwich tube with Miura-pattern foldcore over the origami-pattern counterparts, nested tube, and traditional honeycomb sandwich tube with CFRP or aluminium-made cores. Therefore, the structural parameters optimisation of the Miura pattern tube was carried out by the Response Surface method (RSM) and a design strategy for increasing the energy absorption capacity was found. The findings offer guidance for designing high-specific energy absorption tubular structures for future advanced engineering applications.

Keywords

CFRP foldcore sandwich tube transverse mechanical properties structural parameters optimization design strategy energy absorption performance

Introduction

The transverse energy absorption properties of composite tubes play a crucial role in various engineering applications, energy-absorbing tubes are designed to mitigate impact forces during collisions to safeguard passengers, to fabricate lightweight, high-strength components for the enhancement of both the safety and fuel efficiency of aircraft, and to be incorporated into building structures for the improvement of seismic performance, in the fields of automotive, aerospace and civil engineering respectively. Origami structures, characterized by their lightweight, high strength, and tunability, can be designed as promising applications in healthcare, aviation, mechanical, and civil engineering.^1–5 Incorporating tubular structures as energy absorption devices has been widely used in the automobile industry.^6–14 Origami-inspired tubular structures, compared to conventional tubular structures,^15–19 offer predefined deforming patterns through the creases, leading to a lower initial peak force and steady impact load compared with conventional straight tubes, hence their energy absorption capacity could be enhanced due to the structural advancement,^20,21 as the majority of the material could be involved in the energy-absorbing process.²² Composite materials in structures exhibit excellent properties^23–25 in areas such as seismic resistance,²⁶ wave absorption,²⁷ energy saving,^28,29 energy absorbing,^30,31 heating insulation³² and thermal load,³³ making them applicable across various fields including vehicle industry,³⁴ life sciences³⁵ and mechanical engineering.^36,37 There are different patterns for origami-inspired structures,³⁸ including the full-diamond,^39,40 trapezoid,⁴¹ diamond,^42–46 Miura,^{21,22,38–51} Tachi–Miura,⁵² Waterbomb,^52,53 and Kresling^54–56 patterns. Over the years, origami-inspired tubular structures used as energy absorption devices have been extensively studied. Yasuda et al.⁵² found that the stress concentrations at the vertices could cause energy dissipation when studying the Tachi-Miura polyhedron (TMP) tubes. Ma et al.⁵⁷ introduced a kite-shaped pattern in the design of thin-walled tubes and studied its crushing behaviours numerically, which shows a superior energy absorption ability than conventional square tubes. Lee et al.⁵⁸ incorporated curved creases into the surface of circular tubes as pre-embedded features, enabling precise control over the buckled shape, and resulting in marginal improvements in energy absorption. Ciampaglia et al.⁵⁹ investigated the CFRP origami crash box under axial impact and showed that the fracture-triggering effect is better when more inclined planes are at the top. Due to the ability of sandwich structures to further enhance the energy absorption capacity^60–64 of both the structure and the material, these origami-inspired tubular structures are often employed as folded cores and assembled into sandwich structures by integrating the foldcore⁶⁵ with the inner and outer plates, thereby significantly improving energy absorption capabilities.^66,67

However, current studies have primarily focused on the axial mechanical properties of origami-inspired sandwich tubes.^16,68,69 Since the working conditions could be complicated in real-time, with the energy shock transmitted from different directions,^17,18,70,71 one needs to also consider the energy absorption capacity in the transverse direction, as the cases of under-sea pipes and the fuselage of the aircraft.^72–75 Currently, little research focuses on the transverse mechanical property of the origami-inspired sandwich tube structures as energy absorption devices,^19,76 and the effects of different origami patterns on the energy-absorbing capacity in the transverse direction remain unknown. Therefore, in this paper, we investigate the transverse mechanical performance of energy-absorbing composite sandwich tubes with origami foldcores under two loading conditions, that is, the three-point bending-like local transverse compression and the structural transverse compression. The designing process of each unit cell of the foldcores is detailed, and the FE analysis is carried out and validated by experiments. The effects of different parameters on the transverse mechanical properties are investigated, then a structural parameter optimization process of the foldcore pattern with the best performance is provided and a design reference is given. Note that two loading conditions here in the study correspond to two scenarios, the transverse concentrated force at the local area and the overall structural compression in the transverse direction, these two scenarios are considered for applications such as the road stake at the highway express to provide the buffering effect for the crashing vehicles, and the undersea pipe which withstands the overall transverse compression during its lifespan. Therefore, the specimen designed for the local compression condition is a “three-point bending”-like test specimen, and hereinafter, this loading condition is phrased as the “three-point bending” test for simplicity and clarity purposes.

The paper is organised as follows. First, the numerical models are discussed in Section Experimental setup and numerical model, with two loading conditions (i.e., the three-point bending and the transverse compression) considered. Since the boundary condition of the three-point bending is a more complicated one in two loading cases, the experiments of this loading condition are carried out to validate the numerical model. Then, the design processes for eight origami-inspired foldcores are demonstrated in Design process of the geometric foldcore configurations Section. Further, the parametric study of the composite sandwich tubes with eight origami-inspired foldcore patterns is conducted under both loading conditions in The results of the parameter study Section, and as one of the main novelties in this paper, the main parameter is identified and the optimization process is carried out and the design strategy is given. Finally, a conclusion is given.

This study primarily investigates the incorporation of origami foldcore into composite tubes to enhance their transverse energy absorption performance. The main novelties of this work are fourfold: 1. The geometric design method and geometric constraint of various origami-based foldcore are proposed systematically; 2. The numerical modelling technique for sandwich structures with foldcore is set up and validated; 3. A parametric study on the effects of various foldcore configurations on the transverse mechanical properties of the sandwich structures is studied and main parameters are identified; 4. A design strategy of sandwich structures with foldcore on the transverse mechanical properties is provided. Therefore, this work systematically investigates and provides a reference for designing composite sandwich tubes with origami cores, filling the research gap in the transverse energy absorption performance of tubular origami sandwich structures.

Experimental setup and numerical model

Material and experimental setup

The sandwich tubes with foldcores are made from carbon fibre-reinforced polymer (CFRP) material and comprise two cylindrical inner and outer tubes with a foldcore in between, as shown in Figure 1. Both cylindrical tubes and the foldcore were manufactured using four plies [0°]₄ of T700 (Shanxi Zhongsheng Tianze Co Ltd, China) and T300 (Shanxi Zhongsheng Tianze Co Ltd, China) woven carbon/epoxy prepregs respectively, where 0° in the ply stacking sequence indicates the axial direction (direction 1 shown in Figure 3) of the sandwich tube, and the CFRP material properties are from our previous work,³³ as shown in Table 1. Note that the material 1 direction of the cylindrical inner/outer tubes and foldcore is along the axial direction of the tube, while 2 and three material directions are oriented circumferentially and radially.^39,77–79

Figure 1.

(a) Internal view of the full diamond specimen, (b) complete schematic diagram of the full diamond specimen, (c) schematic diagram illustrating the geometric relationships of the full diamond configuration.

Table 1.

Material properties of two CFRPs.³⁹

Description variable	Variable	CFRP T700	CFRP T300
Density /(kg/m³)	ρ	1600	1690
Poisson’s ratio	ν₁₂	0.043	0.04
Longitudinal modulus /(GPa)	E₁	61.95	50.8
Transverse modulus /(GPa)	E₂	59.6	44.9
In-plane shear modulus /(GPa)	G₁₂	4.5	3.48
Longitudinal tensile strength /(MPa)	X_1t	1169	530.1
Longitudinal compressive strength /(MPa)	X_1c	935	549.2
Transverse tensile strength /(MPa)	X_2t	1210	401.2
Transverse compressive strength /(MPa)	X_2c	941	449.4
In-plane shear strength /(MPa)	S	140	111.5

The assembly is shown in Figure 2. The inner and outer straight cylindrical tubes were constructed through the hot press moulding technique, as shown in Figure 2(a), and the fabrication process of inner and outer tubes involves manufacturing a polycarbonate inner mould and two half-cylindrical aluminium outer moulds (secured by bolts), laying out the prepregs and curing at room temperature. The foldcore was manufactured by vacuum bag technology with female and male moulds, as shown in Figure 2(b), and the fabrication process includes machining the aluminium moulds, laying out the dry woven fabrics, securing two moulds with bolts, injecting the epoxy resin within the vacuum bag, and curing at room temperature. The assembling process involves cleaning, drilling 3 mm diameter through-holes at the contact points between the foldcore and the inner tube, polishing the 5- ${m m}^{2}$ surrounding area of each hole with a solvent, and applying adhesive for fastening. The details of the manufacturing and assembling process are included in our previous work.³⁹

Figure 2.

Schematic diagram of the mold of the straight tube and the foldcore. (a) inner and outer straight tubes mold to manufacture the straight tubes by hot press moulding technique, (b) the full-diamond pattern foldcore mold to manufacture the foldcores by vacuum bag technology, (c) the image of the full-diamond mold, (d) the image of the sandwich tube specimen with full-diamond foldcore.

The dimension of the specimen is shown in Figure 1(b) and 1(c). The radius of the inner and outer tube denotes $R_{i}$ and $R_{o}$ , respectively. The wall thickness for both the inner and outer tubes was determined as $t_{i}$ and $t_{o}$ , respectively. The height of the CFRP sandwich tube is $H$ . In this study, all CFRP sandwich tubes featured an outer radius $R_{o}$ of 42.43 mm, an inner radius $R_{i}$ of 38.41 mm, and uniform wall thicknesses for both the inner and outer tubes, each measuring $t_{i}$ = $t_{o}$ = 0.96 mm (0.24 mm per layer). The foldcore thickness $t_{f}$ was consistent at 0.8 mm (0.2 mm per layer), and the height $H$ of the tubes was 150 mm. These parameters form the design spaces for foldcores.

Numerical model

The element type for the foldcores and inner and outer tubes was SC8R, and the cohesive elements between the layers used COH3D8 elements to capture the delamination failure mode during the deformation process. The thickness of cohesive elements is 0.01 mm, with the material properties that are derived from our previous work³⁹ specified in Table 2. Note that since both studies utilised the same batch of specimens and the obtained material properties have been validated in our prior work, they are directly employed in this study. The average element mesh size for all components was 3 mm due to the convergence analysis. Tie constraints are applied to the contact points between the contact regions of three parts.

Table 2.

Material mechanical properties of cohesive elements.³⁹

Description variable	Variable	Value
Modulus in the normal direction /(MPa)	E_nn	3000
Modulus in the first shear direction /(MPa)	E_ss	1154
Modulus in the second shear direction /(MPa)	E_tt	1154
The original thickness of the cohesive element /(mm)	T₀	0.01
Peak traction stress in the normal direction /(MPa)	$t_{n}^{0}$	20
Peak traction stress in the first shear direction /(MPa)	$t_{s}^{0}$	25
Peak traction stress in the second shear direction /(MPa)	$t_{t}^{0}$	25
Critical fracture energies in the normal direction /(N/mm)	$G_{n}^{c}$	0.249
Critical fracture energies in the first shear direction /(N/mm)	$G_{s}^{c}$	0.733
Critical fracture energies in the second shear direction /(N/mm)	$G_{t}^{c}$	0.733

The FE numerical models and the boundary conditions of both loading cases are shown in Figure 3, three parts are tied together using the Tie constraint, as shown in Figure 3(a). The boundary conditions for the three-point bending test were set as follows in Figure 3(b): face-to-face contact was applied between the loading platen and the outer tube. Tie constraints are applied to the contact surfaces between the clamping ends and the bottom support and between two clamping ends and the outer tube. The bottom supports were fully constrained. Global general contact was applied for other regions. The loading platen, clamping ends, and bottom support were all defined as rigid bodies. During the three-point bending process, a downward displacement $s$ = 8 mm of the loading head is applied. Note that the value of displacement is set to derive the bending stiffness in the three-point bending test. The boundary conditions for transverse plate compression are relatively simple compared to the three-point bending test. As shown in Figure 3(c), two parallel rigid plates at a distance of the diameter ( $2 R_{o}$ ) of the outer tube are at the top and the bottom of the sandwich tube. The bottom plane was fixed, forming a tie contact with the outer tube, while the top plane was in surface-to-surface contact with the outer tube and constrained with a downward displacement of $s$ = 29.7 mm (70% value of the radius of the outer tube $R_{o}$ ).

Figure 3.

Details of FE model of the foldcore sandwich tube under the three-point bending test and transverse compression test. (a) Contact relationships in the finite element model of the full diamond configuration, each part is made by four plies of CFRP with the orientation being $0 °$ , where material 1 direction is aligned with the axis of the tube. The insets show that the numerical model of the cylindrical tubes and the foldcore are built by 4-layer shell element with a 0.01 mm thickness cohesive element layer between each of the foldcore shell layers. (b) the finite element model for the three-point bending experiment; (c) the finite element model for the transverse plate compression experiment.

Validation of FE model

Due to the complexity of the finite element boundary conditions in the three-point bending condition, only this experiment was employed to validate the finite element model. As shown in Figure 3(b), the advantage of the setup of the three-point bending experimental tests is to maintain a linear contact between the half-ring-shaped loading head and the sandwich tube, thereby reducing local stress concentration. The fitting type between the two clamping ends and the outer tube is an interference fit. The rationale behind this design is to avoid stress concentration through the surface-to-surface contact formed by two clamping ends of the fixture. The experimental tests were conducted on a MTS universal testing machine (CMT-5105, MTS, maximum load capacity: 50 kN). During the experiment, the loading rate of the head was set to 1.0 mm/min (ASTM D790 standard). The lower two support heads were fixed, while the upper head was compressing downward.

The experimental process of the three-point bending test is illustrated in Figure 4(a). As the loading head applies load, the outer tube of the sandwich structure is first subjected to compression. Notably, at s = 4 mm, a significant bulging phenomenon occurs in the area near the loading head with a slight decrease of reaction force, indicating the local buckling failure of the carbon fibre outer tube and localized matrix shear failure of the foldcore. After that, the overall structure continues to bear the load during the densification of the foldcore, with the reaction force gradually increasing. At s = 8 mm, the entire sandwich tube remains stable without catastrophic failure, although the buckling failure region has expanded and the foldcore exhibited significant shear failure at the loading head and at the bonding interfaces with the outer tube, it still effectively supports the load. The compressive failure of the sandwich tube and the force-displacement results from the experiments and FE models are depicted in Figure 4(b), which exhibit good agreement. Local buckling and crushing occur in the contact area with the heading load, and some longitudinal cracks appear in the inner tube of the sandwich structure (along the height of the sandwich tube). However, there is no global buckling. And the simulation results accurately replicate the post-bending deformation and the indentation shape. Note that there is currently a trend in experimental testing to conduct only one test to evaluate each design condition for economic purposes. Naturally, the available instrumentation and equipment enable a more accurate measurement of the dimensions, material properties, loads and displacements, decreasing the scatter in the results. Here, experiments of two specimens are carried out, and the scatter in the results is considered to be acceptable, indicating the validation of the numerical model of the sandwich structures with foldcore.

Figure 4.

(a) Comparison between the process of three-point bending experiment and finite element simulation and main damage model of foldcore (shear damage) of the corresponding state of the loading process, (b) comparison of force-displacement curves between three-point bending experiment and finite element simulation.

Two parameters are introduced to assess the transverse energy absorption performance of the composite sandwich tubes: the average specific energy absorption (SEA) and the crushing force efficiency (CFE), as expressed below.^{17,18,39,40,42}

S E A = \frac{\int_{0}^{δ_{c}} F (x) d x}{m}

(1)

where

δ_{c}

is the displacement of the loading head,

F (x)

is the corresponding load force, and

m

is the mass of the crushed section of the specimen. Note that the SEA value has taken the mass into account; as the design space is fixed, this is a normalized value. A higher SEA value indicates a better energy absorption capacity.³³

C F E = \frac{P_{m e a n}}{P_{\max}}

(2)

where

P_{m e a n}

is the mean crushing force, and

P_{\max}

is the peak loading force. A higher value of the CFE indicates a steady process along the deformation process.

Design process of the geometric foldcore configurations

The parametric study investigates the effects of foldcore configurations and main parameters on the energy absorption performance using the validated numerical model. In this section, the design process of 8 different origami-inspired foldcores is introduced. The selected foldcore configurations are: straight/curved-creased axial Miura (SAM/CAM), straight/curved-creased circumferential Miura (SCM/CCM), straight/curved-creased diamond (SD/CD), straight/curved-creased Kresling (SK/CK). As shown in Figure 5(b), the foldcore tubes are designed by assembling unit cells into a basic module (a ring-shape layer) and translating the basic module along the centre axis to build the whole model, all the parameters of the foldcore tubes below are described in the global cylindrical coordinate system, with the original point $O$ being the centre of the bottom plane and the $z$ -axis is the central axis of the tube, $r$ and $θ$ denote the radius and the angular position. Note that hereinafter, a layer of circumferentially arrayed unit cells is called a “module”, and all the foldcores have the same structural thickness defined as $R_{o} - R_{i}$ and height $H$ .

Figure 5.

The design process of the straight/curved-creased axial Miura (SAM/CAM) pattern-based foldcore. (a) the SAM foldcore, which consists of $M$ modules (layers), with an overall height of $H$ , (b) a basic module of SAM, the number of the unit cell in the circumferential direction is $N$ , (c) a unit cell of SAM, with the height of $h$ , the green and purple (yellow) nodes are on the outer (inner) tube, the unit cell is mirror symmetric about the plane which is in blue colour; the nodes could be translated in the central axis direction are in the same colours; the angular displacement increment of the lines $l$ and $k$ denotes $α .$ (d) a unit cell of CAM, which substitutes the originally straight lines in SAM to splines (shown in red colour). (e) The CAM foldcore.

Straight/curve-creased axial miura foldcore

Figure 5(a) is a schematic diagram of the straight-creased axial Miura (SAM) foldcore (the inner and outer tubes are not shown), which is made by stacking the SAM basic module (Figure 5(b)) along the tube axis. Eight parameters in total could define the unit cell and the whole SAM foldcore: $M, N, H, R_{i}, R_{o}, h, α$ and $φ$ , where $M$ , $N$ and $H$ denote the number of layers of modules of the foldcore, the number of circumferential unit cells in one basic module and the total height of the foldcore, respectively; $h, α$ and $φ$ are the height of the unit cell, the circumferential angular displacement across by a single unit cell and the in-plane twist angle, respectively. Hence, the geometric relationship between the above five parameters is expressed in equations (3) and (4), where the twist angle $φ$ is an independent design parameter. Figure 5(c) shows the unit cell of the SAM foldcore, which is composed of four parallelograms of equal size, and it is geometrically symmetrical along the mirror plane (shown as the “symmetric plane” in blue colour), so designing the geometry of the two parallelograms at the bottom determines the SAM unit cell. The design process of the unit cell is illustrated in Figure 5(c), nodes whose positions could be derived by translating along the axial direction of the tube are shown in the same colours. There are three points (green, yellow, and purple) on the bottom plane, the green point and the purple point are located on the outer tube wall, and the yellow point is located on the inner tube wall, the central angle in the bottom plane across the two sides denotes $α$ , which is determined by the number of circumferential units, $N$ ( $N$ is an integer), and the length of the line $l$ on the bottom surface is equal to the length of the line $k$ . By translating a distance of $h / 2$ along $z$ axis with a twist angle $φ$ of these three nodes at the bottom, three corresponding points (in black colour) at the symmetric plane are derived, and by connecting the corresponding nodes and conducting a mirror symmetry, a basic unit cell of the SAM is designed. To ensure the geometric constraint, the equation (4) of $α$ and $N$ holds. As shown in Figure 5(d), the construction of the curved axial Miura is based on the straight axial Miura, with the corresponding straight lines (coloured in red) replaced with the spline. Note that the spline is drawn in a way that the tangent of the spline at three vertices is parallel to the central axis of the tube ( $z$ -axis). After obtaining the CAM unit cell, the design process of the foldcore tube is identical to that of the straight one, as shown in Figure 5(e).

h = \frac{H}{M}

(3)

α = \frac{360 °}{N}

(4)

Straight/curve-creased circumferential miura foldcore

As shown in Figure 6, the circumferential Miura and the axial Miura are similar in geometry but different in orientation. Figure 6(a)and (b) respectively show a complete straight-creased circumferential Miura (SCM) foldcore (inner and outer tubes are not shown) and a basic module, $M$ , $N$ represent the number of basic modules contained in the circumferential Miura foldcore and the number of unit cells contained in a basic module, respectively. The SCM unit cell, as shown in Figure 6(c), is a mirror-symmetrical configuration with a symmetric plane in blue colour, note that all mirror symmetric nodes are in the same colours. Determining the left half of the circumferential Miura is enough to determine a complete SCM unit cell. The central angle occupied by an SCM unit cell is $α$ which is determined by equation (5).

α = \frac{360 °}{N}

(5)

(M + \frac{1}{2}) h = H

(6)

Figure 6.

The design process of the straight/curved-creased circumferential Miura (SCM/CCM) pattern-based foldcore. (a) SCM foldcore, which consists of $M$ modules (layers), with an overall height of $H$ , (b) a basic module of SCM, the number of the unit cell in the circumferential direction is $N$ , (c) a unit cell of SCM, with the height of $h$ , the green and purple (yellow) nodes are on the inner (outer) tube, the unit cell is mirror symmetric about the plane which is in blue colour; the nodes could be translated in the central axis direction are in the same colours; the angular displacement increment of the unit cell denotes $α .$ (d) a unit cell of CCM, which substitutes the originally straight lines in SCM to splines (shown in red colour). (e) The CCM foldcore.

According to the angle $α$ , the vertices of the circumferential Miura unit cell can be determined on the inner and outer tube walls, and the purple and green (yellow and black) points are on the inner (outer) tube wall. The axial distance between two purple, green and yellow points are $h$ , $h$ and $h / 2$ , respectively. It is worth noting that the distance between the top nodes and the bottom nodes of the circumferential Miura foldcore along the tube axis is $M \times h + (1 / 2) \times h$ . Therefore, the relationship between $h$ and $M$ is different from that of the axial Miura, as expressed in equation (6). The curved-creased circumferential Miura (CCM) unit cell is hence designed by replacing the straight creases with splines (as shown in red lines in Figure 6(d)), and the tangents of each spline at the three vertices are parallel to the central axis, the configuration of the CCM foldcore is illustrated in Figure 6(e).

Straight/curve-creased diamond foldcore

The straight-creased Diamond (SD) foldcore is composed of identical trapezoids. As shown in Figure 7(a), $α, β, φ$ and $h$ determine the unit cell of the SD, where $α$ and $β$ denote the circumferential angular displacement increment across by long and short edge, respectively. $φ$ and $h$ denote the in-plane twist angle and the height of the unit cell. For $N$ circumferential units, the characteristic of the diamond shape determines that the twist angle $φ$ is not independent of $α$ and $β$ , as shown in equations (7)–(10).

R_{o} \cos \frac{α}{2} = R_{i}

(7)

α + β = \frac{360 °}{N}

(8)

h = \frac{H}{M}

(9)

φ = \frac{α + β}{2}

(10)

Figure 7.

The design process of the straight/curved-creased diamond (SD/CD) pattern-based foldcore. The angle across the circumferential direction of the short edge and the long edge denotes $β$ and $α$ , respectively. (a) The geometric relationship between the CD/SD foldcore and the inner and outer tubes, (b) half of a basic module of SD, (c) a basic module of SD, (d) the SD foldcore, (e) a basic module of CD, (f) the CD foldcore.

The geometric design process is as follows. As shown in Figure 7(c), the top and the bottom outlines of a module are polygons of $2 N$ edges (as $N$ = 6 in Figure 7, so it is a 12-sided polygon), the vertices of edges are located on the outer tube wall, the inner tube is tangent to the midpoint of each long edge. Hence, the outer tube is the circumscribed circle of the polygon, and the inner tube is the inscribed circle of the polygon. The polygon is translated for a certain distance of $h / 2$ along the central axis of the tube and rotated for a certain twist angle $φ$ , $φ$ and $h$ are determined by equations (9), and (10). As shown in Figure 7(b). The diamond basic module is also a mirror-symmetrical configuration (the symmetric plane is shown in blue colour in Figure 7(c)), and the complete diamond module is obtained. The curved-creased diamond (CD) is derived by replacing the straight creases in the SD with a spline curve, where the tangents of spline lines at the three vertices (yellow, purple, and green) are parallel to the tube centre axis, as shown in Figure 7(d). By stacking the basic modules of SD or CD along the tube axis, the SD foldcore or CD foldcore could be obtained, as shown in Figures 7(e) and 7(f), with the total height of foldcore $H$ and the number of the basic modules $M$ .

Straight/curved-creased kresling foldcore

The straight-creased Kresling (SK) foldcore exhibits similarities to the diamond foldcore in its design process, involving translation, rotation, and connection of a polygon along the central axis of the tube because of the geometric constraint. As shown in Figure 8(a), the outer tube corresponds to the circumscribed circle of the polygon. $N$ and $α$ denote the number of sides and the central angle occupied by each side of the Kresling polygon, respectively (i.e., $N$ = 8 in Figure 8). Upon translating the polygon for a certain distance $h / 2$ along the central axis of the tube and rotating it for a twist angle $φ$ , the corresponding points are connected (green-green, purple-purple, green-purple). The inner tube is tangent to the midpoint of line $l$ . The relationships among $α$ , $N$ , $φ$ , the inner tube radius $R_{i}$ , and the outer tube radius $R_{o}$ are given by equations (11), and (12), and $h$ is determined by equation (13). As shown in Figure 8(b), the curved-creased Kresling (CK) can be obtained by projecting line $l$ of SK onto the outer tube wall. As a result, the inner tube is no longer tangent to the midpoint of line $l$ but instead intersects the midpoint of each side of the Kresling polygon, the cross section of the inner tube is now equivalent to the inscribed circle of the polygon. $φ$ becomes an independent design parameter, and the relationship between $R_{o}$ , $R_{i}$ , and $α$ is expressed by equation (14). By repeating the above process, corresponding SK and CK foldcores can be obtained. To facilitate parametric studies, the configurations obtained from repeating the process twice are considered as a Kresling basic module, as shown in Figure 8(c)–(d), the Kresling foldcore contains $M$ (i.e., $M$ = 3 in the figure) Kresling basic modules, and the total height of Kresling foldcore is denoted by $H$ .

α = \frac{360 °}{N}

(11)

R_{o} \cos \frac{α + φ}{2} = R_{i}

(12)

h = \frac{H}{M}

(13)

R_{o} \cos \frac{α}{2} = R_{i}

(14)

Figure 8.

The design process of the Kresling pattern-based foldcore. (a) The geometric relationship between the Straight-creased Kresling (SK) foldcore and the inner and outer tubes. The inner tube is inscribed at the middle point of the diagonal line at the height of $h / 4$ , whereas the nodes are all on the outer tube. The twist angle denotes $φ$ , which is the angle across the circumferential direction of green nodes on the outer tube wall, the angle across the circumferential direction of the unit cell denotes $α$ , which is determined by the number of the unit cell $N$ in the module. (b) Half of a basic module of curved-creased Kresling (CK), the inner tube is now inscribed at the middle of the point of the horizontal line. (c) SK foldcore, (d) CK foldcore.

The results of the parameter study

In this section, a parametric study is carried out to investigate the effects of foldcores patterns and geometric parameters on the transverse mechanical properties of composite sandwich tubes under three-point bending and transverse compression conditions. Results of straight-creased axial Miura (SAM), curved-creased axial Miura (CAM), straight-creased circumferential Miura (SCM), curved-creased circumferential Miura (CCM), straight-creased diamond (SD), curved-creased diamond (CD), straight-creased Kresling (SK), curved-creased Kresling (CK) are shown. Representative schematic diagrams of each configuration are illustrated in Figure 9. All the results of SEA and CFE values with geometric parameters of unit cells are shown in Tables 3–6 for three-point bending tests and Tables 7–10 for transverse compression tests.

Figure 9.

Representative schematic diagrams of eight distinct configurations employed for parametric study. The “zigzag” line is defined above, the number of N and M influences the zigzag line and further affects the energy absorption capacity of the sandwich tube structures, which will be detailed in the later section.

Table 3.

The results of Curved/Straight Diamond (CD/SD) in three-point bending test.

Model (M)	N	M	Mass/(g)	P_max/(kN)	P_mean/(kN)	CFE	SEA/(J/g)
CD-N73	7	3	186	15.88	9.08	0.572	0.485
CD-N63	6	3	186	15.39	9.30	0.604	0.488
CD-N53	5	3	185	15.45	8.50	0.550	0.456
CD-N74	7	4	186	16.02	10.09	0.630	0.499
CD-N75	7	5	186	16.19	10.37	0.640	0.519
SD-N73	7	3	185	14.77	8.96	0.607	0.474
SD-N63	6	3	186	13.02	8.03	0.617	0.420
SD-N53	5	3	185	15.64	8.59	0.549	0.457
SD-N74	7	4	186	15.36	9.54	0.621	0.488
SD-N75	7	5	186	15.97	9.01	0.564	0.490

Table 4.

The results of Curved/Straight Axial Miura (CAM/SAM) in three points bending test.

Model	N	M	φ	Mass/(g)	P_max/(kN)	P_mean/(kN)	CFE	SEA/(J/g)
CAM-N24M8φ5	24	8	5	183	20.14	11.08	0.550	0.637
CAM-N24M8φ8	24	8	8	185	24.14	11.03	0.457	0.613
CAM-N24M8φ10	24	8	10	186	22.25	10.54	0.474	0.589
CAM-N24M7φ10	24	7	10	185	22.19	9.53	0.429	0.537
CAM-N24M6φ10	24	6	10	184	22.85	9.76	0.427	0.554
CAM-N20M6φ10	20	6	10	181	19.25	9.69	0.504	0.554
CAM-N16M6φ10	16	6	10	180	17.77	10.37	0.584	0.574
SAM-N24M8φ5	24	8	5	183	19.79	11.51	0.582	0.637
SAM-N24M8φ8	24	8	8	184	20.01	11.57	0.578	0.641
SAM-N24M8φ10	24	8	10	186	19.74	10.218	0.518	0.576
SAM-N24M8φ15	24	8	15	191	20.89	10.06	0.481	0.540
SAM-N24M8φ20	24	8	20	196	19.87	9.66	0.486	0.486
SAM-N24M7φ10	24	7	10	185	20.79	9.84	0.473	0.562
SAM-N24M6φ10	24	6	10	184	20.17	12.16	0.603	0.663
SAM-N20M6φ10	20	6	10	181	18.43	10.85	0.589	0.611
SAM-N16M6φ10	16	6	10	179	18.35	10.01	0.546	0.573

Table 5.

The results of Curved/Straight Circumferential Miura (CCM/SCM) in three-point bending test.

Model (M)	N	M	Mass/(g)	P_max/(kN)	P_mean/(kN)	CFE	SEA/(J/g)
CCM-N248	24	8	182	21.61	10.70	0.495	0.598
CCM-N247	24	7	181	23.22	11.00	0.474	0.650
CCM-N246	24	6	180	23.54	11.07	0.470	0.621
CCM-N206	20	6	178	20.82	10.77	0.517	0.607
CCM-N166	16	6	175	19.59	9.60	0.490	0.573
SCM-N248	24	8	181	19.42	10.41	0.536	0.577
SCM-N247	24	7	180	20.43	10.77	0.527	0.632
SCM-N246	24	6	178	22.00	10.94	0.493	0.610
SCM-N206	20	6	176	19.97	10.52	0.527	0.593
SCM-N166	16	6	175	18.76	9.80	0.522	0.566

Table 6.

The results of Curved/Straight Kresling (CK/SK) in three-point bending test.

Model	N	M	φ	Mass/(g)	P_max/(kN)	P_mean/(kN)	CFE	SEA/(J/g)
CK-M3φ5	/	3	5	176	15.19	8.95	0.589	0.492
CK-M3φ10	/	3	10	176	15.05	8.71	0.579	0.488
CK-M3φ15	/	3	15	177	14.07	8.45	0.601	0.470
CK-M4φ10	/	4	10	177	15.10	8.70	0.575	0.480
CK-M5φ10	/	5	10	179	16.00	9.61	0.579	0.502
SK-N10M3	10	3	/	175	13.61	8.45	0.621	0.465
SK-N11M3	11	3	/	176	13.99	8.43	0.603	0.462
SK-N12M3	12	3	/	176	13.86	8.80	0.635	0.482
SK-N12M4	12	4	/	177	14.41	8.68	0.602	0.486
SK-N12M5	12	5	/	177	14.73	9.35	0.635	0.510

Table 7.

The results of Curved/Straight Diamond (CD/SD) in plate compression test.

Model	N	M	Mass/(g)	P_max/(kN)	P_mean/(kN)	CFE	SEA/(J/g)
CD-N7_M3	7	3	186	4.49	2.68	0.597	0.487
CD-N6_M3	6	3	186	5.43	3.26	0.600	0.603
CD-N5_M3	5	3	185	4.07	2.44	0.600	0.439
CD-N7_M4	7	4	186	5.18	3.53	0.681	0.662
CD-N6_M4	6	4	176	7.26	3.90	0.537	0.804
CD-N5_M4	5	4	175	8.00	3.73	0.466	0.783
CD-N7_M5	7	5	186	6.75	3.95	0.586	0.749
CD-N6_M5	6	5	176	8.17	4.03	0.494	0.793
CD-N5_M5	5	5	175	8.07	3.86	0.478	0.797
SD-N7_M3	7	3	185	4.44	3.05	0.686	0.556
SD-N6_M3	6	3	186	4.45	2.63	0.592	0.482
SD-N5_M3	5	3	185	3.51	2.32	0.661	0.407
SD-N7_M4	7	4	186	4.84	3.23	0.668	0.607
SD-N6_M4	6	4	175	4.74	3.22	0.679	0.618
SD-N5_M4	5	4	175	6.76	3.38	0.500	0.670
SD-N7_M5	7	5	186	6.31	3.18	0.504	0.601
SD-N6_M5	6	5	176	5.55	3.65	0.657	0.698
SD-N5_M5	5	5	175	6.23	3.48	0.558	0.700
SD-N7_M8	7	8	178	6.14	3.94	0.641	0.758

Table 8.

The results of Curved/Straight Axial Miura (CAM/SAM) in plate compression test.

Model	N	M	φ	Mass/(g)	P_max/(KN)	P_mean/(KN)	CFE	SEA/(J/g)
CAM-N16M6φ10	16	6	10	180	11.88	5.45	0.459	1.061
CAM-N20M6φ10	20	6	10	181	17.37	5.77	0.332	1.105
CAM-N24M6φ10	24	6	10	184	19.52	6.32	0.324	1.223
CAM-N16M7φ10	16	7	10	179	17.22	5.95	0.345	1.161
CAM-N20M7φ10	20	7	10	182	18.38	5.80	0.315	1.077
CAM-N24M7φ10	24	7	10	185	21.11	6.78	0.321	1.256
CAM-N16M8φ10	16	8	10	180	12.72	5.29	0.416	1.062
CAM-N20M8φ10	20	8	10	183	19.36	6.52	0.337	1.262
CAM-N24M8φ10	24	8	10	186	21.51	6.97	0.324	1.318
CAM-N24M8φ8	24	8	8	185	22.67	6.54	0.288	1.198
CAM-N24M8φ5	24	8	5	183	17.39	5.60	0.322	1.023
SAM-N16M6φ10	16	6	10	179	13.49	4.73	0.351	0.934
SAM-N20M6φ10	20	6	10	181	19.93	6.01	0.302	1.149
SAM-N24M6φ10	24	6	10	184	18.99	5.91	0.311	1.101
SAM-N16M7φ10	16	7	10	179	13.49	4.82	0.357	0.959
SAM-N20M7φ10	20	7	10	182	17.38	5.67	0.326	1.085
SAM-N24M7φ10	24	7	10	185	19.58	6.34	0.324	1.165
SAM-N16M8φ10	16	8	10	179	14.50	4.90	0.338	0.968
SAM-N20M8φ10	20	8	10	182	17.22	5.95	0.345	1.142
SAM-N24M8φ10	24	8	10	186	20.68	5.91	0.286	1.123
SAM-N24M8φ5	24	8	5	183	16.63	5.82	0.350	1.045
SAM-N24M8φ8	24	8	8	184	19.48	6.09	0.312	1.129
SAM-N24M8φ15	24	8	15	191	25.39	8.76	0.345	1.643
SAM-N24M8φ20	24	8	20	196	28.97	9.79	0.338	1.780

Table 9.

The results of Curved/Straight Circumferential Miura (CCM/SCM) in plate compression test.

Model (M)	N	M	Mass/(g)	P_max/(KN)	P_mean/(KN)	CFE	SEA/(J/g)
CCM-N248	24	8	182	13.08	5.15	0.394	1.021
CCM-N247	24	7	181	11.35	5.16	0.455	1.023
CCM-N246	24	6	180	10.38	4.69	0.452	0.979
CCM-N208	20	8	180	11.74	5.40	0.460	1.075
CCM-N207	20	7	179	11.25	5.31	0.472	1.073
CCM-N206	20	6	178	9.45	4.69	0.496	0.953
CCM-N168	16	8	178	10.52	5.61	0.533	1.115
CCM-N167	16	7	177	10.48	5.21	0.497	1.058
CCM-N166	16	6	175	9.97	4.88	0.489	1.022
SCM-N248	24	8	181	13.58	5.64	0.456	1.148
SCM-N247	24	7	180	12.26	4.92	0.402	1.011
SCM-N246	24	6	178	10.02	4.56	0.455	0.957
SCM-N208	20	8	179	11.58	5.58	0.482	1.093
SCM-N207	20	7	177	9.94	4.54	0.456	0.966
SCM-N206	20	6	176	10.09	4.81	0.477	1.034
SCM-N168	16	8	177	11.48	5.11	0.445	0.992
SCM-N167	16	7	176	9.60	5.19	0.541	1.048
SCM-N166	16	6	175	9.75	4.73	0.485	1.007
SCM-N78	7	8	174	9.58	5.52	0.576	1.106

Table 10.

The results of Curved/Straight Kresling (CK/SK) in plate compression test.

Model	N	M	φ	Mass/(g)	P_max/(KN)	P_mean/(KN)	CFE	SEA/(J/g)
CK-M3φ5	/	3	5	176	5.86	3.32	0.566	0.668
CK-M3φ10	/	3	10	176	5.92	3.41	0.576	0.682
CK-M3φ15	/	3	15	177	6.05	3.46	0.572	0.693
CK-M4φ5		4	5	177	7.78	3.51	0.451	0.704
CK-M4φ10	/	4	10	177	7.10	3.71	0.523	0.779
CK-M4φ15	/	4	15	178	7.87	4.58	0.582	0.913
CK-M5φ5	/	5	5	178	9.51	4.92	0.518	0.997
CK-M5φ10	/	5	10	179	8.18	4.49	0.549	0.913
CK-M5φ15	/	5	15	180	9.84	5.37	0.545	1.055
SK-N10M3	10	3	/	175	3.41	2.30	0.673	0.435
SK-N10M4	10	4	/	175	5.61	3.53	0.628	0.706
SK-N10M5	10	5	/	176	5.85	3.85	0.658	0.751
SK-N11M3	11	3	/	176	3.91	2.31	0.592	0.457
SK-N11M4	11	4	/	176	5.66	3.76	0.663	0.724
SK-N11M5	11	5	/	177	6.33	3.92	0.620	0.777
SK-N12M3	12	3	/	176	3.75	2.36	0.629	0.479
SK-N12M4	12	4	/	177	4.47	2.55	0.571	0.543
SK-N12M5	12	5	/	177	5.22	2.72	0.520	0.604

The results of the parameter study of foldcores under three-point bending

Figure 10(a)–(d) present the simulation results of the force-displacement curves for ten different $N$ , $M$ combinations of sandwich tubes with straight diamond (SD) and curved diamond (CD) foldcores under three-point bending condition. The variations of the specific energy absorption (SEA) and the crushing force efficiency (CFE) are plotted in Figures 10(e) and (f). With the increase of $M$ , the energy absorption performance of both SD and CD sandwich tubes improves, when $M$ increases from three to four and 5, the SEA of SD sandwich tubes increases by 5.27% and 9.49%, and the CFE increases by 2.95% and 3.38%, respectively; for CD sandwich tubes, the SEA increases by 2.89% and 7.01%, and the CFE increases by 10.14% and 11.89% respectively. Overall, curved diamond exhibits higher specific energy absorption (SEA) capacity compared to straight diamond. However, SD demonstrates less fluctuation in the reactionary force, leading to a more stable bending process compared to the result of CD, this indicates a higher CFE value for SD models.

Figure 10.

The three-point bending results of the parametric study of the sandwich tube structure with straight/curved-creased diamond foldcore. Force displacement curves for (a) models N7M5, N7M4, and N7M3, and (b) models N7M3, N6M3, and N5M3 of the straight-creased diamond foldcore sandwich tube, with $N$ and $M$ denoting the number of circumferential units and the number of layers of modulus, respectively. (c) and (d) are force-displacement curves for the corresponding curved-creased diamond-pattern counterparts that are shown in (a) and (b), respectively. (e) and (f) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) of both straight/curved-creased diamond foldcore models that have been shown in (a-d), with the inset demonstrating the deformed configurations of foldcore.

Next, the force-displacement curves of nine models with straight-creased axial Miura (SAM) foldcore and seven models of curved axial Miura (CAM) foldcore are presented in Figure 11(a)–(f), respectively. As shown in Figure 11(g), an increase in the twist angle $φ$ resulted in a significant decline in both SEA and CFE for both SAM and CAM foldcore tubes. As $φ$ increased from $5 °$ to $20 °$ , the SEA and CFE values of SAM foldcore sandwich tubes decreased by 23.7% and 16.49%, respectively. Similarly, as $φ$ increased from $5 °$ to $10 °$ , the SEA and CFE of CAM foldcore sandwich tubes decreased by 7.54% and 13.82%, respectively. As illustrated in Figure 11(h), the energy absorption performance of SAM decreases with the increase of the number of modules $M$ . When $M$ changes from six to 7, both the specific energy absorption (SEA) and crushing force efficiency (CFE) of SAM decrease by approximately 15%. However, $M$ has little impact on the SEA value of the CAM foldcore and can even cause an increase in its CFE value. When $M$ increases from six to 8, the CFE of the CAM foldcore model rises by 11.01%. Furthermore, as depicted in Figure 11(i), with $N$ increasing from 16 to 20 and 24, the SEA of SAM foldcore sandwich tubes increased by 6.63% and 15.71%, while the CFE value increased by 7.88% and 10.44%. However, CAM foldcore sandwich tubes exhibited an opposite trend, with CFE value rapidly decreasing by 13.70% and 26.88%, and a slight decrease in SEA value. This indicates that increasing $N$ can enhance the energy absorption performance of SAM foldcore.

Figure 11.

The three-point bending results of the parametric study of the sandwich tube structure with straight/curved-creased axial Miura foldcore. Force displacement curves for (a) models N24M8φ5, N24M8φ8, N24M8φ10, N24M8φ15, and N24M8φ20, and (b) models N24M8φ10, N24M7φ10, and N24M6φ10, and (c) models N24M6φ10, N20M6φ10, and N16M6φ10 of the Straight-creased axial Miura foldcore sandwich tube, with $N$ , $M$ and $φ$ denoting the number of circumferential units, the number of layers of modulus, and in-plane twist angle respectively. (d) models N24M8φ5, N24M8φ8, N24M8φ10 of the curved-creased axial Miura foldcore sandwich tube, (e) and (f) are force-displacement curves for the corresponding curved-creased axial Miura-pattern counterparts that are shown in (b) and (c), respectively. (g), (h), and (i) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) of both straight/curved-creased axial Miura foldcore models that have been shown in (a-f), with the inset demonstrating the deformed configurations of foldcore.

The force-displacement curves of five different parameter configurations for straight/curved-creased circumferential Miura (SCM/CCM) foldcore tubes are plotted in Figure 12. From Figures 12(a) and 12(c), it can be observed that the variation in $M$ affects the location of the peak force for both SCM and CCM configurations. As shown in Figures 12(b) and 12(d), the change in $N$ influences the magnitude of the peak force for SCM and CCM configurations. From Figure 12(e), the SEA of both SCM and CCM configurations increases with increasing $M$ . For SCM configuration, the CFE increases by 6.90% and 8.72% as $M$ changes from six to seven and 8, respectively. From Figure 12(f), as $N$ increases, both SCM and CCM configurations show an overall upward trend. When $N$ changes from 16 to 20 and 24, the SEA of SCM (CCM) increases by 4.77% and 7.77% (5.93% and 8.38%), respectively. However, compared to $N$ = 16, the CFE of SCM and CCM foldcores with $N$ = 24 decreased by 5.56% and 4.08%, respectively. This indicates that increasing the number of circumferential unit cells $N$ benefits the energy absorption capacity of SCM and CCM foldcores; however, too many unit cells may lead to an increase in the fluctuation of the reactive force. Moreover, for the circumferential Miura pattern, the three-point bending performance is more sensitive to the parameter $M$ than to $N$ . The results of SCM/CCM foldcores under three-point bending are shown in Table 5.

Figure 12.

The three-point bending results of the parametric study of the sandwich tube structure with straight/curved-creased circumferential Miura foldcore. Force displacement curves for (a) models N24M8, N24M7, and N24M6, and (b) models N24M6, N20M6, and N16M6 of the Straight-creased circumferential Miura foldcore sandwich tube, with $N$ and $M$ denoting the number of circumferential units and the number of layers of modulus, respectively. (c) and (d) are force-displacement curves for the corresponding curved-creased circumferential Miura-pattern counterparts that are shown in (a) and (b), respectively. (e) and (f) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) of both straight/curved-creased circumferential Miura foldcore models that have been shown in (a)-(d), with the inset demonstrating the deformed configurations of foldcore.

Figure 13 presents the force-displacement curves of straight/curved-creased Kresling foldcore sandwich tubes with different parameters. As shown in Figure 13(b), the peak force of SK increases with the increase of $M$ . The peak force grows from 13.86 kN for $M$ = 3 to 14.41 kN for $M$ = 4, and 14.73 kN for $M$ = 5. Similarly, Figure 13(d) demonstrates that CK also exhibits the same behaviour, with the peak force increasing from 14.07 kN for $M$ = 3 to 15.10 kN for $M$ = 4, and 16.00 kN for $M$ = 5. Furthermore, as depicted in Figure 13(e), the increase in $M$ results in the increase of SEA for SK. As $M$ increases from three to four and 5, the SEA increases by 0.83% and 5.81%. The relationship depicted in Figure 13(g) indicates that the increase in the in-plane twist angle $φ$ leads to a decrease in the CFE of CK. As $φ$ increases from 5 $°$ to 10 $°$ and 15 $°$ , the SEA of CK decreases by 0.81% and 4.47%, respectively. Similar to the results of SAM, this suggests that the rotation angle $φ$ of this type of foldcore configuration should not be too large and should be controlled around $5 °$ .

Figure 13.

The three-point bending results of the parametric study of the sandwich tube structure with straight/curved-creased Kresling foldcore. Force displacement curves for (a) models N10M3, N11M3, and N12M3, and (b) models N12M3, N12M4, and N12M5 of the Straight-creased Kresling foldcore sandwich tube, with $N$ and $M$ denoting the number of circumferential units and the number of layers of modulus, respectively. (c) models M3φ10, M4φ10, and M5φ10, and (d) models M3φ5, M3φ10, and M3φ15 of the curved-creased Kresling foldcore sandwich tube, with $φ$ denoting the in-plane twist angle. (e), (f), and (g) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) of both straight/curved-creased Kresling foldcore models that have been shown in (a-d), with the inset demonstrating the deformed configurations of foldcore.

For the three-point bending tests, the bending stiffness and the energy absorption capacity (SEA) of the Miura patterns are higher than the diamond and Kresling counterparts. One main reason might be the fact that the contact region between the Miura foldcore and tubes is formed by zigzag lines, whereas the contact region between the diamond/Kresling foldcore and tubes is mostly formed by straight lines. However, the crushing force efficiency (CFE) value of diamond and Kresling foldcores is generally higher than that of circumferential and axial Miura foldcores, which indicates a steadier and less fluctuation of the force during the loading process.

The damage modes of the sandwich tubular structures with diamond, circumferential Miura, axial Miura, and Kresling foldcores under three-point bending conditions exhibit similarities, are shown in Appendix A.1-A.4. Structurally, none of the foldcores experienced global buckling failure. Instead, localized areas in contact with the indenter displayed a certain degree of tension/ compression matrix damage and shear damage. The main energy absorption mechanisms under the condition of compression for each foldcore material will be discussed in detail in Section 4.2.

The results of the parameter study of foldcores under plate compression

In this section, a parametric study of the aforementioned models under plate compression is conducted, and all results are presented in Tables 7–10. Note that the energy absorption capacity is the main investigated index under the transverse compression condition. Figure 14(a) to 14(d) illustrate the simulation results of force-displacement curves for eleven different combinations of $N$ and $M$ in sandwich tubes employing straight-creased diamond (SD) and curved-creased diamond (CD) foldcores subjected to plate compression. As shown in Figure 14(e), with an increase in $M$ , both CD and SD exhibit an upward trend in specific energy absorption (SEA) under plate compression, similar to the behaviour observed in the three-point bending tests. With $M$ increasing from three to 5, the average of SD’s SEA increases by 18.47%, while CD, influenced more significantly by $M$ , demonstrates average of SEA increases of 27% when $M$ increases from three to 5. with $N$ increases, SD also displays improved energy absorption performance. With $N$ increasing from five to six and 7, the SEA value of the SD foldcore increases by 18.43% and 36.61%, respectively. Furthermore, CD demonstrated superior lateral energy absorption potential compared to SD, with an overall SEA of CD being 15.24% higher than that of SD.

Figure 14.

The plate compression results of the parametric study of the sandwich tube structure with straight/curved-creased diamond foldcore. Force displacement curves for (a) models N7M5, N7M4, and N7M3, and (b) models N7M3, N6M3, and N5M3 of the Straight-creased diamond foldcore sandwich tube, with $N$ and $M$ denoting the number of circumferential units and the number of layers of modulus, respectively. (c) and (d) are force-displacement curves for the corresponding curved-creased diamond-pattern counterparts that are shown in (a) and (b), respectively. (e) and (f) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) variations with respect to $M$ of both straight/curved-creased diamond foldcore models for additional configurations beyond those presented in (a)-(d), which illustrate only partial foldcore models.

Appendix A.5.-A.6 present the damage modes of the SD and CD foldcores under plate compression conditions, respectively. The structural failure mode is characterized by global buckling, with the compressed cross-sectional shapes exhibiting an elliptical form. This indicates that the sandwich tubes with diamond foldcore do not undergo shear deformation in the xy-plane. For the CD foldcore, the primary material damage occurs at the intersection points where it connects with the outer tube and in the collapse regions where the outer tube deforms plastically and contacts the core, leading to matrix damage in tension/compression, and shear damage. In contrast, the SD core exhibits a smaller and less severe damage area, but lower energy absorption, indicating that the curves effectively transfer the load, allowing for more efficient utilization of the foldcore material and enhancing the energy absorption capacity of the structure.

Figure 15(a)–(f) present the force-displacement curves of straight-creased axial Miura (SAM) and curved-creased axial Miura (CAM) under different parameters. It can be observed that the AM configuration generally exhibits higher peak forces, with force-displacement curves typically showing two higher peaks, where the second peak is slightly lower than the first one, followed by a gradual reduction and plateauing of the reactive forces. Figure 15(g) reveals that increasing the in-plane twist angle $φ$ leads to an upward trend in the specific energy absorption (SEA) for both SAM and CAM. As $φ$ increases from 5 to 20, the specific energy absorption of the SAM model increases by 70.33%, as $φ$ increases from 5 to 10, the specific energy absorption of the CAM model increases by 28.84%. This behaviour differs significantly from their performance in the three-point bending tests, where an increase in $φ$ results in decreased energy absorption per unit mass, indicating that larger $φ$ makes the AM sandwich tubes more resistant to compression but less resistant to bending. Furthermore, it can be observed that increasing $φ$ has a relatively small effect on crushing force efficiency (CFE). As shown in Figures 15(g) and (h), within a certain range, increasing $N$ leads to higher SEA for both SAM and CAM foldcores but reduces their CFE.

Figure 15.

The plate compression results of the parametric study of the sandwich tube structure with straight/curved-creased axial Miura foldcore. Force displacement curves for (a) models N24M8φ5, N24M8φ8, N24M8φ10, N24M8φ15, and N24M8φ20, and (b) models N24M8φ10, N24M7φ10, and N24M6φ10, and (c) models N24M6φ10, N20M6φ10, and N16M6φ10 of the Straight-creased axial Miura foldcore sandwich tube, with $N$ , $M$ and $φ$ denoting the number of circumferential units, the number of layers of modulus, and in-plane twist angle respectively. (d) models N24M8φ5, N24M8φ8, N24M8φ10 of the curved-creased axial Miura foldcore sandwich tube, (e) and (f) are force-displacement curves for the corresponding curved-creased axial Miura-pattern counterparts that are shown in (b) and (c), respectively. (g) and (h) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) variations with respect to $M$ of both straight/curved-creased axial Miura foldcore models for additional configurations beyond those presented in (a)-(d), which illustrate only partial foldcore models.

The damage modes of the SAM and CAM foldcores are detailed in Appendix A.7.-A.8. Structurally, the failure modes are characterized by global buckling accompanied by partial delamination. Due to their geometric characteristics, shear deformation occurs in the cross-section during compression. The primary damage modes of the foldcore material include matrix damage in tension/compression and shear damage. The main areas of material damage are the zigzag regions in contact with the outer tube (along the axial direction of the tube), while there is almost no damage along the zigzag line direction (circumferential direction of the tube). This is because the presence of the zigzag lines helps to distribute the load transferred from the outer tube across the entire core, significantly enhancing the energy absorption efficiency of the material and structure.

Figure 16(a)–(d) reveal the force-displacement curves under transverse compression of CM foldcore, which share a similar pattern with AM foldcore but they generally exhibit significantly lower peak forces. As seen in Figures 16(a) and 16(c), increasing $M$ leads to an increase in the first peak (usually the largest peak), which is generally undesirable in energy absorption device structures. However, Figure 16(e) indicates that increasing $M$ enhances the specific energy absorption per unit mass for SCM (e.g., increasing $M$ from six to 8 results in a 7.74% increase in average SEA) with a similar impact on CCM. Therefore, to optimize the plate compression performance of SCM, it is essential to balance peak force and SEA by carefully considering the value of $M$ . For CCM, large values of $M$ are not desirable as they can increase peak forces. Furthermore, Figures 16(b) and (d) illustrate that changing values of $N$ do not significantly alter the magnitude of peak forces but rather affect the location of these peaks. Additionally, increasing $N$ leads to a slight decrease in CFE value for both SCM and CCM foldcores, as shown in Figure 16(f).

Figure 16.

The plate compression results of the parametric study of the sandwich tube structure with straight/curved-creased circumferential Miura foldcore. Force displacement curves for (a) models N24M8, N24M7, and N24M6, and (b) models N24M6, N20M6, and N16M6 of the Straight-creased circumferential Miura foldcore sandwich tube, with $N$ and $M$ denoting the number of circumferential units and the number of layers of modulus, respectively. (c) and (d) are force-displacement curves for the corresponding curved-creased circumferential Miura-pattern counterparts that are shown in (a) and (b), respectively. (e) and (f) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) variations with respect to $M$ of both straight/curved-creased circumferential Miura foldcore models for additional configurations beyond those presented in (a)-(d), which illustrate only partial foldcore models.

Appendix A.9.-A.10 present the damage modes of the SCM and CCM foldcores. Similar to the AM foldcores, their geometric characteristics result in shear deformation of the core cross-section during compression, contributing to energy absorption. The primary material damage modes are matrix damage in tension/compression, and shear damage. The presence of the zigzag lines helps to distribute the load transferred from the outer tube across the entire core and to the inner tube, enhancing the energy absorption capability of the structure.

As shown in Figure 17(a)–(d), the force-displacement curves for the Kresling model exhibit a distinct pattern compared to the Miura-pattern models but are similar to the diamond-pattern models. These curves do not feature exceptionally high peak forces maintain a relatively constant level after the initial peak and exhibit a significant decline in the latter half of the compression process. For straight-creased Kresling (SK), increasing both $M$ and $N$ results in an increase in this peak, with $M$ having a more significant impact. Conversely, for curved-creased Kresling (CK), increasing $M$ leads to an increase in this peak, while the influence of $φ$ is relatively smaller. Figure 17(e) demonstrates that increasing $M$ substantially enhances the SEA for both SK and CK models, which is same with SD and CD. Increasing $M$ from three to four or five results in a respective increase of 43.91% and 55.51% (17.28% and 45.13%) in average SEA value for SK (CK) foldcore, respectively. Overall, CK exhibits superior energy absorption capabilities compared to SK. CK exhibits an average SEA 35.21% higher than SK, a trend also observable within the diamond configuration.

Figure 17.

The plate compression results of the parametric study of the sandwich tube structure with straight/curved-creased Kresling foldcore. Force displacement curves for (a) models N10M3, N11M3, and N12M3, and (b) models N12M3, N12M4, and N12M5 of the straight-creased Kresling foldcore sandwich tube, with $N$ and $M$ denoting the number of circumferential units and the number of layers of modulus, respectively. (c) models M3φ10, M4φ10, and M5φ10, and (d) models M3φ5, M3φ10, and M3φ15 of the curved-creased Kresling foldcore sandwich tube, with $φ$ denoting the in-plane twist angle. (e) and (f) are the results of specific energy absorption (SEA) and crushing force efficiency (CFE) variations with respect to $M$ of both straight/curved-creased diamond foldcore models for additional configurations beyond those presented in (a)-(d), which illustrate only partial foldcore models.

Appendix A.11.-A.12 show the damage modes of the SK and CK foldcores. It can be observed that during compression, the CK core exhibits shear deformation in the cross-section, contributing to energy absorption. The damage extent of the CK foldcore is also greater than that of the SK foldcore, demonstrating that the CK foldcore more effectively transfers the load throughout the entire foldcore structure. The SEA results also confirm that the energy absorption performance of the CK foldcore is superior to that of the SK foldcore.

Based on the results of the parametric study, it was found that the energy absorption capability of different core structures is primarily influenced by two factors: 1. The ability of the geometric configuration to transfer the load to the inner tube, e.g., the supporting structures of the zigzag lines formed by the geometric characteristics of the Miura foldcore; 2. The ability of the geometric configuration to distribute the load to other areas of the core structure, that is, whether the majority of the material could participate into the deformation process.

Sandwich tubes with Miura configurations exhibit better specific energy absorption (SEA) capability per unit mass under both loading conditions, compared to their counterparts. One could assume that the primary reason is that the Miura configuration efficiently transfers the load to the inner tube via the zigzag lines, thus making efficient use of the sandwich structure characteristics. Additionally, the Miura core itself undergoes shear deformation within the cross-section during compression, which effectively distributes the load throughout the core, thereby maximizing the material’s participation in energy absorption.

In contrast, the diamond and Kresling configurations, upon receiving the load transferred by the outer tube, lack the direct zigzag line mechanism to transfer the load to the inner tube as seen in the Miura configuration. Instead, they can only transfer the load to the inner tube through their planar elements, resulting in significantly reduced load transfer efficiency and consequently lower energy absorption performance. And, the curved configurations of the diamond and Kresling foldcores can better enhance their ability to distribute the load to other parts of the core, thereby improving their transverse energy absorption capabilities.

Optimization of the energy absorption capability of SAM and SCM

Overall, the sandwich tube structures with Miura-patterned foldcore could provide more stiffness and a higher energy absorption capacity than other foldcore counterparts due to the essence of its structural geometry. Then, the question arises as to whether the assumption that we have earlier on the effect of zigzag lines in contact region on the energy absorption capacity. Here, we chose the straight-creased axial Miura (SAM) and straight-creased circumferential Miura (SCM) pattern to investigate the influence of unit cell types on the lateral energy-absorbing performance of sandwich tubes under transverse compression.

Based on the shape of the projected rectangle area (whose sides are the axial height $h$ and the distance across the circumferential direction) of the SCM unit in Figure 18(b), the SCM configuration can be categorized into three types: Type I (a “belt” shape), Type II (a “square” shape), and Type III (a “tower” shape). The unit cell of the straight-creased axial Miura (SAM) foldcores is defined in a similar way as Type I, II and III, as shown in Figure 18(a)–(b), with a $90 °$ tipping orientation difference for SAM and SCM unit cell, that is, the Type I unit cell of SCM (as shown in Figure 18(b)) could be derived by rotating the Type I unit cell of SAM (as shown in Figure 18(a)) $90 °$ counterclockwise. As $N$ and $M$ increase, the unit cell area gradually decreases, leading to an increase in SEA. Yet Type I and Type II of both axial and circumferential Miura demonstrate superior overall energy absorption performance compared to Type III.

Figure 18.

The optimization design of the sandwich tube with Miura-pattern foldcore. The illustration of various types of (a) straight-creased axial Miura (SAM), (b) straight-creased circumferential Miura (SCM) configuration sandwich tubes. Three types of the unit cell of the foldcore are defined by their shapes of projected area: Type I (a “belt” shape), Type II (a “square” shape), and Type III (a “tower” shape), with a $90 °$ tipping orientation difference for SAM and SCM unit cell. The effects of $M$ and $N$ on the specific energy absorption (SEA) value of (c) straight-creased axial Miura (SAM), (d) straight-creased circumferential Miura (SCM) under plate compression. Results in (a-b) is included in (c-d). The functions of two fitting surfaces in (c) and (d) are shown in Appendix A.3. The optimized results of Miura patterns based on the fitting surfaces could be derived.

To further quantify the effects of $M$ and $N$ on the SEA for both Miura patterns, a large number of data points within the range of $6 \leq N \leq 60$ and $6 \leq M \leq 40$ are selected and the structural parameters optimization was carried out by response surface method (RSM),⁸⁰ with results shown in Figure 18(c)–(d) and listed in Appendix A.13-A.14, the results in Figure 18(a)–(b) are also included in Figure 18(c)–(d). Using the RSM is capable of performing both constrained multi-objective optimizations. The RSM effectively reduces the number of experiments and simulations required, and facilitates the analysis of the variance in simulation results, allowing for a detailed examination of how different parameters influence the energy absorption performance of Miura foldcore composite sandwich tubes.²⁵ Note that the low bound of $N$ and $M$ is chosen based on the purpose of aesthetics, where the configurations of the foldcore resemble tube-like structures, whereas the upper bound is chosen to maintain the proper ratio of the thickness-to-length of the facet so that the whole foldcore is a thin-walled structure. We then utilized quadratic surface fitting to fit the data points to derive the optimal solution for SEA, which is formulated as follows:

\begin{array}{l} m a x i m i z e : S E A (M, N) \\ s . t . 6 \leq M \leq 40 \\ 6 \leq N \leq 60 \end{array}

(15)

Several evaluation parameters, Residual Sum of Squares (

R S S

), Total Sum of Squares (

T S S

) and coefficient of determination

R_{s q u a r e}

, are used to evaluate the accuracies of response surface fitting, as shown below.

R_{s q u a r e} = 1 - \frac{R S S}{T S S} = 1 - \frac{\sum {(S E A_{F E M} - S E A_{F i t t i n g})}^{2}}{\sum {(S E A_{F E M} - \bar{{S E A}_{F E M}})}^{2}}

(16)

where

S E A_{F E M}

is the finite element simulation results of

S E A

S E A_{F i t t i n g}

is the surface fitting results of SEA,

\bar{{S E A}_{F E M}}

is the average value of finite element simulation results of SEA.

R S S

is the sum of the squares of the residuals, lower which indicates a better fit of the model to the data;

T S S

is the sum of the squares of the differences;

R_{s q u a r e}

ranges from 0 to 1, indicating the model’s goodness of fit. (Table 11).

Table 11.

Results of the evaluation parameters for the fitted surfaces.

	RSS	$R_{s q u a r e}$
SAM	0.3640	0.8873
SCM	0.4161	0.9203

Based on the results of the fitting surface, the maximum value of predicted specific energy absorption (

S E A

) for SAM occurs at

N = 60

M = 38

, with a value of 2.20, whereas the corresponding finite element simulation result is 2.07, resulting in a relative error of 6.28%. For SCM, the maximum value of

S E A

appears at

N = 60

M = 40

, with a value of 2.40. The corresponding finite element result is 2.33, resulting in a relative error of 3.00%. The fitted surface functions are provided in Appendix A.15. To validate the fitting surface, we selected individual data points and the results are shown in Table 12, where relative error characterizes the magnitude of the difference in predictions, with a smaller value indicating higher accuracy in the predictions. It indicates that the fitting surfaces could predict well within an agreeable range. As depicted in Figure 18(c) and (d), it was calculated that the partial correlation coefficient¹ for SAM with respect to parameter

N

and

M

was 0.821 and 0.695 respectively; For SCM, the partial correlation coefficient for parameter

N

was 0.578 and for the parameter

M

was 0.917.

Table 12.

The validation results of the fitted surfaces for SAM and SCM.

	N	M	SEA (fitting)	SEA (FEM)	Relative error (%)
SAM	35	25	1.638	1.511	8.41
SCM	35	25	1.642	1.540	6.62

Therefore, due to differences in the direction of the zigzag lines (as defined in Figure 9), axial (circumferential) Miura displays higher sensitivity to parameter $N$ ( $M$ ), that said, for either AM or CM, densifying the unit in a single zigzag line, that is, increasing the number of “zigzags” units within a single zigzag line, has better effect to increase the SEA value, compared to increasing the number of zigzag lines. The zigzag line is better designed to lay out axially to provide better energy absorption capacity (i.e., the SEA of circumferential Miura is generally better than axial Miura, given that the number of the Miura units is the same). This provides a design strategy for the Miura pattern, which becomes one of the main novelties provided in this study. The results shed light on the intricate relationship between geometric parameters and energy absorption characteristics in Miura configurations, highlighting the importance of carefully selecting $N$ and $M$ to optimize the lateral energy-absorbing performance of sandwich tubes.

Comparison and discussion

Finally, the Miura-pattern foldcore is compared to the traditional honeycomb core with the same unit cell dimension and the nested tube for the energy absorption capacity. As shown in Figure 19(a), the SCM configuration with the optimized transverse compression energy absorption performance ( $N$ = 60, $M$ = 40) was selected for comparison with two traditional honeycomb sandwich tubes (with two different core materials) with identical unit cell dimensions and nested tubes. CFRP T300 and aluminium 2024⁸² are used as the core material of two honeycomb sandwich tubes respectively, CFRP T300 is used as foldcore material for the SCM sandwich tube and the middle tube of the nested tube, and CFRP T700 is used for all the inner/outer sandwich tubes and nested tubes. The force-displacement curves are depicted in Figure 19(b). The results (as shown in Table 13) indicate that the SCM sandwich tube exhibits superior energy absorption performance compared to the traditional honeycomb structure and nested tube, its specific energy absorption (SEA) is 2.9, 3.9 and 8 times that of the composite honeycomb, aluminium honeycomb and nested tube respectively. Additionally, its crushing force efficiency (CFE) is 31.1% (93.1%) higher than that of the honeycomb sandwich tube with the composite (aluminium) core. This demonstrates that incorporating origami structures as foldcore into sandwich structures significantly enhances the transverse energy absorption performance and force consistency of the sandwich tubes.

Figure 19.

(a) Schematic representation of nested tube, two sandwich tubes with honeycomb core (one core material being CFRP T300 and the other being Aluminium) and straight-creased circumferential Miura foldcores sandwich tube with identical unit cell dimensions, the inner and outer tubes of the honeycomb and circumferential Miura sandwich tube structures are not shown for clarity of the figure. (b) force-displacement curves of sandwich tubes with SCM foldcore, composite Honeycomb foldcore, aluminium Honeycomb foldcore and nested tube under transverse compression testing.

Table 13.

Comparative results of SCM sandwich tubes and same-sized Honeycomb sandwich tubes under plate compression.

Model	N	M	Mass/(g)	P_max/(KN)	P_mean/(KN)	CFE	SEA/(J/g)
SCM-N60M40	60	40	242	26.02	15.31	0.588	2.332
Honeycomb-composite	60	40	272	13.40	5.68	0.424	0.797
Honeycomb-Al⁸²	60	40	362	20.93	6.02	0.288	0.604
Nested tube	/	/	174	11.50	1.64	0.142	0.292

As shown in Figure 20, the damage modes of the foldcore material in the circumferential Miura (CM) structure and the intermediate tube in the nested tube structure demonstrate global buckling. However, the CM structure’s matrix material did not exhibit significant damage, whereas the intermediate tube of the nested tube structure experienced minor matrix damage and shear failure at the interfaces with the inner and outer tubes.

Figure 20.

Comparison of tension/compression matrix damage and shear damage in the circumferential Miura N60M40 foldcore and the intermediate tube of the nested tube, the figure of the damage mode is the ending state of the loading process.

Based on the parametric study analysis, the CM foldcore transfers the load received by the outer tube along the zigzag lines, engaging most of the structure and material in the energy absorption process. The absence of extensive material damage in the CM N60M40 might be attributed to the high density of the Miura unit cells, which disperses the energy without reaching the failure threshold of the material. In contrast, despite the nested tube structure showing only minor localized material damage, the significantly lower load-displacement curve in Figure 19 indicates that most of the material did not contribute to energy absorption compared with the CM foldcore structure.

This underscores the advantages of the origami core: it increases structural support (enhancing stiffness) and transfers the load to the inner tube. The folds can be seen as predefined deformation paths, allowing a greater portion of the material to resist deformation and thereby contributing to the energy absorption process.

Conclusion

This paper presents a comprehensive study of the effects of composite sandwich tube structures with 8 different origami-inspired foldcores (diamond, Kresling, circumferential Miura, axial Miura and their curved-crease counterparts) on the transverse mechanical properties, that is, the energy absorption performance. Specimens are manufactured, and three-point bending-like local compression experiments are carried out to validate the FE numerical. The design process of the foldcore is detailed, and its main parameters are identified. The parametric study and damage mode analysis of the transverse mechanical properties is conducted, the main parameters are studied to understand the effects on the energy absorption capacity, and the comparisons between the SCM foldcore structures with honeycomb core counterparts and nested tubes are made.

The results show that the specific energy absorption (SEA) of Miura configurations is generally higher than diamond and Kresling configurations, exhibiting superior energy absorption performance (higher SEA values): In three-point bending (transverse compression) tests, the average SEA values of the Miura configuration is 0.591 J/g (1.098 J/g), which was 22.9% (78.8%) higher than the average SEA value of 0.481 J/g (0.614 J/g) for diamond and Kresling configurations. This is because the Miura configuration effectively transfers external loads to both the inner tube and other parts of the foldcore, thereby better utilizing the geometric characteristics of the Miura pattern-based structure as the foldcore. The crushing force efficiency (CFE) values of Miura are generally lower than those of diamond and Kresling patterns, which indicates that foldcores with diamond and Kresling patterns have steadier force-displacement curves during deformation: In three-point bending (transverse compression) tests, the CFE value of the Miura configuration is 0.513 (0.471), which was 14.4% (20.8%) lower than the average CFE value of 0.599 (0.595) for diamond and Kresling configurations. For diamond and Kresling sandwich tube structures, it is worth noting that $M$ , which represents the number of basic modules, stands out as a prominent influencing parameter. Increasing the number of basic modules ( $M$ ) effectively enhances their bending and compression energy absorption capacities. Moreover, under plate compression conditions, the energy absorption performance of the diamond and Kresling curved configurations surpasses that of their straight-core counterparts. This is because the curved configurations are more effective in distributing the load throughout the foldcore structure, thereby enhancing the material utilization efficiency. For both circumferential and axial Miura foldcores, since the zigzag lines are across the region of the foldcore and connect the inner and outer tube, which form support and transfer load effectively during the deformation process. Yet, the curved Miura does not show significant enhancement of energy absorption capacity compared to the straight Miura counterpart, hence indicating that the geometric characteristics of zigzag line supports of Miura could be more effective than changing the straight crease line to curved ones. Therefore, as one of the main novelties in this paper, based on the results, the essence of increasing the energy absorption capacity of the Miura pattern is hence found: increasing the zigzag units within a single zigzag line is better than increasing the number of zigzag lines. Further, choosing the circumferential Miura pattern (for the same number of Miura units, circumferential Miura is generally better than axial Miura, due to the zigzag line orientation) could effectively enhance the energy absorption capacity, which could be used as the design strategy of the Miura pattern foldcore for future energy absorption applications. Finally, the CFRP sandwich tubes with Miura foldcore pattern demonstrated significantly superior energy absorption performance compared to traditional sandwich tube structures with honeycomb cores and nested tube. Specifically, the specific energy absorption (SEA) of the composite SCM sandwich tube is higher than its counterparts, demonstrating its potential to replace traditional honeycomb structures in future transverse compression-resistant applications.

The work in this paper investigates the quasi-static transverse mechanical properties of sandwich tube structures with foldcores. The design method is systematically provided, and the parametric study is carried out to identify the key factors for different foldcore patterns. Critical insights into the behaviour of these sandwich tube structures with foldcores were revealed by the findings of the study, the Miura pattern foldcore is optimized and a design strategy is given. Hence, the work in this paper could provide a reference for future designs when the transverse mechanical properties are considered, especially for applications/ designs such as under-sea pipes, pressure containers, functional structures (e.g., fuselage in aircraft), etc. Yet the behaviours under impact are not included in the scope of this research. Future avenues would be to investigate the effect of fibre orientation and thermal effects on the transverse modulus and crashworthiness of the composite sandwich tubular structures with foldcores. Also, we will consider exploring the use of new manufacturing techniques and materials with better mechanical properties to produce tubular origami-based foldcore at a lower cost with better performance, which could potentially enable us to conduct extensive physical experiments for further study. Additionally, in conjunction with the work of Li et al.⁸³ and Wang et al.,³⁶ our study may offer new potential directions for Origami-based TPMS structures.

Footnotes

Acknowledgements

Haitao Ye completed the research at Shanghai Jiao Tong University and has moved to City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, since completing the research. The authors would like to thank for all the comments received.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial support from the National Natural Science Foundation of China 52373293 and 51408357 are acknowledged.

ORCID iD

Keyao Song

Note

Appendix

Figure A.1.

The damage mode of diamond configuration foldcore material under three-point bending.

Figure A.2.

The damage mode of axial Miura configuration foldcore material under three-point bending, the figure of the damage mode is the ending state of the loading process.

Figure A.3.

The damage mode of circumferential Miura configuration foldcore material under three-point bending, the figure of the damage mode is the ending state of the loading process.

Figure A.4.

The damage mode of Kresling configuration foldcore material under three-point bending, the figure of the damage mode is the ending state of the loading process.

Figure A.5.

The damage mode of straight-diamond configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.6.

The damage mode of curved-diamond configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.7.

The damage mode of Straight-axial Miura configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.8.

The damage mode of curved-axial Miura configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.9.

The damage mode of straight-circumferential Miura configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.10.

The damage mode of curved-circumferential Miura configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.11.

The damage mode of straight-Kresling configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Figure A.12.

The damage mode of curved-Kresling configuration foldcore material under plate compression, the figure of the damage mode is the ending state of the loading process.

Table 14.

The finite element analysis results of specific energy absorption (SEA) for SAM under plate compression.

Configuration	N	M	SEA (J/g)
SAM	10	10	0.63
	20	20	1.23
	20	30	1.33
	20	40	1.36
	24	8	1.05
	24	15	1.25
	30	12	1.23
	30	20	1.43
	30	30	1.88
	34	39	1.63
	40	20	1.52
	45	25	1.66
	46	30	2.38
	48	30	2.19
	50	10	1.37
	60	23	2.00
	60	38	2.07

Table 15.

The finite element analysis results of specific energy absorption (SEA) for SCM under plate compression.

Configuration	N	M	SEA (J/g)
SCM	16	6	1.01
	16	7	1.05
	16	8	0.99
	20	6	1.03
	20	7	0.97
	20	8	1.09
	20	19	1.51
	20	20	1.72
	20	30	1.56
	20	40	1.75
	24	6	0.96
	24	7	1.01
	24	8	1.15
	24	15	1.47
	27	25	1.50
	30	20	1.39
	30	30	1.65
	30	40	2.16
	32	20	1.34
	37	38	2.04
	40	20	1.45
	40	30	1.93
	40	40	2.39
	48	30	1.90
	50	15	1.34
	50	35	2.01
	60	23	1.92
	60	40	2.33

Table 16.

The fitting surface functions of SAM and SCM.

SAM	$S E A (M, N) = - 0.0004 N^{2} - 0.0010 M^{2} + 0.0004 N M + 0.0382 N + 0.0572 M - 0.2190$
SCM	$S E A (M, N) = - 0.0002 M^{2} + 0.0004 N M + 0.0004 N + 0.0275 M + 0.8214$

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