A novel star auxetic honeycomb with enhanced in-plane crushing strength

Thin–Walled Structures 149 (2020) 106623

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A novel star auxetic honeycomb with enhanced in-plane crushing strength Lulu Wei , Xuan Zhao *, Qiang Yu , Guohua Zhu School of Automobile, Chang’an University, Xi’an, 710064, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Auxetic honeycomb Deformation model Crushing strength Densification strain Negative Poisson’s ratio

Auxetic honeycombs exhibit low weight, shear stiffness, and excellent energy absorption capacity and thus have great potential for achieving the requirements of crashworthiness and lightweight in automotive fields. This work presents a novel auxetic structure called the star-triangular honeycomb (STH), in which the horizontal and vertical ligaments of the star honeycombs (SH) are replaced with triangular structures. The dynamic crushing behaviors of the STH under three different crushing velocities were investigated using 1D shock theory. The results show that the STH has a more obvious negative Poisson’s ratio effect than the SH and that transverse contraction mainly occurs in the first plateau stage. Theoretical models were deduced based on the collapse mechanism of the typical unit revealed by numerical simulation for STH crushing strength prediction. The theoretical predictions agreed well with the simulation results, and two different plateau stresses appeared under low-velocity crushing. In addition, the influences of the STH geometric parameters and crushing velocity on the energy-absorbing capacity and densification strain were systematically explored. The parameter analysis indi­ cated that the effects of the cell-wall thickness and incline angle on the dynamic response and energy absorption capacity of the STH under low-and medium-velocity crushing are more significant than those under high-velocity crushing. Moreover, the STH showed better energy absorption capacity than the SH. Thus, this design is expected to provide a novel means of improving the mechanical properties of honeycombs.

1. Introduction Recently, the pursuit of lightweight design and safety in the aero­ space, automobile, packaging, and human protection industries has driven the need for new lightweight materials to meet higher perfor­ mance demands [1,2]. As typical honeycomb metamaterials, auxetic materials have drawn extensive attention due to their excellent perfor­ mance, including lightweight [3], high shear modulus [4,5], indentation resistance [6,7], and excellent acoustic [8] and energy absorption properties [9–11]. These superior characteristics are attributable to the negative Poisson’s ratio (NPR) effect of auxetic materials, which expand (contract) when they are stretched (compressed) [12,13]. The me­ chanical properties of auxetic materials are affected by their micro­ structure, matrix materials, and relative density, among which the microstructure is the most critical factor [14]. Therefore, considerable work has been conducted to improve the mechanical properties of the auxetic honeycomb by designing new topological structures. After auxetic material was first manufactured by Lakes [13] in 1987, various topological structures of auxetic honeycombs were designed and studied, such as arrow structures [14], re-entrant auxetic structures [15,

16], and chiral honeycombs [17], among which the double arrow and re-entrant structures have superior mechanical properties [18]. Scarpa et al. [19] revealed the relationships between the geometric and me­ chanical properties of auxetic honeycombs via finite element analysis. Furthermore, Masters and Evans [20] predicted the Young’s modulus, shear modulus and Poisson’s ratio of the re-entrant auxetic honeycomb using plastic deformation theory. Theocaris et al. [21] explored the ef­ fects of the material properties and microstructure on the Poisson’s ratio by numerical homogenization approach. Gao [22,23] analyzed the ef­ fects of cell-wall angles and thicknesses on the dynamic crushing behavior of a 3D double-arrowed structure. The results showed that increasing the cell-wall angle, cell-wall thickness and number of cells could effectively improve the energy absorption capacity of the 3D double-arrowed structure. In fact, the dynamic crushing performance is an essential parameter for evaluating the material mechanical properties of auxetic honey­ combs, especially in the aerospace and automotive industries. Therefore, researchers have studied the dynamic deformation mode and crushing €nig and Stronge performance of auxetic honeycombs extensively. Ho [24] derived an analytical formula for the critical crushing velocity

* Corresponding author. E-mail address: [email protected] (X. Zhao). https://doi.org/10.1016/j.tws.2020.106623 Received 29 September 2019; Received in revised form 13 December 2019; Accepted 18 January 2020 Available online 29 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.

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under a uniaxial load using “wave trapping” theory. Based on this the­ ory, Zhao et al. [25] discussed the deformation mode and established a theoretical model for the dynamic crushing strength of double-arrowed auxetic structures in the z- and x-directions. Qiao et al. [26,27] inves­ tigated the quasi-static and dynamic crushing deformations of both uniform and functionally graded double-arrowed auxetic structures and established theoretical formulas for the plateau stresses under low- and high-velocity loads. Simultaneously, Zhang et al. [28] analyzed the dynamic crushing response of re-entrant auxetic honeycombs with different cell-wall angles and crushing velocities, showing that the crushing strength is proportional to the square of the crushing velocity. Moreover, Hu et al. [29] predicted the Poisson’s ratio and plateau stress of re-entrant auxetic honeycombs via theoretical analysis and numerical simulation. In addition, a dynamic sensitivity index was proposed to assess the enhancement capacity of the plateau stress of the honeycombs caused by the crushing velocity. More recently, Qi et al. [30] deduced the effective Poisson’s ratio and crushing strength of tetra-chiral hon­ eycombs under quasi-static and dynamic crushing via theoretical anal­ ysis and numerical simulation. Among the traditional honeycomb structures, the triangular honey­ comb has a unique structural stability, which makes it excellent me­ chanical properties. Gu et al. [31] measured the tensile strength and Young’s modulus of the triangular honeycomb and evaluated the ac­ curacy of the current measurement methods used for mechanical properties. Liu et al. [32] systematically revealed the influences of the micro-structure and arrangements on the energy-absorbing capacity and deformation mechanism of triangular honeycombs. They found that triangular honeycombs have higher crushing strength and excellent energy-absorbing capacity compared to other honeycomb structures. To enhance the crushing strength of triangular honeycombs, hierarchical triangular honeycombs were constructed to explore the collapse modes and deformation mechanisms under in-plane uniaxial compression. A similar method involving hierarchical triangular honeycombs was employed by Sun[33], who found that hierarchical triangular honey­ combs have enhanced crushing strength compared to traditional hex­ agonal honeycombs, especially under low-velocity crushing. Generally, the above-mentioned analyses indicated that the triangular structure can effectively improve the mechanical properties of honeycombs. To enhance the in-plane stiffness and strength of general auxetic porous structures, many new methods have been conceived [34]. For example, sinusoidal, rhombic and vertical ligaments have been embedded into the classical auxetic structure to improve the stiffness, strength, and energy absorption capacity of the honeycombs [35–38]. Wang et al.[39,40] proposed two types of novel auxetic structures by adding a re-entrant configuration and an arrow structure into the star-shaped structure and predicted the plateau stress and Young’s modulus. They found that the crushing strength and energy absorption capacity were significantly improved, but the NPR effect was weakened. Thus, considerable work has been performed with the objective of designing a new auxetic structure with enhanced in-plane crushing strength, but the structures developed thus far have the drawback of a reduced NPR effect. To the knowledge of the authors, the further application of auxetic materials has been hindered because of the compromise between mechanical properties and auxetic performances [35]. Consequently, extensive research is necessary to study the dy­ namic crushing deformation behaviors of auxetic honeycombs further, to design a novel auxetic structure to achieve both excellent mechanical properties and a strong NPR effect simultaneously. In this study, we propose a novel star-triangular auxetic structure by replacing the ligaments in the star-shaped structure to improve the en­ ergy absorption capacity. The deformation behaviors of the startriangular honeycomb (STH) and star honeycomb (SH) under different crushing velocities are studied using 1D shock theory and numerical simulation. Analytical formulas are deduced to predict the crushing strength of the STH under both low- and high-velocity loading, then verified by conducting numerical simulations. Moreover, numerical

simulations are performed to reveal the effects of the geometric pa­ rameters and crushing velocity on the STH crashworthiness. Finally, the mechanical properties of the STH and SH were compared. This research is expected to improve understanding of the dynamic crushing behaviors and crushing strength of auxetic honeycombs and to provide design guidance for auxetic structures. 2. In-plane dynamic crushing behaviors 2.1. Star-triangular structure 2.1.1. Geometric description The star-shaped auxetic structure is composed of four arrow struc­ tures, horizontal and vertical ribs, which are periodically arranged to form an SH. Fig. 1(a) shows a star-shaped periodic auxetic structure. A novel auxetic structure can be obtained by replacing the horizontal ribs of the star-shaped structure with a triangular-shaped structure, creating what is referred to as an STH, as shown in Fig. 1(b). The cell-wall lengths of the star-shaped and triangular-shaped structures are equal in the STH. Four geometric parameters are critical: l is the length of the inclined cell wall, θ is the angle between the cell wall and vertical direction, t is the thickness of the cell wall, and h is the length of the horizontal and ver­ tical ribs. The mechanical properties of the auxetic honeycomb structure depend primarily on the relative density, and the weight of this parameter exceeds those of the other parameters. According to the geometric parameters of the auxetic structure, the relative densities of the SH and STH can be described as follows [41,42]:

ρSH ¼

ρ*SH t ð1 þ 2l=hÞ ¼ ρs h �1 þ pffi2ffi l sinðπ=4

ρSTH ¼

�2 θÞ

ρ*STH t ð3 þ sin θÞ ¼ h cosð2 cos θ sin θÞ ρs

(1)

(2)

Where ρSH and ρSTH are the relative densities of SH and STH, respec­ tively. ρ*SH and ρ*STH are the densities of SH and STH, respectively. ρs is the density of the matrix material. 2.1.2. Finite element models To reveal the dynamic deformation behavior and mechanical prop­ erties of the STH systematically, a finite element model was established using the explicit code of the nonlinear finite element software package LS-DYNA. In this model, the STH is located between the upper and lower rigid plates to simulate in-plane dynamic crushing. The rigid bottom plate is fixed, and the rigid upper plate is crushed in the y-direction with a constant crushing velocity [43,44]. Three crushing modes can be defined according to the crushing velocity: low-, medium-, and high-velocity modes. In this model, the length of the cell is l ¼ 5 mm, the angle is θ ¼ 30� , the length of the horizontal and vertical ribs is h ¼ 4 mm, the out-of-plane thickness of the STH b is 0.5 mm, and the cell wall thickness t is determined based on the relative density. Fig. 1 depicts the in-plane dynamic crushing models of the SH and STH. The matrix material of the auxetic honeycomb is aluminum alloy 6061O, and it is assumed that the cell wall is elastic-perfectly plastic with Young’s modulus E ¼ 68GPa, yield stress σ s ¼ 80MPa, density ​ ρs ¼ 2700 kg=m3 and Poisson’s ratio γ ¼ 0:33 [45]. The out-of-plane (the z-direction) degrees of freedom of all of the nodes in the honey­ comb were constrained to prevent the honeycomb from undergoing out-of-plane expansion. In addition, failure of the cell materials was ignored during the simulation. To prevent interpenetration of cell walls, automatic single-surface contact was applied to define the cell walls. Additionally, the contact between the honeycomb and two rigid plates was defined as automatic surface-to-surface contact. In this finite element model, both the static and dynamic coefficients of friction were taken to be 0.2. To improve the calculation efficiency while ensuring the accuracy of the numerical simulation, convergence tests were conducted 2

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Fig. 1. Geometric structure and finite element models, (a) Star-shaped structure, (b) Star-triangular structure.

to choose a suitable element size. Fig. 2 shows the force-displacement curves of STH with four different element sizes (0.25 mm, 0.5 mm, 1 mm and 1.5 mm) under dynamic crushing loading. With decreasing element size, the convergence trends of the force-displacement curves are obvious. The results show that using 10 elements for each cell-wall of the STH can balance the computational efficiency and numerical convergence. Thus, four-point Belytschko-Tasy shell elements with sizes of 0.5 mm were adopted for the cell wall mesh. 2.1.3. Model validation To verify the accuracy of the FE models, the same numerical model as in Ref. [39] was established to compare the deformation behavior and crushing stress of the star-shaped honeycomb under low-velocity crushing. Fig. 3 shows the deformation process of the SH under a crushing velocity of 25m/s. It is found that both the numerical and the verification model results exhibit local deformation near the impact end.

Fig. 3. Comparison of deformation modes for the SH between FE model and verification model.

Although the local deformation differs, the global deformation mode of the numerical model is in good agreement with that of the verification model during compression. In addition, the stress-strain curves obtained for the SH from the numerical and verification results are compared in Fig. 4, showing error between the numerical curve and verification re­ sults in the later stage of compression. However, the error is small and can be accepted, indicating high consistency between the simulation and verification model. Therefore, the established FE models were deter­ mined to be reliable and useable for the subsequent study.

Fig. 2. Force-displacement curves of different element sizes. 3

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direction. For two honeycombs of the same relative density, Fig. 5 shows the in-plane crushing deformation processes of the SH and STH in the ydirection under a crushing velocity of 2 m/s. In the initial deformation phase (Fig. 5(b)), the deformation of the STH first appears at the crushing end, consisting mainly of rotation and bending of the cell wall of the star-shaped structure, and rapidly spreads to the entire honey­ comb. The crushed and fixed ends of the honeycombs exhibit a V-shaped deformation band, and the V-shaped opening is oriented towards both ends. With increasing crushing displacement (Fig. 5(c)), the crushed end of the honeycomb still presents a V-shaped deformation band, but the deformation band at the fixed end becomes X-shaped. Every star cell deforms homogeneously with lateral shrinkage along the vertical di­ rection. Thus, boundary necking forms during crushing. In addition, the V-shaped deformation band gradually expands from the crushed end to the fixed end (Fig. 5(d)). Eventually, the STH changes into another classic triangular grid structure (Fig. 5(e)). The triangular grids collapse in a layer-by-layer manner from the fixed end to the crushed end until the honeycomb becomes denser (Fig. 5(f)–(g)). The deformation of the SH first appears at the fixed end. Similarly to the deformation mode of the STH, the rotation and bending of the cell wall of the SH cause transverse shrinkage in the y-direction and exhibit a significant NPR effect (Fig. 5(a)–(d)). Additionally, the horizontal and vertical beams of the SH buckle under uniaxial compression and are expanded towards the x-direction (Fig. 5(e)–(h)), which is similar to the conventional honey­ comb. Compared with the deformation process of the two honeycombs, the deformation of the STH is more uniform, and the problem of pre­ mature densification of the honeycomb can be effectively remedied. Fig. 6 shows the stress-strain curves of the STH under a crushing velocity of 2 m/s. It is evident that the stress-strain curve consists of two stages with the same change pattern but different values. In the first stage, certain inclined cell walls of the star-shaped cell rotate and bend, plastic hinges form at both ends of the cell walls, and the stress oscillates around a constant value and forms a plateau in the stress-strain curve (Fig. 6). In the second stage, the geometric parameters of the triangular grid honeycombs are different from those of STH. The equivalent wall thickness of the inclined cell wall is 3t, and that of the vertical cell wall is still t. As the compression continues, the inclined and vertical cell walls buckle, more plastic hinges appear at both ends of each cell wall and in the middle of the vertical cell walls. Therefore, the peak and plateau stresses are significantly increased, and the plateau stress is approxi­ mately 10 times larger than that in the first phase. After most of the cells are squeezed together, the stress-strain curves increase rapidly, which implies final densification of the honeycomb. (relative density ρ ¼ 0:15).

Fig. 4. Comparison of stress-strain curves for the SH between numerical and verification results.

2.2. Critical crushing velocity The crushing velocity is an important parameter that affects the dynamic crushing deformation behaviors of honeycomb. Under varying crushing velocities, different stress waves are generated at the crushing end of the honeycomb. As the crushing velocity increases and the stress amplitude exceeds the yield stress of the honeycomb, which the defor­ mation begins to change from the overall model to local model. In the critical stage, the crushing velocity that causes plastic deformation is known as the yield velocity or the first critical velocity. Based on the €nig and Stronge [24] defined the analytical ‘wave trapping’ theory, Ho formula for the critical velocity under uniaxial loading, which can be obtained as: Z εcr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vcr1 ¼ σ ’ðεÞ=ρ0 dε (3) 0

Where εcr is the initial strain of the honeycomb (the strain at which the stress reaches the initial peak), ​ σ’ðεÞ ¼ dσ=dε is the Young’s modulus of the honeycomb in linear elastic phase, and ρ0 is the density of the honeycomb; When the relative density ρ ¼ 0:15, the first critical crushing velocity of the SH and STH are 7 m/s and 7.5 m/s, respectively. With the increase in crushing velocity, the local deformation of the honeycomb becomes even more pronounced. The honeycomb collapses from the crushing end in a progressive manner under the influence of the ‘compaction wave’, which is referred to as the ‘steady-wave’ [46]. The critical velocity for the occurrence of the steady-wave in honeycomb is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vcr2 ¼ 2σ0 εd =ρ0 (4)

2.3.2. High-velocity crushing Under a crushing velocity of 100 m/s, the collapse of the STH starts from the crushed end and propagates in a layer-by-layer manner to the fixed end, and an I-shaped deformation band appears at the crushed end, as shown in Fig. 7. With increasing crushing velocity, the inertia effect becomes dominant, and the collapse of the honeycomb first appears at the crushed end. Therefore, the deformation process of the STH is similar to that of the SH. In addition, in the high-velocity model, the honeycombs do not have sufficient time to reach the phase of full shrinkage, so the SH and STH do not form boundary necking. Since the cell walls of the honeycomb collapse in a layer-by-layer manner through a periodic process, the stress of the STH fluctuates around a constant value until the cell walls are entirely packed together and the crushing stress increases rapidly. The high-velocity dynamic crushing stressstrain curves of the honeycombs are shown in Fig. 8.

Where ​ σ 0 is the static yield stress of the honeycomb, εd is the densi­ fication strain; When the relative density ρ ¼ 0:15, the second critical crushing velocity of the SH and STH are 35 m/s, 48 m/s, respectively. 2.3. Dynamic crushing deformation mode In this section, the relative density ​ ρ ​ of the SH and STH is 0.15. In addition, according to the critical velocities of the two honeycombs, the following velocities corresponding to the three crushing modes were applied in the numerical simulation: 2 m/s, 20 m/s, and 100 m/s.

2.3.3. Moderate-velocity crushing Fig. 9 shows the deformation processes of the STH and SH under a crushing velocity of 20 m/s. With increasing crushing velocity, the cells near the crushed end do not have sufficient time for transverse contraction, as in the high-velocity crushing model (Fig. 9(a)). As the

2.3.1. Low-velocity crushing Similar to the conventional auxetic material, the STH and SH exhibit different deformation modes as the crushing velocity increases in the y4

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Fig. 5. The deformation evolutions of the STH and SH under the crushing velocity of 2 m/s (relative density ρ ¼ 0:15).

velocity deformation modes emerged during medium-velocity crush­ ing of the honeycombs, indicating the transition from the low-velocity mode to the high-velocity mode. In addition, the stress-stain curves of the honeycomb include the above two deformation modes in the medium-velocity crushing mode, as shown in Fig. 10. During the initial stage, due to the localized collapse at the crushed end, the stress of the honeycomb displays more significant oscillation. The deformation gradually enters a low-velocity collapse mode, and the stress gradually increases and oscillates around a constant value, which is approximately equal to the plateau stress under low-velocity crushing. The deformation mode of the SH is similar to that of the STH, but the crushing strength is significantly less than that of the STH. 2.4. Deformation modes map As the crushing velocity increases, the deformation modes of the STH changes from the global deformation mode to the local deformation mode. The critical crushing velocities represent the critical transition boundaries between the critical deformation modes for the honeycomb, as described in Section 2.2. The deformation modes are affected by relative density variation. Through numerical simulations of STHs with different relative densities under crushing velocities, empirical formulas for the critical crushing velocities were deduced based on linear regression theory. When the crushing velocity v < vcr1 , the deformation of the honeycomb is described by the global deformation mode (Gmode). When the crushing velocity vcr1 < v < vcr2 , both global and local

Fig. 6. Stress–strain curves of the STH and SH under the crushing velocity of 2 m/s.

dynamic crushing continues, the deformation of the honeycombs ex­ hibits an overall transverse contraction mode similar to that observed under the low-velocity crushing (Fig. 9(b)–(d)), and necking of the boundary is evident (Fig. 9(b)). Therefore, both the low- and high5

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Fig. 7. The deformation evolutions of the STH and SH under the crushing velocity of 100 m/s (relative density ρ ¼ 0:15).

study the crushing strength of the STH, i.e., the plateau stress level of the stress-strain curve of the STH, two plateau phases appear during the collapse process of the STH. Because of the periodic collapse charac­ teristics of the cells in the STH, the deformation processes of a repre­ sentative block with four cells as shown in Fig. 12(a) and (b), and a typical STH unit was employed for theoretical analysis, as depicted in Fig. 12(c) and (d). According to conservation of energy, the work per­ formed by the crushing force is equal to the sum of the energy dissipated by the plastic hinge of the cell walls and the kinetic energy of the typical unit under low-velocity crushing. Firstly, the crushing strength of the first plateau stress phase for the typical unit was studied. At the initial moment of deformation, all of the cell walls remain in the original positions, without deformation or rotation. Fig. 12(c) shows a schematic diagram of the force analysis for the typical unit. The original width of the unit is L0 ¼ 2lð2cosθ sinθ t =2lÞ in the x-direction, and the original height is H0 ¼ 2lcosθ in the y-direction. Thereafter, the typical unit is compressed downward in the y-direction, and cell wall AB rotates around points A and B simultaneously until it overlaps with cell walls AM and BC. At this stage, it is assumed that the length of cell wall AB remains constant at l during the deformation process, and all of the middle portion retains the original configuration. The deformations of cell walls BE, GH, and ML are similar to that of cell wall AB, whereas the other cell walls only move horizontally or vertically without rotation or deformation until the cell changes into a triangular grid, as shown in Fig. 12(b). Therefore, two plastic hinges form at both ends of a cell wall. Eight plastic hinges appear in a typical unit and are highlighted by blue circles in Fig. 12(c). Thus, the work performed by the crushing forces acting on the typical unit during the deformation process E1 is � E1 ¼ σ L1 L0 b H0 Hf 1 ¼4l2 bσ L1 ð2cosθ sinθ t=2lÞðcosθ sinθ 2t=ðlcosθÞÞ

Fig. 8. Stress–strain curves of the STH and SH under the crushing velocity of 100 m/s (relative density ρ ¼ 0:15).

deformation occur within the honeycomb; this deformation mode is known as the transitional deformation mode (T-mode). When the crushing velocity v � vcr2 , the deformation of the honeycomb exhibits the local deformation mode (L-mode). Fig. 11 shows the deformation modes of the STH and SH. From Eqs. (5) and (6), it can be observed that linear relationships occur between the critical crushing velocities and relative densities, and the results are consistent with Fig. 11. Compared with the SH, the STH exhibits a higher critical crushing velocity at the same relative density, which indicates that the STH has better crushing resistance than the SH. vcr1 ¼ 5:6 þ 12:2 ρSTH

(5)

vcr2 ¼ 18:4 þ 196:6 ρSTH

(6)

(7) Where σ L1 is the first plateau stress of the typical unit, L0 is the original width of the typical unit, b is the thickness in the z-direction of STH, H0 is the original height of the typical unit, and Hf1 is the height of trian­ gular grid structure. The kinetic energy generated by a typical unit during the collapse process is

3. Dynamic crushing strength 3.1. Low-velocity crushing strength

Ek ¼ 7ρs btlv2

(8)

During the first deformation phase of a typical unit, the total energy absorbed by the plastic hinges Wp1 is

The plateau stress that occurs during the collapse process of the cells contributes the most to the crushing strength of the STH. Therefore, to 6

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Fig. 9. The deformation evolutions of the STH and SH under the crushing velocity of 20 m/s (relative density ρ ¼ 0:15).

Fig. 10. Stress–strain curves of the STH and SH under the crushing velocity of 20 m/s (relative density ρ ¼ 0:15).

Wp1 ¼ 8Mp1

�π 2

�π � 2θ ¼ 2σys bt2 2

� 2θ

Fig. 11. Deformation modes map of the STH.

angle of the plastic hinge is same as in the first plateau phase. Thus, eight plastic hinges appear at the inclined walls and are marked with red circles in Fig. 12(d). The middle portion of the vertical cell wall also undergoes buckling. Three plastic hinges appear at a vertical cell wall and are marked with green circles. In addition, Fig. 12 shows that ver­ tical cell walls CD and IJ are shared with other cells. The length of OM and FP is half that of CD and IJ, which are separate cell walls. Therefore, it is assumed that three vertical cell walls exist in each representative cell, where the equivalent thickness of cell walls CD and IJ is t/2. The second plateau stress σL2 of the STH can be derived using the same methods described above. An isolated wall can collapse, as shown in Fig. 12(d), and in doing so plastic work π Mp1 per unit depth of the cell wall is performed against the plastic moment Mp1. For complete collapse of the three cell walls, the plastic bending energy Wp2 is

(9)

Where Mp1 is the fully plastic moment of the inclined wall in bending, Mp1 ¼ σys bt 2 =4 in the case of rectangular cross-sectional beams, and σ ys is the yield stress of matrix material. According to conservation of energy, E1 ¼ Wp1 þ Ek , the lowvelocity crushing strength of the STH, i.e., the first plateau stress is � � � � �2 � t σys π2 2θ þ 7t4l ρs v2 l � (10) σ L1 ¼ � �� t 2t 2 2cosθ sinθ cosθ sinθ 2l lcosθ After the cell is changed into a triangular grid structure, the cells collapse in a layer-by-layer manner from the fixed end to the crushed end under the low-velocity crushing load. At this stage, the geometric parameters of the STH have changed. Therefore, the equivalent thick­ ness of the inclined cell wall is 3t, but that of the vertical ligament is still t. During the collapse process of the typical unit, two plastic hinges still appear at both ends of the inclined cell wall, and the relative rotational

Wp2 ¼ πbMp1 þ 2πbMp2

(11)

Where Mp2 is the fully plastic bending moment of the equivalent ver­

tical cell wall, Mp2 ¼ σys bðt=2Þ2 =4. During the collapse process of the equivalent inclined cell walls in a 7

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Fig. 12. Collapse modes of the STH typical unit under low-velocity crushing: (a) deformation processes of the first plateau stress phase, (b) collapse processes of the second plateau stress phase, (c) the initial configuration of the typical unit, (d) the initial configuration and densification configuration of the second plateau stress phase.

typical unit, the total energy absorbed by the plastic hingesWp3 is Wp3 ¼ 8Mp3 θ ¼ 2σ ys bð3tÞ2 θ

(12)

Where Mp3 is the fully plastic bending moment of the equivalent in­

clined cell wall, Mp3 ¼ σys bð3tÞ2 =4. It can be determined from Fig. 12(d) that the work performed by the crushing force E2 is � � � � t � 2t 9t sinθ þ (13) E2 ¼ σL2 L1 b Hf 1 Hf 2 ¼ 4l2 bσ L2 cosθ þ lsinθ lcosθ 2l

Where Hf 2 is the final height of a typical unit, L1 is the original width of the triangular grid structure, L1 ¼ 2lcosθ þ 2t=sinθ. Using the same methods described above, and according to E2 ¼ Wp1 þ Wp3 þ Ek , the second plateau stress ​ σ L2 of the STH can be obtained � �� �2 � � 3π 9θ t 7t þ σ þ ρs v 2 ys 2 l 4l 32 � � σ L2 ¼ � (14) � t 2t 9t cosθ þ lsinθ sinθ þ lcosθ 2l

Fig. 13. The low-velocity crushing strengths under the different relative den­ sities (v ¼ 2 m/s).

To verify the accuracy of the theoretical prediction formula for lowvelocity crushing strength, the theoretical predictions and simulation results for the two plateau stresses of STHs with three different relative densities under a crushing velocity of 2 m/s were compared, as shown in Fig. 13. It is evident that the theoretical predictions agree well with the numerical results. In addition, Table 1 compares additional theoretical and numerical results for the two plateau stresses for STHs and provides the relative errors. Under low-velocity crushing loads, the relative errors between the simulation results and theoretical predictions are <6%, which means that Eqs. (10) and (14) can accurately predict the two plateau stresses.

the low-velocity crushing analysis, two vertically connected unit cells were employed to estimate the high-velocity crushing strength of the STH, as shown in Fig. 14(b). Fig. 14(c) shows the configuration of the typical units at the initial state, with the upper cell joining the densified region while the lower cell remains at the original positions. In addition, the final state of the typical units shown in Fig. 14(d). The lower cell becomes the new front of the densified region, which represents the period at which the cell collapse is finished and the beginning of a new period. Therefore, T0 and Tf are the time of the start and finish of a collapse period, respectively. The σ2 and σ1 are the crushing strength acting on the supported and crushed end of the representative block, respectively, as shown in Fig. 14(b). Moreover, inertial effects become dominant under high-velocity crushing, the cell near the fixed end nearly retains original configuration, such that the σ2 should not be larger than the quasi-static crushing strength σ0. Thus, it is assumed that σ2 is a constant value σ 0 throughout the entire collapse period of the

3.2. High-velocity crushing strength Because the cells of the STH collapse in a layer-by-layer manner with a periodic process, the crushing strength can be predicted by analyzing the periodic collapse of the STH, as shown in Fig. 14(a). Different from 8

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Table 1 The low-velocity crushing strengths of the STH. Relative density ​ρ​

Crushing velocity v (m/s)

0.05

1 3 5 1 3 5 1 3 5 1 3 5

0.1 0.15 0.2

First plateau stress σL1 (MPa)

Relative errors (%)

Theoretical

Numerical

0.0120 0.0127 0.0142 0.0539 0.0555 0.0587 0.1414 0.1441 0.1497 0.2885 0.2927 0.3012

0.0122 0.0133 0.0149 0.0542 0.0554 0.0606 0.1401 0.1476 0.1519 0.2848 0.2804 0.2959

1.438 4.259 5.023 0.475 0.250 3.084 0.904 2.348 1.468 1.285 4.394 1.806

Second plateau stress σL2 (MPa) Theoretical

Numerical

0.1172 0.1186 0.1213 0.4880 0.4908 0.4966 1.1732 1.1777 1.1868 2.1795 2.1859 2.1986

0.1129 0.1144 0.116 0.4765 0.4845 0.5056 1.1928 1.2144 1.2262 2.1637 2.1508 2.1654

Relative errors (%)

3.827 3.666 4.604 2.410 1.310 1.787 1.647 3.022 3.213 0.731 1.631 1.534

Fig. 14. Collapse modes of the STH typical units under high-velocity crushing: (a) collapse processes of the typical units, (b) the initial configuration, (d) the densification configuration.

typical units [48]. It can be observed from Fig. 14(a) that the typical units have almost no obvious expansion in the x-direction. Thus, it is assumed that width L0 of the typical units remains at the original value during the collapse process. According to conservation of momentum, the different between the impulses of σ 1 and σ 2 over a collapse period is equal to the change of the momentum of the typical units in the y-direction [40], thus, Z Tf � T T bL0 ðσ1 σ2 Þdt ¼ bL0 ðσ 1 σ 2 Þ Tf T0 ¼ P1 f þ P2 f PT1 0 PT2 0 T0

T0, respectively. At the initial instant of a collapse period, except for cell walls AB, AM, LM and LH, the other cell walls remain in their initial position without obvious rotation and deformation, as shown in Fig. 14(c). The momentum of the cell I at T0, PT20 is (16)

PT2 0 ¼ 0

Because of the periodic and repeated collapse of the typical units, the momentum of the cell I at T0 is equal to that of the cell II at Tf, i.e., T

PT1 0 ¼ P2 f

(15)

(17)

By substituting Eqs. (16) and (17) into Eq. (14), the change in mo­ mentum can be obtained

T

Where P1f and PT10 represent the momentum of cell I at Tf and T0, T

respectively. P2f and PT20 represent the momentum of cell II at Tf and 9

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Z

Thin-Walled Structures 149 (2020) 106623

Tf

ðσ1

bL0 T0

(18)

T

σ2 Þdt ¼ ΔP ¼ P1 f

Because the cell walls of cell I move with a contain velocity v, considering that adjacent cells in the same layer share the same vertical wall, only half of the thickness for the vertical cell walls CD and IJ was employed to calculate the momentum of the typical units. In addition, the length of cell walls OM and FP is half that of other cell walls, and therefore, their final momentum can be calculated by . Tf T Tf T PCD ¼ PIJf ¼ POM ¼ PFPf ¼ ρs blvt 2 (19) Similar to the above method, the final momentum of other cell walls can be obtained T

T

T

T

T

T

T

T

T

T

T

T

f f PABf ¼ PBCf ¼ PBD ¼ PBEf ¼ PFEf ¼ PFG ¼ ρs blvt

f f f PHG ¼ PHIf ¼ PHJf ¼ PHLf ¼ PLM ¼ PAM ¼ ρs blvt

(20)

Combined with Eqs. (19) and (20), Eq. (18) is expressed as

Fig. 15. The high-velocity crushing strengths with different relative densities, (v ¼ 100 m/s).

(21)

ΔP ¼ 14ρs blvt

Because the height of the typical units change from H0 to Hf with a constant velocity in a collapse period, the collapse period T is T ¼ Tf

T0 ¼

H0

Hf v

2lcosθ ¼ v

7t

Table 2 The high-velocity crushing strengths of the STH.

(22)

Finally, by substituting Eqs. (18), (21) and (22) into Eq. (15), the high-velocity crushing strength, σH , can be derived

σH ¼ σ0 þ

7ρs t=l 7t=lÞð2cosθ

ð2cosθ

2

v sinθÞ

(23)

Since the quasi-static stress σ0 of the typical units should not be larger than the average low-velocity crushing strength ​ σ L , i.e., σ L is the upper limit value of σ0 . In addition, the deformation of the cell near the supported end is similar to that of the cell under low-velocity crushing. A proportional coefficient β was employed to estimate the average lowvelocity crushing strength [40], thus

σ L ¼ βσL1 þ ð1

7ρs t=l 7t=lÞð2cosθ

First plateau stress σH (MPa) Theoretical

Numerical

0.05

80 100 120 80 100 120 80 100 120 80 100 120

0.9721 1.4894 2.1225 2.1921 3.2992 4.6540 3.7569 5.5620 7.7709 5.6339 8.2203 11.3854

0.9613 1.5551 2.1938 2.301 3.3998 4.7715 3.8693 5.7744 7.9228 5.6949 8.1878 11.4744

0.15 0.2

(24)

βÞσL2

ð2cosθ

Crushing velocity v (m/s)

0.1

It can be seen from the stress-strain curves in Fig. 6 that the strain ranges corresponding to the two plateau phases are generally the same. However, the deformation of the cell near the supported end is more similar to the first plateau phase. Thus, β was taken to be 0.6 to calculate the average low-velocity crushing strength, and Eq. (23) is expressed as

σH ¼ σL þ

Relative density ​ρ​

sinθÞ

v2

Relative errors (%)

1.121 4.224 3.252 4.733 2.958 2.462 2.904 3.678 1.917 1.072 0.396 0.776

velocity under dynamic crushing loading [47]. To avoid the randomness of artificial selection, the method of energy absorption efficiency is widely used to determine the densification strain. In addition, the last maximum point on the energy efficiency-strain curve is defined as the initial point of the densification strain [28]. Thus, densification strain ​ εd can be described as:

(25)

dEðεÞ j ¼0 dε ε¼εd

Similar to the analysis of low-velocity crushing strength, a compar­ ison of the theoretical prediction and simulation results with different crushing velocities and relative densities is shown in Fig. 15 and Table 2, demonstrating good agreement and a deviation of less than 5%. Thus, it is shown that Eq. (25) can satisfactorily estimate the plateau stress of STH under high-velocity crushing loads.

(26)

Where EðεÞ is the ratio of the energy absorbed by the cellular material and the nominal stress, is expressed as R ε σðεÞdε EðεÞ ¼ 0 (27) σ ðεÞ

4. Discussions

Fig. 16(a) and (c) show the energy efficiency-strain curves of the STH and SH under different crushing velocities. The densification strains of the STH are greater than that of the SH under the low- and mediumvelocity model, as shown in Table 3. This characteristic is due to the buckling of the horizontal and vertical ligaments of the SH, which cause the star structure to appear to undergo unstable deformation during the compression process, which leads the honeycomb to enter the densifi­ cation stage more easily, as shown in Figs. 5 and 9. In addition, due to the unique structural stability of the triangular structure, the STH col­ lapses layer-by-layer after turning into a triangular grid, thereby entering the densification stage later. However, under the same relative

4.1. Densification strain With the collapse of the cell walls, large compressive strains cause the cell walls to pack together and the stress of the honeycombs to in­ crease rapidly. When the stress-strain curve increases rapidly, the cor­ responding strain is densification strain. Currently, certain controversies exist regarding the definition of densification strain. A subset of scholars thought that the densification strain of cellular materials could be defined as ​ εd ¼ 1 λρ ðλ ¼ 1:4Þ [14]. However, considerable research has shown that the densification strain is affected by the crushing 10

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Fig. 16. Energy efficiency and densification strain. (a–c) Energy efficiency-strain curves of the STH and SH under three different crushing velocities (relative density ​ ρ ¼ 0:15); (d) Effects of the cell-wall thicknesses and crushing velocities on the densification strain. Table 3 Densification strains and SEA of the STH and SH under three crushing models. UV (kJ/m3)

Crushing velocity V (m/s)

εd STH

SH

STH

SH

STH

SH

2 20 100

0.669 0.688 0.748

0.539 0.585 0.751

457 667 4440

213 335 4110

1.104 1.619 10.821

0.526 0.831 10.146

density, the densification strains of the STH and SH tend to be equal with increasing crushing velocity. It can be obtained that the densification strain of the honeycomb is affected by the topological structure and crushing velocity under low- and medium-velocity crushing but is only related to the relative density under high-velocity crushing (v > 100 m/ s). The densification strains of STH with different wall thicknesses under various crushing velocities are shown in Fig. 16(d), which shows that the densification strains increase gradually with the crushing velocity. In contrast, with increasing wall thickness, the densification strain de­ creases gradually. Therefore, the densification strain is related to the topology structure, crushing velocity, and wall thickness of the honeycombs.

Em (kJ/kg)

ESTH ESH ESH 109.89% 94.83% 6.64%

for these results is that the STH changes into a triangular grid structure and the equivalent cell wall thickness increases (Fig. 5). Moreover, the unit energy absorption efficiency of the STH under low-velocity crush­ ing gradually exceeds that of the SH under medium-velocity crushing. However, the inertial effect becomes dominant under high-velocity crushing, and the unit energy absorption efficiency of the honeycombs increases uniformly. The energy absorbed per unit volume (UV ) and unit energy absorption efficiency (Em ) of STH under different crushing modes are shown in Table 3. Notably, the unit energy absorption efficiency of the STH is 109.89% larger than that of the SH under low-velocity crushing (v ¼ 2 m/s), and 94.83% larger than that of the SH under medium-velocity crushing (v ¼ 20 m/s). However, the unit energy ab­ sorption efficiency of STH is only 6.64% larger than that of the SH under high-velocity crushing (v ¼ 100 m/s). Thus, the STH has better energy absorption ability under the low- and medium-velocity crushing models but shows little difference from the SH under the high-velocity crushing model. The unit energy absorption efficiency of the STH with different cellwall thicknesses under various crushing velocities are shown in Fig. 17 (b). It is indicated that the energy absorption capacity of the STH in­ creases with the increase of impact velocity and cell-wall thickness. However, when the crushing velocity v > vcr2 (48 m/s), the influence of the increase in cell-wall thickness on the energy absorption capacity of the STH is gradually weakened. This behavior is attributable to the fact that the inertial effect plays a dominant role under high-velocity crushing, and the cell-wall collapses layer-by-layer, which results in the crushing strength of STH being directly proportional to the square of the crushing velocity. Thus, the energy absorption capacity of STH is not only related to the topological structure, but also proportional to the cell-wall thickness and crushing velocity. By choosing these parameters

4.2. Energy absorption Due to the different geometrical dimensions of the STH and SH, the unit energy absorption efficiency Em was employed to compare their energy absorption capabilities, which can be defined as [28]: R εd σðεÞdε U Em ¼ ¼ 0 (28) m ρρs Where U is the total absorbed energy of the honeycomb, σðεÞ is the Rε nominal stress of the honeycomb, UV ¼ 0 d σ ðεÞdε is the energy absor­ bed per unit volume. Fig. 17(a) shows the absorbed energy per unit mass of the STH and SH under different crushing velocities. The STH has a higher unit energy absorption efficiency than the SH with the same relative density under different crushing velocities. It is especially notable that the Em strain curve of the STH increases significantly at a longitudinal strain εy ¼ 0:33 under the low- and medium-velocity crushing models. The reason 11

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Thin-Walled Structures 149 (2020) 106623

Fig. 17. Energy absorption capacity of honeycombs. (a) Comparison of SEA for the STH and SH under three crushing modes (relative density ​ ρ ¼ 0:15 ​ ); (b) effects of cell-wall thicknesses and crushing velocities on the energy absorption capacity.

properly, selected novel structures can be designed that can further improve the energy absorption capacity. Different auxetic honeycomb structures exhibit different mechanical properties and energy absorption capabilities. The specific energy ab­ sorption (SEA) of the STH is compared with those of other auxetic honeycombs in Fig. 18, where the 3-D auxetic honeycombs have supe­ rior energy absorption capacities compared to the 2-D auxetic honey­ combs, and the novel auxetic honeycombs with the same matrix material have excellent mechanical properties. Therefore, the energy absorption capacity of the honeycombs can be improved obviously by adding the traditional geometry structure into the classical auxetic honeycomb structure [50,51]. Specifically, the SEA of the re-entrant star-shaped honeycomb varies from 0.03 J/cm3 to 3 J/cm3, slightly larger than that of the star-arrowhead honeycomb, which varies from 0.024 J/cm3 to 1.79 J/cm3. For the STH, SEA varies from 0.0546 J/cm3 to 4.383 J/cm3, which is significantly higher than those of other hon­ eycomb structures, especially under low-velocity crushing. This differ­ ence is due to the fact that the load carrying capacity of the triangular structure is more prominent under low-velocity crushing loadings. In addition, metal auxetic honeycombs have superior load carrying ca­ pacities compared to polymer auxetic honeycombs.

4.3. Poisson’s ratio Based on the study of Huang et al. [39,40] on the variation of Pois­ son’s ratio for the auxetic honeycomb with the dynamic crushing pro­ cess, the Poisson’s ratio of the STH and SH under dynamic crushing loading was estimated by numerical simulation. As shown in Fig. 19(a), according to the displacement in the x-direction of the six representative symmetric nodes on both sides of the FE model, the transverse average contraction displacement ​ ΔL and dynamic Poisson’s ratio ​ γ of the honeycomb are obtained as �� ΔL ¼ Ax þ Bx þ Cx þ Dx þ Ex þ Fx þ A’x þ B’x þ C’x þ D’x þ E’x þ F’x 6

(28)

γ¼

εx ΔL=L ¼ εy εy

(29)

Where ΔL is the transverse average contraction displacement of the honeycomb; L is the initial length of the honeycomb in the x-direction; Ax , A’x etc. are the displacements of the symmetric nodes on both sides of the honeycomb in the x-direction. Fig. 19(b) shows the dynamic Poisson’s ratio (DPR) curves of the STH and SH under different crushing velocities. The DPRs of STH are less

Fig. 18. SEA of the STH compared with other auxetic honeycombs [49].

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Thin-Walled Structures 149 (2020) 106623

Fig. 19. Poisson’s ratio of the honeycombs. (a) Schematic diagram of lateral displacement measurement of the honeycombs; (b) dynamic Poisson’s ratios of the STH and SH under different crushing velocities. (relative density ​ ρ ¼ 0:15).

than those of SH under the crushing velocities of 2 m/s and 20 m/s. Therefore, the deformation process of the STH exhibits a more obvious NPR effect than that of SH under the low- and medium-velocity crushing models. Due to the influences of the inertia effect, the cells near the crushed end exhibit obvious non-homogeneous deformation under medium-velocity crushing, as shown in Fig. 9. Therefore, if the longi­ tudinal strain is less than 0.15, the DPR of STH under medium-velocity crushing is obviously larger than that under low-velocity crushing. In particular, the STH transforms into a triangular grid structure during deformation under the low- and medium-velocity crushing models, and the deformation models are similar. Thus, after the longitudinal strain εy > 0:33, the DPR of STH under low-velocity crushing is close to that under medium-velocity crushing. With the increase in crushing velocity, the cells collapse in a layer-by-layer manner from top to bottom, and do not have sufficient time for transverse contraction. Thus, the STH and SH tend to have larger DPR, demonstrating that the lateral contraction is weakened. However, after the longitudinal strain exceeds 0.15, trans­ verse contraction of the STH is still larger than that of SH under highvelocity crushing. In summary, the STH exhibits more obvious NPR ef­ fect than SH under dynamic crushing deformation and still shows lateral contraction under high-velocity crushing.

Fig. 21(a) plots the stress-strain curves of STHs with different cellwall angles under low-velocity crushing. As the incline angle θ de­ creases, the two plateau stresses of the STH obviously increase, but the second plateau stress disappears under the crushing velocity of 100m/s. This is due to the fact that the cell-wall thickness of the honeycomb increases with decreasing the incline angle under the same relative density, which results in an increase in the crushing strength of the STH. Due to the decrease of the incline angle θ, the rotation angle α of the plastic hinge at both ends of the cell-wall increase, as shown in Fig. 20. Thus, the time during which the honeycomb changes from a STH to a triangular grid increases, resulting in an increase in the first plateau stress region (d3 > d2 > d1 ) of STH, as shown in Fig. 21(a). Conversely, decreasing the incline angle causes the height of the triangle structure to decrease in the y-direction. The enhancement effect of the triangular structure on the mechanical properties of the STH is weakened, so the second plateau region decreases or even disappears completely. How­ ever, the variations of incline angle have no significant effect on the crushing strength of the STH under the high-velocity crushing, as shown in Fig. 21 (b). According to the above-mentioned analysis, the influences of the incline angles on the energy absorption capacity of the STH were investigated under low-and medium-velocity crushing, as shown in Fig. 22. It is clearly that the energy absorption change trends of the STHs with different angles are the same under both crushing modes. When the compressive strain εy < 0:33, the energy absorption capacity of the STH increases with increasing incline angle. However, after εy > 0:33, the εy value corresponding to the honeycomb becoming a triangular grid is different due to the increase in the cell-wall angle α, yielding a phe­ nomenon in which the energy absorption curves alternately increase. Finally, as the incline angle decreases, the energy absorption capacity of the STH firstly increases and then decreases. Additionally, the unit en­ ergy absorption efficiency of STH-1 under low-velocity crushing is larger

4.4. Effect of incline angle The characteristics of the cell micro-structure of an STH, i.e., the incline angle and ligament length, affect its dynamic crushing and en­ ergy absorption capacity. From the STH geometry (Fig. 1), it can be seen that the ligament length is determined by the incline angle. Therefore, three micro-structures with different incline angles (θ ¼ 30� , 22.5� , 15� ) were systematically investigated while keeping the cell wall length l and the relative density ρ constant. These micro-structures are depicted in Fig. 20.

Fig. 20. Three different micro-structures of STH. (a) θ ¼ 30� , (b) θ ¼ 22.5� , (c) θ ¼ 15� . 13

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Thin-Walled Structures 149 (2020) 106623

than that of the STH under medium-velocity crushing. Thus, it is evident that varying the incline angle can effectively improve the dynamic crushing responses and energy absorption capabilities of STHs. 5. Conclusions In this study, a novel STH structure in which the horizontal and vertical ligaments of the SH were replaced by triangular structures was developed to improve the in-plane crushing performance. The defor­ mation behaviors of STH under different crushing velocities were studied based on 1D shock theory and numerical simulation. In partic­ ular, when εy � 0:33, the STH exhibited obvious X- and V -shaped deformation bands under low-velocity crushing, then the cells changed into triangular lattices. In addition, an empirical formula for the critical crushing velocity that can distinguish different deformation modes was derived. Three types of deformation modes (G-mode, T-mode, L-mode) for the STH were distinguished under different crushing velocities. Theoretical expressions for the plateau stress were predicted based on the plastic energy consumption theory and conservation of momentum under low- and high-velocity crushing, as shown in Eqs. (9), (13) and (24), which are functions of matrix material, crushing velocity, and geometric topology. Moreover, the theoretical formulas can be used to estimate the crushing strengths of STH accurately. According to the comparative energy absorption analysis, the STH has a higher energy absorption capacity than the SH under dynamic crushing loadings. Specifically, unit energy absorption efficiency of STH is 109.89% greater than that of the SH under low-velocity crushing and 94.83% greater under medium-velocity crushing. However, no signifi­ cant enhancement under high-velocity crushing was observed. The geometric parameters analysis indicated that the energy absorption capacity significantly increases with increasing cell-wall thickness and crushing velocity but firstly increases and then decreases with increasing incline angle. Furthermore, the densification strain of the STH is larger than that of the SH, which increases as the crushing velocity increases or the cell-wall thickness decreases. However, the effects of the geometric parameters on the mechanical properties of the STH become weaker as the crushing velocity increases. In addition, the numerical simulation showed that the STH has a more obvious NPR effect than the SH, which mainly appears in the first plateau stress phase. In summary, the STH can significantly enhance the crushing strength, energy absorption capacity, and densification strain while maintaining the NPR. This design offers a new means of improving the mechanical properties of honeycombs so that they can be more widely used. However, as the matrix material was assumed to be a perfectly elastic-plastic material, the strain rate and buckling in the plastic phase were ignored in this study. In the future, an accurate theoretical model that includes the elastic-plastic properties and buckling should be established to predict the crushing strengths of the STH. Furthermore, the dynamic crushing strengths with different cell-wall length ratios should be studied to evaluate the energy absorption capacities of the STH.

Fig. 21. The effects of the incline angles on crushing strength. (a) v ¼ 2 m/s, (b) v ¼ 100 m/s.

CRediT authorship contribution statement Lulu Wei: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Qiang Yu: Formal analysis, Data curation, Project administration, Funding acquisition, Resources, Supervision. Guohua Zhu: Methodology, Software, Visualization. Acknowledgements This work is supported by National Key R&D Program of China (2017YFC0803904), National Natural Science Foundation of China (51905042).

Fig. 22. The effects of the incline angles on energy absorption capacity.

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Appendix A. Supplementary data

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