Abstract

In order to further improve the bending performance of the traditional re-entrant (RE) honeycomb, a novel auxetic honeycomb architecture, called RE-L honeycomb, was proposed by adding an additional link-wall structure to the RE cell. The bending behaviors of the novel RE-L honeycomb, including the properties under linear elastic deformation and the bending behaviors under large deformation, were comprehensively investigated by the analytical, numerical, and experimental models. Results show that the proposed RE-L honeycomb significantly improves the bending compliance in the x-direction due to the highly flexible performance of the additional structure, where the bending rigidity and the maximum bending force are only 23% and 29.4% of those of the RE honeycomb, respectively. Besides, the additional structure obviously improves the designability and orthotropic properties of the original auxetic honeycomb. In conclusion, the proposed RE-L shows improved bending performance, which deserves more attention in future research and related applications.

Introduction

Honeycomb structures, a kind of ordered cellular structure with lightweight, high strength, and many other excellent properties, have attracted extensive research from scholars during the last decades [13]. As a kind of typical artificial structure, honeycomb structures can be easily designed by adjusting the basic unit cell to obtain different mechanical properties [4]. Especially, by adjusting its concave angle, scholars developed a novel structure, named as re-entrant (RE) honeycomb, which demonstrates an unusual auxetic property, i.e., the so-called negative Poisson's ratio (NPR) effect [5,6]. Since Lakes first synthesized NPR foam [7], NPR materials and structures have attracted widespread concern [8]. It has been demonstrated that, compared with conventional honeycombs, the RE honeycombs have many superiorities in energy absorption [9], impact resistance [10,11], and shear modulus [12], which have been widely used in many fields such as automotive [13], architecture [14], and medical equipment [15]. Considering the outstanding properties of NPR, a great deal of research has been carried out on the RE honeycombs, such as their in-plane elastic constants [16] and energy absorption performance [17] related to compressional properties. However, studies on the analysis and improvement of their out-of-plane bending performance are still limited. The bending behaviors of the honeycombs may have a closed relationship with some potential applications, such as medical stents [18], morphing wings [19,20], soft robotics [21], and other fields requiring specific bending properties. Therefore, it is of great significance to carry out further research on exploring and improving the bending performance of the RE honeycomb.

Mechanical properties of the artificial structures should be specially designated to match the requirements of the engineering applications. For the traditional hexagonal (HEX) and RE honeycomb structures, adjusting the geometry parameters, such as decreasing their cell thickness, can indeed increase their bending compliance, while it may also affect the stiffness in other directions, and produce irreconcilable results. For example, a well-designed stent always requires both high axial flexibility and high radial support performance [22,23], and it is difficult to balance these properties by simply adjusting the geometric parameters. In view of this, it is necessary to solve this problem from the perspective of the innovative design of cell topologies, which has been widely proved to be an effective approach to improve the honeycomb properties [2426]. A series of novel auxetic cells have been proposed to improve their in-plane stiffness, strength, or NPR effect by additional cell walls [27], hierarchical designs [28], hybrid designs [29,30], and gradient designs [31,32]. For the bending properties, however, the relevant innovative designs are relatively limited. Huang et al. [33] proposed a novel zero Poisson’s ratio cellular structure by combining a HEX/RE honeycomb and a thin plate and found that the novel design has great bending compliance and highly tunable mechanical properties. Jiang et al. [34] studied a novel tubular lattice architecture by experiments and simulations and found that the proposed structure exhibits a more compliant behavior compared to the conventional diamond structure. These studies have exhibited huge potential of microstructure topology design in improving structure bending compliance, and developing more novel cells with auxetic effect, enhanced bending performance, and tailorable mechanical properties is greatly expected in the following research.

In order to better guide engineering applications, exploration of the relationships between the microstructures of the honeycombs and their macroscopic mechanical properties is of great significance. Many works have been carried out to clarify the mechanical properties of the honeycombs by the theoretical, numerical, and experimental methods [3537]. Gibson and Ashby [6] gave the in-plane elastic constants of the HEX honeycombs, and some other scholars derived and analyzed the elastic constants of the RE honeycombs [38], star-shaped honeycombs [39], and double arrowhead honeycombs [40]. Nevertheless, the in-plane elastic constants cannot be directly used to characterize or calculate the out-of-plane bending properties [41], and the research on the theoretical expressions of the honeycomb flexural rigidities is still relatively limited. Abd El-Sayed et al. [42] proposed a method for calculating the bending curvature, while it has been proved to be with relatively low precision. Chen [43,44] analyzed the bending and torsional deformations of the cell walls and established a theoretical model to predict the flexural rigidities of the honeycombs, which has been well verified by the numerical method. Systematic analysis of the bending behaviors of the auxetic honeycombs, including the orthotropic performance, parameter influence, and the bending behaviors under large deformation, is still greatly lacking.

In this work, inspired by the link/bridge structure of medical stents, an important factor affecting the axial flexibility of the stents [23,45], a novel auxetic structure called RE-L honeycomb is developed by adding an additional link-wall structure to the traditional RE cell. We first derive the theoretical expression of the equivalent bending rigidities of the traditional RE honeycomb and the proposed RE-L honeycomb. Then, we establish the numerical models to verify the linear elastic deformation behaviors of the honeycombs and to investigative their three-point bending deformation behaviors. Further, by the 3D printing specimens, we carry out the three-point bending experiments to verify the established analytical and numerical models. By using the validated models, a systematic work is carried out to analyze the bending behaviors of the RE and RE-L honeycombs in the two orthogonal directions and the geometric effects on those properties. Through the analysis in this work, we aim to highlight the great superiority of the proposed structure in out-of-plane bending performance and to provide some theoretical basis for its further study and application.

Models and Methods

Geometry.

Figure 1 shows the geometric configurations of the conventional RE honeycomb and the proposed RE-L honeycomb. As indicated in Fig. 1(a), the RE cell is composed of eight cell walls, including four inclined walls, length l0; two major horizontal walls, length l1; and two connecting walls, length l1/2. Besides, it also includes the following geometry parameters: the cell height h, the cell length s1, the angle θ0, the cell wall thickness t, and the out-of-plane width b. To simplify the model, the value of the cell height h is fixed, and the value of the length l1 is set to twice the value of l0. According to the geometric relationship, the following expressions can be obtained as follows: l0 = h/(2sin θ0) and s1 = (2 − cos θ0)h/sin θ0. Hence, geometry of the RE cell can be uniquely defined by the parameters h, θ0, t, and b.

Fig. 1
Geometric configurations: (a) configuration of the RE cell, (b) configuration of the adding structure, and (c) configuration of the proposed RE-L cell
Fig. 1
Geometric configurations: (a) configuration of the RE cell, (b) configuration of the adding structure, and (c) configuration of the proposed RE-L cell
Close modal
The proposed RE-L honeycomb, as shown in Fig. 1(c), is constructed by adding double-V-shaped link-wall structures between two RE cells, and the added structures have the centrosymmetric property to facilitate interconnection during periodic arrangement, as shown in Fig. 1(b). On the basis of the parameters of the original RE structure, two additional parameters including the length l2 and the angle θ1 are specifically defined, as shown in Fig. 1(b). For the RE-L cell, the cell length s2 can be expressed as follows: s2 = s1 + 4l2cos θ1. The relative density of the RE-L honeycomb, i.e., the ratio of actual cell wall volume to its occupied space, can be expressed as follows:
ρr=t(4l0+2l1+4l2)hs2=th4h+4l2sinθ0(2cosθ0)h+4l2sinθ0cosθ1
(1)

Analytical Model of Bending Properties.

For the bending deformation of a plate, three flexural rigidities, Dx, Dy, and D1, are used to relate the applied uniformly distributed moments Mx¯ and My¯ with curvatures 1/ρx and 1/ρy [46]:
Mx¯=Dx1ρx+D11ρy,My¯=Dy1ρy+D11ρx
(2)

The Kirchhoff hypothesis is usually used to estimate the flexural rigidities of a homogeneous plate from its in-plane elastic constants, while its predictive applicability may be greatly reduced for a honeycomb plate [41], and the deformation of cell walls should be considered in the analysis of the bending deformation of a honeycomb plate. Here, we established force analysis models of the RE cell and the proposed RE-L cell, and consequently, derived their flexural rigidities based on the energy equivalence principle, as shown in Fig. 2.

Fig. 2
Analytical model of the equivalent bending rigidities: (a) schematic diagram of the energy equivalence principle, (b) moments loading on the RE cell, (c) moments loading on the representative walls of RE cell, (d) moments loading on the RE-L cell, and (e) moments loading on the representative walls of the adding structure
Fig. 2
Analytical model of the equivalent bending rigidities: (a) schematic diagram of the energy equivalence principle, (b) moments loading on the RE cell, (c) moments loading on the representative walls of RE cell, (d) moments loading on the RE-L cell, and (e) moments loading on the representative walls of the adding structure
Close modal
For a homogeneous plate with in-plane length L1, L2, out-of-plane width b, as shown in Fig. 2(a), its elastic strain energy can be expressed as follows:
ue=120L20L1[Dx(2wx2)2+2D12wx22wy2+Dy(2wy2)2+4Dxy(2wxy)2]dxdy
(3)
where w is the bending deflection; Dx, Dy, and D1 are the three flexural rigidities of the plate; and Dxy is the torsional rigidity. When applying the uniformly distributed bending moment Mx¯ and My¯ to the plate with a fixed point (x0, y0), the theoretical solution of its bending deflection w can be given as follows [44]:
w=C0+C1x2+C2y2
(4)
Here,
C1=DyMx¯D1My¯2(DxDyD12),C2=DxMy¯D1Mx¯2(DxDyD12),C0=C1x02C2y02
(5)

In this work, the center point of the honeycomb structure is set as the coordinate origin, and it is fixed in the analytical model, i.e., (x0, y0) = (0, 0). The horizontal and vertical directions that intersect the coordinate origin are the corresponding x- and y-axis.

Bringing Eq. (4) into Eq. (3), elastic strain energy of the plate only bearing bending moment without torque can be written as follows:
ue=L1L22(DxDyD12)(DyMx¯22D1Mx¯My¯+DxMy¯2)
(6)
For the RE cell, a 1/4 model is used to analyze its bending deformation, as shown in Figs. 2(b) and 2(c). It needs to be noted that the symbol of double arrow indicates a uniformly distributed load at the far end, and the symbol of single arrow indicates a concentrated load. According to Fig. 2(c), the bending moments and the torsional moments on the RE cell walls can be obtained as follows:
M1=M3=12hMx¯,T1=T3=0,M2=12s1sinθ0My¯12hcosθ0Mx¯,T2=12s1cosθ0My¯+12hsinθ0Mx¯
(7)
The relative rotation angle of a cell wall under the bending or torsional moment can be expressed as follows [43]:
θM=12lEstb3M=qMM,θT=12qτl2Est3b2T=qTT
(8)
where Es, t, and b are the Young’s modulus, thickness of the cell wall, and width of the cell wall; qM and qT are defined as the coefficients related to the geometry of the cell wall; and qτ is the torsion coefficient depending on the value of b/l [43], which should meet the condition of 0.1 ≤ b/l ≤ 1.25. Consequently, the relative rotation angle of the cell wall 1, 2, and 3 can be calculated as follows:
θM1=θM3=qM1hMx¯2,θT1=θT3=0,θM2=qM2(s1sinθ0My¯hcosθ0Mx¯)2,θT2=qT2(s1cosθ0My¯+hsinθ0Mx¯)2
(9)
where
qM1=qM3=12l1Estb3,qM2=12l0Estb3,qT2=12qτ2l02Est3b2
(10)
The strain energy of a single wall can be calculated as follows:
u=12θMM+12θTT
(11)
Consequently, the total strain energy of the RE cell can be expressed as follows:
ua1=4(u1+u2+u3)=Nx1Mx¯2+Ny1My¯2+Nxy1Mx¯My¯
(12)
where
Nx1=qM1h2+12qM2h2cos2θ0+12qT2h2sin2θ0,Ny1=12qM2s12sin2θ0+12qT2s12cos2θ0,Nxy1=s1hsinθ0cosθ0(qT2qM2)
(13)
Similarly, for the RE-L, as shown in Figs. 2(d) and 2(e), the bending moments and the torsional moments on the cell walls 4, 5, and 6 of the RE-L cell can be obtained as follows:
M4=M5=M6=hcosθ1Mx¯,T4=T5=T6=hsinθ1Mx¯
(14)
Consequently, the strain energy of the cells can be obtained as follows:
u4=u6=12qM4h2cos2θ1Mx¯2+12qT4h2sin2θ1Mx¯2,u5=12qM5h2cos2θ1Mx¯2+12qT5h2sin2θ1Mx¯2
(15)
where
qM4=12l2Estb3,qT4=12qτ4l22Est3b2,qM5=24l2Estb3,qT5=48qτ5l22Est3b2
(16)
The total strain energy of the RE-L cell can then be obtained as follows:
ua2=4(u1+u2+u3)+u4+u5+u6=Nx2Mx¯2+Ny2My¯2+Nxy2Mx¯My¯
(17)
where
Nx2=qM1h2+12qM2h2cos2θ0+12qT2h2sin2θ0+qM4h2cos2θ1+qT4h2sin2θ1+12qM5h2cos2θ1+12qT5h2sin2θ1,Ny2=12qM2s22sin2θ0+12qT2s22cos2θ0,Nxy2=s2hsinθ0cosθ0(qT2qM2)
(18)
According to the energy equivalence principle ua = ue, the following coefficient correspondence can be obtained as follows:
Nx=shDy2(DxDyD12),Ny=shDx2(DxDyD12),Nxy=shD1DxDyD12
(19)
where the symbol s should be especially designated as s1 or s2 for RE or RE-L honeycomb. Therefore, the flexural rigidities can be obtained as follows:
Dx=2Nysh4NxNyNxy2,Dy=2Nxsh4NxNyNxy2,D1=Nxysh4NxNyNxy2
(20)

Under the given parameter of h = 15 mm, b = 4 mm, t = 0.4 mm, θ0=60deg, θ1=75deg, and l2 = 8 mm, values of partial variables and three flexural rigidities have been given in Table 1.

Table 1

Values of partial variables and three flexural rigidities

VariablesqM1 (N−1mm−1)qM2 (N−1mm−1)qT2 (N−1mm−1)qM4 (N−1mm−1)qM5 (N−1mm−1)qT4 (N−1mm−1)qT5 (N−1mm−1)Dx (N · mm)Dy (N · mm)D1 (N · mm)
RE1.177 × 10−45.883 × 10−50.00293789.893985.24−3496.86
RE-L1.177 × 10−45.883 × 10−50.00295.434 × 10−51.087 × 10−40.00260.0060209.62677.95−146.659
VariablesqM1 (N−1mm−1)qM2 (N−1mm−1)qT2 (N−1mm−1)qM4 (N−1mm−1)qM5 (N−1mm−1)qT4 (N−1mm−1)qT5 (N−1mm−1)Dx (N · mm)Dy (N · mm)D1 (N · mm)
RE1.177 × 10−45.883 × 10−50.00293789.893985.24−3496.86
RE-L1.177 × 10−45.883 × 10−50.00295.434 × 10−51.087 × 10−40.00260.0060209.62677.95−146.659
Combining the aforementioned three flexural rigidities with Eq. (2), the equivalent bending rigidity of the honeycombs when only one of Mx¯ or My¯ acting can be expressed as follows:
EIx=Mx¯1ρx=DxD12Dy,EIy=My¯1ρy=DyD12Dx
(21)

Finite Element Models.

Linear static finite element (FE) models were established through the commercial software abaqus/standard to verify the derived theoretical models. For the RE honeycomb specimen and the RE-L honeycomb specimen, as shown in Figs. 3(a) and 3(b), their center points are fixed to move and rotate, and concentrated moments are applied to each edge node to simulate the distributed bending moments Mx¯ and My¯. The marked points in Figs. 3(a) and 3(b), i.e., P1–P5, are selected to record the deflections of the models at the specific locations. To balance the simulation accuracy and the computational consumption, four-node curved shell elements with five integration points are used to model the cell walls. Sensitivity analyses have been carried out to determine the element size and the cell number, in which the deflections of P1 and P2 are used as the reference. The analysis results show that when the cell height is set to 15 mm, the element size of 1.5 mm is optimal, and the cell number is accordingly set to 5 × 8. Considering that the traditional aluminum material has been widely used in various studies of honeycomb structures [17,47], the material of aluminum, with the Young’s modulus of 69 GPa and Poisson’s ratio of 0.3, is selected as the major material for the simulations.

Fig. 3
Numerical models for simulating the linear elastic bending behaviors of (a) RE honeycomb and (b) RE-L honeycomb
Fig. 3
Numerical models for simulating the linear elastic bending behaviors of (a) RE honeycomb and (b) RE-L honeycomb
Close modal

Further, we established the quasi-static three-point bending FE model through the commercial software abaqus/explicit to study the bending behaviors of the RE honeycomb and the RE-L honeycomb under the large deformation. As shown in Fig. 4, the honeycomb specimen is freely placed on two fixed cylindrical walls, and an additional cylindrical wall, which is placed above the center of the specimen, moves down at a constant velocity. These three cylindrical walls are all set as rigid walls, and the honeycomb specimen is discretized by the shell elements. Besides, three reference points (RPs) are defined for the three rigid cylindrical walls, and the boundary conditions are all imposed through the RPs. The force curves can be obtained by recording the reaction force on RP1. For the above parts, the standard self-contact algorithm is adopted to simulate the potential contact behaviors. The matrix material here is also set as the aluminum with the density of 2700 kg/m3 and the yield strength of 76 MPa to simulate the plastic behaviors under large deformation. Besides, in the experiment analysis, the material of the numerical model is set as the nylon, and the mechanical properties of the nylon are described in the next section. The focus of this simulation is mainly on the bending behaviors, and therefore, the potential material fracture is ignored. For the explicit model, the sensitivity analysis of the mesh size and the loading velocity has been carried out, considering that too rough mesh or too fast loading speed will reduce the accuracy of the results. According to the analysis results, the element size of 1.5 mm under the cell height of 15 mm is considered to be optimal, and the loading velocity is set to 10 mm/s. Besides, the ratio of the kinetic energy to the internal energy, which should not exceed 5%, is monitored to ensure low dynamic effects.

Fig. 4
Numerical models for simulating three-point bending behaviors of (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction
Fig. 4
Numerical models for simulating three-point bending behaviors of (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction
Close modal

Experimental Setup.

In order to further compare the bending behaviors of the RE and RE-L honeycombs structures, and to verify the established FE models, prototypes of the honeycombs were constructed by the multi jet fusion [48,49], a kind of 3D printing process, and groups of quasi-static three-point bending tests were carried out to capture the force–displacement curves. First, for the printing matrix material nylon, a group of tensile tests were carried out to determine its material properties, as shown in Figs. 5(b) and 5(c). A universal testing machine was used to load the specimen, and a group of extensometers and a force sensor were used to capture the tensile displacement and force. Three independent test results are shown in Fig. 5(c), and accordingly, the elastic modulus and the yield strength can be determined as 1.1 GPa and 40 MPa, respectively.

Fig. 5
Setup of the three-point bending experiment: (a) bending test setup, (b) nylon samples tested, (c) stress–strain curves of the tensile tests, (d) and (e) RE honeycomb specimen in x- and y-directions, and (f) and (g) RE-L honeycomb specimen in x- and y-directions.
Fig. 5
Setup of the three-point bending experiment: (a) bending test setup, (b) nylon samples tested, (c) stress–strain curves of the tensile tests, (d) and (e) RE honeycomb specimen in x- and y-directions, and (f) and (g) RE-L honeycomb specimen in x- and y-directions.
Close modal

The three-point bending tests were carried out by a universal testing machine with a pair of three-point bending fixture. The fabricated specimen is symmetrically placed on the bending fixture, and an indenter presses the specimen downward at a uniform speed, as shown in Fig. 5(a). For reducing the impact of dynamic effects, the loading velocity is set to 2 mm/min. The span length of the bending fixture Ls is set to 160 mm, and the maximum deflection is set to 15 mm. The continuous bending reaction force is recorded by a high-precision force sensor. The fabricated specimens are shown in Figs. 5(d)5(g), where the parameters of the honeycombs are unified as h = 15 mm, θ0=60deg, b = 10 mm, and t = 1 mm for the RE and RE-L honeycombs, and θ1=75deg and l2 = 8 mm for the RE-L honeycomb.

Results and Discussion

Analytical Analysis of Bending Properties

Analytical Model Validation.

Figure 6 shows the deformation modes of the RE and RE-L honeycombs under the bending loads. Especially, for the traditional RE honeycomb, the parameter θ0 is set to 120deg, 90deg, and 60deg, and accordingly, the configurations of their unit cells become HEX cell, rectangle (REXT) cell, and RE cell, respectively, as shown in Figs. 6(a)6(c). For the HEX cell, the value of l1 is set to 1.5 times of l0 to avoid excessive cell length. Besides, other geometric parameters are set as follows: h = 15 mm, b = 4 mm, t = 0.4 mm, θ1=75deg, and l2 = 8 mm, and the distributed moment loads of all structures are set as follows: Mx¯=1N and My¯=1N. According to the existing studies [4,6], the Poisson’s ratios of the HEX honeycomb are positive in two orthogonal directions, and the Poisson’s ratios of the REXT honeycomb are equal to zero in one direction and almost negative infinity in another. For the HEX honeycomb, as shown in Fig. 6(a), its deformation mode exhibits a clear saddle shape, i.e., the directions of the bending curvatures are opposite in two orthogonal directions. The deformation mode of the REXT honeycomb exhibits a valley shape, as shown in Fig. 6(b). It can be found that the deflections of the points along the x-axis at the REXT honeycomb, with the maximum value of about 4.2 mm, are significantly larger than that of the points along the y-axis, with the value close to 0. For the RE honeycomb, as shown in Fig. 6(c), its deformation mode exhibits a dome shape, i.e., the directions of the bending curvatures are the same in both the orthogonal directions. Finally, for the RE-L honeycomb, as shown in Fig. 6(d), its deformation is the combination of the valley shape and the dome shape. Especially, the deflections of the points along the x-axis, with the maximum value of about 32.5 mm, are greatly larger than that of the points along the y-axis, while the latter also shows a certain radian, rather than a flat valley like the REXT honeycomb. This may be because although the RE-L honeycomb shows the auxetic properties, causing a dome deformation, its bending rigidity in the x-direction is significantly lower than that in the y-direction, which makes the bending curvature in the x-direction greatly larger than another.

Fig. 6
(Left) Deformation modes of the honeycombs from the numerical model, (middle) deformation modes of the corresponding homogeneous plate from the analytical model, and (right) deflection errors of the selected five points. (a) HEX honeycomb, (b) REXT honeycomb, (c) RE honeycomb, and (d) RE-L honeycomb.
Fig. 6
(Left) Deformation modes of the honeycombs from the numerical model, (middle) deformation modes of the corresponding homogeneous plate from the analytical model, and (right) deflection errors of the selected five points. (a) HEX honeycomb, (b) REXT honeycomb, (c) RE honeycomb, and (d) RE-L honeycomb.
Close modal
It can be found from Fig. 6 that the honeycomb deformation modes from the analytical models are well consistent with those from the numerical models. Further, the deflection error is defined as follows:
Error=|wnumericalwanalyticalwnumerical|
(22)

As shown in Fig. 6, the statistical results of the errors of the deflections are all lower than 12.1%, which quite verifies the established analytical models. Besides, some existing references associated with the deformation modes of the honeycombs also corroborated the results of this work [5052]. In order to further confirm the effectiveness of the established modes, statistical investigation of the deflection errors has also been performed, as shown in Table 2. It can be found that under the given parameter settings, the errors between the numerical model and the analytical model are mainly about 6%, and the maximum error is about 11.1%. The errors here may be from the estimation of the torsion angle of the loaded cell wall in the analytical expressions. Besides, in the  Appendix, the NPR effect of the RE-L was discussed and confirmed by the in-plane numerical analysis and the bending mode analysis.

Table 2

Deflections of the point P1 and P2 from the numerical and analytical models

No.t (mm)θ0()θ1()l2 (mm)P1P2
wanalytical (mm)wnumerical (mm)Errorwanalytical (mm)wnumerical (mm)Error
10.6607580.0360.0390.0640.7250.7940.086
20.6457580.0830.0900.0710.8940.9640.072
30.6757580.0130.0140.0580.6500.7200.097
40.4607580.1210.1270.0402.4352.6040.065
50.8607580.0150.0170.0910.3080.3470.111
60.6604080.0460.0490.0590.6590.7260.092
70.6608080.0350.0370.0620.7000.7620.081
80.6607560.0350.0360.0380.5210.5750.095
90.66075100.0380.0400.0620.9511.0360.082
No.t (mm)θ0()θ1()l2 (mm)P1P2
wanalytical (mm)wnumerical (mm)Errorwanalytical (mm)wnumerical (mm)Error
10.6607580.0360.0390.0640.7250.7940.086
20.6457580.0830.0900.0710.8940.9640.072
30.6757580.0130.0140.0580.6500.7200.097
40.4607580.1210.1270.0402.4352.6040.065
50.8607580.0150.0170.0910.3080.3470.111
60.6604080.0460.0490.0590.6590.7260.092
70.6608080.0350.0370.0620.7000.7620.081
80.6607560.0350.0360.0380.5210.5750.095
90.66075100.0380.0400.0620.9511.0360.082

Parameter Effects.

To further investigate the bending properties of the RE and RE-L honeycombs in two orthogonal directions, the parameter effects on the equivalent bending rigidities have been studied in this section, as shown in Fig. 7. In the analysis, the baseline settings of the parameter h, t, b, θ1, and l2 are 15 mm, 0.6 mm, 8 mm, 60deg, 75deg, and 8 mm, respectively. The uniformly distributed bending moment Mx¯ and My¯ are set to 1 N. As shown in Fig. 7(a), with the parameter θ0 increasing, the equivalent bending rigidity of the RE honeycomb in the x-direction shows a trend of negative correlation, and its rigidity in the y-direction shows a positive correlation. This phenomenon may be because that with the θ0 increasing, the inclined cell wall shows less deformation energy under the moments My¯, which makes the honeycomb more difficult to bend. For the RE-L honeycomb, effects of the θ0 on its equivalent bending rigidities show a similar behavior with that of the RE honeycomb. By comparison, the RE-L honeycomb has a greatly smaller bending rigidity in the x-direction than the RE honeycomb, only 23% of the latter when θ0=60deg, which is mainly because the additional structure enhances its bending flexibility. Besides, it can be found that the rigidity of the RE-L in the y-direction has a small decline compared with the RE honeycomb, which can be because that the additional structure increases the overall length of the RE-L cell. Further, we analyzed the orthotropic behavior of the bending rigidities with the parameter θ0 increasing, as shown in Fig. 7(b). It can be found that increasing the parameter θ0 can help to improve the orthogonality of the two honeycombs. Especially, for the RE-L honeycomb, the bending rigidity exhibits more significant orthogonality, with the maximum value of EIy/EIx up to about 12.4, while that of the RE honeycomb is only about 3.7. In addition, effects of the parameter t are given in Figs. 7(c) and 7(d). As expected, with the parameter t increasing, the bending rigidities of the honeycombs are all increasing with different rates. However, the parameter seems to have a little influence on the orthotropic behavior, as shown in Fig. 7(d), that is, it may not be effective to adjust the parameter t for improved bending properties only in one direction.

Fig. 7
Parameter effects on the equivalent bending rigidities and the orthotropic properties: (a) and (b) parameter θ0, (c) and (d) parameter t, (e) parameter θ1 and (f) parameter l2
Fig. 7
Parameter effects on the equivalent bending rigidities and the orthotropic properties: (a) and (b) parameter θ0, (c) and (d) parameter t, (e) parameter θ1 and (f) parameter l2
Close modal

For the RE-L honeycomb, the effects of its additional in-plane parameter l2 and θ1 have been depicted in Figs. 7(e) and 7(f). Effects of the parameter θ1 on the two rigidities are opposite, that is, with the θ1 increasing, the rigidity in the x-direction has a huge decline with the ratio up to 67%, while the rigidity in the y-direction shows an increasing trend with the ratio about 60.2%. With the parameter l2 increasing, the rigidities in both x- and y-directions have a certain decline, and the former shows a more obvious decrease with the rate about 39.2%. Besides, it can also be found that both of the two parameters have a positive effect on the orthogonality of the RE-L honeycomb. The aforementioned analysis not only demonstrates the improvement of the proposed RE-L honeycomb in the bending compliance but also clarifies the tailorability of its bending property.

Bending Properties Under Large Deformation

Experiment Results.

Figure 8 shows the comparison results of the force–displacement curves from the numerical, experimental, and analytical models. It should be noted that the matrix materials of the different models are all nylon. The analytical curves are obtained by the following expression [53]:
W=FLs348LEI
(23)
where W and F are corresponding to the displacement and force, respectively, and L is the specimen length in the orthogonal direction different from the loading direction. It can be found that within the deflection of 15 mm, the force–displacement curves of the specimens, including the numerical and experimental curves, have a similar trend and are approximately linear, which may be due to the fine elasticity of the selected nylon material. Nevertheless, it is still valuable to use the existing experimental data to verify the numerical and analytical modes. It can be found from Figs. 8(a) and 8(b) that the reaction force of the RE-L honeycomb is significantly lower than that of the RE honeycomb when loading in the x-direction, while in the case of the y-direction loading, as shown in Figs. 8(c) and 8(d), their difference is not clear. Besides, for the RE-L honeycomb, the reaction force difference between the two directions is obvious, which just further verifies the significant orthotropic of the RE-L honeycomb. Taking the average value of two experimental results as a reference, the errors of the curve slopes of the numerical results in the two directions are about 6.5% and 5.4%, respectively, for the RE honeycomb, and 1.1% and 3.2%, respectively, for the RE-L honeycomb. The errors here may be due to the printing deviation of the specimens and test measurement error. Similarly, the errors of the analytical prediction are about 14.3% and 10.5%, respectively, for the RE honeycomb, and 17.7% and 1.6%, respectively, for the RE-L honeycomb. The relatively high errors here are due to different loading strategies of the experimental models and the analytical models. Especially, the three-point bending test adopts a displacement load, while the analytical model adopts a distributed moment load. Nevertheless, the general trends of the curves predicted by the analytical model, especially the difference between the structures, are well consistent with the numerical and experimental results, which further verifies the effectiveness of the established analytical models.
Fig. 8
Comparison results of the force–displacement curves from the analytical, numerical, and experimental models: (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction.
Fig. 8
Comparison results of the force–displacement curves from the analytical, numerical, and experimental models: (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction.
Close modal

Analysis of Three-Point Bending Deformations.

Further, the elastic–plastic performance of the honeycomb structures was thoroughly studied, in which the aluminum material was adopted for the matrix material. Figure 9 shows the deformation processes of the RE honeycomb and the RE-L honeycomb during the three-point bending simulations in the x-direction. As shown in Fig. 9(a), when the indenter displacement δ = 5 mm, deformation of the RE honeycomb mainly occurs near the indenter, where the stress is the highest and gradually decreases with the increase of the distance from the indenter. Besides, compared with the straight walls, the inclined walls have the higher stress value due to their greater deformation. With the indenter further pressed, more straight cell walls start to undergo plastic deformation when the indenter displacement δ = 10 mm and 15 mm. It seems that the cell walls at the upper and lower edges of the honeycomb have higher stress values than the central cell walls, which may be due to the special dome-shaped deformation mode caused by its negative Poisson's ratio effect. Figure 9(b) shows the evolution of the bending deformations of the RE-L honeycomb. It can be observed that significant stress occurs at the additional link structures of the RE-L honeycomb, especially those near the indenter. Different from the RE honeycomb with a global bending mode, the RE-L honeycomb shows a distinct local bending phenomenon. This difference may be due to the compliant characteristic of the additional structure, which makes the bending deformation of the RE-L honeycomb looks more like local indentation. Figure 9(c) gives the deflection curves of the honeycombs bending in the x-direction. It can be found that for the RE honeycomb, the deflection of its central point is always larger than the indenter displacement during the whole bending process. While for the RE-L honeycomb, its central point is almost close to the indenter during the pressing process, and its deflection near the indenter is smaller than that of the RE honeycomb, which also means a higher curvature. Considering that the additional link structure is greatly flexible to produce adequate deformation, the RE-L honeycomb shows a soft behavior in the local position, which makes it closer to the indenter than the RE honeycomb. Besides, research by Jiang et al. [34] also found that the more compliant tubular lattice seems to have similar local behavior, and in contrast, the more rigid structure shows a global behavior, which just further confirms the simulation results demonstrated earlier.

Fig. 9
Stress contour plots during the bending process of the (a) RE honeycomb, (b) RE-L honeycomb in the x-direction, and (c) deflections of the points at the centerlines of the specimens
Fig. 9
Stress contour plots during the bending process of the (a) RE honeycomb, (b) RE-L honeycomb in the x-direction, and (c) deflections of the points at the centerlines of the specimens
Close modal

Figure 10 indicates the bending deformation processes of the RE and RE-L honeycombs in the y-direction. It can be found that, for the RE honeycomb, the deformation stress of the inclined cell walls is always remarkable during the whole pressing process, and meanwhile, the stress of the straight walls is almost negligible, which is different from that in the x-direction shown in Fig. 9(a). It can be explained that when bending in the y-direction, the straight walls of the RE honeycomb hardly bears moment, which makes them few or no deformations. Deformations of the straight and inclined walls of the RE-L honeycomb, as shown in Fig. 10(b), are generally similar to the RE honeycomb, considering that they have the same connection modes in the y-direction. However, behaviors of the additional walls of the RE-L honeycomb seem to have some influence on the overall deformation of the honeycomb. Specifically, the link structure close to the indenter has a significant deformation, which greatly offsets the rigidity in the x-direction. Therefore, the bending curvature in the longitudinal direction is small enough to make the indenter well close to the RE-L specimen, as shown in Fig. 10(c). While for the RE honeycomb, a clear gap between the indenter and the specimen is founded, as shown in Fig. 10(c), and during the whole bending process, the maximum deflections of the specimen are 5.7 mm, 11.6 mm, and 17.4 mm in sequence. The difference between the RE and RE-L honeycomb in the y-direction can be attributed to the influence of the additional structure on the behaviors in another direction.

Fig. 10
Stress contour plots during the bending process of the (a) RE honeycomb, (b) RE-L honeycomb in the y-direction, and (c) deflections of the points in the centerlines of the specimens
Fig. 10
Stress contour plots during the bending process of the (a) RE honeycomb, (b) RE-L honeycomb in the y-direction, and (c) deflections of the points in the centerlines of the specimens
Close modal

Parameter Effects.

Further, a detailed examination of the parameter effects on the bending force of the RE and RE-L honeycomb structures has been carried out. Here, the baseline parameters h, θ0, t, b, θ1, and l2 are set to 15 mm, 60deg, 0.6 mm, 8 mm, 75deg, and 8 mm, respectively, and the matrix material is aluminum. Figure 11 shows the bending forces of the two honeycombs with different θ0 in the x- and y-directions. It can be found that the force–displacement curves generally show a linear rise first and then enter a platform period, which is mainly due to the yield phenomenon of the whole structure. As shown in Figs. 11(a) and 11(b), when loading in the x-direction, the forces of the RE honeycomb and RE-L honeycomb exhibit clear differences. Especially, for the RE honeycomb, the curve with a smaller θ0 seems to have a higher platform. However, for the RE-L honeycomb, the influence of the parameter θ0 is always greatly small. Besides, the forces of the RE-L honeycomb, with the maximum value of about 6.3 N, are significantly lower than those of the RE honeycomb with the maximum values of about 18.1 N, 21.4 N, and 39.1 N, where the maximum value of the former is only about 29.4% of the latter with θ0 of 60deg. As shown in Figs. 11(c) and 11(d), the parameter θ0 has a clear effect on the bending force of the two honeycombs in the y-direction, where with θ0 increasing, the force curves are significantly improved, including the linear part and the platform part.

Fig. 11
Bending forces with different values of the parameter θ0: (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction
Fig. 11
Bending forces with different values of the parameter θ0: (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction
Close modal

Figure 12 shows the effects of the parameter thickness t on the bending force of the RE and RE-L honeycombs. It can be found that t has a dominate effect on the bending force, and with t increasing, the bending forces of the two honeycombs in both x- and y-directions have a significant improvement. Especially, when t changing from 0.4 mm to 0.8 mm, the maximum force of the RE honeycomb in the x-direction is changing from 9.2 N to 38.4 N, with the increase of about 317.4%, and that of the RE-L honeycomb is changing from 2.3 N to 12.3 N, with the increase of about 434.8%. Similarly, in the y-direction, the forces of the two honeycombs increase by 454.7% and 300%, respectively. In essence, the thickness improvement can directly and significantly increase the relative density of the honeycomb structures, and accordingly, the bending compliance of the structure will also greatly decline. However, this effect seems difficult to occur in only one direction, and it may not work when orthotropic bending behavior is required, just as mentioned in Introduction section.

Fig. 12
Bending forces with different values of the parameter t: (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction
Fig. 12
Bending forces with different values of the parameter t: (a) RE honeycomb in the x-direction, (b) RE-L honeycomb in the x-direction, (c) RE honeycomb in the y-direction, and (d) RE-L honeycomb in the y-direction
Close modal

Figure 13 shows the effects of the geometric parameters of the additional structure, i.e., the parameters θ1 and l2. When loading in the x-direction, as shown in Fig. 13(a), improving θ1 from 40deg to 80deg significantly decreases the bending force with the maximum value changing from 10.8 N to 6.6 N, about a 38.9% decrease. Similarly, increasing L2 from 6 mm to 10 mm also has an improvement to the bending compliance, while its effect seems to be weaker than that of the θ1, as shown in Fig. 13(b). This point is very similar to their effects on the linear bending rigidities shown in Figs. 7(e) and 7(f). Further, when loading in the y-direction, the influence of the parameter θ1 and l2 seem to be greatly small, with the maximum force of about 26.8 N, as shown in Figs. 13(c) and 13(d). The additional structure does not provide extra support in the y-direction, and accordingly, the two parameters have little influence on the bending behaviors in this direction. Therefore, it can be considered that adjusting the parameters θ1 and l2 can significantly improve the bending compliance in the x-direction, while the bearing capacity in the y-direction may be affected a little.

Fig. 13
Effects of (a) the parameter θ1 and (b) the parameter l2 on the bending force of the RE-L honeycomb in the x-direction, and effects of (c) the parameter θ1 and (d) parameter l2 on the bending force of the RE-L honeycomb in the y-direction
Fig. 13
Effects of (a) the parameter θ1 and (b) the parameter l2 on the bending force of the RE-L honeycomb in the x-direction, and effects of (c) the parameter θ1 and (d) parameter l2 on the bending force of the RE-L honeycomb in the y-direction
Close modal

Conclusions

In this work, a novel auxetic honeycomb, called as RE-L honeycomb, was developed by adding an additional link-wall structure to the traditional RE cell to improve the bending compliance, and its bending mechanical properties were investigated systematically through the analytical, numerical, and experimental methods. Especially, an analytical model was derived first to predict the equivalent bending rigidities based on the energy equivalence principle. And then, we established numerical models to simulate the linear bending behaviors and the three-point bending behaviors. Three-point bending experiment was also carried out by using the 3D printing specimens to verify the established analytical and numerical models. Detailed parametric studies, including parameters θ0, t, θ1, and l2, were performed to reveal the parameter influence on the bending behaviors as well as the orthotropic characteristics. Some concrete conclusions can be summarized as follows:

  1. The established analytical models can well predict the bending modes and the equivalent bending rigidities of the RE and RE-L honeycombs, which can be further used to guide the engineering application of the honeycomb structures theoretically for required bending properties.

  2. The bending properties of the proposed RE-L honeycomb, including the linear elastic bending rigidities and the bending behaviors under large deformations, have a great tailorability. Especially, increasing the values of θ1 or L2 can significantly improve the bending compliance of the RE-L honeycomb in the x-direction, and decreasing the value of θ0 can significantly improve that in another direction.

  3. The proposed RE-L honeycomb structure can markedly improve the bending compliance and the orthotropic properties of the traditional RE honeycomb, and the further application research in related fields just like an implantable stent is expected in future works.

Funding Data

  • The National Key Research and Development Program of China (Grant No. 2019YFB2006404).

  • Jiangsu Industrial and Information Industry Transformation and Upgrading Project (Grant No. 7602006021).

  • National Science Foundation for Young Scientist of China (Grant No. 52102421).

  • The State Key Laboratory of Automotive Safety and Energy under Project (Grant No. KFZ2202).

Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

Appendix: Negative Poisson's Ratio Effect Verification

To verify the NPR effect of the proposed RE-L honeycomb, an in-plane numerical model is established by the commercial software abaqus/standard. As shown in Fig. 14(a), when loading along the y-direction, the nodes at the lower boundary are fixed along the y-direction, and the nodes at the upper boundary are imposed with a uniform displacement Δuy. Especially, the horizontal wall at the upper boundary is coupled to a feature point to unify their displacement uy. The nodes at the left and right boundaries are coupled in the x-direction to unify their displacement ux. The boundary condition loading along the x-direction can be obtained similarly, as shown in Fig. 14(c). This periodic boundary has been used in our previous study to evaluate the in-plane properties of a novel auxetic honeycomb [49], and it has also been adopted in some similar studies [38,54]. The simulation material and the mesh setting are the same as those of the bending elastic finite element model described in the main text. The Poisson’s ratio can be calculated as follows:
νyx=εxεy=ΔUx/L1ΔUy/L2,νxy=εyεx=ΔUy/L2ΔUx/L1
where νyx and νxy are Poisson’s ratios in the y- and x-directions, respectively; εy and εx are the strain in the y- and x-directions, respectively; ΔUy and ΔUx are the length variation of the structure in the y- and x-directions, respectively; L1 and L2 are the horizontal length and vertical height of the structure, respectively. For comparison, the Poisson's ratio of the RE honeycomb is also calculated by the numerical model and the analytical model [6]. Geometric parameters are set as follows: h = 15 mm, b = 4 mm, t = 0.4 mm, θ1=75deg, and l2 = 8 mm, and the θ0 varies from 45 deg to 75 deg.
Fig. 14
In-plane numerical analysis of the Poisson’s ratio of the RE and RE-L honeycomb: (a) and (b) boundary condition and results in the y-direction and (c) and (d) boundary condition and results in the x-direction
Fig. 14
In-plane numerical analysis of the Poisson’s ratio of the RE and RE-L honeycomb: (a) and (b) boundary condition and results in the y-direction and (c) and (d) boundary condition and results in the x-direction
Close modal

As shown in Figs. 14(b) and 14(d), the numerical results of the RE honeycomb are in good agreement with the corresponding analytical predictions, which just confirms the validity of the established finite element model. It can be found that Poisson’s ratios of both the RE honeycomb and RE-L honeycomb are negative in the y- and x-directions. In the y-direction, the Poisson’s ratio of the RE-L honeycomb varies from −0.417 to −1.455. Especially, when θ0=60deg, νyx of the RE-L is equal to −0.744, and that of the RE honeycomb is −1. In the x-direction, νxy of the RE-L honeycomb varies from −0.189 to −0.038, which is greatly larger than that of the RE honeycomb.

This high orthotropic behavior of the Poisson's ratio of the RE-L honeycomb is related to its deformation mechanism in the x- and y-directions. When loading in the y-direction, the bending deformation of the inclined cell wall is the main deformation mechanism, which can cause lateral contraction of the structure. However, when loading in the x-direction, the bending deformation of the additional structure is the main deformation mechanism, which can hardly cause the strain in the y-direction.

Further, bending modes are also studied to confirm the NPR effect of the RE-L honeycomb. As shown in Fig. 15(a), when the moment My¯=1N is applied to the RE-L honeycomb, an obvious dome-shaped deformation mode is observed, which is related to its NPR effect. However, when the moment Mx¯=1N is applied to the structure, as shown in Fig. 15(b), the structure shows a deformation mode similar to the valley-shaped mode, which is related to its weakened NPR effect. Further, when the moments of My¯ and Mx¯ are considered simultaneously, the deformation mode is also similar to the valley-shaped mode, as shown in Fig. 15(c). This behavior may be because the deformation in the x-direction is significantly larger than that in the y-direction, and therefore, the dome-shaped deformation is covered by the valley-shaped deformation. It also illustrates that the deformation mechanism in two directions makes the structure highly orthotropic.

Fig. 15
Bending deformation modes of the RE-L honeycomb when the moment loading along (a) y-direction, (b) x-direction, and (c) y- and x-directions
Fig. 15
Bending deformation modes of the RE-L honeycomb when the moment loading along (a) y-direction, (b) x-direction, and (c) y- and x-directions
Close modal

Therefore, it can be confirmed that the RE-L honeycomb has a clear NPR effect in the y-direction, while in the x-direction, its deformation shows a significantly weakened NPR effect (almost zero Poisson’s ratio).

References

1.
Yu
,
X.
,
Zhou
,
J.
,
Liang
,
H.
,
Jiang
,
Z.
, and
Wu
,
L.
,
2018
, “
Mechanical Metamaterials Associated With Stiffness, Rigidity and Compressibility: A Brief Review
,”
Prog. Mater. Sci.
,
94
, pp.
114
173
.
2.
Banhart
,
J.
,
2001
, “
Manufacture, Characterisation and Application of Cellular Metals and Metal Foams
,”
Prog. Mater. Sci.
,
46
(
6
), pp.
559
632
.
3.
Qi
,
C.
,
Jiang
,
F.
, and
Yang
,
S.
,
2021
, “
Advanced Honeycomb Designs for Improving Mechanical Properties: A Review
,”
Composites, Part B
,
227
, p.
109393
.
4.
Tatlıer
,
M. S.
,
Öztürk
,
M.
, and
Baran
,
T.
,
2021
, “
Linear and Non-Linear In-Plane Behaviour of a Modified Re-Entrant Core Cell
,”
Eng. Struct.
,
234
, p.
111984
.
5.
Gibson
,
L. J.
,
Ashby
,
M. F.
,
Schajer
,
G. S.
, and
Robertson
,
C. I.
,
1982
, “
The Mechanics of Two-Dimensional Cellular Materials
,”
Proc. R. Soc. A
,
382
(
1782
), pp.
25
42
.
6.
Gibson
,
L. J.
, and
Ashby
,
M. F.
,
1997
,
Cellular Solids: Structure and Properties
,
Cambridge University Press
,
Cambridge, UK
.
7.
Lakes
,
R.
,
1987
, “
Foam Structures With a Negative Poisson's Ratio
,”
Science
,
235
(
4792
), pp.
1038
1040
.
8.
Lakes
,
R. S.
,
2017
, “
Negative-Poisson's-Ratio Materials: Auxetic Solids
,”
Annu. Rev. Mater. Res.
,
47
(
1
), pp.
63
81
.
9.
Francisco
,
M. B.
,
Pereira
,
J. L. J.
,
Oliver
,
G. A.
,
Roque Da Silva
,
L. R.
,
Cunha
,
S. S.
, and
Gomes
,
G. F.
,
2021
, “
A Review on the Energy Absorption Response and Structural Applications of Auxetic Structures
,”
Mech. Adv. Mater. Struct.
,
29
(
27
), pp.
1
20
.
10.
Qi
,
C.
,
Remennikov
,
A.
,
Pei
,
L.
,
Yang
,
S.
,
Yu
,
Z.
, and
Ngo
,
T. D.
,
2017
, “
Impact and Close-in Blast Response of Auxetic Honeycomb-Cored Sandwich Panels: Experimental Tests and Numerical Simulations
,”
Compos. Struct.
,
180
, pp.
161
178
.
11.
Wang
,
T.
,
Li
,
Z.
,
Wang
,
L.
, and
Hulbert
,
G. M.
,
2020
, “
Crashworthiness Analysis and Collaborative Optimization Design for a Novel Crash-Box With Re-Entrant Auxetic Core
,”
Struct. Multidiscipl. Optim.
,
62
(
4
), pp.
2167
2179
.
12.
Scarpa
,
F.
, and
Tomlin
,
P. J.
,
2000
, “
On the Transverse Shear Modulus of Negative Poisson's Ratio Honeycomb Structures
,”
Fatigue Fract. Eng. Mater. Struct.
,
23
(
8
), pp.
717
720
.
13.
Tan
,
H.
,
He
,
Z.
,
Li
,
E.
,
Cheng
,
A.
,
Chen
,
T.
,
Tan
,
X.
,
Li
,
Q.
, and
Xu
,
B.
,
2021
, “
Crashworthiness Design and Multi-Objective Optimization of a Novel Auxetic Hierarchical Honeycomb Crash Box
,”
Struct. Multidiscipl. Optim.
,
64
(
4
), pp.
2009
2024
.
14.
Menon
,
H. G.
,
Dutta
,
S.
,
Krishnan
,
A.
,
Hariprasad
,
M. P.
, and
Shankar
,
B.
,
2022
, “
Proposed Auxetic Cluster Designs for Lightweight Structural Beams With Improved Load Bearing Capacity
,”
Eng. Struct.
,
260
, p.
114241
.
15.
Prithipaul
,
P. K. M.
,
Kokkolaras
,
M.
, and
Pasini
,
D.
,
2018
, “
Assessment of Structural and Hemodynamic Performance of Vascular Stents Modelled as Periodic Lattices
,”
Med. Eng. Phys.
,
57
, pp.
11
18
.
16.
Masters
,
I. G.
, and
Evans
,
K. E.
,
1996
, “
Models for the Elastic Deformation of Honeycombs
,”
Compos. Struct.
,
35
(
4
), pp.
403
422
.
17.
Hou
,
X.
,
Deng
,
Z.
, and
Zhang
,
K.
,
2016
, “
Dynamic Crushing Strength Analysis of Auxetic Honeycombs
,”
Acta Mech. Solida Sin.
,
29
(
5
), pp.
490
501
.
18.
Bhullar
,
S. K.
,
Lekesiz
,
H.
,
Karaca
,
A. A.
,
Cho
,
Y.
,
Willerth
,
S. M.
, and
Jun
,
M. B. G.
,
2022
, “
Characterizing the Mechanical Performance of a Bare-Metal Stent With an Auxetic Cell Geometry
,”
Appl. Sci.
,
12
(
2
), p.
910
.
19.
Bubert
,
E. A.
,
Woods
,
B. K. S.
,
Lee
,
K.
,
Kothera
,
C. S.
, and
Wereley
,
N. M.
,
2010
, “
Design and Fabrication of a Passive 1D Morphing Aircraft Skin
,”
J. Intell. Mater. Syst. Struct.
,
21
(
17
), pp.
1699
1717
.
20.
Heo
,
H.
,
Ju
,
J.
, and
Kim
,
D.-M.
,
2013
, “
Compliant Cellular Structures: Application to a Passive Morphing Airfoil
,”
Compos. Struct.
,
106
, pp.
560
569
.
21.
Mosadegh
,
B.
,
Polygerinos
,
P.
,
Keplinger
,
C.
,
Wennstedt
,
S.
,
Shepherd
,
R. F.
,
Gupta
,
U.
,
Shim
,
J.
,
Bertoldi
,
K.
,
Walsh
,
C. J.
, and
Whitesides
,
G. M.
,
2014
, “
Pneumatic Networks for Soft Robotics That Actuate Rapidly
,”
Adv. Funct. Mater.
,
24
(
15
), pp.
2163
2170
.
22.
Holman
,
H.
,
Kavarana
,
M. N.
, and
Rajab
,
T. K.
,
2021
, “
Smart Materials in Cardiovascular Implants: Shape Memory Alloys and Shape Memory Polymers
,”
Artif. Organs
,
45
(
5
), pp.
454
463
.
23.
Pan
,
C.
,
Han
,
Y.
, and
Lu
,
J.
,
2021
, “
Structural Design of Vascular Stents: A Review
,”
Micromachines.
,
12
(
7
), p.
770
.
24.
Chen
,
Y.
,
Fu
,
M.
,
Hu
,
H.
, and
Xiong
,
J.
,
2022
, “
Curved Inserts in Auxetic Honeycomb for Property Enhancement and Design Flexibility
,”
Compos. Struct.
,
280
, p.
114892
.
25.
Xu
,
N.
,
Liu
,
H.
,
An
,
M.
, and
Wang
,
L.
,
2021
, “
Novel 2D Star-Shaped Honeycombs With Enhanced Effective Young’s Modulus and Negative Poisson’s Ratio
,”
Extreme Mech. Lett.
,
43
, p.
101164
.
26.
Fu
,
M.H.
,
Chen
,
Y.
, and
Hu
,
L.-L.
,
2017
, “
Bilinear Elastic Characteristic of Enhanced Auxetic Honeycombs
,”
Compos. Struct.
,
175
, pp.
101
110
.
27.
Li
,
X.
,
Wang
,
Q.
,
Yang
,
Z.
, and
Lu
,
Z.
,
2019
, “
Novel Auxetic Structures With Enhanced Mechanical Properties
,”
Extreme Mech. Lett.
,
27
, pp.
59
65
.
28.
Tan
,
H. L.
,
He
,
Z. C.
,
Li
,
K. X.
,
Li
,
E.
,
Cheng
,
A. G.
, and
Xu
,
B.
,
2019
, “
In-Plane Crashworthiness of Re-Entrant Hierarchical Honeycombs With Negative Poisson’s Ratio
,”
Compos. Struct.
,
229
, p.
111415
.
29.
Zhang
,
X.
,
Hao
,
H.
,
Tian
,
R.
,
Xue
,
Q.
,
Guan
,
H.
, and
Yang
,
X.
,
2022
, “
Quasi-Static Compression and Dynamic Crushing Behaviors of Novel Hybrid Re-Entrant Auxetic Metamaterials With Enhanced Energy-Absorption
,”
Compos. Struct.
,
288
, p.
115399
.
30.
Xu
,
M.
,
Liu
,
D.
,
Wang
,
P.
,
Zhang
,
Z.
,
Jia
,
H.
,
Lei
,
H.
, and
Fang
,
D.
,
2020
, “
In-Plane Compression Behavior of Hybrid Honeycomb Metastructures: Theoretical and Experimental Studies
,”
Aerosp. Sci. Technol.
,
106
, p.
106081
.
31.
Li
,
C.
,
Shen
,
H.
, and
Wang
,
H.
,
2019
, “
Nonlinear Dynamic Response of Sandwich Beams With Functionally Graded Negative Poisson’s Ratio Honeycomb Core
,”
Eur. Phys. J. Plus
,
134
(
2
).
32.
Jin
,
X.
,
Wang
,
Z.
,
Ning
,
J.
,
Xiao
,
G.
,
Liu
,
E.
, and
Shu
,
X.
,
2016
, “
Dynamic Response of Sandwich Structures With Graded Auxetic Honeycomb Cores Under Blast Loading
,”
Composites, Part B
,
106
, pp.
206
217
.
33.
Huang
,
J.
,
Zhang
,
Q.
,
Scarpa
,
F.
,
Liu
,
Y.
, and
Leng
,
J.
,
2016
, “
Bending and Benchmark of Zero Poisson’s Ratio Cellular Structures
,”
Compos. Struct.
,
152
, pp.
729
736
.
34.
Jiang
,
H.
,
Ziegler
,
H.
,
Zhang
,
Z.
,
Atre
,
S.
, and
Chen
,
Y.
,
2022
, “
Bending Behavior of 3D Printed Mechanically Robust Tubular Lattice Metamaterials
,”
Addit. Manuf.
,
50
, p.
102565
.
35.
Gao
,
Q.
,
Ge
,
C.
,
Zhuang
,
W.
,
Wang
,
L.
, and
Ma
,
Z.
,
2019
, “
Crashworthiness Analysis of Double-Arrowed Auxetic Structure Under Axial Impact Loading
,”
Mater. Des.
,
161
, pp.
22
34
.
36.
Zhao
,
X.
,
Gao
,
Q.
,
Wang
,
L.
,
Yu
,
Q.
, and
Ma
,
Z. D.
,
2018
, “
Dynamic Crushing of Double-Arrowed Auxetic Structure Under Impact Loading
,”
Mater. Des.
,
160
, pp.
527
537
.
37.
Hönig
,
A.
, and
Stronge
,
W. J.
,
2002
, “
In-Plane Dynamic Crushing of Honeycomb. Part I: Crush Band Initiation and Wave Trapping
,”
Int. J. Mech. Sci.
,
44
(
8
), pp.
1665
1696
.
38.
Wang
,
T.
,
Wang
,
L.
,
Ma
,
Z.
, and
Hulbert
,
G. M.
,
2018
, “
Elastic Analysis of Auxetic Cellular Structure Consisting of Re-Entrant Hexagonal Cells Using a Strain-Based Expansion Homogenization Method
,”
Mater. Des.
,
160
, pp.
284
293
.
39.
Meng
,
J.
,
Deng
,
Z.
,
Zhang
,
K.
,
Xu
,
X.
, and
Wen
,
F.
,
2015
, “
Band Gap Analysis of Star-Shaped Honeycombs With Varied Poisson's Ratio
,”
Smart Mater. Struct.
,
24
(
9
), p.
95011
.
40.
Qiao
,
J.
, and
Chen
,
C. Q.
,
2015
, “
Analyses On the In-Plane Impact Resistance of Auxetic Double Arrowhead Honeycombs
,”
ASME J. Appl. Mech.
,
82
(
5
), p.
051007
.
41.
Chen
,
D. H.
,
2011
, “
Bending Deformation of Honeycomb Consisting of Regular Hexagonal Cells
,”
Compos. Struct.
,
93
(
2
), pp.
736
746
.
42.
Abd El-Sayed
,
F.
,
Jones
,
R.
, and
Burgess
,
I. W.
,
1979
, “
Theoretical Approach to the Deformation of Honeycomb Based Composite-Materials
,”
Composites
,
10
(
4
), pp.
209
214
.
43.
Chen
,
D. H.
,
2011
, “
Equivalent Flexural and Torsional Rigidity of Hexagonal Honeycomb
,”
Compos. Struct.
,
93
(
7
), pp.
1910
1917
.
44.
Chen
,
D.
,
2011
, “
Flexural Rigidity of Honeycomb Consisting of Hexagonal Cells
,”
Acta Mech. Sin.
,
27
(
5
), pp.
840
844
.
45.
Azaouzi
,
M.
,
Makradi
,
A.
, and
Belouettar
,
S.
,
2013
, “
Numerical Investigations of the Structural Behavior of a Balloon Expandable Stent Design Using Finite Element Method
,”
Comput. Mater. Sci.
,
72
, pp.
54
61
.
46.
Timoshenko
,
S.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
McGraw-Hill Book Company
,
New York, Toronto and London
.
47.
Ruan
,
D.
,
Lu
,
G.
,
Wang
,
B.
, and
Yu
,
T. X.
,
2003
, “
In-Plane Dynamic Crushing of Honeycombs—A Finite Element Study
,”
Int. J. Impact Eng.
,
28
(
2
), pp.
161
182
.
48.
Zhou
,
Y.
,
Pan
,
Y.
,
Gao
,
Q.
, and
Sun
,
B.
,
2023
, “
In-Plane Quasi-Static Crushing Behaviors of a Novel Reentrant Combined-Wall Honeycomb
,”
ASME J. Appl. Mech.
,
90
(
5
), p.
051002
.
49.
Zhou
,
Y.
,
Pan
,
Y.
,
Chen
,
L.
,
Gao
,
Q.
, and
Sun
,
B.
,
2022
, “
Mechanical Behaviors of a Novel Auxetic Honeycomb Characterized by Re-Entrant Combined-Wall Hierarchical Substructures
,”
Mater. Res. Express
,
9
(
11
), p.
115802
.
50.
Alderson
,
A.
, and
Alderson
,
K. L.
,
2007
, “
Auxetic Materials
,”
Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
,
221
(
4
), pp.
565
575
.
51.
Evans
,
K. E.
, and
Alderson
,
A.
,
2000
, “
Auxetic Materials: Functional Materials and Structures From Lateral Thinking
,”
Adv. Mater.
,
12
(
9
), pp.
617
628
.
52.
Alderson
,
A.
,
Alderson
,
K. L.
,
Chirima
,
G.
,
Ravirala
,
N.
, and
Zied
,
K. M.
,
2010
, “
The In-Plane Linear Elastic Constants and Out-of-Plane Bending of 3-Coordinated Ligament and Cylinder-Ligament Honeycombs
,”
Compos. Sci. Technol.
,
70
(
7
), pp.
1034
1041
.
53.
Huang
,
J.
,
Zhang
,
Q.
,
Scarpa
,
F.
,
Liu
,
Y.
, and
Leng
,
J.
,
2017
, “
Shape Memory Polymer-Based Hybrid Honeycomb Structures With Zero Poisson’s Ratio and Variable Stiffness
,”
Compos. Struct.
,
179
, pp.
437
443
.
54.
Lu
,
Z.
,
Li
,
X.
,
Yang
,
Z.
, and
Xie
,
F.
,
2016
, “
Novel Structure With Negative Poisson’s Ratio and Enhanced Young’s Modulus
,”
Compos. Struct.
,
138
, pp.
243
252
.