## 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 [1–3]. 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 [24–26]. 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 [35–37]. 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 *l*_{0}; two major horizontal walls, length *l*_{1}; and two connecting walls, length *l*_{1}/2. Besides, it also includes the following geometry parameters: the cell height *h*, the cell length *s*_{1}, the angle *θ*_{0}, 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 *l*_{1} is set to twice the value of *l*_{0}. According to the geometric relationship, the following expressions can be obtained as follows: *l*_{0} = *h*/(2sin *θ*_{0}) and *s*_{1} = (2 − cos *θ*_{0})*h*/sin *θ*_{0}. Hence, geometry of the RE cell can be uniquely defined by the parameters *h*, *θ*_{0}, *t*, and *b*.

*l*

_{2}and the angle

*θ*

_{1}are specifically defined, as shown in Fig. 1(b). For the RE-L cell, the cell length

*s*

_{2}can be expressed as follows:

*s*

_{2}=

*s*

_{1}+ 4

*l*

_{2}cos

*θ*

_{1}. 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:

### Analytical Model of Bending Properties.

*D*

_{x},

*D*

_{y}, and

*D*

_{1}, are used to relate the applied uniformly distributed moments $Mx\xaf$ and $My\xaf$ with curvatures 1/

*ρ*

_{x}and 1/

*ρ*

_{y}[46]:

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.

*L*

_{1},

*L*

_{2}, out-of-plane width

*b*, as shown in Fig. 2(a), its elastic strain energy can be expressed as follows:

*w*is the bending deflection;

*D*

_{x},

*D*

_{y}, and

*D*

_{1}are the three flexural rigidities of the plate; and

*D*

_{xy}is the torsional rigidity. When applying the uniformly distributed bending moment $Mx\xaf$ and $My\xaf$ to the plate with a fixed point (

*x*

_{0},

*y*

_{0}), the theoretical solution of its bending deflection

*w*can be given as follows [44]:

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., (*x*_{0}, *y*_{0}) = (0, 0). The horizontal and vertical directions that intersect the coordinate origin are the corresponding *x*- and *y*-axis.

*E*

_{s},

*t*, and

*b*are the Young’s modulus, thickness of the cell wall, and width of the cell wall;

*q*

_{M}and

*q*

_{T}are defined as the coefficients related to the geometry of the cell wall; and $q\tau $ 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:

*u*

_{a}=

*u*

_{e}, the following coefficient correspondence can be obtained as follows:

*s*should be especially designated as

*s*

_{1}or

*s*

_{2}for RE or RE-L honeycomb. Therefore, the flexural rigidities can be obtained as follows:

Under the given parameter of *h* = 15 mm, *b* = 4 mm, *t* = 0.4 mm, $\theta 0=60deg$, $\theta 1=75deg$, and *l*_{2} = 8 mm, values of partial variables and three flexural rigidities have been given in Table 1.

Variables | q_{M1} (N^{−1}mm^{−1}) | q_{M2} (N^{−1}mm^{−1}) | q_{T2} (N^{−1}mm^{−1}) | q_{M4} (N^{−1}mm^{−1}) | q_{M5} (N^{−1}mm^{−1}) | q_{T4} (N^{−1}mm^{−1}) | q_{T5} (N^{−1}mm^{−1}) | D_{x} (N · mm) | D_{y} (N · mm) | D_{1} (N · mm) |
---|---|---|---|---|---|---|---|---|---|---|

RE | 1.177 × 10^{−4} | 5.883 × 10^{−5} | 0.0029 | — | — | — | — | 3789.89 | 3985.24 | −3496.86 |

RE-L | 1.177 × 10^{−4} | 5.883 × 10^{−5} | 0.0029 | 5.434 × 10^{−5} | 1.087 × 10^{−4} | 0.0026 | 0.0060 | 209.62 | 677.95 | −146.659 |

Variables | q_{M1} (N^{−1}mm^{−1}) | q_{M2} (N^{−1}mm^{−1}) | q_{T2} (N^{−1}mm^{−1}) | q_{M4} (N^{−1}mm^{−1}) | q_{M5} (N^{−1}mm^{−1}) | q_{T4} (N^{−1}mm^{−1}) | q_{T5} (N^{−1}mm^{−1}) | D_{x} (N · mm) | D_{y} (N · mm) | D_{1} (N · mm) |
---|---|---|---|---|---|---|---|---|---|---|

RE | 1.177 × 10^{−4} | 5.883 × 10^{−5} | 0.0029 | — | — | — | — | 3789.89 | 3985.24 | −3496.86 |

RE-L | 1.177 × 10^{−4} | 5.883 × 10^{−5} | 0.0029 | 5.434 × 10^{−5} | 1.087 × 10^{−4} | 0.0026 | 0.0060 | 209.62 | 677.95 | −146.659 |

### 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 $Mx\xaf$ and $My\xaf$. 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.

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/m^{3} 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.

### 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.

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 *L*_{s} 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, $\theta 0=60deg$, *b* = 10 mm, and *t* = 1 mm for the RE and RE-L honeycombs, and $\theta 1=75deg$ and *l*_{2} = 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 *θ*_{0} 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 *l*_{1} is set to 1.5 times of *l*_{0} to avoid excessive cell length. Besides, other geometric parameters are set as follows: *h* = 15 mm, *b* = 4 mm, *t* = 0.4 mm, $\theta 1=75deg$, and *l*_{2} = 8 mm, and the distributed moment loads of all structures are set as follows: $Mx\xaf=1N$ and $My\xaf=1N$. 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.

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 [50–52]. 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.

No. | t (mm) | $\theta 0(\u2218)$ | $\theta 1(\u2218)$ | l_{2} (mm) | P_{1} | P_{2} | ||||
---|---|---|---|---|---|---|---|---|---|---|

w_{analytical} (mm) | w_{numerical} (mm) | Error | w_{analytical} (mm) | w_{numerical} (mm) | Error | |||||

1 | 0.6 | 60 | 75 | 8 | 0.036 | 0.039 | 0.064 | 0.725 | 0.794 | 0.086 |

2 | 0.6 | 45 | 75 | 8 | 0.083 | 0.090 | 0.071 | 0.894 | 0.964 | 0.072 |

3 | 0.6 | 75 | 75 | 8 | 0.013 | 0.014 | 0.058 | 0.650 | 0.720 | 0.097 |

4 | 0.4 | 60 | 75 | 8 | 0.121 | 0.127 | 0.040 | 2.435 | 2.604 | 0.065 |

5 | 0.8 | 60 | 75 | 8 | 0.015 | 0.017 | 0.091 | 0.308 | 0.347 | 0.111 |

6 | 0.6 | 60 | 40 | 8 | 0.046 | 0.049 | 0.059 | 0.659 | 0.726 | 0.092 |

7 | 0.6 | 60 | 80 | 8 | 0.035 | 0.037 | 0.062 | 0.700 | 0.762 | 0.081 |

8 | 0.6 | 60 | 75 | 6 | 0.035 | 0.036 | 0.038 | 0.521 | 0.575 | 0.095 |

9 | 0.6 | 60 | 75 | 10 | 0.038 | 0.040 | 0.062 | 0.951 | 1.036 | 0.082 |

No. | t (mm) | $\theta 0(\u2218)$ | $\theta 1(\u2218)$ | l_{2} (mm) | P_{1} | P_{2} | ||||
---|---|---|---|---|---|---|---|---|---|---|

w_{analytical} (mm) | w_{numerical} (mm) | Error | w_{analytical} (mm) | w_{numerical} (mm) | Error | |||||

1 | 0.6 | 60 | 75 | 8 | 0.036 | 0.039 | 0.064 | 0.725 | 0.794 | 0.086 |

2 | 0.6 | 45 | 75 | 8 | 0.083 | 0.090 | 0.071 | 0.894 | 0.964 | 0.072 |

3 | 0.6 | 75 | 75 | 8 | 0.013 | 0.014 | 0.058 | 0.650 | 0.720 | 0.097 |

4 | 0.4 | 60 | 75 | 8 | 0.121 | 0.127 | 0.040 | 2.435 | 2.604 | 0.065 |

5 | 0.8 | 60 | 75 | 8 | 0.015 | 0.017 | 0.091 | 0.308 | 0.347 | 0.111 |

6 | 0.6 | 60 | 40 | 8 | 0.046 | 0.049 | 0.059 | 0.659 | 0.726 | 0.092 |

7 | 0.6 | 60 | 80 | 8 | 0.035 | 0.037 | 0.062 | 0.700 | 0.762 | 0.081 |

8 | 0.6 | 60 | 75 | 6 | 0.035 | 0.036 | 0.038 | 0.521 | 0.575 | 0.095 |

9 | 0.6 | 60 | 75 | 10 | 0.038 | 0.040 | 0.062 | 0.951 | 1.036 | 0.082 |

#### 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*, *θ*_{1}, and *l*_{2} are 15 mm, 0.6 mm, 8 mm, $60deg$, $75deg$, and 8 mm, respectively. The uniformly distributed bending moment $Mx\xaf$ and $My\xaf$ are set to 1 N. As shown in Fig. 7(a), with the parameter *θ*_{0} 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 *θ*_{0} increasing, the inclined cell wall shows less deformation energy under the moments $My\xaf$, which makes the honeycomb more difficult to bend. For the RE-L honeycomb, effects of the *θ*_{0} 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 $\theta 0=60deg$, 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 *θ*_{0} increasing, as shown in Fig. 7(b). It can be found that increasing the parameter *θ*_{0} 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 *EI*_{y}/*EI*_{x} 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.

For the RE-L honeycomb, the effects of its additional in-plane parameter *l*_{2} and *θ*_{1} have been depicted in Figs. 7(e) and 7(f). Effects of the parameter *θ*_{1} on the two rigidities are opposite, that is, with the *θ*_{1} 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 *l*_{2} 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.

*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.

#### 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.

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.

#### 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*, *θ*_{0}, *t*, *b*, *θ*_{1}, and *l*_{2} 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 *θ*_{0} 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 *θ*_{0} seems to have a higher platform. However, for the RE-L honeycomb, the influence of the parameter *θ*_{0} 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 *θ*_{0} of $60deg$. As shown in Figs. 11(c) and 11(d), the parameter *θ*_{0} has a clear effect on the bending force of the two honeycombs in the *y*-direction, where with *θ*_{0} increasing, the force curves are significantly improved, including the linear part and the platform part.

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.

Figure 13 shows the effects of the geometric parameters of the additional structure, i.e., the parameters *θ*_{1} and *l*_{2}. When loading in the *x*-direction, as shown in Fig. 13(a), improving *θ*_{1} 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 *L*_{2} 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 *θ*_{1}, 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 *θ*_{1} and *l*_{2} 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 *θ*_{1} and *l*_{2} can significantly improve the bending compliance in the *x*-direction, while the bearing capacity in the *y*-direction may be affected a little.

## 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 *θ*_{0}, *t*, *θ*_{1}, and *l*_{2}, 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:

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.

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

*θ*_{1}or*L*_{2}can significantly improve the bending compliance of the RE-L honeycomb in the*x*-direction, and decreasing the value of*θ*_{0}can significantly improve that in another direction.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

*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 Δ

*u*

_{y}. Especially, the horizontal wall at the upper boundary is coupled to a feature point to unify their displacement

*u*

_{y}. The nodes at the left and right boundaries are coupled in the

*x*-direction to unify their displacement

*u*

_{x}. 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:

*ν*

_{yx}and

*ν*

_{xy}are Poisson’s ratios in the

*y*- and

*x*-directions, respectively; $\epsilon y$ and $\epsilon x$ are the strain in the

*y*- and

*x*-directions, respectively; Δ

*U*

_{y}and Δ

*U*

_{x}are the length variation of the structure in the

*y*- and

*x*-directions, respectively;

*L*

_{1}and

*L*

_{2}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, $\theta 1=75deg$, and

*l*

_{2}= 8 mm, and the

*θ*

_{0}varies from 45 deg to 75 deg.

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 $\theta 0=60deg$, *ν*_{yx} of the RE-L is equal to −0.744, and that of the RE honeycomb is −1. In the *x*-direction, *ν*_{xy} 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 $My\xaf=1N$ 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 $Mx\xaf=1N$ 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 $My\xaf$ and $Mx\xaf$ 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.

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).