# Dynamic Crushing Analysis of Hexagonal-Kagome Combined Honeycomb with Negative Poisson’s Ratio

Authors: Wei Min Wang; Zhi Liang Yang; Jun Hu
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

To enhance the mechanical performances of traditional honeycombs, a novel combined honeycomb (HKH) was proposed based on the regular hexagonal honeycomb (RHH) and Kagome honeycomb (KH). A systematic investigation of the in-plane dynamic crushing behaviors of the honeycombs was conducted via the finite element method, and the crashworthiness characteristics of the three honeycombs were examined. Using one-dimensional shock wave theory and least squares fitting, a fitting formula of the plateau stress was obtained, and the deformation mechanism of the HKH was determined based on the elastic buckling of the cell wall. The results showed that the HKH exhibited a higher plateau stress than the RHH for any velocity impact, while the negative Poisson’s ratio effect was more significant than that of the KH for low-velocity impacts (the peak Poisson’s ratios of the HKH and KH were −0.312 and −0.286, respectively). Furthermore, the effects of the impact velocity and relative density on the energy absorption and auxetic performances were explored. In this study, a novel design was proposed by combining various cell elements to obtain better crashworthiness, providing a new concept to promote the development of lightweight materials.


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## Body

## 1. Introduction

Honeycombs have been applied universally in many fields, such as the military, aerospace, mechanical engineering, and the automobile industries, owing to their high strength and energy absorbing properties [1–9]. Many studies [10–13] have shown that a honeycomb with a negative Poisson’s ratio effect can exhibit more unique deformation behaviors and higher shear capacities and fracture resistance compared to conventional honeycombs under dynamic crushing. Hence, a considerable amount of research has been focused on the design of novel honeycombs that unfold to achieve a negative Poisson’s ratio effect to increase their crashworthiness. Their mechanical performances can be determined directly by the topological structure of the honeycomb, and tiny differences in the topological structures have a remarkable influence on their performance. For in-plane dynamic crushing, the crashworthiness of honeycombs is enhanced significantly by changing the topological structure of the honeycomb such that various unit cells are combined [14]. Therefore, this is a popular research area.Through ongoing research, conventional honeycombs with unit cells of regular hexagons [15], triangles [16], Kagome cells [17], and circles [18] have been studied extensively to explore the performances under specific conditions, such as out-of-plane compression and explosions. However, it is important for novel honeycombs to be designed to pursue higher performances over traditional honeycombs under dynamic crushing. For instance, Wu et al. [19] classified honeycomb materials into two categories based on the differences of the stress-strain curves under dynamic crushing after numerous experiments. The main characteristics of the first category were lower initial peak stresses and more stable plateau stages of the stress-strain curves. When honeycombs in the first category, such as hexagonal ones, undergo deformation, the cell walls generally bend. However, the Kagome honeycomb, as a representative of the second category, undergoes the opposite behavior during the initial peak stress and plateau stages, with the unit cells being dominated by stretching under in-plane impacts. In addition, Wu et al. [20] proposed a fresh concept of changing the angle of the connection between the hexagon and triangle based on the topological structure of the Kagome honeycomb to enhance its energy absorbing capacity and crashworthiness. In the meantime, Lu et al. [21] designed a novel combined honeycomb in which the circular cell was put into the center of the star unit cell to explore an efficient route to improve the properties of the honeycomb, in which the deformation modes were analyzed using a theoretical method and simulation. With this trending concept, Wei et al. [22] created a new honeycomb by changing the connection type between the star unit cells, and the energy absorption of the traditional star honeycomb was compared with that of the new honeycomb. The results showed that connection mode plays a key role in performance.The two categories of honeycombs have their own characteristics, and many researchers have investigated them adequately. However, the concept of combining the different categories into a single honeycomb has rarely been presented. This was explored in the current study, and the results are meaningful for developing lightweight materials. In this study, a novel combined honeycomb (HKH) was developed. At first, the deformation modes of the HKH were investigated and compared with the Kagome honeycomb (KH) and the regular hexagonal honeycomb (RHH) under different impact velocities (V  = 3, 10, and 100 m/s). Subsequently, the deformation mechanism of the HKH was explained by the elastic buckling of the cell walls to understand the deformation behavior of the HKH under low and mid-velocity impacts. Based on a uniform relative density, the plateau stresses and specific energy absorption (SEA) values of the three honeycombs (HKH, KH, and RHH) were analyzed. The effects of relative density and impact velocity on the performance indices were investigated, and a map relating crashworthiness, relative density, and impact velocity was created. Finally, HKHs and KHs with negative Poisson’s ratios were examined, and the peak Poisson’s ratio was used to represent the effects of the three honeycombs under different impact velocities.


## 2. Modeling Description

### 2.1. Geometric Configurations of Three Honeycombs

As shown in Figure1, the novel combined honeycomb (i.e., the HKH) can be established by placing a KH inside an RHH. To allow the various honeycombs to be compared, the lengths and widths of the three cell elements used in the impact simulations were kept the same size. Furthermore, three geometric parameters for the HKH were interpreted: the length of the external cell wall l1, the length of the internal cell wall l2, and the thickness of the cell wall t.Figure 1
Design concept for novel combined honeycomb (HKH) and the geometric configuration of the honeycomb.Relative density is a vital index used to evaluate reliability when comparing the results between the HKH, KH, and RHH. For the KH and RHH, the relative densities ΔρΚΗ and ΔρRΗΗ can be defined, respectively, as follows [23]:(1)ΔρKH=ρ∗KHρs=3tl2,ΔρRHH=ρ∗RHHρs=23tl1,where ρKH∗ and ρRHH∗ denote the density of the KH and RHH, respectively, and ρs is the density of the matrix material.Based on the relative density formula for honeycomb materials, the relative density of the HKH can be expressed as follows [24]:(2)ΔρHKH=ρ∗HKHρs=∑litiL1L2=1716l1+3312l2tL1L2,where ΔρΗΚΗ and ρHKH∗ represent the relative density of the HKH and the density of the HKH, respectively, and L1 and L2 are the horizontal and vertical lengths of the honeycomb model, respectively. In this study, the same relative density (Δρ = 4.4%) was used for the three honeycombs, and the corresponding thicknesses of the HKH, KH, and RHH were determined to be 0.02, 0.04, and 0.1 mm. The different relative densities of the honeycomb were obtained based on the modified cell wall thickness using the above formula.


### 2.2. Finite Element Analysis

In the present investigation, the dynamic crushing behaviors and crashworthiness of the three honeycombs (HKH, KH, and RHH) under in-plane impacts were analyzed using the nonlinear finite element software ABAQUS/Explicit. The local structures of the honeycomb specimens are shown in Figure2. Simulations with the finite element model of honeycombs with dimensions of L1×L2×Z were conducted, and the numbers of unit cells in the x- and y-directions were set to 24 and 23, respectively, to capture the deformation behavior of the honeycomb under vertical crushing. The honeycomb specimen was placed between the rigid plates, in which the bottom rigid plate was fixed and the top rigid plate moved downward at a constant velocity. The impact velocity v was increased from 3 to 100 m/s, allowing for three deformation modes to occur (low-, mid-, and high-velocity modes), to investigate the crashworthiness of the honeycomb. To simplify the simulation of the impact experiments, the out-of-plane thickness of the honeycomb specimen was 1 mm, the external cell wall thickness was l1 = 2.7 mm, and the internal cell wall thickness was l2 = 4.7 mm.Figure 2
Finite element models of three honeycombs under in-plane impact.The matrix material of the honeycomb specimen was selected as aluminum, which was assumed to be an elastic–perfectly plastic material. The attributes of the matrix material are presented in Table1. To ensure the convergence and calculation accuracy, the unit cells were meshed using shell elements (S4R) with sizes of 1 mm, and five integration points were defined in the direction of the thickness. Furthermore, the surface-to-surface contact was set to describe the behavior between the honeycomb specimen and the rigid plate under dynamic crushing and prevent the initial penetration, in which the frictional coefficient was set to 0.5. Self-surface contact was applied for the individual cell elements within the honeycomb specimen to guarantee that the behavior between cells was smooth and frictionless. When the simulation was analyzed, the freedom of the fixed rigid plate was restricted, and the impact rigid plate only moved in the y-direction. To prevent the out-of-plane buckling under dynamic crushing, the out-of-plane displacements of all the nodes in the honeycomb specimen were constrained.Table 1
Attributes of honeycomb and rigid plate.
Material typeMass densityρ (kg/m3)Young’s modulusE (GPa)Poisson’s ratioμYield stressσs (MPa)Aluminum2700690.376Rigid plate78002100.3235


### 2.3. Crashworthiness Indicator

To intuitively evaluate the crashworthiness and energy absorbing performances of the HKH, KH, and RHH, various indicators were used: the plateau stressσp, the densification strain εd, and the SEA. These were obtained from the stress-strain curves and contrasted.The plateau stressσp and densification strain εd, as fundamental indices, not only directly represent crashworthiness but also can be used for follow-up studies. Based on the one-dimensional shock wave theory, the shock wave model can be expressed as follows [25]:(3)σp=σ0+ρ∗εdV2,σ0 represents the static platform stress, and the impact velocity is defined by V.As the impact velocity was gradually increased, the plateau stressσp [26] and the densification strain εd [27] were calculated as follows:(4)σp=∫εwεpσεdεεp−εw,dEεdε|ε=εd=0,Eε=∫0εσεdεσε,where εp is the plateau stain, and εw and σ(ε) represent the initial strain of the strain-stress curve (the strain at which the stress reached the initial peak) and the nominal stress, respectively. The densification strain εd is defined as the last maximum point on the energy efficiency-strain curve, in which E(ε) is the energy absorbing efficiency of the honeycomb (the ratio of the energy absorption to the nominal stress).The SEA is an important indicator that visually describes the energy absorption per unit mass, which can be calculated as follows [28]:(5)SEA=EAM=∫0εdσεdεΔρ⋅ρs,where EA denotes the total energy absorption during the impact process, and M is the total mass of the honeycomb. A higher SEA usually indicates that an excellent energy absorbing capacity was achieved.


### 2.4. Finite Element Model Verification

To guarantee the accuracy and reliability of the simulation, the finite element model used in the subsequent analysis was verified. An analogous model to that in [29] was established for the RHH under the different velocity impacts, in which the other boundary conditions were the same as those in the reference. The comparison of the deformation behavior between the finite element model and the results in [29] under an impact velocity of V = 14 m/s is shown in Figure 3. The discrepancies of the deformation were small, and identical buckling bands appeared under the same strain. Subsequently, the force-displacement curve and the corresponding investigation are shown in Figure 4(a). Based on the impact velocity of V = 3.5 m/s, the height of force and the tendency of the force-displacement curve are close to [29]. To better prove the accuracy of simulation, the comparison of plateau stress between the finite element model and the results is presented in Figure 4(b). The gap of plateau stress is tiny under the various impact velocity and the value of finite element model can be perfectly fitted by equation (4). Therefore, the method used to build a finite element model in the present investigation was precise and effective and could be used for subsequent study.Figure 3
Comparison of deformation between the present finite element model and the results in [29].Figure 4
The comparison between finite element model and [29]. (a) The force-displacement curve of finite element model, (b) the result of finite element and [29].
(a)(b)


## 2.1. Geometric Configurations of Three Honeycombs

As shown in Figure1, the novel combined honeycomb (i.e., the HKH) can be established by placing a KH inside an RHH. To allow the various honeycombs to be compared, the lengths and widths of the three cell elements used in the impact simulations were kept the same size. Furthermore, three geometric parameters for the HKH were interpreted: the length of the external cell wall l1, the length of the internal cell wall l2, and the thickness of the cell wall t.Figure 1
Design concept for novel combined honeycomb (HKH) and the geometric configuration of the honeycomb.Relative density is a vital index used to evaluate reliability when comparing the results between the HKH, KH, and RHH. For the KH and RHH, the relative densities ΔρΚΗ and ΔρRΗΗ can be defined, respectively, as follows [23]:(1)ΔρKH=ρ∗KHρs=3tl2,ΔρRHH=ρ∗RHHρs=23tl1,where ρKH∗ and ρRHH∗ denote the density of the KH and RHH, respectively, and ρs is the density of the matrix material.Based on the relative density formula for honeycomb materials, the relative density of the HKH can be expressed as follows [24]:(2)ΔρHKH=ρ∗HKHρs=∑litiL1L2=1716l1+3312l2tL1L2,where ΔρΗΚΗ and ρHKH∗ represent the relative density of the HKH and the density of the HKH, respectively, and L1 and L2 are the horizontal and vertical lengths of the honeycomb model, respectively. In this study, the same relative density (Δρ = 4.4%) was used for the three honeycombs, and the corresponding thicknesses of the HKH, KH, and RHH were determined to be 0.02, 0.04, and 0.1 mm. The different relative densities of the honeycomb were obtained based on the modified cell wall thickness using the above formula.


## 2.2. Finite Element Analysis

In the present investigation, the dynamic crushing behaviors and crashworthiness of the three honeycombs (HKH, KH, and RHH) under in-plane impacts were analyzed using the nonlinear finite element software ABAQUS/Explicit. The local structures of the honeycomb specimens are shown in Figure2. Simulations with the finite element model of honeycombs with dimensions of L1×L2×Z were conducted, and the numbers of unit cells in the x- and y-directions were set to 24 and 23, respectively, to capture the deformation behavior of the honeycomb under vertical crushing. The honeycomb specimen was placed between the rigid plates, in which the bottom rigid plate was fixed and the top rigid plate moved downward at a constant velocity. The impact velocity v was increased from 3 to 100 m/s, allowing for three deformation modes to occur (low-, mid-, and high-velocity modes), to investigate the crashworthiness of the honeycomb. To simplify the simulation of the impact experiments, the out-of-plane thickness of the honeycomb specimen was 1 mm, the external cell wall thickness was l1 = 2.7 mm, and the internal cell wall thickness was l2 = 4.7 mm.Figure 2
Finite element models of three honeycombs under in-plane impact.The matrix material of the honeycomb specimen was selected as aluminum, which was assumed to be an elastic–perfectly plastic material. The attributes of the matrix material are presented in Table1. To ensure the convergence and calculation accuracy, the unit cells were meshed using shell elements (S4R) with sizes of 1 mm, and five integration points were defined in the direction of the thickness. Furthermore, the surface-to-surface contact was set to describe the behavior between the honeycomb specimen and the rigid plate under dynamic crushing and prevent the initial penetration, in which the frictional coefficient was set to 0.5. Self-surface contact was applied for the individual cell elements within the honeycomb specimen to guarantee that the behavior between cells was smooth and frictionless. When the simulation was analyzed, the freedom of the fixed rigid plate was restricted, and the impact rigid plate only moved in the y-direction. To prevent the out-of-plane buckling under dynamic crushing, the out-of-plane displacements of all the nodes in the honeycomb specimen were constrained.Table 1
Attributes of honeycomb and rigid plate.
Material typeMass densityρ (kg/m3)Young’s modulusE (GPa)Poisson’s ratioμYield stressσs (MPa)Aluminum2700690.376Rigid plate78002100.3235


## 2.3. Crashworthiness Indicator

To intuitively evaluate the crashworthiness and energy absorbing performances of the HKH, KH, and RHH, various indicators were used: the plateau stressσp, the densification strain εd, and the SEA. These were obtained from the stress-strain curves and contrasted.The plateau stressσp and densification strain εd, as fundamental indices, not only directly represent crashworthiness but also can be used for follow-up studies. Based on the one-dimensional shock wave theory, the shock wave model can be expressed as follows [25]:(3)σp=σ0+ρ∗εdV2,σ0 represents the static platform stress, and the impact velocity is defined by V.As the impact velocity was gradually increased, the plateau stressσp [26] and the densification strain εd [27] were calculated as follows:(4)σp=∫εwεpσεdεεp−εw,dEεdε|ε=εd=0,Eε=∫0εσεdεσε,where εp is the plateau stain, and εw and σ(ε) represent the initial strain of the strain-stress curve (the strain at which the stress reached the initial peak) and the nominal stress, respectively. The densification strain εd is defined as the last maximum point on the energy efficiency-strain curve, in which E(ε) is the energy absorbing efficiency of the honeycomb (the ratio of the energy absorption to the nominal stress).The SEA is an important indicator that visually describes the energy absorption per unit mass, which can be calculated as follows [28]:(5)SEA=EAM=∫0εdσεdεΔρ⋅ρs,where EA denotes the total energy absorption during the impact process, and M is the total mass of the honeycomb. A higher SEA usually indicates that an excellent energy absorbing capacity was achieved.


## 2.4. Finite Element Model Verification

To guarantee the accuracy and reliability of the simulation, the finite element model used in the subsequent analysis was verified. An analogous model to that in [29] was established for the RHH under the different velocity impacts, in which the other boundary conditions were the same as those in the reference. The comparison of the deformation behavior between the finite element model and the results in [29] under an impact velocity of V = 14 m/s is shown in Figure 3. The discrepancies of the deformation were small, and identical buckling bands appeared under the same strain. Subsequently, the force-displacement curve and the corresponding investigation are shown in Figure 4(a). Based on the impact velocity of V = 3.5 m/s, the height of force and the tendency of the force-displacement curve are close to [29]. To better prove the accuracy of simulation, the comparison of plateau stress between the finite element model and the results is presented in Figure 4(b). The gap of plateau stress is tiny under the various impact velocity and the value of finite element model can be perfectly fitted by equation (4). Therefore, the method used to build a finite element model in the present investigation was precise and effective and could be used for subsequent study.Figure 3
Comparison of deformation between the present finite element model and the results in [29].Figure 4
The comparison between finite element model and [29]. (a) The force-displacement curve of finite element model, (b) the result of finite element and [29].
(a)(b)


## 3. Results and Discussion

### 3.1. Deformation Modes

As the impact velocity increased, the deformation modes could be divided into low-velocity, mid-velocity, and high-velocity impact modes. To investigate the effect of impact velocity on crashworthiness and energy absorption, the deformation modes of the HKH were considered for three impact velocities (3, 10, and 100 m/s), and the two substructures (KH, RHH) were compared.For the dynamic crushing of the honeycomb under an impact velocity ofV = 3 m/s, the locations of the buckling band and weak layers in each honeycomb varied (Figure 5(a)). At the initial time of dynamic crushing, the HKH first exhibited a “V”-shaped buckling band near the fixed rigid plate, and a “necking” phenomenon occurred when the unit cells on both sides of the bottom layer began to rotate toward the inner segment due to the free-edge effect. In contrast to the KH, in which an inverted “V”-shaped buckling band appeared near the middle layers initially, the integral compression process of the HKH was more stable and uniform during the crushing. Under the low-velocity impact, the RHH began to exhibit a double “X”-shaped buckling band (ε = 0.293), and the time at which the buckling band appeared was later than that of the HKH. Subsequently, the rotary velocity of the unit cell on the both sides was higher than that on the middle part, a “‿“-shaped buckling band of the HKH was present, and the cellular core was not compacted. The KH was compressed downward with a “⁀“-shaped buckling band, and the cellular core was fully dense. Under the same conditions, the HKH became dense when the nominal strain ε = 0.421, while KH and RHH were still in the elastic stage. In Figure 5(b), the strain-stress curves of three honeycombs exhibited an elastic-plateau-densification stage, because the deformation behavior of the HKH was more stable and uniform. Meanwhile, the deformation velocity was higher than that of the substructure under dynamic crushing. The densification stage of HKH was longer with the comparison of KH and RHH.Figure 5
Dynamic crushing behaviors of three honeycombs under an impact velocity ofV = 3 m/s: (a) deformation maps and (b) strain-stress curves.
(a)(b)As the impact velocity increased (V = 10 m/s), three honeycombs were destroyed near the impact rigid plate, in which various buckling bands could be produced. As shown in Figure 6(a), the behavior could be expressed as follows: the cell walls of the HKH oriented in the y-direction first underwent the same elastic flexing as the KH under a larger axial pressure, and then the external cell walls started to wrap around the end node of the cell element. The shrinkage in the cross direction was produced by the overall instability of the cells. As the nominal strain increased, the “V”-shaped buckling band of the HKH appeared near the impact rigid plate and the “necking” phenomenon was transferred from the bottom layer to the top layer. In the meantime (ε = 0.08), the KH and HKH produced similar crushing bands, in which the deformation amplitude of the KH was unclear under the mid-velocity impact. When the impacted rigid plate gradually moved downward (ε = 0.230), the deformation velocity of the HKH on the both sides was higher owing to the free-edge effect, and as a result, the local “⁀“-shaped buckling band was generated. Under the same nominal strain, the RHH formed a “V”-shape buckling band, and the deformation characteristics were represented by layer-by-layer deformation, in which the deformation degree was less than that of the HKH. In the transitional mode, the deformation behavior of the HKH was similar to that of the KH, but the amplitude of the HKH deformation was more significant. Meanwhile, the deformation velocity of the HKH was higher than that of the substructure (RHH), and different buckling bands appeared.Figure 6
Dynamic crushing behaviors of three honeycombs under an impact velocity ofV = 10 m/s: (a) deformation maps and (b) strain-stress curves.
(a)(b)When the impact velocity was increased toV = 100 m/s, the inertial effect and stress wave were significantly enhanced. Under this condition, “I”-shaped buckling bands appeared in all the honeycombs (Figure 7(a)). Once the impact side touched the HKH, the unit cell had time to rotate, and the cell wall directly flexed. As the strain increased (ε = 0.281), the negative Poisson’s ratio effects of the HKH and KH were markedly weakened, and the three honeycombs collapsed downward layer by layer. Furthermore, when the honeycomb structure was gradually compressed (ε = 0.517), the local buckling band of the HKH was transformed from an “I”-shaped buckling band to an inverted “K”-shaped buckling band. As shown in Figure 7(b), an oscillation appeared in the strain-stress curve under the high-velocity mode, and the value of the plateau stage was effectively enhanced.Figure 7
Dynamic crushing behaviors of three honeycombs under an impact velocity ofV = 100 m/s: (a) deformation maps and (b) strain-stress curves. Deformation mechanism of novel combined honeycomb (HKH).
(a)(b)Under the condition in which the relative density of the honeycomb was small (Δρ < 10%) and the cell wall was slender, the elastic buckling of the unit cell occurred before the plastic collapse [30]. When the honeycomb near the impact side began to compress downward, cell walls AE and BF were subjected to larger axial pressures, as shown in Figure 8(a). Then, AE and BF experienced elastic instability, and the whole unit cells began to expand in the x-direction. Plentiful plastic hinges of the cell wall were produced and dissipated energy (the red circles in Figure 8). As shown in Figure 8(b), the AE and BF were rotated counterclockwise and clockwise around the points A and B, respectively, owing to the buckling deformation of the cell wall. When the degree of compression increased, cell wall AB started to bulge upward because the points A and B moved downward, creating an opposite rotation. In the inside of the HKH, the cell wall kept a fixed length and flexibility. When the axial loading continued, the critical force of cell wall AD was less compared to that of cell wall AE. Therefore, the deformation of AD was less than that of cell wall AE under dynamic crushing. In the topological structure of the HKH, the length of cell wall AC was less than that of AE and AD as a result of the critical force of AC being higher. Initially, the axial pressure was not sufficient to induce the instability of the AC under low- and mid-velocity impacts. In the location of point A, cell walls AE, AD, and AB all underwent anticlockwise winding but the rotation direction of cell wall AC was unclear in the initial stage of crushing. As the plastic hinge of cell wall AE absorbed more energy, the degree of compression was more notable. Subsequently, the axial force of point A increased to cause the anticlockwise winding of AC (Figure 8(c)). Based on the same principle, a similar deformation occurred on both sides of the honeycomb, and similar winding behaviors occurred at the nodes. As the bending of the cell wall around the end of the rods increased, visible deformation of cell walls AE and BF occurred. Meanwhile, a negative Poisson’s ratio effect of the HKH occurred as the distance between the points A and B gradually decreased. When cell walls AE and BF reached the ultimate elastic strength, cell wall EF overlapped with DE. Finally, the unit cell began to collapse, and as a result, the densification of the honeycomb could be produced under dynamic crushing.Figure 8
Deformation mechanism of HKH under the low- and mid-velocity impacts.
(a)(b)(c)


### 3.2. Plateau Stress

As is well known, the plateau stress can be used to evaluate the crashworthiness of the honeycomb, and it is an important factor for the energy absorption. According to one-dimensional shock wave theory [31], the mathematical model of the plateau stress and impact velocity is given by equation. (4). Based on the above mathematical model, three fitting formulas were obtained by fitting the plateau stress-impact velocity data of the three honeycombs, and the mathematical model was perfectly fitted to prove the accuracy of this simulation (as shown in Figure 9). Under the quasi-static impact, the difference of the plateau stresses between the HKH and KH was small. However, as the impact velocity gradually increased, rate of increase of the plateau stress for KH was higher than that of the HKH. During the crushing process, the uniform deformation of the HKH demonstrated that the deformation degree of the complex cell was smaller, and the bending degree of the cell wall was lower. Thus, many plastic hinges had limited abilities to absorb energy under the same conditions. The number of plastic hinges that were generated in the cell walls of the RHH was smaller, and the energy absorption capacity was limited. Therefore, the height of the plateau stage was lower compared with those of the HKH and KH from the strain-stress curves (Figures 5–7). As the impact velocity increased, the differences of the plateau stresses between the RHH and HKH and between the RHH and KH also increased because the multiple plastic hinges were produced in the bending cell walls of the HKH and KH.Figure 9
Plateau stresses of three honeycombs: (a) regular hexagonal honeycomb (RHH), (b) HKH, and (c) Kagome honeycomb (KH).
(a)(b)(c)The difference of the plateau stress between the HKH and RHH was defined as Δσ. To thoroughly investigate the regularity of the plateau stress, the differences for various relative densities are shown in Figure 10. In this study, the sensitivity of the HKH and RHH to the relative density and impact velocity can be reflected by Δσ. As shown in Figure 10, the difference of the plateau stress increased approximately linearly with the impact velocity, in which the slope of the fitting formula was enhanced from 0.0058 to 0.0165 with the relative density increasing from Δρ = 4.4% to Δρ = 8.8%. The impact velocity and relative density had an influence on Δσ. When the impact velocity and relative density improved, Δσ significantly increased, and the influence of the relative density was more significant. When determining Δσ, the plateau stress of the HKH was more easily affected by the relative density and impact velocity than the RHH.Figure 10
Differences of plateau stress for (a) Δρ = 4.4% and (b) Δρ = 8.8%.
(a)(b)


### 3.3. Specific Energy Absorption (SEA)

In this section, the specific energy absorption is compared to determine the crashworthiness of the novel combined honeycomb. For comparison, the value of the SEA corresponding to the densification strainεd was selected from the SEA-strain curve. The densification strain was so important for the SEA that it could result in a range of SEA values. A higher εd usually indicates that the honeycomb required more time to absorb energy under the dynamic impact. According to the above study, εd can be obtained from the energy efficiency-strain curve (Figure 11(a)), and the densification strains of the three honeycombs at various impact velocities are shown in Figure 11(b). The impact velocity influenced the densification strain, which changed rapidly under the low- and mid-velocity modes. As the impact velocity increased, the inertial effect gradually dominated the deformation behavior of the honeycomb. For example, similar buckling bands occurred with the HKH, RHH, and KH. Then, under the high-velocity mode, the densification strains of the three honeycombs remained around a fixed value (εd = 0.9). Because the damage forms of the unit cells were different, i.e., the HKH underwent uniform flexion and the KH and RHH underwent layer-by-layer flexing, the densification strains were lower than those of the KH and RHH.Figure 11
Energy efficiency and densification strain: (a) energy efficiency-strain curve of HKH under an impact velocityV = 3 m/s and (b) densification strain of three honeycombs with different impact velocities.
(a)(b)The SEA values of the three honeycombs at different velocities are shown in Figure12. The SEA of the RHH was lower than that of the HKH under all modes because the extra rods in the topological structure of the HKH produced more plastic hinges to absorb energy. Compared with the KH, a design with extra rods was introduced to allow the SEA of the HKH to be stronger than the KH, with ε < 0.7, under the low- and mid-velocity impacts. The difference of the SEA values between the HKH and KH was small (ε < 0.4) under the high-velocity mode. To directly indicate the improvement of the HKH for the RHH, the SEA values with the different impact velocities and relative densities are shown in Table 2. As the impact velocity and relative density increased, the SEA values of the HKH and RHH were significantly enhanced. The absorbed energy of the HKH was 137.52% larger than that of the RHH under the mid-velocity impact (V = 35 m/s), and the absorbed energy of the HKH was 25.70% larger than that of the RHH under a low-velocity impact (V = 3 m/s). Different from previous studies, because the second plateau stage only appeared in the strain-stress curves of the mid- and high-velocity modes (as shown in Figure 7(b)), the absorbed energy of the HKH was 54.73% larger than that of the RHH under the high-velocity impact, and the degree of improvement was stronger compared to that of the low-velocity mode. Similar to the plateau stress, the rate of increase of the SEA was related to the relative density. The rate of increase of the SEA increased from 137.52% to 209.30% as the relative density increased from Δρ = 4.4% to Δρ = 8.8% under the mid-velocity impact.Figure 12
Specific energy absorption (SEA) curves of three honeycombs with different impact velocities.Table 2
SEA values of RHH and HKH with different impact velocities and relative densities.
Impact velocityv (m/s)Relative densities Δρ (%)SEAHKH (kJ/kg)SEARHH (kJ/kg)SEAHKH−SEARHHSEARHH34.40.6310.50225.70%8.81.4970.90066.30%354.43.5961.514137.52%8.86.8052.200209.30%1004.415.1539.79354.73%8.823.80811.111114.27%


### 3.4. Negative Poisson’s Ratio

When many honeycombs experience an in-plane impact, they usually expand to both sides. According to the difference in topological structures, the partial honeycombs underwent transverse shrinking due to the unit cell being rotated inwards under the dynamic crushing, which will generate a negative Poisson’s ratio (NPR) effect. The deformation behaviors of a honeycomb that exhibited an NPR are more sophisticated, and these honeycombs possess more practical performance advantages than the traditional honeycomb [32]. The dynamic Poisson’s ratio v can be defined as follows [33]:(6)ν=−εxεy=−ΔL/L1εy=−∑i=06ΔAi−ΔBi6εL1,where εx and εy are the nominal strain in the x- and y-directions, respectively, ΔL is defined as the shrinking displacement of the honeycomb in the x-direction, and L1 is the width of the honeycomb specimen. To obtain a more accurate ΔL value, it was calculated by taking the average of the shrinking displacements of the six nodes in Figure 13, in which ∆Ai and ∆Bi are the shrinking displacements of points Ai and Bi in the x-direction, respectively.Figure 13
Schematic diagram of node selection.The dynamic Poisson’s ratios of the HKH and KH under different impact velocities are presented in Figure14(a). Under all the impact velocities, the HKH and KH exhibited NPRs. The NPR was not evident as the impact velocity was gradually increased. In contrast to the KH, which maintained an NPR under any impact velocity, the HKH underwent a process in which the dynamic Poisson’s ratio was transformed from negative to positive (when the nominal strain ε = 0.8) under the low- and mid-velocity modes. The NPRs under high-velocity impacts for the HKH and KH were less apparent than for the other two deformation modes but steadily approached a fixed value (v = −0.1). As shown in Figure 14(b), the peak Poisson’s ratio of the RHH was most easily affected by the impact velocity, and it decreased continuously as the impact velocity increased. Under the same condition, the peak Poisson’s ratio-strain curves of three honeycombs varied rapidly under low-velocity impacts, and the HKH exhibited a stronger NPR and higher velocity sensitivity than the KH under the low-velocity mode. After the combination of cell elements, the HKH exhibited a strong NPR effect, and its auxetic property was stronger than that of the KH, which was more susceptible to impact forces.Figure 14
Map of dynamic Poisson’s ratio: (a) dynamic Poisson’s ratio-stain curve of the HKH and KH with different impact velocities and (b) peak Poisson’s ratio of three honeycombs with different impact velocities.
(a)(b)


## 3.1. Deformation Modes

As the impact velocity increased, the deformation modes could be divided into low-velocity, mid-velocity, and high-velocity impact modes. To investigate the effect of impact velocity on crashworthiness and energy absorption, the deformation modes of the HKH were considered for three impact velocities (3, 10, and 100 m/s), and the two substructures (KH, RHH) were compared.For the dynamic crushing of the honeycomb under an impact velocity ofV = 3 m/s, the locations of the buckling band and weak layers in each honeycomb varied (Figure 5(a)). At the initial time of dynamic crushing, the HKH first exhibited a “V”-shaped buckling band near the fixed rigid plate, and a “necking” phenomenon occurred when the unit cells on both sides of the bottom layer began to rotate toward the inner segment due to the free-edge effect. In contrast to the KH, in which an inverted “V”-shaped buckling band appeared near the middle layers initially, the integral compression process of the HKH was more stable and uniform during the crushing. Under the low-velocity impact, the RHH began to exhibit a double “X”-shaped buckling band (ε = 0.293), and the time at which the buckling band appeared was later than that of the HKH. Subsequently, the rotary velocity of the unit cell on the both sides was higher than that on the middle part, a “‿“-shaped buckling band of the HKH was present, and the cellular core was not compacted. The KH was compressed downward with a “⁀“-shaped buckling band, and the cellular core was fully dense. Under the same conditions, the HKH became dense when the nominal strain ε = 0.421, while KH and RHH were still in the elastic stage. In Figure 5(b), the strain-stress curves of three honeycombs exhibited an elastic-plateau-densification stage, because the deformation behavior of the HKH was more stable and uniform. Meanwhile, the deformation velocity was higher than that of the substructure under dynamic crushing. The densification stage of HKH was longer with the comparison of KH and RHH.Figure 5
Dynamic crushing behaviors of three honeycombs under an impact velocity ofV = 3 m/s: (a) deformation maps and (b) strain-stress curves.
(a)(b)As the impact velocity increased (V = 10 m/s), three honeycombs were destroyed near the impact rigid plate, in which various buckling bands could be produced. As shown in Figure 6(a), the behavior could be expressed as follows: the cell walls of the HKH oriented in the y-direction first underwent the same elastic flexing as the KH under a larger axial pressure, and then the external cell walls started to wrap around the end node of the cell element. The shrinkage in the cross direction was produced by the overall instability of the cells. As the nominal strain increased, the “V”-shaped buckling band of the HKH appeared near the impact rigid plate and the “necking” phenomenon was transferred from the bottom layer to the top layer. In the meantime (ε = 0.08), the KH and HKH produced similar crushing bands, in which the deformation amplitude of the KH was unclear under the mid-velocity impact. When the impacted rigid plate gradually moved downward (ε = 0.230), the deformation velocity of the HKH on the both sides was higher owing to the free-edge effect, and as a result, the local “⁀“-shaped buckling band was generated. Under the same nominal strain, the RHH formed a “V”-shape buckling band, and the deformation characteristics were represented by layer-by-layer deformation, in which the deformation degree was less than that of the HKH. In the transitional mode, the deformation behavior of the HKH was similar to that of the KH, but the amplitude of the HKH deformation was more significant. Meanwhile, the deformation velocity of the HKH was higher than that of the substructure (RHH), and different buckling bands appeared.Figure 6
Dynamic crushing behaviors of three honeycombs under an impact velocity ofV = 10 m/s: (a) deformation maps and (b) strain-stress curves.
(a)(b)When the impact velocity was increased toV = 100 m/s, the inertial effect and stress wave were significantly enhanced. Under this condition, “I”-shaped buckling bands appeared in all the honeycombs (Figure 7(a)). Once the impact side touched the HKH, the unit cell had time to rotate, and the cell wall directly flexed. As the strain increased (ε = 0.281), the negative Poisson’s ratio effects of the HKH and KH were markedly weakened, and the three honeycombs collapsed downward layer by layer. Furthermore, when the honeycomb structure was gradually compressed (ε = 0.517), the local buckling band of the HKH was transformed from an “I”-shaped buckling band to an inverted “K”-shaped buckling band. As shown in Figure 7(b), an oscillation appeared in the strain-stress curve under the high-velocity mode, and the value of the plateau stage was effectively enhanced.Figure 7
Dynamic crushing behaviors of three honeycombs under an impact velocity ofV = 100 m/s: (a) deformation maps and (b) strain-stress curves. Deformation mechanism of novel combined honeycomb (HKH).
(a)(b)Under the condition in which the relative density of the honeycomb was small (Δρ < 10%) and the cell wall was slender, the elastic buckling of the unit cell occurred before the plastic collapse [30]. When the honeycomb near the impact side began to compress downward, cell walls AE and BF were subjected to larger axial pressures, as shown in Figure 8(a). Then, AE and BF experienced elastic instability, and the whole unit cells began to expand in the x-direction. Plentiful plastic hinges of the cell wall were produced and dissipated energy (the red circles in Figure 8). As shown in Figure 8(b), the AE and BF were rotated counterclockwise and clockwise around the points A and B, respectively, owing to the buckling deformation of the cell wall. When the degree of compression increased, cell wall AB started to bulge upward because the points A and B moved downward, creating an opposite rotation. In the inside of the HKH, the cell wall kept a fixed length and flexibility. When the axial loading continued, the critical force of cell wall AD was less compared to that of cell wall AE. Therefore, the deformation of AD was less than that of cell wall AE under dynamic crushing. In the topological structure of the HKH, the length of cell wall AC was less than that of AE and AD as a result of the critical force of AC being higher. Initially, the axial pressure was not sufficient to induce the instability of the AC under low- and mid-velocity impacts. In the location of point A, cell walls AE, AD, and AB all underwent anticlockwise winding but the rotation direction of cell wall AC was unclear in the initial stage of crushing. As the plastic hinge of cell wall AE absorbed more energy, the degree of compression was more notable. Subsequently, the axial force of point A increased to cause the anticlockwise winding of AC (Figure 8(c)). Based on the same principle, a similar deformation occurred on both sides of the honeycomb, and similar winding behaviors occurred at the nodes. As the bending of the cell wall around the end of the rods increased, visible deformation of cell walls AE and BF occurred. Meanwhile, a negative Poisson’s ratio effect of the HKH occurred as the distance between the points A and B gradually decreased. When cell walls AE and BF reached the ultimate elastic strength, cell wall EF overlapped with DE. Finally, the unit cell began to collapse, and as a result, the densification of the honeycomb could be produced under dynamic crushing.Figure 8
Deformation mechanism of HKH under the low- and mid-velocity impacts.
(a)(b)(c)


## 3.2. Plateau Stress

As is well known, the plateau stress can be used to evaluate the crashworthiness of the honeycomb, and it is an important factor for the energy absorption. According to one-dimensional shock wave theory [31], the mathematical model of the plateau stress and impact velocity is given by equation. (4). Based on the above mathematical model, three fitting formulas were obtained by fitting the plateau stress-impact velocity data of the three honeycombs, and the mathematical model was perfectly fitted to prove the accuracy of this simulation (as shown in Figure 9). Under the quasi-static impact, the difference of the plateau stresses between the HKH and KH was small. However, as the impact velocity gradually increased, rate of increase of the plateau stress for KH was higher than that of the HKH. During the crushing process, the uniform deformation of the HKH demonstrated that the deformation degree of the complex cell was smaller, and the bending degree of the cell wall was lower. Thus, many plastic hinges had limited abilities to absorb energy under the same conditions. The number of plastic hinges that were generated in the cell walls of the RHH was smaller, and the energy absorption capacity was limited. Therefore, the height of the plateau stage was lower compared with those of the HKH and KH from the strain-stress curves (Figures 5–7). As the impact velocity increased, the differences of the plateau stresses between the RHH and HKH and between the RHH and KH also increased because the multiple plastic hinges were produced in the bending cell walls of the HKH and KH.Figure 9
Plateau stresses of three honeycombs: (a) regular hexagonal honeycomb (RHH), (b) HKH, and (c) Kagome honeycomb (KH).
(a)(b)(c)The difference of the plateau stress between the HKH and RHH was defined as Δσ. To thoroughly investigate the regularity of the plateau stress, the differences for various relative densities are shown in Figure 10. In this study, the sensitivity of the HKH and RHH to the relative density and impact velocity can be reflected by Δσ. As shown in Figure 10, the difference of the plateau stress increased approximately linearly with the impact velocity, in which the slope of the fitting formula was enhanced from 0.0058 to 0.0165 with the relative density increasing from Δρ = 4.4% to Δρ = 8.8%. The impact velocity and relative density had an influence on Δσ. When the impact velocity and relative density improved, Δσ significantly increased, and the influence of the relative density was more significant. When determining Δσ, the plateau stress of the HKH was more easily affected by the relative density and impact velocity than the RHH.Figure 10
Differences of plateau stress for (a) Δρ = 4.4% and (b) Δρ = 8.8%.
(a)(b)


## 3.3. Specific Energy Absorption (SEA)

In this section, the specific energy absorption is compared to determine the crashworthiness of the novel combined honeycomb. For comparison, the value of the SEA corresponding to the densification strainεd was selected from the SEA-strain curve. The densification strain was so important for the SEA that it could result in a range of SEA values. A higher εd usually indicates that the honeycomb required more time to absorb energy under the dynamic impact. According to the above study, εd can be obtained from the energy efficiency-strain curve (Figure 11(a)), and the densification strains of the three honeycombs at various impact velocities are shown in Figure 11(b). The impact velocity influenced the densification strain, which changed rapidly under the low- and mid-velocity modes. As the impact velocity increased, the inertial effect gradually dominated the deformation behavior of the honeycomb. For example, similar buckling bands occurred with the HKH, RHH, and KH. Then, under the high-velocity mode, the densification strains of the three honeycombs remained around a fixed value (εd = 0.9). Because the damage forms of the unit cells were different, i.e., the HKH underwent uniform flexion and the KH and RHH underwent layer-by-layer flexing, the densification strains were lower than those of the KH and RHH.Figure 11
Energy efficiency and densification strain: (a) energy efficiency-strain curve of HKH under an impact velocityV = 3 m/s and (b) densification strain of three honeycombs with different impact velocities.
(a)(b)The SEA values of the three honeycombs at different velocities are shown in Figure12. The SEA of the RHH was lower than that of the HKH under all modes because the extra rods in the topological structure of the HKH produced more plastic hinges to absorb energy. Compared with the KH, a design with extra rods was introduced to allow the SEA of the HKH to be stronger than the KH, with ε < 0.7, under the low- and mid-velocity impacts. The difference of the SEA values between the HKH and KH was small (ε < 0.4) under the high-velocity mode. To directly indicate the improvement of the HKH for the RHH, the SEA values with the different impact velocities and relative densities are shown in Table 2. As the impact velocity and relative density increased, the SEA values of the HKH and RHH were significantly enhanced. The absorbed energy of the HKH was 137.52% larger than that of the RHH under the mid-velocity impact (V = 35 m/s), and the absorbed energy of the HKH was 25.70% larger than that of the RHH under a low-velocity impact (V = 3 m/s). Different from previous studies, because the second plateau stage only appeared in the strain-stress curves of the mid- and high-velocity modes (as shown in Figure 7(b)), the absorbed energy of the HKH was 54.73% larger than that of the RHH under the high-velocity impact, and the degree of improvement was stronger compared to that of the low-velocity mode. Similar to the plateau stress, the rate of increase of the SEA was related to the relative density. The rate of increase of the SEA increased from 137.52% to 209.30% as the relative density increased from Δρ = 4.4% to Δρ = 8.8% under the mid-velocity impact.Figure 12
Specific energy absorption (SEA) curves of three honeycombs with different impact velocities.Table 2
SEA values of RHH and HKH with different impact velocities and relative densities.
Impact velocityv (m/s)Relative densities Δρ (%)SEAHKH (kJ/kg)SEARHH (kJ/kg)SEAHKH−SEARHHSEARHH34.40.6310.50225.70%8.81.4970.90066.30%354.43.5961.514137.52%8.86.8052.200209.30%1004.415.1539.79354.73%8.823.80811.111114.27%


## 3.4. Negative Poisson’s Ratio

When many honeycombs experience an in-plane impact, they usually expand to both sides. According to the difference in topological structures, the partial honeycombs underwent transverse shrinking due to the unit cell being rotated inwards under the dynamic crushing, which will generate a negative Poisson’s ratio (NPR) effect. The deformation behaviors of a honeycomb that exhibited an NPR are more sophisticated, and these honeycombs possess more practical performance advantages than the traditional honeycomb [32]. The dynamic Poisson’s ratio v can be defined as follows [33]:(6)ν=−εxεy=−ΔL/L1εy=−∑i=06ΔAi−ΔBi6εL1,where εx and εy are the nominal strain in the x- and y-directions, respectively, ΔL is defined as the shrinking displacement of the honeycomb in the x-direction, and L1 is the width of the honeycomb specimen. To obtain a more accurate ΔL value, it was calculated by taking the average of the shrinking displacements of the six nodes in Figure 13, in which ∆Ai and ∆Bi are the shrinking displacements of points Ai and Bi in the x-direction, respectively.Figure 13
Schematic diagram of node selection.The dynamic Poisson’s ratios of the HKH and KH under different impact velocities are presented in Figure14(a). Under all the impact velocities, the HKH and KH exhibited NPRs. The NPR was not evident as the impact velocity was gradually increased. In contrast to the KH, which maintained an NPR under any impact velocity, the HKH underwent a process in which the dynamic Poisson’s ratio was transformed from negative to positive (when the nominal strain ε = 0.8) under the low- and mid-velocity modes. The NPRs under high-velocity impacts for the HKH and KH were less apparent than for the other two deformation modes but steadily approached a fixed value (v = −0.1). As shown in Figure 14(b), the peak Poisson’s ratio of the RHH was most easily affected by the impact velocity, and it decreased continuously as the impact velocity increased. Under the same condition, the peak Poisson’s ratio-strain curves of three honeycombs varied rapidly under low-velocity impacts, and the HKH exhibited a stronger NPR and higher velocity sensitivity than the KH under the low-velocity mode. After the combination of cell elements, the HKH exhibited a strong NPR effect, and its auxetic property was stronger than that of the KH, which was more susceptible to impact forces.Figure 14
Map of dynamic Poisson’s ratio: (a) dynamic Poisson’s ratio-stain curve of the HKH and KH with different impact velocities and (b) peak Poisson’s ratio of three honeycombs with different impact velocities.
(a)(b)


## 4. Conclusions

In this study, a novel design that combined a regular hexagonal honeycomb (RHH) and a Kagome honeycomb (KH) was proposed to enhance the crashworthiness and energy absorption of traditional honeycombs. Using the nonlinear finite element software ABAQUS/Explicit, the deformation behaviors of the three honeycombs (HKH, KH, and RHH) were investigated under the in-plane impact, and the mechanical capacities of three honeycombs were compared. The key results are summarized as follows:(1)
The HKH and KH exhibited a “necking” shrinkage phenomenon, and the deformation characteristics of the HKH were uniform and stable. Under low- and mid-velocity impacts, the deformation velocity of the HKH was higher than that of the RHH, and the deformation amplitude of the HKH was larger than that of the KH. Furthermore, the second plateau stage of the HKH only appeared for the mid- and high-velocity impacts.(2)
The plateau stress of the HKH was higher and more easily affected by the variations of the impact velocity and the relative density than that of the RHH. Owing to the bending degree of the cell walls, the differences between the plateau stresses of the HKH and KH were small under the low- and mid-velocity impacts. Owing to the complex cells of the HKH, the densification strain was lower in the HKH, and under a high-velocity impact, a similar densification strain occurred in the three honeycombs. The rates of increase of the SEA values of the RHH and HKH were the lowest in the low-velocity mode. Subsequently, the SEA values of the HKH were closer to those of the KH under the low- and mid-velocity impact modes when the nominal strainε reached 0.7.(3)
The HKH and KH exhibited multiple deformation modes that generated negative Poisson’s ratios. Under low-velocity impacts, the negative Poisson’s ratio of the HKH was greater than that of the KH. By combining various unit cells, the HKH exhibited excellent auxetic properties, and the peak Poisson’s ratio was strongly influenced by the impact velocity.



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*Source: 1411885-2022-06-23.xml*