In-Plane Dynamic Cushioning Performance of Concave Hexagonal Honeycomb Cores

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Introduction
Porous solid honeycomb cores are widely used in aerospace, automobile transportation, construction engineering, product packaging, and other felds because of their good cushioning performance when subjected to impact and vibration.Te honeycomb structure also has an excellent energy absorption capacity, sound absorption capacity, and heat absorption capacity [1,2].Porous solid honeycomb cores can be divided into regular type and irregular type according to diferent geometric topological structures.Regular-type structures include regular hexagon, triangle, circle, and concave hexagon.Concave hexagonal honeycomb is a typical honeycomb structure, and it is the material with negative Poisson's ratio.Negative Poisson's ratio materials have strong application value; they have good industrial application prospects in engineering biomedicine, aerospace, construction, navigation, and other felds [3][4][5][6][7].As a typical porous cushioning material with negative Poisson's ratio, concave hexagonal honeycomb cores (CHCs) have many unique advantages and have become one of the indispensable porous cushioning materials.As for cushioning materials, energy absorption performance characterization and cellular structure optimization are currently the research focus of scholars at home and abroad.Accurate characterization of energy absorption and cushioning performance can guide the optimization design of material structure and expand the application feld of materials, which is of great signifcance for CHCs.
Domestic and foreign scholars have carried out a series of studies on CHCs with negative Poisson's ratio structure.Li et al. [8] compared the concave hexagonal honeycomb structure with the regular quadrilateral and hexagonal honeycomb structures, and the research showed that under the same impact load, the concave hexagonal honeycomb structure has higher compression modulus, yield strength, and surface specifc energy absorption, the energy absorption efect is optimal, and the shock wave attenuation characteristics are better than other structures.Li et al. [9] compared the macromechanical responses of four honeycomb structures under quasistatic compression.Te results show that the honeycomb structure with negative Poisson's ratio gets extra support due to the shrinkage of the matrix, which enhances the stifness and energy absorption properties.Te abovementioned studies show that CHCs have excellent energy absorption performance, and it is of great signifcance to further quantify the infuence of its structural parameters and impact velocity on the energy absorption performance.
Shen et al. [10,11] studied and analyzed the dynamic performance of hexagonal honeycomb of aluminum substrate under diferent impact velocities.On this basis, the dynamic response of CHCs is studied deeply.Zhang et al. [12] studied the dynamic impact crushing behavior of opencell aluminum foam with negative Poisson's ratio efect through fnite element numerical simulation and discussed their platform stress, specifc energy absorption, and deformation mode.Hu et al. [13] studied the infuence of diferent concave forms of concave triangular honeycomb on the platform stress, specifc energy absorption value, and deformation mode under the action of axial impact through numerical simulation.Te result show that the less the number of concave edges, the more obvious the negative Poisson's ratio efect.Te structural platform with unilateral internal and internal concave has higher stress, longer stress response time, and better energy absorption efect.Ma et al. [14] compared the platform stress and energy absorption of the concave triangular honeycomb structure with diferent number of concave edges and angles under impact at different velocities.Te results show that the stress and energy absorption of the structure with only concave bottom edge are greater than those of the structure with three concave bottom edge.With the increase of impact velocity, the stress and energy absorption capacity of the concave triangular honeycomb are increased.Zhao et al. [15] used the fnite element software ABAQUS to analyze the in-plane impact characteristics and deformation modes of three similar concave hexagonal honeycomb structures at low, medium, and high speed in the concave direction.Alomarah et al. [16] proposed a new structure with a negative Poisson's ratio and its in-plane mechanical properties were improved.Te Poisson's ratio of honeycomb can be changed by adjusting the new cylindrical structure.It can be seen from the abovementioned research that the current research on CHCs with negative Poisson ratio mostly focuses on the impact velocity and the number of concave edges on the platform stress and energy absorption.In fact, the cellular structural parameters such as wall thickness, side length, and expansion angle of CHCs can afect their mechanical properties.
It is common for honeycomb core materials to bear loads such as impact and collision in practical applications, so accurate characterization of dynamic cushioning performance is particularly important.For CHCs, there is little research on this aspect.At present, the energy absorption cushioning performance characterization methods mainly include the cushioning curve method, capacity absorption curve method, energy absorption diagram method, Janssen factor method, and Rusch curve method [17].Te cushioning coefcient curve characterization can consider many factors such as force area and thickness.Further introducing dynamic cushioning coefcient can more accurately and truly refect the cushioning performance of honeycomb materials in practical applications, and it can also overcome the shortcomings of static parameters in characterizing the bufering performance of honeycomb materials.
To sum up, in this paper, based on the fnite element numerical analysis method (FEM), the in-plane high-speed crushing process of CHCs is simulated, and aluminum is selected as the matrix material.Trough relevant postprocessing software, the dynamic cushioning coefcient curve and the minimum dynamic cushioning coefcient of CHCs were obtained based the results of FEM.Te infuence law of related structural parameters and impact velocities on the minimum cushioning coefcient was analyzed.It provides the theoretical basis for the structure optimization, performance improvement, and further application of CHCs.

Model Description
2.1.Finite Element Model.Sun et al. [18] framed the FE model for the in-plane impact of CHCs by following the previous FE investigations about the mechanics of cellular materials.Te similar full-scale FE model is used here.Te model for in-plane dynamic cushioning analysis is shown in Figure 1.ANSYS/LS-DYNA software is employed here to simulate the in-plane cushioning performance of CHCs along the x 1 direction.Te number of cells in the x 1 and x 2 directions are 11 and 15, respectively.Te specimen of CHCs is placed between the upper pressing plate (P1) and the support plate (P2), both of which are rigid.Te mass of the upper pressing plate is large enough to ensure that the specimens are crushed.When the model is loaded, the support rigid plate (P2) is fxed, and the upper rigid plate (P1) impacts the specimen along the direction x 1 at a constant speed v.
Te bottom end of the honeycomb structure is bound to the fxed-end rigid body, and the left and right sides are free in plane.Te displacement constraint of the honeycomb structure in the x 3 direction is 0 to ensure that the honeycomb always meets the plane strain state during the impact process.Te honeycomb structure was meshed with square Belytschko Shell163 elements with fve integration points and an element edge length of 0.3 mm.Te entire model defnes single face frictionless automatic contact, and the honeycomb body and two rigid plates are defned as surface-to-surface automatic contact, with a friction coefcient of 0.02.Te single surface automatic contact is set among the cells in the honeycomb in case the structure penetrates each other during crushing [19].

Shock and Vibration
Te impact velocity is shown in Table 1.Following Sun and Zhang [20], the bilinear strain-hardening model (see Figure 2(b)) is used to represent the constitutive relationship of basis material, which is typically aluminum.Te mechanical properties of basis material are shown in Table 2. Tis model is suited for modeling isotropic and kinematic hardening plasticity with the option of including rate efects.It is a very cost-efective model and is available for shell, solid, and beam elements.Gao [21,22] and Tan et al. [23] also used the bilinear strain-hardening material model in their research studies.
Te structure of CHCs is shown in Figure 2(a).Te cell structural parameters of CHCs are side length (l), width (h), wall thickness (t), expansion angle (q), depth (b), edge length ratio (h/l), and the ratio of cell wall thickness to edge length (t/l).l of all specimen is equal to 3 mm.b of all specimen is equal to 10 mm.Te other parameters were divided into two groups.Te frst group was used to study the infuence of diferent wall thickness on the in-plane cushioning performance of CHCs.Te second group is used to study the infuence of diferent expansion angles on the in-plane cushioning performance of CHCs.Te specifc parameters are shown in Tables 3 and 4.

Model Verifcation.
In order to ensure the reliability of the model, the present FE model is shown in Figure 1, which is similar to the models used by Ruan et al. used by Ruan et al. [24], Zheng et al. [25], Li et al. [26], Ali et al. [27], Liu and Zhang [28], and Sun and Zhang [20].
In addition, to verify the accuracy and reliability of the fnite element algorithm and model, the in-plane dynamic mechanical properties of a hexagonal honeycomb structure are simulated using the loading conditions described in Reference [19], where all degrees of freedom are constrained on the bottom side of the specimen, while the other edges are free in plane.Te impact plate collides with the specimen at a constant speed.As shown in Figure 3, the simulation results are basically consistent with the research results in Reference [19] and can efectively reproduce the formation of "W" deformation bands in the dynamic impact process of concave hexagonal honeycomb structures, which proves the reliability of the modeling and algorithm in this paper.

Dynamic Cushioning Coefficient
According to the cushioning performance analysis theory, the cushioning coefcient is the ratio of the applied force to the deformation energy per unit thickness.Te cushioning coefcient curve can be used to characterize the cushioning properties of materials, which can consider many factors such as the force area and thickness.Te dynamic cushioning coefcient can refect the energy absorption properties of materials in practical applications more accurately and truly, and it can overcome the defciency of statics parameters characterization.It is of great reference significance for the application and optimization design of CHCs.

Optimal Unit Volume Energy Absorption Point.
According to the "cushioning coefcient-maximum stress curve" method, the cushioning coefcient can be calculated by the following three formulas: where C is the cushioning coefcient, dimensionless.e is the energy absorption per unit volume of the bufer material (g/ cm 2 ).ε is strain, and F is the force (KN).A is the bearing area of the sample (cm 2 ).Te force and displacement curves F − u of CHCs obtained by the fnite element method are shown in Figure 4.According to the characteristics of the curve, it was frst simplifed to get the simplifed curve as shown in Figure 5 and then standardized to get the corresponding stress-strain curve.Obviously, the obtained stress-strain curve is also a four stage.Shock and Vibration

Stress
Strain Te corresponding simplifed four-section e − ε curve is as follows: Defne dynamic cushioning coefcient C � σ/e.Terefore, the simplifed "dynamic cushioning coefcient-strain" relationship model is as follows: Te typical "dynamic cushioning coefcient-strain" curve of CHCs can be obtained by MATLAB programming, as shown in Figure 6.
According to the abovementioned method, dynamic cushioning coefcient-strain curves of all samples can be obtained, and their morphology is similar to that of Figure 6.
Tere is a minimum point on each dynamic cushioning Te quantity absorption value e D is defned as "optimal unit volume energy absorption."Terefore, the point D is called the "optimal unit volume energy absorption point."Te smaller the minimum dynamic bufer coefcient value, the higher the energy absorption efciency of the material, that is, the better the cushioning performance of the material.Terefore, the point D is called the "optimal unit volume energy absorption point."Te smaller the minimum dynamic cushioning coefcient value, the higher the energy absorption efciency of the material, that is, the better the cushioning performance of the material.

Te Value of Minimum Dynamic Cushioning Coefcient (MDCC).
As mentioned above, the density energy absorption value can be obtained by FEM, and then divided by the volume of the sample, the density energy absorption per unit volume can be calculated, that is, the optimal energy absorption per unit volume e D .Te MDCC can be calculated by the following formula.
As mentioned above, the densifed energy absorption value can be obtained by FEM.Te densifed energy absorption per unit volume, i.e., the optimal energy absorption per unit volume e D , is equal to the densifed energy absorption value divided by the volume of the sample.Te value of MDCC can be calculated by the following formula.
Based on FEM calculation results, the calculated value of MDCC of each sample are listed in Tables 5-10.It can be seen from the table that the values of MDCC are greater than and close to 1, which can be theoretically explained from the following analysis.According to equation ( 5), the value of e D for optimal unit volume energy absorption is shown in the following.
Te stress corresponding to e D is the average stress in the platform area, namely, the dynamic peak stress σ p , which corresponds to the MDCC.As can be seen from the impact force-displacement curve of CHCs (see Figure 4), the initial displacement in the linear elastic stage is very small, and the corresponding energy absorption is very small.ε 0 is approximately equal to 0, so the formula of MDCC is as follows: Teoretically, the value of MDCC can be approximated as the reciprocal of the densifcation strain.Also, since e D is less than 1, the value of MDCC are all greater than 1.According to the results of numerical analysis, e D ranges from 0.70 to 0.95 and tends to 1 with the increase of impact velocity.Terefore, the value of MDCC is going to be close to 1.

Infuence of the Expansion Angle on MDCC.
Under diferent impact loads, when other structural parameters are fxed, the MDCC value of CHCs with diferent expansion angles are listed in Tables 5-7 and Figures 7-9.Te results show that the value of MDCC decreases with the increase of the expansion angle at the same impact velocity.

Shock and Vibration
When other structural parameters are fxed and the expansion angle varies from 10 °to 80 °, the relation curve between the relative density and the expansion angle of the CHCs can be obtained according to equation (11), as shown in Figure 10.Te results show that with the increase of the expansion angle, the relative density gradually decreases and the attenuation decreases.When the expansion angle reaches about 80 °, the relative density hardly decreases.At a given impact velocity, the densifcation strain tends to increase with the decrease of relative density.It can also be seen from equation (10) that the value of MDCC is approximately inversely proportional to the densifcation Shock and Vibration strain.Terefore, at a given impact velocity, when other structural parameters are constant, the value of MDCC of CHCs will decrease with the increase of the expansion angle.Tis is also proved by the calculation results in Tables 5-7.

Infuence of Cell Wall Tickness to Edge Length (t/l Ratio) on MDCC. At diferent impact velocities and when other structural parameters are constant, the value of MDCC of
CHCs with diferent t/l ratio is listed in Tables 8-10, corresponding to Figures 11-13.It shows that the value of MDCC increases with the increase of the t/l ratio at the same impact velocity.
According to equation (11), the relation curve between the relative density and t/l ratio of CHCs is shown in Figure 14.As can be seen from the fgure, when other structural parameters are constant, the relative density increases with the increase of the t/l ratio, and the densifcation strain    Shock and Vibration decreases with the increase of relative density.According to the analysis theory of the simplifed model, the relationship between MDCC and densifcation strain is approximately inversely proportional.Terefore, under a given impact velocity load, the value of MDCC will increase with the increase of the t/l ratio.Tis is also proved by the calculation results in Tables 8-10.

Infuence of Impact Velocity on Cushioning Performance and Deformation
Modes.From Tables 5 to 10, it can be seen that when all structural parameters are constant, the value of MDCC of CHCs decreases with the increase of impact velocity.Tis physical phenomenon can be explained by the deformation modes of CHCs under diferent impact velocities.
Te deformation modes of CHCs are obtained by simulation calculation under low speed, medium speed, and high impact load, as shown in Figures 15-17.For the quasistatic deformation mode at low speed (Figure 15), the deformation mode of each part of CHCs is uniform, and densifcation of all parts occurs almost simultaneously.At the frst stage, a local collapse zone in the shape of "W" frst appears near the pressure plate.When the deformation of

Shock and Vibration
CHCs is less than 20%, it can be clearly seen that "M"-shaped deformation zone appears at the support plate, and with the further deepening of the impact end, "W" collapse zone, the shape of "double W".When the deformation propagates to some extent, the cells in the center of the specimen begin deformation appearing in the indistinct "X"-shaped form with extension deformation on both sides (see Figure 15(d)).All collapse bands evolve gradually until the top and bottom local deformation bands touch and are absolutely crushed to densifcation (see Figures 15(e) and 15(f)).
Under the medium-speed impact (see Figure 16), the collapse zone of CHCs slowly begins to converge towards the upward direction of the pressure plate, accompanied by the collapse of the lower part.Such a deformation mode also determines that with the increase of impact velocity, the strain at the arrival of densifcation will be gradually larger.In this stage, most deformation stages of CHCs will produce a "W"-shaped deformation belt near the pressure plate, and only in the later stage, a cross-sloping deformation belt can be produced.
Under the high-speed impact (see Figure 17), the deformation of CHCs only occurs near the upper pressure plate, in the shape of "−".No inclined deformation belt is generated in the whole process.When the upper pressure plate is almost completely close to the lower support plate, the whole material will enter the densifcation stage.It can also be seen that when the deformation mode of "−" occurs, the densifcation strain of the sample will increase to a certain value and will not change with the further increase of impact velocity.
According to the above analysis, when the structural parameters are constant, before the dynamic deformation mode occurs, the densifcation strain of CHCs also increases with the increase of impact velocity.According to formula  (12), under the premise of fxed structural parameters, the value of MDCC of CHCs decreases with the increase of impact velocity.
When t � 0.05 mm and h/l � 2, the deformation modes of CHCs with diferent expansion angles under diferent impact velocity loads are shown in Figure 18(a).It can be found that when the expansion angle increases from 10 °to 80 °, the critical conversion velocity between deformation modes hardly depends on the expansion angle.When h/l � 2 and θ � 30 °, the deformation modes of CHCs with diferent wall thickness and side length ratios under diferent impact velocity loads are shown in Figure 18(b).Te critical velocity v 1 of the transformation from "M" to "W" is almost independent of the wall thickness and side length ratio, and it is shown as a horizontal straight line.Te critical velocity v 2 of the transition from "W" to "one" is related to the wall thickness side length ratio.
As σ ∝ (t/l) 2 , ρ ∝ t/l [31], the critical deformation mode conversion velocity is v ∝ (t/l) 1/2 , and it can be obtained by [32].When h/l = 2 and θ = 30 °, based on the fnite element numerical analysis results, the empirical formula of the change of the critical transformation velocity of the deformation mode with the wall thickness side length ratio can be obtained by the least square method as follows:     12 Shock and Vibration

Conclusion
In this paper, the dynamic cushioning coefcient is used to characterize the cushioning performance of CHCs.According to the characteristics of dynamic stressstrain curves, a simplifed energy absorption model was proposed.Based on the simplifed mode, the value of MDCC of CHCs is obtained to characterize the cushioning performance.It is a methodological innovation.
Te infuence of structural parameters and impact velocity on the MDCC is analyzed.When the structural parameters are constant, the MDCC of CHCs decreases with the increase of the impact velocity, that is, the greater the impact velocity, the higher the energy absorption efciency of CHCs and the better the cushioning performance.When the impact velocity and other parameters are constant, the MDCC of CHCs increases with the increase of the wall thickness side length ratio and decreases with the increase of the expansion angle, that is, the greater the wall thickness side length ratio, the worse the cushioning performance of CHCs, and the greater the expansion angle, the better the cushioning performance of CHCs.
It is concluded that the deformation modes of CHCs under dynamic impact load can be divided into three types.Te frst type is "M"-type collapse at low speed impact, the Shock and Vibration second type is "W"-type transition deformation mode, and the third type is the "I-shaped" deformation mode during high-speed impact.According to the fnite element simulation results, the transformation velocity of deformation mode is obtained, and the empirical expression between the transformation velocity and the structural parameters of honeycomb element is obtained.
Te abovementioned conclusions are based on numerical and theoretical research felds, but it still provides guidance and basis for the practical engineering application of CHC in the future.In practical applications, the dynamic impact load may also be from out-plane direction, the matrix materials are also diverse, and the actual production process may cause some structural defects.Terefore, in the future research, the impact of diferent plane impact loads on the cushioning performance of CHCs can be carried out, the efects of different matrix materials on the cushioning properties of CHCs can be compared and summarized, and the infuence of structural defects on the cushioning properties of CHCs can be analyzed.Tese studies have important guiding signifcance for the safe and reasonable use of CHCs.

Figure 14 :
Figure 14: ρ − t/l curve of the concave hexagonal honeycomb cores with fxed h/l and θ.

Table 4 :
Te second group of confguration parameters.

Table 5 :
C D for diferent at diferent v (h/l � 2.0 and t � 0.05 mm).