In-plane crushing response and energy absorption of two different arranged circular honeycombs

Circular honeycombs have been widely used in many fields due to their excellent mechanical properties recently, however, circular honeycombs with different arrangements have been rarely studied. In this paper, two circular honeycombs with different arrangements were designed and fabricated using a 3D printer. Both honeycombs were investigated by experiment and finite element (FE) analysis to compare their performance. It is worth noting that both honeycombs possess the same relative density. The accuracy of the FE models was validated by comparing them with experimental results and relevant reports. Subsequently, a series of numerical studies were conducted to analyse the in-plane dynamic crushing behaviour and energy absorption characteristics at different impact velocities. Based on different experiment deformation modes were identified from the observation of results respectively. Additionally, continuous circle honeycomb (CCH) exhibits a higher reaction force but is prone to fracture. On the other hand, spacing circle honeycomb (SCH) possesses a lower reaction force but offers greater stability due to its ability to flex and release force. As a result, SCH can effectively absorb energy and demonstrates superior crushing capacity compared to CCH. To investigate and compare the plateau stress and energy absorption of these honeycombs, the FE method was used, which also involved a detailed analysis of the impact velocity and relative density of the honeycomb. It was observed that the crushing stress and energy absorption of SCH were higher than those of CCH with the same impact velocity and relative density. According to the one dimensional shock wave theory, empirical formulas for two circular honeycombs to predict the plateau stress are given respectively with the maximum error lower than 12%. This paper aims to offer valuable insights for the design of different configurations, with the goal of improving the crashworthiness and energy absorption capacity of a specific honeycomb structure.


Introduction
Porous materials have attracted considerable research interest due to their excellent mechanical properties, such as high stiffness, Young's modulus and better energy absorption capacity.L J Gibson [1] and A G Evans [2] first studied and prepared the properties of porous materials.Circular honeycomb is a significant type of porous material that finds promising applications in many engineering fields, such as aviation, packaging, construction, blast [3], vehicle protection [4,5] and so on.In recent years, many scholars have carried out a great deal of research to improve the properties of the material, through experimental [6] and numerical study [7], theory [8], uniaxial and biaxial compression, in-plane [9] and out-plane impact.
In the experiment study, Hu et al [5] conducted in-plane uniaxial and equi-biaxial compression experiment on the circular polycarbonate honeycombs.A special test rig was designed to carry out the in-plane equi-biaxial compression tests.Papka and Kyriakides [10] developed a testing device that allowed the polycarbonate circular honeycombs to be compressed simultaneously and independently in two orthogonal directions.The effect of the biaxiality ratio on the energy absorption capacity of circular polycarbonate honeycombs was comprehensively discussed.J Chung and A M Waas [11] analysed the crushing response of a polycarbonate circular cell honeycomb to in-plane biaxial loading by static and dynamic experiments.Static experiments corresponding to several different biaxial loading conditions under displacement control are carried out in two different in-plane directions respectively.In addition, dynamic experiments related to various impact forces under in-plane biaxial loading in orthogonal directions are conducted.The crushing response of a polycarbonate circular honeycomb in plane biaxial loading is analysed by J Chung and A M Waas [12].The result showed different collapsed modes under different biaxial loading histories.Experimental measurements indicate how the magnitude of the collapse load varies and how the load-displacement responses.
Through the application of numerical methods, X Yang and X Xi [13] proposed a novel circular-celled honeycomb by incorporating the petal-shaped structure into the regular circular cell honeycomb, and investigated the in-plane crushing behaviour and energy absorption capacity under different impact velocities through the nonlinear finite element software LS-DYNA.In-plane uniaxial and equi-biaxial compression tests by a special designed test facility were carried out the in-plane quasi-statically on polycarbonate circular honeycombs by L L Hu and T X Yu [5].The deformation behaviours of circular polycarbonate honeycombs under uniaxial compression were quantitatively described by tracking the variations of the cell parameters.
By using the theoretical analysis, Chung and Waas [14] derived the equivalent elastic mechanics of circular and elliptic honeycombs and analyzed the sensitivities of honeycomb stiffness in the in-plane properties.T P Gotkhindi and K R Y Simha [15] explored the prospects of bundled tubes and circular honeycombs in a general diamond array structure, effective transverse Young's moduli and Poisson's ratio for thick or thin diamond array structure were obtained theoretically.Karagiozova and Yu [16] analysed the post-collapse behaviour and strain characteristics of ductile circular honeycombs under in-plane uniaxial and biaxial loadings.S D Papka and S Kyriakides [17] derived a theory for the in-plane compression of polycarbonate circular honeycomb.However, for average crushing strains above about 10%, their effect was relatively small and the calculated responses were in good agreement with the experimental ones.
Some researchers have studied the behaviour of the circular honeycomb under uniaxial and biaxial compression.Papka and Kyriakides [18] presented the force-displacement response of polycarbonate with circular close-packed cells under the in-plane uniaxial compression experimentally and analytically.As a result of the rate dependence of the material, the initiation and propagation stresses increase as the rate of crushing of the honeycomb is increased.Aiming to improve the performance of polycarbonate circular honeycombs, T X Yu and Z Y Gao [19] studied the deformation mechanics of cellular materials with different topology when subjected to multi-axial loadings.
There are also some researches on in-plane and out-plane impact of circle honeycombs.C Waas [20] carried out static and dynamic experiment of a polycarbonate circular honeycomb subjected to in-plane biaxial loading through numerical analysis using the finite element method.The biaxial in-plane crushing processes of circle honeycomb are investigated.L L Hu et al studied the dynamic crushing of circular-celled honeycombs under out-of-plane impact within a wide range of impact velocity by using both theoretical and numerical methods, and it is verified by the experiments.
It is evident from the above studies that scholars have studied the circular honeycomb through various techniques and methods, but they have all conducted their studies on CCH.Few articles have been written to study the different arrangement of honeycomb structures.In this paper, on the basis of CCH honeycomb structure, a new circular honeycomb SCH is designed and proposed with the same density to compare.These two structures were fabricated using Stereo Lithography Appearance (SLA) 3D printing and tested using an MTS machine.Some FE simulations of different parameters are carried out.Then, a discussion is made to understand the effect on the dynamic crushing and energy absorption response of honeycomb.Based on the onedimensional shock wave theory [1,21], the relation between the dynamic plateau stresses of circular honeycomb with different densities and impact velocities is established.It is clear that a better understanding the influence of configuration on the impact of circle honeycomb would be essentially critical to the energy absorbing design and lightweight.

Geometry of two different arranged circular honeycombs
Two types of circular honeycomb structures are formed by circular rings in both horizontal and vertical directions.The schematic diagram of the in-plane setup is shown in figure 1. Continuous circle honeycomb (CCH) is formed by the contact arrangement of units in the horizontal and vertical directions shown in figure 1(a).Figure 1(b) manifests spacing circle honeycomb (SCH) assembled by the contact arrangement of units in the degree of 45 with the horizontal and vertical directions.It can also be seen that there are intervals between neighbouring rings in the horizontal and vertical directions.The dimensions of CCH and SCH in both horizontal and vertical directions are L 1 and L 2 respectively, while the out-of-plane width is b.Geometric sizes of the honeycomb units can be uniquely defined by three independent variables r, R and t, as indicated in figures 1(a) and (b), where r is the inner radius of the circle, R is the outer radius of the circle and t is the thickness of the units, Two adjacent circles are connected by the thickness t of the honeycomb units, and the unit wall is assumed to be uniform with thickness t.The samples are printed by 3D method as shown in figures 1(c) and (d), respectively.In figure 1, both forms of honeycombs have the same size and are made up of 100 arcs.It is assumed that each unit is made up of four identical arcs.Only the spatial arrangement is different.Thus, the relative density of the two circular honeycomb structures is basically consistent.According to the theory of porous materials, the relative density of the circular honeycomb can be calculated by the ratio of the actual volume of representative units to their three-dimensional spatial volume.Therefore, the relative density can be given by the following equation, where, V RV represents the actual volume size of representative cells, while V Total represents the threedimensional spatial volume.

Experimental test of two different arranged circular honeycombs
The two proposed circular honeycomb configurations are relatively complicated and not easily manufactured by traditional machining methods.Therefore, samples were fabricated by using 3D printer (JX-345, CREAKITY) with SLA technology, as shown in figure 2(a).The whole size of the machine is 745 * 455 * 585 mm, while the printing platform size is 345 * 195 * 335 mm.In addition, the printing accuracy of the 3D printer is 0.1 mm.Width L 1 = 95 mm, height L 2 = 95 mm, out-of-plane breadth b = 20 mm, and cell wall thickness t = 1.0 mm, which corresponding to circular radius r = 9 mm and circular radius R = 10 mm, respectively, were the same for both samples.ABS resin serves as the samples' structural component.Using a universal testing apparatus (CMT 4304), we conducted a quasi-static compression test on 3Dprinted circular honeycomb samples, as shown in figure 2(b).The specimen was placed on a flat steel plate that was fixed in bottom and then compressed by a top steel plate at a loading speed of 5 mm min −1 , as indicated in figures 2(c) and (d).A sensor in the loading plate captured data on the crushing force, while the deformation of the specimen was recorded using a high-definition Nikon digital camera.

Finite element modelling
The in-plane impact finite element model of circular honeycomb used is shown in figure 3(a).A sample is located in the middle of two plates with free ends on both the left and right sides.Additionally, all out-of-plane displacements of the sample are limited to ensure a plane strain state.The upper plate moves in the y-direction at a constant speed, while the lower plate is fixed to the bottom.In the baseline FE model, the corresponding values of r = 9 mm, R = 10 mm, t = 1 mm, and b = 20 mm were adopted for the unit, respectively.
The ABAQUS/EXPLICIT [22] finite element method was employed for the numerical modelling work.The material of unit wall was assumed to be elastic and perfectly plastic with a mass density of 1.04 g cm −3 , a Young's modulus of 2.5 GPa, a Poisson's ratio of 0.3, and a yield stress of 32 MPa.In order to have better convergence and computational correctness, five integration points through the thickness of the components was established.Additionally, unit walls of the honeycomb were simulated using four-node doubly curved shell elements (S4R).Through a convergence research, each wall was characterized as a single self-contact surface, and all surfaces that may come into contact during the compression process were defined as self-contact surfaces.The contact friction between the two plates and the sample was set to 0.3 [23,24].The generic contact algorithm was employed for contact between the sample and the two plates.Figure 3(b) illustrates the results between the experiment and the different FE mesh sizes.It can be observed that there is a certain error between the FE results and the experimental results.When the strain ε is less than 0.5, the results are essentially the same, but when the strain ε is greater than 0.5, the experimental force increases sharply, closely followed by mesh 1.0, mesh 2.0 and mesh 0.75, respectively.Due to the influence of 3D printing manufacturing error, defect and different curing time on the sample.There is a certain deviation from the experimental results, but the result of mesh size of 1.0 is more close to the experimental result.

Crushing force of two different arranged circular honeycombs
This section aims to study the quasi-static compression behaviours of two different proposed circular honeycombs, including the deformation pattern and the force.The force-displacement history of the sample measured during the quasi-static compression test is depicted in figure 4. In addition, snapshots of sample deformation during the compressing process along the vertical direction at the same moment were also shown in figure 4. Distinct deformation forms and force curves were observed between CCH and SCH samples, exhibiting several different phases in the compression process.
First of all, the vertical displacements of both honeycombs ranged from 0 to 7.5 mm, indicating that they were primarily in the elastic phase of the compression process.Both honeycombs exhibited predominantly elastic deformation, with the force being transmitted from the top to the bottom of the honeycomb.During this process, both honeycombs displayed uniform deformation throughout.The force curves also exhibited an inclined shape.However, when the displacement reached 7.5 mm, the forces applied to CCH and SCH were 1200 N and 500 N, respectively.At this point, it can be concluded that CCH can withstand a maximum force that is 2.4 times greater than that of SCH.Secondly, as the compression continues, the sample undergoes yielding by transferring the bent circle.During this yielding phase, it is worth noting that the circle of CCH adjacent to impact plate experiences significant deformation.At a displacement of 16 mm, the circle begins to fracture, resulting in a sharp decrease in force to 350 N. In contrast to the CCH samples, the SCH samples do not exhibit the same level of compression as the CCH samples.Unlike the CCH sample, the SCH sample exhibited no signs of fracture in this phase, and the force applied to the SCH remained relatively constant at approximately 450 N. Following this, the plateau phase promptly ensued.The subsequent layer of arcs in the fractured circles of the CCH sample began to bend and subsequently break, resulting in a wave-like pattern of force curve increase and then decrease.The CCH sample repeated this layer-by-layer process until it was compressed against the fixed plate.The maximum force of second wave was lower than the first wave due to the decrease in the number of circle layers subjected to the force.Meanwhile, the arcs of the SCH sample continued to bend but did not break layer by layer, leading to a relatively stable applied force during the process.Finally, as the arcs of the neighbouring layers began to make contact with one another, the density phase was reached, and the force on the SCH sample experienced a sudden rise due to the contact between these neighbouring layers.
Based on experimental observations, it can be concluded that the CCH honeycomb has the better ability to withstand force approximately 2.4 times than the SCH honeycomb during the elasticity phase.However, the CCH honeycomb is more susceptible to fractures due to its material and structure in the follow-up phase.On the other hand, the SCH honeycomb, although it can endure less force than CCH, demonstrates greater stability by releasing the force through the bending of circles.Consequently, the samples treated with resin indicate that the SCH honeycomb possesses a more reliable capacity to absorb energy and withstand pressure compared to the CCH honeycomb.

Failure mechanism of two different arranged circular honeycombs
The failure mechanisms of two organized circular honeycombs subjected to quasi-static compression, which result in distinct damage mechanisms are shown in figure 5.The top and lower deformations of the CCH honeycomb remain essentially the same for displacements ranging from 0 to 7.5 mm during elastic phase, as illustrated in figure 4(a).However, when compressed to 16 mm in the yield phase, the CCH honeycomb near the impact plate experiences significant deformation and even fracture, especially at the M and N junction region as seen in figure 5(a).This resulted in the fracture of the first layer of arcs at the impact plate at positions a, b, c, and d, as shown in figure 5(b), and the following morphology could not be maintained.Failure to preserve morphology upon fracture, the following yielding stage then moves on to the second layer breaking at the same places.
On the other hand, the SCH deformation is shown to be predominantly bending during this compression process.Under the compression displacement from 0 to final in figure 4(b), the dominant deformation mechanism observed in figure 5(d) is bending of the circle walls.As a result, the arcs on the left (AC) and right (BD) sides of the two connected circular arc cells are the main bending place of the circular honeycomb structure.The arcs at the top (AB) and bottom (CD) begin to straighten at the same time, but there is no fracture of the SCH arc is observed during compression.

Simulations and validation
Through experimentation, the quasi-static responses of the SLA fabricated circular honeycomb were examined in considering the local strain achieved by the unit wall and the resultant danger of failure.The two types of circular honeycombs were built using generalized metallic material so that the unbroken deformation mode could be observed.In order to conduct a complete investigation of the crushing process of such a circular honeycomb with various parameters, the numerical models in this part used the metallic material without failure.According to table 1, the generalized metallic substance that makes up the circular honeycomb was considered to be rate-independent elastic perfect-plastic with a yield strength of 130 MPa, a Young's modulus of 70 GPa, a mass density of 2700 kg m −3 , and a Poisson's ratio of 0.3.Other numerical parameters are the same as in the preceding FE model.
In order to validate the numerical model, FE model of the SCH was performed to simulate the quasi-static compressing result and the experimental result are shown in figure 3 in the present study.The numerically produced deformation patterns of the specimen at different strain levels can be observed in figure 6.The stressstrain curve of the specimen estimated during the quasi-static compression simulation is shown in figure 7(b), which matches the results from article [25] (figure 7(a)) very well.The following studies examine the effects of impact velocity on the stress and strain of numerical calculations during the crushing process.The nominal stress and strain curves of CCH and SCH are produced by simulation with the same relative density (Δρ = 0.165).Figure 8 shows that under these circumstances the stress on the specimen increases as the velocity increases.In addition, the nominal stress at 1 m s −1 and 15 m/s remains essentially unchanged.It can be seen the stress and strain curves in figure 8(a) exhibit less fluctuations, the curves in figure 8(b) exhibit a lot of fluctuations, though.The stress of SCH is higher than that of CCH also can be seen in figure (8).

Effect of velocity on energy absorption
Under external loads, energy obeys the first law of thermodynamics, which is denoted by Formula (2).In the equation, Ei stands for the material's internal energy, Ek for its kinetic energy, Ef for contact friction loss, Ew for the work produced by the external load, and Eqb for the energy lost as a result of the surrounding medium's damping.The external work is primarily transformed into kinetic energy and internal energy absorbed by the impact object in Formula (2), leaving out the energy of friction loss and the energy of damping dissipation of surrounding media.As a result, the sum of these two quantities is taken as the total energy absorbed by the specimen.
The curve of total energy with strain in this paper is shown in figure 9.It can be concluded from the results in figure 9 that when the relative density unchanged, the ability of the specimen to absorb energy during compression increases with the increase of velocity for CCH and SCH.Additionally, in figures 9(a) and (b), the ability of the two types of specimen to absorb energy is essentially the same at velocities of 1 m s −1 and 15 m s −1 .Finally, we may draw the conclusion that SCH's capacity for absorbing energy is somewhat greater than CCH's.By altering the impact velocity and unit design, energy absorption may be increased.

Effect of velocity on energy distribution coefficient
The internal energy distribution coefficient Φ (the percentage of internal energy in total absorbed energy) is established in order to analyse the energy absorption distribution of two types of specimen under in-plane impact.The equation as follows: The effect of impact velocity on the internal energy distribution coefficient Φ during specimen impact was studied.Figure 10 shows the variation of Φ with nominal strain when the relative density of the specimen is constant (Δρ = 0.165) and the impact velocity is different.It is evident from the figure that impact velocity has a significant influence on the internal energy distribution coefficient Φ.As the impact velocity increases, the value of Φ gradually decreases from approximately 1.0 at low impact velocity (V = 1 m s −1 ) to 0.55 at high impact velocity (V = 120 m s −1 ).This suggests that at impact velocities below the second critical impact velocity, the honeycomb mainly absorbs internal energy.As the impact velocity increases, the proportion of absorbed internal energy decreases due to the increased inertial effect.Additionally, it is observed that the internal energy distribution coefficient Φ is slightly higher for SCH than for CCH.Therefore, it can be concluded that SCH absorbs more energy than CCH at different impact velocities.

Effect of velocity and density on stress
During the compression process of the specimen, there is a stage where the force remains stable and this stage is phase III as shown in figure 7.This phase is known as the platform stage, and the force obtained during this stage is called the platform force.The platform force per unit area is defined as the platform stress, which is an important indicator for describing the dynamic response characteristics of honeycombs.It can be calculated using formulas (4) and (5), Formula (4) defines ε 1 as the initial strain, which represents the corresponding strain value when the initial stress reaches the platform stress and stabilizes.As a result, the value of ε 1 is expected to be very small.On the other hand, ε D reflects the dense strain occurring due to contact between adjacent cell walls within the specimen.
In Formula (5), the value of F(ε) is determined by evaluating the average value of the force exerted on the upper impact plate within the platform area, which is obtained through simulations.Meanwhile, Lx denotes the length of the specimen in the x-axis direction, and L z represents the length of the specimen in the vertical direction of the xy plane.where, σ ys stands for the yield stress of the matrix material, Δρ represents the ratio of the designed honeycomb, such as CCH or SCH relative to the matrix material, ρ ys represents the density of the matrix material.Formula (6) and the information in table 2 are used to fit the three CCH and SCH curves, respectively and to derive the equations for the six curves in figure 11, which are then used to solve the formula's parameters m.The platform stress calculation formula is changed by fitting the value after the findings are verified and it is discovered that the value of m conforms to the linear distribution by observation.Formulas (9) and (10) are the modified versions, respectively, when the value of ε D set as 0.88.

( )
Figure 11 shows the comparison between the platform stresses obtained from simulation results and the fitting curves of CCH and SCH under different relative densities.It is observed that the stress increases with higher relative densities or velocities, and the relationship between them follows a quadratic curve.The formula curves are in good agreement with the simulation results.When the relative density of SCH (Δρ) is 0.247 and its velocity is 60 m s −1 , the maximum error between the formula curve and the simulation result is only 12%.At lower relative densities, the agreement between the curves and the simulation results is even better.Therefore, the correction formula is valid and the derived results demonstrate the rationality of the formula.

Conclusion
This study investigated the quasi-static compression responses of two circularly organized honeycomb structures, both experimentally and numerically.Firstly, 3D-printed samples of CCH and SCH were fabricated and underwent quasi-static uniaxial compression tests.The results revealed that CCH and SCH exhibited different deformation modes, failure mechanisms, and reaction forces.The comparison of the literature and experimental results was carefully examined to ensure the accuracy of the numerical calculations.The effects of velocity, including relative density, on the crushing behaviours of CCH and SCH were also thoroughly studied.
The key findings from the numerical calculations and experimental testing can be summarized as follows: The crushing force of CCH and SCH exhibited four stages during the compression process, including the elastic phase, the failure or yield phase, the plateau phase, and the densification phase.The crushing force obtained in the FE analysis was found to be in good agreement with that of the experimental tests and other literature.CCH has a greater reaction force but is more prone to fracture, while SCH has a smaller reaction force but is more stable because it can bend and release the force.So SCH can absorb energy stably and have better ability to crush than CCH.
The impact velocity had the highest effect on the crushing stress, energy absorption, and energy distribution coefficient of CCH and SCH.Under the same density, stress and energy absorption increased with an increase in impact velocity, while the energy distribution coefficient decreased with an increase in velocity.Furthermore, the thickness of the cell wall had a similar effect on stress, and it was found to be related to the density.The higher the density, the higher the stress obtained.Finally, it was observed that the crushing stress and energy absorption of SCH were higher than those of CCH with the same impact velocity and relative density.Based on onedimensional shock wave theory, an empirical formula was obtained by thoroughly analysing the FE results, with a maximum relative error of 12%.

Figure 1 .
Figure 1.The schematic diagrams and samples: (a) The schematic diagram of CCH; (b) The schematic diagrams of SCH; (c) The sample of CCH; (d) The sample of SCH.

Figure 2 .
Figure 2. The machine of fabricate and test: (a) JX-345 3D printer, CREAKITY; (b) Quasi-static compressing test; (c) The test experiment of CCH; (d) The test experiment of SCH.

Figure 3 .
Figure 3.The FE simulation of circular honeycomb: (a) The schematic diagram of simulation module; (b) The comparison of FE simulation and experiment results.

Figure 4 .
Figure 4. Experiment results of two different circular honeycomb: (a) The experiment result of CCH; (b) The experiment result of SCH.

Figure 5 .
Figure 5. Failure mechanism of two different circle honeycomb: (a) The external force state of CCH; (b) The fracture point of SCH; (c) The external force state SCH; (d) The deformation of SCH.

Figure 7 .
Figure 7.The comparison of stress-strain curve: (a) The stress-strain curve from [25]; (b) The stress-strain curve in CCH.

Figure 8 .
Figure 8.The stress-strain comparison of CCH and SCH: (a) The stress-strain curve of CCH in different velocities; (b) The stressstrain curve of SCH in different velocities.

Figure 9 .
Figure 9.The energy comparison of CCH and SCH: (a) The energy of CCH in different velocity; (b) The energy of SCH in different velocity.

Figure 10 .
Figure 10.The energy ratio comparison of CCH and SCH: (a) The energy ratio of CCH in different velocities; (b) The energy ratio of SCH in different velocities.

Figure 11 .
Figure 11.The comparison between simulation results and formula curves: (a) Simulation results and formula curves of CCH; (b) Simulation results and formula curves of SCH.

Table 1 .
Engineering parameters of aluminium.