Study on the Effect of Nodal Configuration on the Mechanical Properties of Hexa-Ligamentous Chiral Honeycombs

Abstract

To investigate the effect of nodal configuration on the mechanical properties of hexachiral honeycombs, three specimens, namely, a standard honeycomb, a thickened-node honeycomb, and a filled-node honeycomb, were prepared using 3D-printing technology. Several quasi-static compression tests were performed, which revealed that nodal reinforcement can inhibit nodal aberrations during ligament winding, thus facilitating the “rotational” mechanism and improving the negative Poisson’s ratio properties of the honeycomb. Experiments performed using the finite element method showed that nodal reinforcement mainly played a role in the stage of stress rise, and the role of nodal filling was more significant than that of nodal thickening. Low-strain standard honeycombs presented the highest specific absorption energy. However, the specific absorption energy of the filled-node honeycomb and the thickened-node honeycomb exceeded that of the standard honeycomb at a strain of 0.71. The conclusions presented herein can aid in the optimal design of honeycombs and contribute to the design of protective structures.

1. Introduction

Protective structures are crucial for naval ships as they can safeguard the ships against explosions and impact loads. Modern ships are generally equipped with a combination of gas and liquid tank protection on the sides of the ship. How to endow these protective structures with special features, such as light weight, is a popular research topic. Porous materials, such as foam and honeycombs, can be engineered to possess these properties. Yang et al. [1] proposed a macroscopic honeycomb-side protection structure and analyzed the simulation results, concluding that the honeycomb protection structure had good impact resistance. An auxetic honeycomb is a structure that has a negative Poisson’s ratio due to its special deformation mechanism [2]. Conventional honeycombs exhibit lateral expansion or contraction behavior under uniaxial compression or tension, while an auxetic honeycomb exhibits a negative Poisson’s ratio (NPR). In recent years, research on the mechanical properties and energy absorption characteristics of negative Poisson’s ratio materials has been very popular. Such research mainly involves improving energy absorption [3,4,5] and mechanical properties [6,7,8,9].

With the development of manufacturing technology, more and more new honeycomb structures are being used in a wide range of applications, such as for deformed wing structures [10], smart fasteners, replacement vessels, and curved door panels [11]. The cellular configuration of a honeycomb can significantly affect its mechanical properties [12,13], so investigating the relationship between the cellular configuration and the macroscopic mechanical behavior of a honeycomb is of great interest. A large number of cytosolic configurations have been studied by scholars [14,15,16,17,18,19,20], such as hexagonal, triangular, square, circular, concave-angled hexagonal, chiral cytosolic, and star-shaped cytosolic configurations. However, these traditional cytosolic configurations cannot fully meet the increasingly complex functional requirements in this field, and there is an urgent need for materials with better mechanical properties, better energy absorption characteristics, and a wider range of Poisson’s ratios for specific application scenarios [21]. Therefore, optimized designs based on classical honeycomb structures are constantly being proposed.

The negative Poisson’s ratio properties of honeycomb structures are induced by two main mechanisms, namely, an inner concave mechanism and a rotational mechanism [22]. Due to the unique mechanical properties of negative Poisson’s ratio honeycombs, they have been the focus of a great deal of research. Prall et al. [23] investigated the in-plane mechanical properties of six-ligament chiral honeycombs using both theoretical analysis and experimental studies. Qi et al. [24] investigated the in-plane response of a four-ligament chiral honeycomb under quasi-static and dynamic loading conditions and derived theoretical equations for platform stresses applicable under quasi-static and dynamic loads. Dong et al. [25] experimentally investigated the effect of the thickness of a concave-angled hexagonal honeycomb on deformation and found that the deformation patterns of thin-walled honeycombs differed significantly from those of thick-walled honeycombs. Zhu et al. [26] established a chiral honeycomb model filled with cell elements in nodes and studied it using simulation methods. The results showed that this configuration improved the honeycomb’s overall mechanical properties. Evidently, increasing the thickness of the nodes and filling them had an impact on their overall mechanical behavior. However, research on the effects of node thickening and node filling on mechanical properties is limited and thus deserves more attention.

The aim of this study was to investigate the effect of node thickening and node filling on mechanical properties and to compare them using a standard six-line chiral honeycomb. Quasi-static compression tests were carried out on three honeycombs with different nodes in order to investigate the similarities and differences in their deformation patterns, mechanical properties, and energy absorption characteristics.

2. Specimen Preparation and Testing

2.1. Geometry Description and Fabrication

Three different configurations of six-ligament chiral honeycombs were developed: (a) standard fovea, (b) thickened node, and (c) filled node honeycombs. The standard honeycomb consists of a node connected to six ligaments in a periodic arrangement. It has 10 columns (9 columns in even rows) in the transverse direction and 11 rows in the longitudinal direction, with transverse and vertical lengths of 211.2 mm and 203.6 mm, respectively [27]. The node-thickened honeycomb was developed by thickening the nodes of a standard honeycomb, thus effectively improving the honeycomb’s overall impact resistance. The node-filled honeycomb is a standard honeycomb whose nodes are filled with small chiral cells. The six ligaments of the cell are connected to the ligaments at the nodes for transferring the load.

The above three honeycombs were prepared using 3D-printing technology for quasi-static compression tests with a thickness of 25 mm. The honeycomb material employed was 316L stainless steel, and its material parameters are shown in Table 1. The honeycomb specimens were fabricated using selective-laser-melting (SLM) technology, which was widely applied alongside 3D-printing technology because of the wide range of alloys and rapid fabrication enabled by this technique. During the manufacturing process, a high-power laser was used to completely melt the surface layer of the metal powder and process it layer-by-layer until the sample was completed. To avoid defects in the material structure and maximize the performance potential of the material, the specimens were surface-ground. All the fabricated specimens are shown in Figure 1, which shows their good molding quality.

The detailed geometric parameters of the honeycombs are shown in Figure 2 and Table 2, where r₁ is the node radius, r₂ is the filled node radius, t₁ is the ligament thickness, t₂ is the node thickness, l₁ is the ligament length, and l₂ is the filled ligament length.

2.2. Quasi-Static Compression Test

The compressor selected for the test was a WDW-300 (see Figure 3) microcomputer-controlled electronic testing machine, with a maximum test force of 300 kN and a measurement accuracy of ±5%. This testing machine is mainly composed of two parts: an operating system and a testing machine. The operating system controls the compression speed of the testing machine and stores the relevant data during a test, and the testing machine is responsible for the compression or tension at a given time. To investigate the effect of the nodal configurations on the mechanical properties of the six-ligament chiral honeycombs, constant velocity compression tests at 1 mm/min were carried out on the three honeycombs. As a result, the compressive displacement, reaction force, and deformation behavior observed during the tests were recorded.

3. Experimental Results

3.1. Macroscopic Compression Deformation

The macroscopic compression deformation processes applied to the three different honeycomb configurations are shown in Figure 4. It can be observed that the three honeycombs exhibit similar overall deformation patterns under quasi-static compression, with each of them exhibiting a double V-shaped necking deformation pattern. This is because the applied stress on the six-ligament chiral honeycomb exceeds the elastic strength limit, causing the individual ligament to wrap around the node and rotate clockwise, with the ligaments on the left and right sides wrapping first and eventually spreading toward the center until all nodes are in full contact.

All three honeycombs showed significant deformation at a nominal strain of 0.098; most of this strain was observed in the lower left and upper right sections of the honeycombs. However, the double V-shaped deformation zone in the standard honeycomb was slightly smaller than that of the other two honeycombs. When the nominal strain reached 0.196, the double V-shaped deformation zone increased significantly in all three honeycombs. Although the deformation area expanded from both sides toward the middle, the increase in the deformation area in the standard honeycomb was not as obvious as that of the other two honeycombs. When the nominal strain reached 0.295, as the ligament continued to expand, the force from the double-V deformation was transmitted from both sides. The standard honeycomb possessed the largest unconsolidated zone and the lowest degree of ligament entanglement. When the specimens were compressed further up to a strain of 0.393, the ligaments of all three honeycombs were fully wound, and the subsequent deformation mode was nodal extrusion. Due to the lower node stiffness of the standard honeycomb, the deformation speed of both the thickened-node honeycomb and the node-filled honeycomb was greater than that of the standard honeycomb. Nodal buckling and a slowed ligament-winding process were observed during compression for the standard honeycomb. The increase in nodal stiffness due to nodal strengthening accelerates the ligament entanglement process under certain circumstances. Moreover, the deformation processes of the honeycomb are closely related to the forms of nodal strengthening.

3.2. Mechanical Properties

The stress–strain curves of the standard honeycomb under quasi-static compression are presented in Figure 5a, which shows three different stages, including the initial linear elasticity stage, the plateau stage, and the stress elevation stage. In the early stage of compression, the whole honeycomb exhibits linear elastic deformation. The ligaments and nodes do not deform significantly, and the stress grows linearly with the strain, which is manifested as the initial linear elasticity stage in the stress–strain curves. As compression proceeds, the ligaments on both sides yield, and plastic hinges form at the node connections, causing the nodes to rotate. They intertwine with each other and propagate this deformation behavior uniformly to neighboring cytosolic elements. In this phase, stress is kept nearly constant to form a platform, which manifests as the platform stage in the curve. A curl-winding behavior of the cellular element leads to the “necking” phenomenon, which is a typical negative Poisson’s ratio behavior. As the specimen is compressed further, the ligament is almost completely curled, and the nodes are in close contact with each other. The whole structure shows overall rigidity, for which the stress increases sharply in the curve.

A comparison of the stress–strain curves for the standard honeycomb, the thickened-node honeycomb, and the filled-node honeycomb is given in Figure 5b. It can be seen that the overall trend is consistent for all the structures and composed of three phases. However, the plateau stress and densification strain of the standard honeycomb are significantly greater than those of the thickened-node and filled-node honeycombs. The platform stress of the thickened-node honeycomb was 73.2% that of the standard honeycomb, and its platform strain was 96.1% that of the standard honeycomb; the platform stress of the filled-node honeycomb was 75.4% that of the standard honeycomb, and its platform strain was 91.5% that of the standard honeycomb. The nodes of the standard honeycomb deformed during the ligament entanglement phase to absorb energy, whereas little deformation occurred for the nodes of the thickened-node honeycomb and filled-node honeycomb, resulting in a larger plateau stress in the standard honeycomb. The nodal deformation of the standard honeycomb slowed down the degree of ligament winding; thus, the ligament bending deformation of the standard honeycomb was significantly less than that of the thickened-node and filled-node honeycombs, resulting in the standard honeycomb having the largest plateau strain. Moreover, the difference in densification strain between the thickened-node honeycomb and the filled-node honeycomb is small, which is the reason why the deformation mode for both structures was only ligament curling.

3.3. Poisson’s Ratio

Poisson’s ratio is one of the most important indicators of the mechanical behavior of cellular materials. It describes the deformation characteristics of a material when it is subjected to external loads. The negative Poisson’s ratio characteristic of the six-ligament chiral honeycomb mainly occurs in the platform stage, which is mainly investigated in this paper via the experimental results. The cross-sectional strain was calculated by dividing the average value of the transverse deformation displacement at the nodal edges on both sides of the honeycomb by the original transverse length value [28]. The transverse strain ${\epsilon }_x$, longitudinal strain ${\epsilon }_y$, and Poisson’s ratio $\nu$ of the honeycomb were calculated as follows. Figure 6 shows the measurement locations of the points for calculating Poisson’s ratio.

(1)$\Delta \overline{x}=\frac{\underset{i=1}{\overset{6}{\Sigma }}(A_{ix}+B_{ix})}{6}$

(2)${\epsilon }_x=\frac{\Delta \overline{x}}{X}$

(3)${\epsilon }_y=\frac{\Delta y}{Y}$

(4)$\nu =-\frac{{\epsilon }_x}{{\epsilon }_y}$

Figure 7 illustrates the variation in the Poisson’s ratio values during the compression process. It can be seen that significant differences exist in the Poisson’s ratio values of the three kinds of honeycombs. Overall, the trend was the same for all three structures. Poisson’s ratio increases with increasing nominal strain. The negative Poisson’s ratio effect was more evident for filled-node honeycombs than thickened-node honeycombs and was weakest for standard honeycombs. Moreover, as shown in Figure 7, the lateral shrinkage of the thickened honeycomb under compression is significantly greater than that of the standard honeycomb (shaded in blue in the figure). It should be noted that the deformation of the filled-node honeycomb was similar to that of the thickened-node honeycomb, so only the thickened-node honeycomb was used in this section for image overlap comparison. The compression displacement of the thickened honeycomb at its nodes was completely caused by the winding deformation of the ligaments, while the compression displacement of the standard honeycomb was induced by both ligament deformation and nodal yielding, thus preserving a certain deformation margin for the ligaments, while the other two honeycombs did not undergo nodal yielding due to their nodal stiffness. The other two types of honeycombs exhibited no nodal yielding due to their increased nodal stiffness. The main reason for this phenomenon is that nodal strengthening increases the stiffness of a node and enhances the ‘rotational’ deformation mechanism; thus, an increasing negative Poisson’s ratio effect is achieved.

4. Numerical Simulation

4.1. Numerical Method and Validation

With the development of computer technology and computational mechanics, the use of simulation software for numerical studies has gradually become an accepted research method by mainstream scholars. Due to the high strength of the honeycombs employed in this study, it was difficult to fully compress the honeycombs until densification. Thus, a simulation was conducted, and its results are presented in this section. In this paper, the finite element software ABAQUS was selected to carry out in-plane compression numerical simulations for the standard honeycomb, the thickened-node honeycomb, and the filled-node honeycomb. The material parameters used in the finite element model were an ideal elastic–plastic principal structure with a density of 7980 kg/m³, a Young’s modulus of 205 Gpa, a Poisson’s ratio of 0.3, and a yield strength of 742 MPa. The detailed settings of the computational model are shown in Figure 8. Two discrete rigid plates were arranged at the top and bottom positions, with the lower plate fixed and the upper plate compressed vertically at a constant speed of 0.001 m/s. A general degree of contact was defined between the honeycomb itself and the upper and lower rigid plates with a friction coefficient of 0.20. The S4R shell element type was selected. When the cell size is reduced, the equivalent force tends to converge; at the same time, for the consideration of accuracy and computational efficiency, we chose a 0.5 mm mesh for finite element calculation. The overall mesh size of the structure was 0.5 mm, and 20,832 elements were generated. Mesh sensitivity analysis was conducted to ensure the selected mesh size, which is appropriate for calculation, was employed. The out-of-plane displacement of the model was constrained to ensure its in-plane compression state.

In Figure 9, the simulation stress–strain curves as well as the compression processes are compared with those of the experimental results. It can be seen that both structures show a similar double-V deformation pattern. Moreover, the stress–strain curves are also almost the same, showing good consistency. The comparison results validate the reliability of the numerical simulation method used in this study.

4.2. Analysis of Deformation Process

The simulated deformation processes of the standard, thickened-node, and filled-node honeycombs are illustrated in Figure 10, Figure 11 and Figure 12, respectively. The deformation of all three kinds of honeycombs was consistent with the deformation presented in the experimental results. The main structural difference between the three honeycombs pertained to the nodes, and a change in node configuration will affect the deformation of the ligaments when winding. After a certain amount of ligament winding, a node collapse behavior will appear. At this stage, the influence of node configuration is more obvious, where the deformation of the thickened-node honeycomb and filled-node honeycomb is more uniform when compared with that of the standard honeycomb. Moreover, no localized node collapse occurred for the two-node configurations.

When the compression strain was 0.491, the stress of the two node configurations was higher than that of the standard honeycomb. At this stage, while the difference between these two types of honeycombs was not significant, the nodes at the top and bottom of the standard honeycomb were significantly deformed. This was due to the greater nodal stiffness of these two types of honeycomb node configurations, allowing them to resist compression. When the compression strain reached 0.589, the stress of the filled-node honeycomb was higher than that of the thickened-node honeycomb despite the deformation patterns being similar. When the compression strain reached 0.688, the stress of the filled-node honeycomb was significantly greater than that of the thickened-node honeycomb. The nodes of the node-filled honeycomb are almost invisible, while significant deformation can be observed in the nodes at the top and bottom ends of the thickened-node honeycomb. When the specimens were compressed further to a strain of 0.786, all three types of honeycombs were densified, with the filled-node honeycomb presenting the highest stress, the thickened-node honeycomb presenting the next highest stress, and the standard honeycomb presenting the lowest stress. This result indicates that node stiffness can have a significant impact on the compressive performance of cellular structures and that the deformation process varies under different reinforcement methods.

4.3. Mechanical Performance

Figure 13a presents the simulated results regarding quasi-static compression for the standard honeycomb. The curve characteristics of the initial linear elasticity phase and the plateau phase are consistent with the experimental results. In the stress rise stage, the ligament is almost completely curled, and the nodes are in close contact. As the specimen is compressed further, the nodes of the upper and lower sides first undergo compression collapse; then, the overall nodes yield until they are destroyed.

Figure 13b compares the simulated compression stress–strain curves of the three kinds of honeycomb structures. It can be seen that the thickened-node honeycomb mainly starts to exhibit affected mechanical properties after the plateau stage, for which the difference is not significant. In the stress enhancement stage, the average stress of the standard honeycomb is 25.1 MPa, and the average stress of the thickened-node honeycomb is 58.6 MPa. A 133.5% increase in stress was obtained because the thickened nodes are more resistant to yielding during mutual extrusion, and the overall stiffness of the thickened-node honeycomb is higher than that of the standard honeycomb at this stage. The average stress of the filled-node honeycomb was 74.3 MPa, which is 196.0% higher than that of the standard honeycomb. This is because the filled cells inside the nodes play a supporting role, significantly increasing the node stiffness of the filled-node honeycomb.

5. Discussion

5.1. Energy Absorption

Energy absorption characteristics are important properties of honeycomb materials, and they are critical for the design and application of impact-resistant honeycomb structures. Based on analyzing the compression deformation and mechanical properties of the honeycombs, the energy absorption characteristics of standard honeycomb materials, thickened-node honeycomb materials, and filled-node honeycomb materials were compared, and the specific energy absorptions (SEA) of the three kinds of honeycombs were investigated. Specific energy absorption is defined as follows

(5)$SEA=\frac{U}{m}=\frac{{\displaystyle \int \sigma (\epsilon )\mathrm{d}\epsilon }}{\rho \overline{\rho }}$

where U is the total absorbed energy of the honeycomb, m is the mass of the honeycomb, $\sigma (\epsilon )$ is the nominal stress of the honeycomb, $\rho$ is the density of the base material, and $\overline{\rho }$ is the relative density of the honeycomb.

As shown in Figure 14, the specific absorption energy of the three honeycombs increases linearly during the initial linear elasticity and plateau phases, with the standard honeycomb outperforming the thickened-node and filled-node honeycombs. This is because the relative density of the standard honeycomb is less than that of the thickened-node and filled-node honeycombs and the reinforcement of the node is not involved in deformation energy absorption at this time. However, there are both localized nodal-crushing and ligament-wrapping mechanisms in the standard honeycomb and, essentially, only one ligament wrap mechanism in the thickened-node and filled-node honeycombs. The combination of the above factors leads to the fact that nodal strengthening of the honeycomb node reduces the efficiency of energy absorption in the early stage. After the first platform stage, the effect of nodal reinforcement begins to manifest itself gradually, and the differences between the two node-reinforced honeycombs and the standard honeycomb gradually decrease. Particularly, when the nominal strain is 0.71, equivalent specific energy absorption can be obtained. Afterwards, the SEA of the two nodal-reinforced honeycombs gradually exceeds that of the standard honeycomb, and the role of the node starts to become prominent. Moreover, as the nodal stiffness of the node-filled honeycomb is greater than that of the thickened-node honeycomb, the specific energy absorption of the nodal-filled honeycomb under high strain is higher than that of the thickened-node honeycomb.

5.2. Comparison of Node Enhancement Effects

In this paper, to analyze the internal mechanisms that induce the differences between the modified honeycombs and the standard honeycomb, two different nodal reinforcement configurations were designed. The main effect of the strengthening of the nodes is an increase in the stress level in the extrusion phase of the nodes and an enhancement of the negative Poisson’s ratio effect of the whole structure.

Zhu et al. [26] designed a honeycomb configuration based on the definition of a multilevel honeycomb (see Figure 15), whose main feature was that the cytokinetic elements in the nodes of the base structure were filled. The difference in the mechanical properties between the proposed structure and the base structure under quasi-static compression was investigated using numerical simulation methods. It was observed that the filling of the nodes with cytosolic elements can reduce the distortion of the nodes during the ligament-winding process, which can help to promote the “rotation” mechanism and strengthen the overall negative Poisson’s ratio effect.

In this paper, the overall negative Poisson’s ratio characteristics of a thickened-node honeycomb and a filled-node honeycomb were compared. It was found that both node-strengthening methods improve the overall negative Poisson’s ratio characteristics. The reason for this improvement is that a greater stiffness of node configurations reduces the distortion of the nodes in the ligament-winding stage and promotes the rotational mechanism of chiral honeycombs (see Figure 16). Moreover, when compared with the thickened-node honeycomb, the internal cytosol of the filled-node honeycomb is more connected to the six ligaments on the node, and the inhibition of the node’s distortion is better, leading to a greater improvement in the negative Poisson’s ratio. This suggests that the negative Poisson’s ratio performance of chiral honeycombs can also be improved via nodal thickening in addition to the node filling.

6. Conclusions

To investigate the effect of node configuration on the six-ligament honeycomb, an in-plane compression test and its numerical simulation were conducted to study the macroscopic deformation, mechanical properties, and energy absorption characteristics of a standard honeycomb, a thickened-node honeycomb, and a filled-node honeycomb. The main conclusions drawn are as follows:

• (1). Nodal thickening can improve the negative Poisson’s ratio characteristics of honeycombs. The thickened-node honeycomb presented a greater wall thickness than that of the standard honeycomb, and this property suppressed the distortion of the node in the ligament-winding stage. Meanwhile, the cell elements inside the node played a supporting role for the nodes, preventing their distortion, of the filled-node honeycomb.

• (2). Mechanical behaviors of the three kinds of honeycombs included an initial linear elasticity stage, a plateau stage, and a stress rise stage. The effect of nodal differences on deformation was reflected in both the plateau stage and the stress increase stage. The effect of node differences on the stress state was mainly reflected in the stress rise stage. The stress enhancement of the filled-node honeycomb was more obvious than that of the thickened-node honeycomb.

• (3). The energy absorption of both the thickened-node honeycomb and the nodal-filled honeycombs was weaker than that of the standard honeycomb in the inelastic phase and platform phase and exceeded that of the standard honeycomb in the stress rise phase. This is because the strengthening of the nodes began to play a role in the stress rise stage and had no significant effect before that, while the mass rose due to the strengthening of the nodes leading to a decrease in the specific energy absorption.

• (4). Through experimental and simulation studies, it was determined that the filled-node honeycomb had better energy-absorbing properties and more prominent negative Poisson’s ratio characteristics at high strains, granting it better application prospects in relation to its lightweight and impact-resistant structure and greater significance for the design of the new protective structures for ships and marine structures.

Author Contributions

Z.W.: Investigation, Validation, Writing—original draft, Formal analysis. G.C. and W.C.: Supervision, Conceptualization, Methodology, Software, Validation, Formal analysis, Writing—original draft. X.C. and C.B.L.: Writing—review and editing, Validation. X.L.: Writing—review & editing, Supervision. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The authors declare that the data presented in this study are available on request.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Honeycomb specimens: (a) standard honeycomb; (b) thickened-node honeycomb; (c) filled-node honeycomb.

Figure 2. Honeycomb geometry: (a) standard honeycomb; (b) thickened-node honeycomb; (c) filled-node honeycomb.

Figure 3. WDW-300 microcomputer-controlled electronic universal testing machine.

Figure 4. Compression deformation modes of three different honeycomb configurations. (a) [Forumla omitted. See PDF.] = 0.098. (b) [Forumla omitted. See PDF.] = 0.196. (c) [Forumla omitted. See PDF.] = 0.295. (d) [Forumla omitted. See PDF.] = 0.393.

Figure 5. Stress–strain curves of honeycomb structures: (a) Standard honeycomb stress–strain curve; (b) stress–strain comparison.

Figure 6. Boundary measurement points for calculating Poisson’s ratio.

Figure 7. Variation of the Poisson’s ratio during the compression of the honeycomb structures.

Figure 8. Quasi-static compression finite element model. (a) Simulation settings. (b) Comparison of simulation results for different mesh sizes.

Figure 9. Comparison of test results with simulation results.

Figure 10. Deformation process of the standard honeycomb under different compression strains: (a) [Forumla omitted. See PDF.] = 0.098; (b) [Forumla omitted. See PDF.] = 0.196; (c) [Forumla omitted. See PDF.] = 0.295; (d) [Forumla omitted. See PDF.] = 0.393; (e) [Forumla omitted. See PDF.] = 0.491; (f) [Forumla omitted. See PDF.] = 0.589; (g) [Forumla omitted. See PDF.] = 0.688; (h) [Forumla omitted. See PDF.] = 0.786.

Figure 10. Deformation process of the standard honeycomb under different compression strains: (a) [Forumla omitted. See PDF.] = 0.098; (b) [Forumla omitted. See PDF.] = 0.196; (c) [Forumla omitted. See PDF.] = 0.295; (d) [Forumla omitted. See PDF.] = 0.393; (e) [Forumla omitted. See PDF.] = 0.491; (f) [Forumla omitted. See PDF.] = 0.589; (g) [Forumla omitted. See PDF.] = 0.688; (h) [Forumla omitted. See PDF.] = 0.786.

Figure 11. Deformation process of the thickened-node honeycomb under different compression strains: (a) [Forumla omitted. See PDF.] = 0.098; (b) [Forumla omitted. See PDF.] = 0.196; (c) [Forumla omitted. See PDF.] = 0.295; (d) [Forumla omitted. See PDF.] = 0.393; (e) [Forumla omitted. See PDF.] = 0.491; (f) [Forumla omitted. See PDF.] = 0.589; (g) [Forumla omitted. See PDF.] = 0.688; (h) [Forumla omitted. See PDF.] = 0.786.

Figure 12. Deformation process of the filled-node honeycomb under different compression strains: (a) [Forumla omitted. See PDF.] = 0.098; (b) [Forumla omitted. See PDF.] = 0.196; (c) [Forumla omitted. See PDF.] = 0.295; (d) [Forumla omitted. See PDF.] = 0.393; (e) [Forumla omitted. See PDF.] = 0.491; (f) [Forumla omitted. See PDF.] = 0.589; (g) [Forumla omitted. See PDF.] = 0.688; (h) [Forumla omitted. See PDF.] = 0.786.

Figure 13. Simulated compression stress–strain curves. (a) Simulated compression stress–strain curve for the standard honeycomb. (b) Comparison of simulated compression stress–strain curves.

Figure 14. Specific energy absorption curves of the three different kinds of honeycombs.

Figure 15. The hexachiral structure proposed by Zhu [26], and a comparison of the effects on the Poisson’s ratio. (a) Basic structure. (b) Proposed structure. (c) Effect on Poisson’s ratio under average strain for different hexachiral structures.

Figure 16. Comparison of cytosolic deformation of the three kinds of honeycomb structures. (a) Standard. (b) Node-thickened. (c) Node-filled.

Table 1

Material parameters.

Materials	Density (kg/m3)	Young’s Modulus (GPa)	Poisson’s Ratio
316L steel	7980	205	0.3
Yield strength (MPa)	Tensile strength (MPa)		Failure strain
742	1042		0.33

Table 2

Honeycombs’ geometric parameters.

Honeycomb Configuration	r1 (mm)	r2 (mm)	t1 (mm)	t2 (mm)	l1 (mm)	l2 (mm)
Standard honeycomb	5		1.2	1.2	20
Node-thickened	5		1.2	2.4	20
Node-filled	5	2.5	1.2		20	$2.5\sqrt{3}$

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