J. Cent. South Univ. (2020) 27: 621-628

DOI: https://doi.org/10.1007/s11771-020-4321-2

Crashworthiness of innovative hexagonal honeycomb-like structures subjected to out-of-plane compression

WANG Zhong-gang(王中钢)^{1, 2}, SHI Chong(施冲)², DING San-san(丁叁叁)¹, LIANG Xi-feng(梁习锋)²

1. CRRC Qingdao Sifang Co., Ltd., Qingdao 266111, China;

2. Key Laboratory of Traffic Safety on Track (Ministry of Education), School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Seeking for innovative structures with higher mechanical performance is a continuous target in railway vehicle crashworthiness design. In the present study, three types of hexagonal reinforced honeycomb-like structures were developed and analyzed subjected to out-of-plane compression, namely triangular honeycomb (TH), double honeycomb (DH) and full inside honeycomb (FH). Theoretical formulas of average force and specific energy absorption (SEA) were constructed based on the energy minimization principle. To validate, corresponding numerical simulations were carried out by explicit finite element method. Good agreement has been observed between them. The results show that all these honeycomb-like structures maintain the same collapsed stages as conventional honeycomb; cell reinforcement can significantly promote the performance, both in the average force and SEA; full inside honeycomb performs better than the general, triangular and double schemes in average force; meanwhile, its SEA is close to that of double scheme; toroidal surface can dissipate higher plastic energy, so more toroidal surfaces should be considered in design of thin-walled structure. These achievements pave a way for designing high-performance cellular energy absorption devices.

Key words: honeycomb-like structure; crashworthiness; basic fold element; structural reinforcement

Cite this article as: WANG Zhong-gang, SHI Chong, DING San-san, LIANG Xi-feng. Crashworthiness of innovative hexagonal honeycomb-like structures subjected to out-of-plane compression [J]. Journal of Central South University, 2020, 27(2): 621-628. DOI:https://doi.org/10.1007/s11771-020-4321-2.

1 Introduction

Train collision causes heavy casualties and huge economic losses. Exploiting train’s highly crashworthy devices is an effective way. Cellular honeycomb structure has been considered the main potential energy absorption material in engineering application circle [1-3], due to its excellent ratio in weight to cost and stable plateau stage, as shown in Figure 1(a). In recent years, besides kinds of lightweight structures [4-9], filled [10], embedded [11], tandem [12], hierarchical [13-15] and multi-cell patterns [16-18] were continually developed.

Although fruitful innovative structures were proposed, further investigation on this topic needs to be deeply conducted, particularly for its superiority in economy, simple as well as behavior stability. From the previous study [19-20], it has been confirmed that collapse mode of honeycomb cell mainly happens near foil wall when undergoing out-of-plane compression. After walls’folding, some considerable empty zone in the center of each cell can be observed clearly. Developing innovative honeycomb-like structure in reinforcing each cell could be as an effective way of sufficiently reusing the foil wall distribution, so as to increase the energy absorption capacities.

Figure 1 Honeycomb application (a) and novel structures (b-e)

In this study, three types of reinforced hexagonal honeycomb-like structures were theoretically and numerically analyzed subjected to out-of-plane compression, namely, triangular honeycomb, double honeycomb and full inside honeycomb, based on conventional general honeycomb. They are abbreviated as TH, DH, FH and GH, respectively, as shown in Figures 1(b) and (e) below.

2 Theoretical analyses

2.1 Geometric characterization

For any piece of honeycomb-like core block, it consists of lots of basic units. For instance, GH, it contains many “Y” sections (see Figure 2(a)). In each “Y” section, it can be described with the cell foil thickness t, cell edge length h, cell edge width l, and tilt angle θ. This study only discusses the case of h=l, θ=π/3. For DH and FH, the edge length of inside small hexagonal cell is exactly half of the original one. That means that the inside reinforced hexagons locate in the middle of original cell.

According to geometrical configuration, the weight M_c of the foilin red dashed triangular area (see Figure 1(b)) can be calculated as

                    (1)

in which ρ₀is the density of foil material and T is the total height of specimen. Its volume V_c is

Figure 2 Scheme of basic unit:

              (2)

Hence, the relative density ρ_cg can be calculated as ρ_cg=M_c/V_cand is simplified as

                            (3)

Following the same rule, the relative density ρ_ct, ρ_cd and ρ_cf respectively for TH, DH and FH can be expressed as

                            (4)

                            (5)

                            (6)

Observing the basic units, there are three types of typical elements mentioned, including “Y” pattern (type I), “*” pattern (type II) and “Ψ”pattern (type III), as shown in Figure 2. Their lengths involved in each element are different from each other.

2.2 Theoretical formulas

As widely known, for a basic unit, when undergoing out-of-plane compression, the dissipation region of localized energy contains the toroidal surface E₁, horizontal hinge line E₂, and inclined hinge line E₃. In each fold, the plastic energy in 2H crush distance can be respectively calculated respectively as:

                       (7)

                              (8)

                        (9)

where H is half-wave length; the unknown temporary parameter b is the small radius of the toroidal shell; c represents the wall length of each fold; M₀ is the fully plastic bending moment calculated as M₀=σ₀t²/4, in which σ₀ stands for the yield strength of foil (in the case of idealized elastic-plastic material behavior); I₁(f₀)=1.05, I₃(f₀)=2.39 and f₀=π/3. Generally, the energy dissipation can be regarded as the sum of the former three parts by calculating all localized regions listed in Table 1.

Table 1 Localized regions

Considering the full fold element subjected to out-of-plane static compression, energy dissipations can be calculated based on the energy minimal principle.

1) For the conventional honeycomb (GH), there is only “Y” pattern element (see Figure 2(a)), and its plastic internal energy E^g can be expressed by following equation and numbers listed in Table 1, as follows:

                 (10)

while the external work W_ext can be calculated as:

W_ext=2P_mgH_g                                            (11)

where P_mg is the average force in 2H_g folding distance. Therefore, according to the energy balance principle, the external work must be equal to the internal dissipation, i.e. W_ext=E^g. Thus, the equation can be constructed as:

          (12)

The minimal energy principle tells that the least P_mg leads to collapse happening. Hence, H_g and b_g can be determined by letting P_mg/H_g=0, P_mg/b_g=0. Consequently, H_g and P_mg can be obtained respectively as:

                        (13)

                      (14)

In this fold, the unit weight of foil walls is M_g=3ρ₀htH_g. Hence, the specific energy absorption s_mg in 2H_g distance can be obtained as follows:

          (15)

2) For TH, it can also be regarded as the sum of specific elements marked in Figure 2(b). This basic unit contains a “*” type element. In addition, the plastic energy E^dp of the three marginal points should be added. Similarly, the total energy of the basic unit E^t can be calculated as:

E^t=E^*+3E^dp                                              (16)

in which

 (17)

          (18)

where H_t is the half-wave length; b_t is the unknown temporary parameter. Substituting Eqs. (17) and (18) into Eq. (16), E^t can be re-written as:

 (19)

Thus, the equation can be built as

         (20)

Also letting P_mt/H_t=0, P_mt/b_t=0, H_t and P_mt can be solved as:

                        (21)

                     (22)

For TH structure, its unit weight of foil walls is M_g=4.5ρ₀htH_t.Hence, the SEA s_mt can be expressed as:

           (23)

3) For DH, it can be divided into 3 type I elements and 1 type II element. They dissipate energy E^yand E^*, respectively. The plastic energy absorption E^d in 2H_d can be calculated as:

E^d=E^*+3E^y                                               (24)

where

 (25)

 (26)

Likewise, b_d is the unknown temporary parameter. Substituting Eq. (25) and (26) back into Eq. (24), then E^d can be re-written as:

     (27)

Likewise, the equation can be built as:

           (28)

In the same manner, the half wave-length and the mean crushing force can be expressed as follows:

                        (29)

                     (30)

As the same as the former derivation, its unit weight of foil walls is M_d=4.5ρ₀htH_d. Hence, the specific energy absorption s_md can be obtained as:

          (31)

4) For FH, this type of unit can be divided into 3 type III elements and 1 type II one. Besides the energy consumed by type III (E^Ψ) and type II (E^*), as the former, the total dissipated energy E^f also contains the part consumed by three marginal points (E^dq):

E^f=E^*+3E^Ψ+3E^dq                                        (32)

in which

    (33)

 (34)

 (35)

where H_f is the half wave-length; b_f is the temporary parameter. Hence, the total energy E^f can be summed by all parts in Eqs. (32)-(35) as:

   (36)

Thus, the equation of average force, P_mf, related to H_f and b_f can be constructed as:

         (37)

Similarly, P_mf and H_f can be solved as:

                        (38)

                     (39)

and the specific energy absorption s_mf of this unit (M_f=6ρ₀htH_f) can be obtained:

          (40)

Table 2 summaries all the results obtained in the theoretical derivation for folding units presented in Figure 2.

3 Numerical simulations

3.1 Finite element model

For each numerical honeycomb specimen, the finite element model was built with commercial software package LS-DYNA[21]. Their cells were meshed with BT4 (Belytschko-Tsay-4node) shell elements and the mesh size was designed as 0.5 mm. Each model had 11 cells in L-direction and 11 cells in W-direction (see Figure 3), which completely satisfies the requirement of the minimum cell number (see Ref. [22]). For each cell, t=0.06 mm, h=l=4 mm. They were 40 mm in T-direction (see Figure 3). The foil material was AL 5052H18, and it was treated as an idealized elastic-plastic constitution model, with the mechanical properties: density 2680 kg/m³, elastic module 69.3 GPa, Poisson ratio 0.33, yield stress 215 MPa. Automatic single surface contact algorithm was applied to avoiding penetration between cell walls. The specimen was placed on a static rigid plate as a boundary and crushed by another moving rigid plate at a constant velocity 36 km/s, to meet the standard for railway vehicle crashworthiness design [23], as shown in Figure 3.

Table 2 Theoretical results

Figure 3 Loading situation

3.2 Validation

General honeycomb structure was employed representatively to validate the numerical finite element model, with the same t=0.06 mm, h=l=4 mm. Here, the top rigid plate moves at a constant speed 0.5 m/s. Corresponding compression history of nominal stress versus strain as well as theoretical derivation result (substituting corresponding parameters back into Eq. (14), the result is 1.0327 MPa) is given in Figure 4. Good agreement between them adequately validates the finite element model.

3.3 Results

Figure 5 shows the numerical results of the response force versus displacement of these four structures (GH, TH, DH and FH).

Figure 4 FEM model validation

Figure 5 Resultant curves of different structures

From the curves shown in Figure 5, it can be clearly seen that, the reinforced honeycomb-like structures maintain the same four stages with conventional one, including elastic, initial collapsed, progressively folding and densification. Figure 6 gives the result of the average force in plateau stage and SEA. As the bar heights indicate that, compared with GH, significant increments have been achieved for TH, DH and FH. The average force increases from 14.14 kN of GH, to 21.11 kN of TH. 29.15 kN of DH and 42.05 kN of FH, respectively. Meanwhile, the SEA increases from 16.62 kJ/kg of GH, to 17.73 kJ/kg of TH, 24.56 kJ/kg of DH and 24.72 kJ/kg of FH, accordingly. These indicate evident increments through cell reinforcement.

Figure 6 Increment of present structures

4 Discussion

4.1 Correlation analyses

SEA correlation was illustrated to validate the theoretical analyses. Differing from the theoretical equations derived under static situation, the numerical simulation was carried out under impact situation. Dynamic effect might influence the result. At the same time, the plastic energy dissipation is related to fold process, and heavily influenced by the folding ratio (usually be considered 0.73-0.75). Therefore, relative ratios are concentrated for these two cases. Figure 7 shows the SEA ration of TH, DH and FH compared with GH. The heights as well as values marks on the top of the bars sufficiently validate the outstanding agreement between theoretical derivation and numerical simulation.

4.2 Structural effect

From Table 2, it can be observed that, for these four types of structures, GH, TH, DH and FH, the average force and SEA have the same expression forms, but with different coefficients. The reasons for these mainly lie in their different half-wave lengths.

Figure 7 SEA ratio increments comparison

To illustrate these, Figure 8 shows the trends of half-wave length, the average force and SEA of the present structures. From figure, it can be seen that, FH has shorter half-wave length compared with GH, but performs the biggest average force and SEA. Both the DH and FH are stronger than GH and TH, because they have shorter half-wave lengths.

Figure 8 Influence of half-wave length on force and SEA

4.3 Formulas

For TH and DH, they have the same lengths of foil walls in each basic unit, also have the same relative densities. However, from Figures 5-8, it can be clearly found that the DH scheme is much better than TH, not only in average force, but also in SEA. This mainly owes to the folding localized energies from the toroidal surface and inclined hinge lines in each basic unit, because they have the same horizontal hinge lines.

From Eqs. (17)-(19) and Eqs. (24)-(27), E^t and E^d can be re-written as:

     (41)

 (42)

Now that they have the same cell length of DH and TH, meeting

                   (43)

It can be easily found that the coefficient of E^d in Eq. (42) is larger than that of E^t in Eq. (41). This is the reason why E^d is much larger than E^t. It indicates that toroidal surface can dissipate higher plastic energy. In design of lightweight energy absorption devices, more toroidal surfaces are recommended.

5 Conclusions

1) All these honeycomb-like structures maintain the same collapsed stages as conventional honeycomb structure. Based on these, they can be served as potential lightweight energy absorption devices.

2) Half-wave length of folding element heavily influences the mechanical performance. A honeycomb-like structure with shorter half-wave length has higher energy absorption capacity. Compared with general honeycomb, cell reinforcement can significantly promote the performance, not only in average force, but also in SEA. The reinforcement largely shortens half-wave length of the structure.

3) Compared with other structures, the half-wave length of FH is the shortest, but its average force and SEA are the biggest. DH and FH are both higher than GH and TH in load bearing ability. Considering the energy absorption capacity and lightweight, double reinforcement honeycomb scheme is recommended.

4) Toroidal surface can dissipate higher plastic energy. In design of lightweight energy absorption devices, more toroidal surfaces should be considered.

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(Edited by YANG Hua)

中文导读

新型六角类蜂窝结构的异面压缩吸能特性

摘要：探寻更优力学性能的轻质新颖结构一直是轨道车辆耐撞性设计持续追求的目标。为此，分析了三种增强型类蜂窝结构的异面压缩吸能特性，包括三角增强型、双层增强型和全填充增强型。基于最小能量原理构建了各型增强蜂窝结构的平均力、比吸能的理论表达式，并采用显式有限元数值模拟方法验证，结果吻合较好。所得结果表明：此三型类蜂窝结构维持了与常规蜂窝结构一致的压缩历程； 胞元增强可显著提升类蜂窝结构的平台载荷与比吸能；相比于其它增强形式，全填充增强型类蜂窝结构与双层增强型结构具有相当的比吸能，但其平台载荷更高；折角单元环形曲面可耗散更多的塑性能，应考虑增加更多折角单元以提升耗能水平。所得结果为轻质多孔高性能能量吸能装置设计提供参考。

关键词：类蜂窝结构；耐撞性；基本折叠单元；结构增强

Foundation item: Projects(51875581, 51505502) supported by the National Natural Science Foundation of China; Projects(2017M620358, 2018T110707) supported by China Postdoctoral Science Foundation; Project(kq1905057) supported by the Training Program for Excellent Young Innovators of Changsha, China

Received date: 2019-06-09; Accepted date: 2019-08-19

Corresponding author: SHI Chong, PhD Candidate; Tel: +86-731-82655294; E-mail: shichong@csu.edu.cn; ORCID: 0000-0001-9302- 3399