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Crashworthiness of bionic fractal hierarchical structures
Accepted Manuscript Crashworthiness of bionic fractal hierarchical structures
Yong Zhang, Jin Wang, Chunhui Wang, Yi Zeng, Tengteng Chen PII: DOI: Reference:
S0264-1275(18)30643-9 doi:10.1016/j.matdes.2018.08.028 JMADE 7326
To appear in:
Materials & Design
Received date: Revised date: Accepted date:
6 March 2018 13 August 2018 14 August 2018
Please cite this article as: Yong Zhang, Jin Wang, Chunhui Wang, Yi Zeng, Tengteng Chen , Crashworthiness of bionic fractal hierarchical structures. Jmade (2018), doi:10.1016/ j.matdes.2018.08.028
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ACCEPTED MANUSCRIPT Crashworthiness of bionic fractal hierarchical structures Yong Zhanga, Jin Wanga, Chunhui Wangb, Yi Zenga, Tengteng Chena a College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China b School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, Australia
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Abstract: This paper presents a fractal hierarchical hexagon structure inspired by bionic structures, such as spider webs, as a novel lightweight energy absorber.
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Experimental testing and computational modeling are carried out to characterize the
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effects of fractal order and geometrical shapes on the crashworthiness of this new structure compared to a simple hierarchical structure. The computational results reveal
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that both simple hierarchical and fractal structures can present significant improvement in energy absorption over single wall nonhierarchical structure. Furthermore, fractal configuration, geometrical parameters and order have important effect on the energy
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absorption efficiency of fractal hierarchical structures, with the 2nd order design being the optimal for a given mass. The findings of this research offer a new route for
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designing novel lightweight energy absorbers with improved crash protection against
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impact.
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Lightweight.
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Keywords: Crashworthiness; Fractal hierarchical structure; Energy absorption;
Corresponding Author Yong Zhang: Tel: +86-592-6162 595; Email: [email protected];
ACCEPTED MANUSCRIPT 1 Introduction The drive towards electrical vehicles calls greater innovation in lightweight energy absorbing structures to provide crash protection at the minimum weight. A broad range of new designs of thin-walled structures with different geometrical cross-sections have been proposed to absorb more energy than single-walled structures. Among them, the
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multi-cell column structures have been found to absorb more energy than the equivalent single walled structure of the same mass. Tran et al. [1] developed theoretical models
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for three types of multi-cell triangular tubes based on super folding element theory, and
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the feasibility of the theoretical models was validated by numerical simulations. Mahmoodi et al. [2] theoretically and numerically investigated the energy absorption
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behavior of tapered multi-cell tubes and found that the increase of the taper angle and the wall thickness could improve the crashworthiness of the structures. Zhang et al. [3]
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performed quasi-static axial compression tests on multi-cell columns with different sections and found that multi-cell columns offered better crushing characteristics than single cell columns. Nia and Parsapour [4] compared the energy absorption capacities
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of simple and multi-cell thin-walled tubes with triangular, square, hexagonal and
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octagonal sections. The results showed that the energy absorption capacity of multi-cell sections was greater than that of simple sections. Xiang et al. [5] compared the energy
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absorption capacities of polygonal tubes, multi-cell tubes and honeycombs. Their results suggested that the energy absorption of multi-cell tubes depended on the number
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of cells. Fang et al. [6] studied the crushing behaviors of multi-cell members subjected to oblique impact loads. The results showed that the increase in cell number was beneficial to the energy absorption. In addition. Wu et al. [7] performed the multiobjective optimal design for a five-cell tube to maximize its specific energy absorption and minimize peak crushing force. Yang et al. [8] optimized the multi-cell tubular structures with pre-folded origami patterns. These results showed that quintuple-cell origami tubes absorbed the highest amount of energy with significantly reduced initial peak force. Recently, biological structures with hierarchical (Fig. 1(a) [9], (b) [10]) and fractal
ACCEPTED MANUSCRIPT (Fig. 1(c) [11]) characteristics have attracted growing attention to enhance the mechanical properties of lightweight materials and structures. Kooistra et al. [12] investigated the transverse compression mechanisms of hierarchical corrugated core sandwich panels, and found that second order trusses had higher compressive and shear collapse strengths than their equivalent mass first order counterparts. Fan et al. [13]
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demonstrated the effectiveness of hierarchical honeycomb structure in improving the stiffness and plastic collapse strength of thin-walled structures. Zheng et al. [14] found
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that hierarchical square honeycombs using ductile woven textile composites had higher specific energy absorption than those of metallic honeycombs. Chen et al. [15] studied
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the hexagonal, kagome, and triangular hierarchical honeycomb metamaterials and found that hierarchical designs could effectively improve the heat resistance and load-
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carrying capacity of regular honeycomb structures. Zhang et al. [16] found that hierarchical honeycombs more efficiently distributed the material across the network
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than traditional hexagonal honeycombs. Oftadeh et al [17] constructed a fractalappearing cellular metamaterial by replacing each three-edge vertex of a base
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hexagonal network with a smaller hexagon. Their results showed that the in-plane
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stiffness for a mass increases with the order of hierarchy. In addition, hierarchical designs can also improve the weight-efficiency of thinwalled tubular structures. Sun et al.[18] found that hierarchical triangular lattice
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structures possessed three to four times higher mean crushing force than single-cell and multi-cell lattice structures. Li et al. [19] investigated the crushing mechanisms of
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hierarchical hexagonal tubes by numerical analysis and found that hierarchical topology greatly improved the energy absorption of thin-walled hexagonal tubes. Luo et al. [20] examined the lateral crushing behaviors of hierarchical quadrangular tubes and found it greatly increased the mean crushing force of single wall quadrangular tubes. Wang et al. [21] developed a hybrid hierarchical structure based on the Koch fractal principles, and found that the 2nd order hybrid Koch absorbers outperformed a wide range of multicell structures of the same mass.
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Fig. 1 Typical hierarchical biological structures:(a) bird-of-paradise [9], (b) spongy bone [10]and
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fractal biological structures: (c) nautilus [11].
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Motivated by these promising findings, in this paper we propose a novel thinwalled structure inspired by the spiral wheel-shaped (orb-web) construction of spider
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webs. This design, denoted fractal hierarchical hexagon structure, is compared with a simple hierarchical hexagon structure. Section 2 describes these two biologically
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inspired designs, numerical models of simple hierarchical and fractal hierarchical structures, and validation of the computational models by experiments. A comparative
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study is presented in Section 3 to investigate the advantage of the fractal hierarchical
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design over the simple, non-fractal hierarchical design. Lastly, using the validated numerical model, sensitivity and parametric design are performed by varying the fractal
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order and geometries.
2 Model and Experiment
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2.1 Geometric models The geometric characteristics of bioinspired hierarchical and fractal hierarchical thin-walled structures are illustrated in Fig. 2, where Fig. 2(a) and Fig. 2(b) show the simple hierarchical construction of spiral spider web and the evolution fractal hierarchical design, respectively. Firstly, hexagon bionic hierarchical sections are established based on the biological geometry of spider web, in which high order hierarchical structures (HS) are formed by adding a regular hexagon at the center of the low order hierarchical structures and connecting the corresponding vertices; the length of the linking elements is denoted as d1 and the wall thickness of the hexagonal structure
ACCEPTED MANUSCRIPT is t. The diameter of the circumscribed circle of the bionic structure is denoted as D. The three hierarchical structures with different orders are named HS1, HS2 and HS3, respectively. Next, a new fractal structure (FS) is obtained by replacing the each of primary hexagonal vertices of the simple hierarchical structures with sub-hexagons and resultant
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fractal designs are named as FS1, FS2 and FS3, respectively. The side length of subhexagon and linking distance are denoted as b and d2, respectively, as shown in Fig.2
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(b). The wall thickness of fractal structures is also denoted as t.
Based on a typical energy absorber of passenger vehicle, as shown in Fig. 2(d)
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[22], a schematic diagram of the bionic thin-walled tubes under axial dynamic loading is presented in Fig. 2(c). In the following numerical investigations, the circumscribed
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circle diameter D of bionic structures is kept at 90 mm, the length L of the columns is 160 mm, and the wall thickness t will be varied. The columns will be axially loaded by
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tube is constrained by a rigid wall.
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an impactor with a mass of 500 kg at a velocity of v=10 m/s. The non-impacted end of
Fig. 2 The evolution of bionic structures: (a) biological geometry of spider web, (b) cross-sections of hierarchical and fractal structures, (c) typical vehicle anti-collision structure [22], and (d)
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2.2 Numerical models and crashworthiness indicators Finite element (FE) models were developed for the bionic thin-walled structures, both simple hierarchical and fractal hierarchical structures, as illustrated in Fig. 3, using
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the nonlinear finite element software LS-DYNA 971. The thin-walled tubes were modeled using Belytschko-Tsay four-node shell elements [23]. To maintain a
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reasonable balance between the computational cost and accuracy of the numerical
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results, the element size of 1.0 mm x1.0 mm was found to be sufficient for to achieve convergence using shell elements. The elastic-plastic material model MAT_24 and
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rigidity material model MAT_20 in LS-DYNA were adopted to model the thin-walled tube and the rigid clamps, respectively. An automatic surface-to-surface contact was
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chosen to simulate the contact between the thin walled tube and rigid clamps. An automatic single-surface contact was adopted for all the thin walls to avoid mutual interpenetration of folding lobes. The static and dynamic frictional coefficients defined
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in these contacts were 0.3 and 0.2, respectively [24].
Fig. 3 The finite element models of thin-walled structures.
ACCEPTED MANUSCRIPT To effectively evaluate the crashworthiness performance of thin-walled structures, several crashworthiness indicators are defined here. They are total energy absorption (EA), specific energy absorption (SEA), peak crushing force (PCF), and crash load efficiency (CLE), respectively. The total energy absorption (EA) [25] during the process of compression is calculated as
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𝑑
EA= ∫ 𝐹(𝑥) dx
(1)
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0
where F(x) denotes the crushing force, x is the instantaneous crushing displacement,
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and d is the effective crushing displacement. Herein, the value of d is kept at 110 mm.
structure of unit mass. It is defined by:
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The specific energy absorption (SEA) [26] denotes the energy absorbed by a EA (2) M where M is the total mass of the structure. SEA is a key indicator to estimate the energy
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SEA =
absorption efficiency of structures with different materials and weights. Therefore, the
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higher the SEA, the better is the energy absorption performance of the structure.
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The peak crushing force PCF [27], i.e., the maximum of F(x), is another important crashworthiness indicator for energy absorbers. Since a high PCF often leads to a high
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deceleration, potentially causing severe injury or even death of occupant, PCF should be reduced or constrained to certain extent from the safety design viewpoint.
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The mean crushing force (MCF) [27] represents the average crushing strength, and is expressed as
MCF =
𝐸𝐴(𝑑) 𝑑
(3)
where 𝐸𝐴(𝑑) is the total energy absorbed when the absorbed is crushed over a distance 𝑑. Finally, the crushing load efficiency (CLE) [28] indicates the constancy of the crushing force during impact, and it is defined as the ratio between the mean load (MCF) and the initial peak force (PCF),
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MCF PCF
(4)
Obviously, the higher the CLE value, the better is the force uniformity for an energy absorber.
2.3 Experimental tests and validation of FE model Quasi-static crushing tests were carried out to characterize the energy absorption
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behavior of bionic structures. The data also served to validate the finite element models. The material of the bionic structures is aluminum alloy A6061-O, with the following
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mechanical properties: density ρ=2700kg/m3, Young's modulus E=68.2GPa, and
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Poisson's ratio μ=0.3. The stress-strain response of the aluminum alloy was measured using the MTS 322 material testing machine at a displacement rate of 5 mm/min. The
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true stress-strain data of A6061-O was presented in Fig. 4. This aluminum was known
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carried out at higher strain rates.
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to be insensitive to the strain rate in the range of interest [29], so no further tests were
Fig. 4 True stress-strain curve of A6061-O.
To validate the computational model outlined in section 2.2, two specimens, denoted as HS2 and FS2 as shown in Fig.5 (a), were manufactured by the wire electrical discharge machining (WEDM) technique from blocks of aluminum alloy 6061-O. This
ACCEPTED MANUSCRIPT WEDM technique used a continuous moving electrode wire to generate pulse spark discharge to etch away the material. The top views of the two specimens were shown in Fig.5 (b). The circumscribed circle diameters of HS2 and FS2 were 70 mm and 90 mm, respectively. The lengths of HS2 and FS2 were 110 mm and 160 mm, respectively. Meanwhile, the length of sub-hexagon b and link distances of d1 and d2 were 5 mm, 15
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mm and 5 mm, respectively, and the wall thickness t of HS2 and FS2 was 0.8 mm. Quasistatic compressive tests were performed in an MTS universal testing machine shown in
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Fig. 5(c). The specimens were placed between two flat plates. During the test, the top plate (pressure head) of the machine moved downward at a constant speed of 5 mm/min.
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The crushing load and displacement were collected automatically. When the compressive displacements reached 70 mm for HS2 and 110 mm for FS2, respectively,
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the tests were terminated and the specimens were unloaded for examination.
Fig. 5 Experimental specimens: (a) 3D view of specimens, (b) top view of specimens, (c) universal testing machine.
Fig.6 shows the collapse modes of the specimens tested together with the predictions of the corresponding FE models at different crushing displacements. It is
ACCEPTED MANUSCRIPT clear from Fig. 6(b) that FS2 undergoes a regular stable collapse in every corner, which is the desired energy absorption mode. During the crushing process, its axial crushing progressed from the bottom end, with regular folds stacking onto the previous folds. By contrast, the deformation of HS2 starts from the top end, and the folding is much less uniform than the FS2.
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Comparisons between the experimental results and the computational model predictions shown in Fig.6 confirm that the FE models have well captured the
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deformation characteristics at different crushing displacements. To further examine the crushing deformation in detail, the tested specimens are cut into two equal halves, so
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the interior of the progressive collapse modes are visible, as presented in Fig. 7. The sub-hexagons of FS2 form very regular and uniform folds, and the wavelength of the
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folds is quite short. By contrast, the collapse wavelength of HS2 is larger than the FS2, which suggests that the sub-hexagons promote the progressive folding. Meantime, the
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interior views of both specimens are very similar to the predictions of the FE models. In addition, Fig. 8 shows the corresponding crushing force-displacement curves of HS2
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and FS2, indicating a periodic fluctuation at regular intervals as collapse progresses.
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This result provides further evidence that the thin-walled structures underwent stable and progressive deformation. The load-displacement curves obtained from the computational simulation are in good agreement with the experiment results. Therefore,
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the FE models have now been validated and can be applied to investigate the
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crashworthiness of new designs, which are described in the following section.
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Fig. 6 Comparison of the deformation modes between the experimental and numerical results: (a)
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HS2, (b) FS2.
Fig. 7 The internal views between the experimental and numerical results: (a) HS2, (b) FS2.
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Fig. 8 Comparison of impacting force-displacement curves: (a) HS2 (b) FS2.
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3 Comparative analysis for simple hierarchical and fractal
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hierarchical structures
To assess the effects of the order of hierarchy and fractal state on the
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crashworthiness of the simple hierarchical structures (HS1, HS2, HS3) and fractal
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hierarchical structures (FS1, FS2, FS3) presented in Fig. 1, computational simulations are carried out of these structures with four different mass (and hence different wall thickness). The mass of these two types of thin-walled structures determines the
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material distribution, which then affects the energy absorption. The corresponding wall thickness to attain the desired mass is also listed in Table 1.
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Table 1 The thicknesses of thin-walled structures with different mass.
Cross section HS1
M1 (kg)
t (mm)
M2 (kg)
t (mm)
M3 (kg)
t (mm)
M4 (kg)
t (mm)
0.13
1.16
0.17
1.44
0.20
1.73
0.24
2.02
0.13
0.58
0.17
0.72
0.20
0.87
0.24
1.01
0.13
0.43
0.17
0.54
0.20
0.65
0.24
0.76
HS2
HS3
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0.13
0.80
0.17
1.00
0.20
1.20
0.24
1.40
0.13
0.43
0.17
0.54
0.20
0.65
0.24
0.76
0.13
0.33
0.17
0.41
0.20
0.49
0.24
0.57
FS2
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FS3
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Table 2 presents the crashworthiness indicators of the different structures examined. The results show that for all the cross-section shapes, both the SEA and PCF
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increase with the mass, as clearly seen in Fig. 9 for the SEA curves of different structures. More importantly, for the simple hierarchical structures, the SEA is near
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linear increase with the mass. Furthermore, the 3rd order design (HS3) seems to yield a close SEA value to the 2nd order design (HS2), implying that there is a diminishing
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improvement in specific energy absorption beyond the 2nd order. However, for the fractal hierarchical structures, the 3rd order design (FS3) seems to still offer substantial
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improvement over the 2nd order design (FS2), although the gain is less than that of 2nd
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order design over the 1st order design.
ACCEPTED MANUSCRIPT Table 2 SEA and PCF of all structures. Cross section
M1(kg) SEA PCF (kJ/kg) (kN)
M2(kg) SEA PCF (kJ/kg) (kN)
M3(kg) SEA PCF (kJ/kg) (kN)
M4(kg) SEA PCF (kJ/kg) (kN)
HS1 11.19
41.76
12.91
53.63
14.20
65.32
15.72
76.93
11.76
36.46
13.72
47.70
15.29
59.96
17.02
72.34
11.98
35.22
14.02
45.27
15.63
56.02
19.58
41.22
22.32
54.27
22.29
36.24
24.08
46.39
23.51
35.47
25.14
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HS2
HS3
67.72
66.26
25.14
78.59
26.43
57.36
27.64
69.75
27.52
58.40
28.91
70.58
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17.21
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FS3
24.30
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FS2
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FS1
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46.84
Fig. 9 SEA of different simple hierarchical and fractal hierarchical structures.
The crushing force-displacement curves of HS and FS structures with mass=0.202
ACCEPTED MANUSCRIPT kg are presented in Fig. 10. The fluctuation of crushing force is more pronounced in low level hierarchical structures, such as HS1 and FS1, as indicated by the large folding wavelength. This behavior is undesirable for energy absorbers. As the hierarchical level increases, however, the fluctuation in the force-displacement curve decreases, indicating that higher level hierarchical structures are better in terms of crush force
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efficiency. The corresponding deformation shapes are presented in Fig. 11. It is seen that as the hierarchical level increases, both HS and FS structures display more
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progressive folding lobes, and the folding wavelength decreases with the hierarchy
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order and the fractal order, i.e., 𝜆HS1 > 𝜆HS2 > 𝜆HS3 and 𝜆𝑆𝐹𝑆1 > 𝜆𝑆𝐹𝑆2 > 𝜆𝑆𝐹𝑆3 . These results reveal that the hierarchical principle improves energy absorption.
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However, the force-displacement curves of the 2nd order (HS2, FS2) and 3rd order structures (HS3, FS3) are fairly similar, indicating that they have near load bearing
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capacity and energy absorption. This is consistent with the similarity between the folding wavelengths of the 2nd order and the 3rd order structures. Therefore, it be concluded that hierarchical organization improves crashworthiness of thin-walled
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structures, with the 2nd order hierarchical structures being the optimum considering the
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gain versus geometrical complexity.
Fig. 10 Impacting force-displacement of different hierarchical structures: (a) HS structures, (b) FS structures.
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Fig. 11 Deformations and section views of different structures: (a) HS structures, (b) FS structures.
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It is important to note that FS structures offer a dramatic enhancement in SEA
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compared to the HS counterparts, as seen in Fig. 9. The percentages of increase in SEA for different mass are presented in Table. 3, indicating the amount of increase ranging
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between 59.92% and 96.24%. Therefore, the fractal geometry offers huge potential in improving the energy absorption capability of thin-walled structures.
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Table 3 Increasing rate of SEA for FS structures and corresponding HS structures. ΔSEA (FS1 vs HS1)
ΔSEA (FS2 vs HS2)
ΔSEA (FS3 vs HS3)
M1=0.1348kg
74.98%
89.54%
96.24%
M2=0.1685kg
72.88%
75.51%
79.32%
M3=0.2022kg
71.12%
72.86%
76.07%
M4=0.2359kg
59.92%
62.39%
67.94%
*∆𝑆𝐸𝐴 =
(𝑆𝐸𝐴𝑆𝐹𝑆 −𝑆𝐸𝐴𝐻𝑆 ) 𝑆𝐸𝐴𝐻𝑆
× 100%
ACCEPTED MANUSCRIPT To further illustrate the improvement offered by the fractal design over the simple hierarchical counterpart, Fig.12 presents the force-displacement curves for FS and HS structures with the same mass of 0.202 kg. It is clear that at the same hierarchical order, a fractal structure offers much higher load bearing capacity and more energy absorption than the corresponding simple hierarchical structure. Furthermore, the 2nd and 3rd order
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fractal hierarchical structures show more progressive folding lobes than the corresponding simple hierarchical structures. From the deformation shapes shown in
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Fig. 12, strain concentration occurs mainly at the corners of structures. In the case of fractal structures, the strain also concentrates at the corners of sub-hexagon structures
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thus the sub-hexagons trigger more progressive deformations, contributing to better
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energy absorption performance of the fractal structures.
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Fig. 12 Force-displacement curves and deformation modes of hierarchical and fractal structures: (a) 1st structures, (b) 2nd structures, (c) 3rd structures.
The PCF values listed in Table. 2 also confirm that an increase in mass causes the PCF to increase. The reason is that higher mass means thicker walls, increasing the
ACCEPTED MANUSCRIPT stiffness of the thin-walled structures. The values of CLE for different structures are plotted in Fig.13. It is clear that the CLE values for both structures increase with the hierarchical order. More importantly, the CLE values of fractal structures are much greater than that of corresponding simple hierarchical structures, indicating that fractal
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organization improves crush force uniformity, which is desirable for energy absorbers.
Fig. 13 CLE of all structures with different mass.
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Overall, the 2nd fractal hierarchical structures (FS2) has higher energy absorption than other structures examined in this paper. Therefore, FS2 is chosen as a candidate in following parametric design.
4 Parametric design of FS2 The crashworthiness behavior of the FS2 highly depends on the geometric and fractal parameters. Therefore, a parametric design is necessary to explore the influence on crashworthiness under dynamic loading.
ACCEPTED MANUSCRIPT 4. 1 The effect of fractal configuration The location of the sub-hexagons, i.e., the fractal configuration, may influence the energy absorption characteristics of the fractal structures. Fig. 14 presents three fractal configurations of sub-hexagon in vertex of 2nd order FS, which are referred as inner sub-hexagon, central sub-hexagon, and outer sub-hexagon, respectively. Applying
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these three configurations to the three hierarchical geometries gives nine types of fractal designs listed in Table 4. Dynamic impacting simulations of the nine fractal structures
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are performed to investigate their crushing behaviors and to identify the optimum
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candidate for further detailed analysis. The side length b of sub-hexagon and the linking distance d2 shown in Fig. 2(b) are set as 5 mm for all nine variants, and total mass is
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kept constant.
Fig. 14 Different fractal configurations for FS2.
Table 4 Fractal structures with different fractal configurations.
Cross section
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Fractal type
Name
Fractal type
Cross section
Name
Fractal type
Cross section
Name
HOHI
HCHI
HIHI
HOHC
HCHC
HIHC
HOHO
HCHO
HIHO
The results of SEA, PCF and CLE of the nine fractal structures are summarized in
ACCEPTED MANUSCRIPT Table 5. It can be seen that the fractal configuration has an important effect on the energy absorption, with the HIHC design having the highest SEA, more than 33.2% than the HOHI design that yields the minimum energy absorption. The HIHC design features inner sub-hexagons in the first layer and central sub-hexagons in the second layer. This particular design produces both the maximum SEA (29.21 kJ/kg) and the
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maximum CLE, while the PCF is close to other configurations. Therefore, HIHC is the optimal fractal configuration among the nine options, and thus will be further evaluated
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in following to quantify the effects of other geometrical parameters, such as the
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thickness of the cell wall.
Table 5 SEA, PCF and CLE of different fractal structures. PCF (kN)
SEA
PCF
(kJ/kg)
(kN)
27.84
74.42
0.85
21.94
70.42
0.70
27.2
73.63
0.84
29.21
75.04
0.88
CLE
Cross
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SEA (kJ/kg)
section
Deformation
CLE
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Deformation
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Cross section
73.99
0.73
27.43
75.77
0.82
24.71
70.37
0.79
24.03
75.16
0.72
70.16
0.77
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23.77
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23.87
4.2 The effect of geometrical parameters Based on the finding presented in Section 4.1 that the fractal structures HIHC is the best design for crashworthiness, further parametric analysis is performed to characterize the effects of the cell walls on crashworthiness of HIHC, such as the side length b of sub-hexagon, the linking distance d2 shown in Fig. 2(b). To avoid geometric interference, the range of parameter b and d are set as 4 mm-7 mm.
ACCEPTED MANUSCRIPT Fig. 15(a) depicts the 3D surface of SEA of HIHC with different geometrical parameters (b and d2). It is clear that increasing the value of d2 only yields a slight increase in SEA, indicating that SEA is not sensitive to the linking distance d2. However, the side length b of the sub-hexagons shows a strong effect on SEA: increasing b causes SEA to rise first and then fall, reaching the maximum value at b=6 mm. This is further
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illustrated in Fig. 16. At a given crushing displacement, the force is highest at b=6 mm.
(a)
(b)
Fig. 15 SEA and PCF of HIHC with different geometrical parameters: (a) SEA, (b) PCF.
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100
b=4mm
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b=5mm
b=6mm
b=7mm
80 70
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Impacting Force (kN)
90
60 50 40 30
20 0
10
20
30
40
50
60
70
80
Displacement （ mm)
90
100 110
Fig. 16 Force-displacement curve of HIHC with d2=6 mm.
In addition, PCF values shown in Fig. 15(b) also display a similar phenomenon with SEA. It is seen that PCF is not sensitive for d2, however, PCF gradually increases with the b increasing. Therefore, it can be believed that geometrical parameter b plays
ACCEPTED MANUSCRIPT an important role in the crashworthiness of HIHC.
6 Conclusions This paper presents a comprehensive investigation of the crashworthiness of simple hierarchical and fractal hierarchical structures using experimental tests and
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computational modelling techniques. Several conclusions are drawn from the results: 1. Bionic designs of simple hierarchical and fractal hierarchical configurations
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offer significant improvement in energy absorption capabilities over the
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hexagonal structure (HS1).
2. The energy absorption of the fractal hierarchical design is better than the simple
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hierarchical design, the 2nd order fractal hierarchical structure (FS2) has the optimum crashworthiness considering the gain versus geometrical complexity.
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3. Both fractal configuration and geometry have important effect on the energy absorption of the 2nd order fractal structure, in which the length of sub-hexagon
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Acknowledgments
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has strong sensitivity on the energy absorption.
This work is supported by The National Natural Science Foundation of China (51675190), Program for New Century Excellent Talents in Fujian Province University,
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Fujian Province Natural Science Foundation of China (2015J01204), Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of
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Huaqiao University (ZQN-PY202).
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ACCEPTED MANUSCRIPT woven lattice composites. Compos. Part B: Eng. 43 (2012) 1516-1522. [15] Y.Y. Chen, Z. Jia, L.F. Wang, Hierarchical honeycomb lattice metamaterials with improved thermal resistance and mechanical properties, Compos. Struct. 152 (2016) 395-402. [16] Y. Zhang, M.H. Lu, C.H. Wang, G.Y. Sun, G.Y. Li, Out-of-plane crashworthiness
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honeycomb sandwich cylindrical column under axial crushing loads, Thin-Walled Struct. 48(2010) 9-18. [24] Y. Zhang, X. Xu, G.Y. Sun, X.M. Lai, Q. Li, Nondeterministic optimization of tapered sandwich column for crashworthiness, Thin-walled Struct. 122 (2018) 193207. [25] X. Xu, Y. Zhang, J. Wang, F. Jiang, C.H. Wang, Crashworthiness design of novel hierarchical hexagonal columns, Compos. Struct. 194 (2018) 36-48. [26] Y. Zhang, G.Y. Sun, G.Y. Li, Z. Luo, Q. Li, Optimization of foam-filled bitubal structures for crashworthiness criteria, Mater. Des. 38 (2012) 99-109.
ACCEPTED MANUSCRIPT [27] M.I.M. Sofi, A review on energy absorption of multi cell thin walled structure, Journal of Advanced Review on Scientific Research 16(2015) 18-24. [28] Y. Zhang, G.Y. Sun, X.P. Xu, G.Y. Li, Q. Li, Multiobjective crashworthiness optimization of hollow and conical tubes for multiple load cases, Thin-Walled Struct. 82 (2014) 331-342.
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[29] G. Zheng, S.Z. Wu, G.Y. Sun, G.Y. Li, Q. Li, Crushing analysis of foam-filled single and bitubal polygonal thin-walled tubes, Int. J. Mech. Sci. 87 (2014) 226-
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Graphical Abstract
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Highlights
Thin-walled structures with hierarchical and fractal cross section present great
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potential to improve energy absorption. The high order fractal designs outperform the corresponding simple hierarchical
Fractal configurations and geometry have remarkable influence on the energy
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absorption of fractal structure.
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designs with the same mass.
Yong Zhang, Jin Wang, Chunhui Wang, Yi Zeng, Tengteng Chen PII: DOI: Reference:
S0264-1275(18)30643-9 doi:10.1016/j.matdes.2018.08.028 JMADE 7326
To appear in:
Materials & Design
Received date: Revised date: Accepted date:
6 March 2018 13 August 2018 14 August 2018
Please cite this article as: Yong Zhang, Jin Wang, Chunhui Wang, Yi Zeng, Tengteng Chen , Crashworthiness of bionic fractal hierarchical structures. Jmade (2018), doi:10.1016/ j.matdes.2018.08.028
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ACCEPTED MANUSCRIPT Crashworthiness of bionic fractal hierarchical structures Yong Zhanga, Jin Wanga, Chunhui Wangb, Yi Zenga, Tengteng Chena a College of Mechanical Engineering and Automation, Huaqiao University, Xiamen, China b School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, Australia
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Abstract: This paper presents a fractal hierarchical hexagon structure inspired by bionic structures, such as spider webs, as a novel lightweight energy absorber.
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Experimental testing and computational modeling are carried out to characterize the
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effects of fractal order and geometrical shapes on the crashworthiness of this new structure compared to a simple hierarchical structure. The computational results reveal
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that both simple hierarchical and fractal structures can present significant improvement in energy absorption over single wall nonhierarchical structure. Furthermore, fractal configuration, geometrical parameters and order have important effect on the energy
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absorption efficiency of fractal hierarchical structures, with the 2nd order design being the optimal for a given mass. The findings of this research offer a new route for
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designing novel lightweight energy absorbers with improved crash protection against
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impact.
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Lightweight.
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Keywords: Crashworthiness; Fractal hierarchical structure; Energy absorption;
Corresponding Author Yong Zhang: Tel: +86-592-6162 595; Email: [email protected];
ACCEPTED MANUSCRIPT 1 Introduction The drive towards electrical vehicles calls greater innovation in lightweight energy absorbing structures to provide crash protection at the minimum weight. A broad range of new designs of thin-walled structures with different geometrical cross-sections have been proposed to absorb more energy than single-walled structures. Among them, the
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multi-cell column structures have been found to absorb more energy than the equivalent single walled structure of the same mass. Tran et al. [1] developed theoretical models
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for three types of multi-cell triangular tubes based on super folding element theory, and
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the feasibility of the theoretical models was validated by numerical simulations. Mahmoodi et al. [2] theoretically and numerically investigated the energy absorption
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behavior of tapered multi-cell tubes and found that the increase of the taper angle and the wall thickness could improve the crashworthiness of the structures. Zhang et al. [3]
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performed quasi-static axial compression tests on multi-cell columns with different sections and found that multi-cell columns offered better crushing characteristics than single cell columns. Nia and Parsapour [4] compared the energy absorption capacities
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of simple and multi-cell thin-walled tubes with triangular, square, hexagonal and
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octagonal sections. The results showed that the energy absorption capacity of multi-cell sections was greater than that of simple sections. Xiang et al. [5] compared the energy
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absorption capacities of polygonal tubes, multi-cell tubes and honeycombs. Their results suggested that the energy absorption of multi-cell tubes depended on the number
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of cells. Fang et al. [6] studied the crushing behaviors of multi-cell members subjected to oblique impact loads. The results showed that the increase in cell number was beneficial to the energy absorption. In addition. Wu et al. [7] performed the multiobjective optimal design for a five-cell tube to maximize its specific energy absorption and minimize peak crushing force. Yang et al. [8] optimized the multi-cell tubular structures with pre-folded origami patterns. These results showed that quintuple-cell origami tubes absorbed the highest amount of energy with significantly reduced initial peak force. Recently, biological structures with hierarchical (Fig. 1(a) [9], (b) [10]) and fractal
ACCEPTED MANUSCRIPT (Fig. 1(c) [11]) characteristics have attracted growing attention to enhance the mechanical properties of lightweight materials and structures. Kooistra et al. [12] investigated the transverse compression mechanisms of hierarchical corrugated core sandwich panels, and found that second order trusses had higher compressive and shear collapse strengths than their equivalent mass first order counterparts. Fan et al. [13]
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demonstrated the effectiveness of hierarchical honeycomb structure in improving the stiffness and plastic collapse strength of thin-walled structures. Zheng et al. [14] found
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that hierarchical square honeycombs using ductile woven textile composites had higher specific energy absorption than those of metallic honeycombs. Chen et al. [15] studied
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the hexagonal, kagome, and triangular hierarchical honeycomb metamaterials and found that hierarchical designs could effectively improve the heat resistance and load-
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carrying capacity of regular honeycomb structures. Zhang et al. [16] found that hierarchical honeycombs more efficiently distributed the material across the network
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than traditional hexagonal honeycombs. Oftadeh et al [17] constructed a fractalappearing cellular metamaterial by replacing each three-edge vertex of a base
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hexagonal network with a smaller hexagon. Their results showed that the in-plane
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stiffness for a mass increases with the order of hierarchy. In addition, hierarchical designs can also improve the weight-efficiency of thinwalled tubular structures. Sun et al.[18] found that hierarchical triangular lattice
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structures possessed three to four times higher mean crushing force than single-cell and multi-cell lattice structures. Li et al. [19] investigated the crushing mechanisms of
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hierarchical hexagonal tubes by numerical analysis and found that hierarchical topology greatly improved the energy absorption of thin-walled hexagonal tubes. Luo et al. [20] examined the lateral crushing behaviors of hierarchical quadrangular tubes and found it greatly increased the mean crushing force of single wall quadrangular tubes. Wang et al. [21] developed a hybrid hierarchical structure based on the Koch fractal principles, and found that the 2nd order hybrid Koch absorbers outperformed a wide range of multicell structures of the same mass.
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Fig. 1 Typical hierarchical biological structures:(a) bird-of-paradise [9], (b) spongy bone [10]and
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fractal biological structures: (c) nautilus [11].
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Motivated by these promising findings, in this paper we propose a novel thinwalled structure inspired by the spiral wheel-shaped (orb-web) construction of spider
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webs. This design, denoted fractal hierarchical hexagon structure, is compared with a simple hierarchical hexagon structure. Section 2 describes these two biologically
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inspired designs, numerical models of simple hierarchical and fractal hierarchical structures, and validation of the computational models by experiments. A comparative
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study is presented in Section 3 to investigate the advantage of the fractal hierarchical
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design over the simple, non-fractal hierarchical design. Lastly, using the validated numerical model, sensitivity and parametric design are performed by varying the fractal
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order and geometries.
2 Model and Experiment
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2.1 Geometric models The geometric characteristics of bioinspired hierarchical and fractal hierarchical thin-walled structures are illustrated in Fig. 2, where Fig. 2(a) and Fig. 2(b) show the simple hierarchical construction of spiral spider web and the evolution fractal hierarchical design, respectively. Firstly, hexagon bionic hierarchical sections are established based on the biological geometry of spider web, in which high order hierarchical structures (HS) are formed by adding a regular hexagon at the center of the low order hierarchical structures and connecting the corresponding vertices; the length of the linking elements is denoted as d1 and the wall thickness of the hexagonal structure
ACCEPTED MANUSCRIPT is t. The diameter of the circumscribed circle of the bionic structure is denoted as D. The three hierarchical structures with different orders are named HS1, HS2 and HS3, respectively. Next, a new fractal structure (FS) is obtained by replacing the each of primary hexagonal vertices of the simple hierarchical structures with sub-hexagons and resultant
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fractal designs are named as FS1, FS2 and FS3, respectively. The side length of subhexagon and linking distance are denoted as b and d2, respectively, as shown in Fig.2
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(b). The wall thickness of fractal structures is also denoted as t.
Based on a typical energy absorber of passenger vehicle, as shown in Fig. 2(d)
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[22], a schematic diagram of the bionic thin-walled tubes under axial dynamic loading is presented in Fig. 2(c). In the following numerical investigations, the circumscribed
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circle diameter D of bionic structures is kept at 90 mm, the length L of the columns is 160 mm, and the wall thickness t will be varied. The columns will be axially loaded by
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tube is constrained by a rigid wall.
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an impactor with a mass of 500 kg at a velocity of v=10 m/s. The non-impacted end of
Fig. 2 The evolution of bionic structures: (a) biological geometry of spider web, (b) cross-sections of hierarchical and fractal structures, (c) typical vehicle anti-collision structure [22], and (d)
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2.2 Numerical models and crashworthiness indicators Finite element (FE) models were developed for the bionic thin-walled structures, both simple hierarchical and fractal hierarchical structures, as illustrated in Fig. 3, using
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the nonlinear finite element software LS-DYNA 971. The thin-walled tubes were modeled using Belytschko-Tsay four-node shell elements [23]. To maintain a
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reasonable balance between the computational cost and accuracy of the numerical
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results, the element size of 1.0 mm x1.0 mm was found to be sufficient for to achieve convergence using shell elements. The elastic-plastic material model MAT_24 and
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rigidity material model MAT_20 in LS-DYNA were adopted to model the thin-walled tube and the rigid clamps, respectively. An automatic surface-to-surface contact was
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chosen to simulate the contact between the thin walled tube and rigid clamps. An automatic single-surface contact was adopted for all the thin walls to avoid mutual interpenetration of folding lobes. The static and dynamic frictional coefficients defined
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in these contacts were 0.3 and 0.2, respectively [24].
Fig. 3 The finite element models of thin-walled structures.
ACCEPTED MANUSCRIPT To effectively evaluate the crashworthiness performance of thin-walled structures, several crashworthiness indicators are defined here. They are total energy absorption (EA), specific energy absorption (SEA), peak crushing force (PCF), and crash load efficiency (CLE), respectively. The total energy absorption (EA) [25] during the process of compression is calculated as
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𝑑
EA= ∫ 𝐹(𝑥) dx
(1)
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where F(x) denotes the crushing force, x is the instantaneous crushing displacement,
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and d is the effective crushing displacement. Herein, the value of d is kept at 110 mm.
structure of unit mass. It is defined by:
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The specific energy absorption (SEA) [26] denotes the energy absorbed by a EA (2) M where M is the total mass of the structure. SEA is a key indicator to estimate the energy
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SEA =
absorption efficiency of structures with different materials and weights. Therefore, the
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higher the SEA, the better is the energy absorption performance of the structure.
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The peak crushing force PCF [27], i.e., the maximum of F(x), is another important crashworthiness indicator for energy absorbers. Since a high PCF often leads to a high
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deceleration, potentially causing severe injury or even death of occupant, PCF should be reduced or constrained to certain extent from the safety design viewpoint.
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The mean crushing force (MCF) [27] represents the average crushing strength, and is expressed as
MCF =
𝐸𝐴(𝑑) 𝑑
(3)
where 𝐸𝐴(𝑑) is the total energy absorbed when the absorbed is crushed over a distance 𝑑. Finally, the crushing load efficiency (CLE) [28] indicates the constancy of the crushing force during impact, and it is defined as the ratio between the mean load (MCF) and the initial peak force (PCF),
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MCF PCF
(4)
Obviously, the higher the CLE value, the better is the force uniformity for an energy absorber.
2.3 Experimental tests and validation of FE model Quasi-static crushing tests were carried out to characterize the energy absorption
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behavior of bionic structures. The data also served to validate the finite element models. The material of the bionic structures is aluminum alloy A6061-O, with the following
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mechanical properties: density ρ=2700kg/m3, Young's modulus E=68.2GPa, and
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Poisson's ratio μ=0.3. The stress-strain response of the aluminum alloy was measured using the MTS 322 material testing machine at a displacement rate of 5 mm/min. The
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true stress-strain data of A6061-O was presented in Fig. 4. This aluminum was known
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carried out at higher strain rates.
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to be insensitive to the strain rate in the range of interest [29], so no further tests were
Fig. 4 True stress-strain curve of A6061-O.
To validate the computational model outlined in section 2.2, two specimens, denoted as HS2 and FS2 as shown in Fig.5 (a), were manufactured by the wire electrical discharge machining (WEDM) technique from blocks of aluminum alloy 6061-O. This
ACCEPTED MANUSCRIPT WEDM technique used a continuous moving electrode wire to generate pulse spark discharge to etch away the material. The top views of the two specimens were shown in Fig.5 (b). The circumscribed circle diameters of HS2 and FS2 were 70 mm and 90 mm, respectively. The lengths of HS2 and FS2 were 110 mm and 160 mm, respectively. Meanwhile, the length of sub-hexagon b and link distances of d1 and d2 were 5 mm, 15
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mm and 5 mm, respectively, and the wall thickness t of HS2 and FS2 was 0.8 mm. Quasistatic compressive tests were performed in an MTS universal testing machine shown in
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Fig. 5(c). The specimens were placed between two flat plates. During the test, the top plate (pressure head) of the machine moved downward at a constant speed of 5 mm/min.
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The crushing load and displacement were collected automatically. When the compressive displacements reached 70 mm for HS2 and 110 mm for FS2, respectively,
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the tests were terminated and the specimens were unloaded for examination.
Fig. 5 Experimental specimens: (a) 3D view of specimens, (b) top view of specimens, (c) universal testing machine.
Fig.6 shows the collapse modes of the specimens tested together with the predictions of the corresponding FE models at different crushing displacements. It is
ACCEPTED MANUSCRIPT clear from Fig. 6(b) that FS2 undergoes a regular stable collapse in every corner, which is the desired energy absorption mode. During the crushing process, its axial crushing progressed from the bottom end, with regular folds stacking onto the previous folds. By contrast, the deformation of HS2 starts from the top end, and the folding is much less uniform than the FS2.
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Comparisons between the experimental results and the computational model predictions shown in Fig.6 confirm that the FE models have well captured the
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deformation characteristics at different crushing displacements. To further examine the crushing deformation in detail, the tested specimens are cut into two equal halves, so
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the interior of the progressive collapse modes are visible, as presented in Fig. 7. The sub-hexagons of FS2 form very regular and uniform folds, and the wavelength of the
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folds is quite short. By contrast, the collapse wavelength of HS2 is larger than the FS2, which suggests that the sub-hexagons promote the progressive folding. Meantime, the
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interior views of both specimens are very similar to the predictions of the FE models. In addition, Fig. 8 shows the corresponding crushing force-displacement curves of HS2
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and FS2, indicating a periodic fluctuation at regular intervals as collapse progresses.
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This result provides further evidence that the thin-walled structures underwent stable and progressive deformation. The load-displacement curves obtained from the computational simulation are in good agreement with the experiment results. Therefore,
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the FE models have now been validated and can be applied to investigate the
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crashworthiness of new designs, which are described in the following section.
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Fig. 6 Comparison of the deformation modes between the experimental and numerical results: (a)
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HS2, (b) FS2.
Fig. 7 The internal views between the experimental and numerical results: (a) HS2, (b) FS2.
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Fig. 8 Comparison of impacting force-displacement curves: (a) HS2 (b) FS2.
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3 Comparative analysis for simple hierarchical and fractal
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hierarchical structures
To assess the effects of the order of hierarchy and fractal state on the
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crashworthiness of the simple hierarchical structures (HS1, HS2, HS3) and fractal
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hierarchical structures (FS1, FS2, FS3) presented in Fig. 1, computational simulations are carried out of these structures with four different mass (and hence different wall thickness). The mass of these two types of thin-walled structures determines the
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material distribution, which then affects the energy absorption. The corresponding wall thickness to attain the desired mass is also listed in Table 1.
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Table 1 The thicknesses of thin-walled structures with different mass.
Cross section HS1
M1 (kg)
t (mm)
M2 (kg)
t (mm)
M3 (kg)
t (mm)
M4 (kg)
t (mm)
0.13
1.16
0.17
1.44
0.20
1.73
0.24
2.02
0.13
0.58
0.17
0.72
0.20
0.87
0.24
1.01
0.13
0.43
0.17
0.54
0.20
0.65
0.24
0.76
HS2
HS3
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0.13
0.80
0.17
1.00
0.20
1.20
0.24
1.40
0.13
0.43
0.17
0.54
0.20
0.65
0.24
0.76
0.13
0.33
0.17
0.41
0.20
0.49
0.24
0.57
FS2
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FS3
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Table 2 presents the crashworthiness indicators of the different structures examined. The results show that for all the cross-section shapes, both the SEA and PCF
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increase with the mass, as clearly seen in Fig. 9 for the SEA curves of different structures. More importantly, for the simple hierarchical structures, the SEA is near
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linear increase with the mass. Furthermore, the 3rd order design (HS3) seems to yield a close SEA value to the 2nd order design (HS2), implying that there is a diminishing
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improvement in specific energy absorption beyond the 2nd order. However, for the fractal hierarchical structures, the 3rd order design (FS3) seems to still offer substantial
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improvement over the 2nd order design (FS2), although the gain is less than that of 2nd
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order design over the 1st order design.
ACCEPTED MANUSCRIPT Table 2 SEA and PCF of all structures. Cross section
M1(kg) SEA PCF (kJ/kg) (kN)
M2(kg) SEA PCF (kJ/kg) (kN)
M3(kg) SEA PCF (kJ/kg) (kN)
M4(kg) SEA PCF (kJ/kg) (kN)
HS1 11.19
41.76
12.91
53.63
14.20
65.32
15.72
76.93
11.76
36.46
13.72
47.70
15.29
59.96
17.02
72.34
11.98
35.22
14.02
45.27
15.63
56.02
19.58
41.22
22.32
54.27
22.29
36.24
24.08
46.39
23.51
35.47
25.14
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HS2
HS3
67.72
66.26
25.14
78.59
26.43
57.36
27.64
69.75
27.52
58.40
28.91
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FS3
24.30
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FS1
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46.84
Fig. 9 SEA of different simple hierarchical and fractal hierarchical structures.
The crushing force-displacement curves of HS and FS structures with mass=0.202
ACCEPTED MANUSCRIPT kg are presented in Fig. 10. The fluctuation of crushing force is more pronounced in low level hierarchical structures, such as HS1 and FS1, as indicated by the large folding wavelength. This behavior is undesirable for energy absorbers. As the hierarchical level increases, however, the fluctuation in the force-displacement curve decreases, indicating that higher level hierarchical structures are better in terms of crush force
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efficiency. The corresponding deformation shapes are presented in Fig. 11. It is seen that as the hierarchical level increases, both HS and FS structures display more
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progressive folding lobes, and the folding wavelength decreases with the hierarchy
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order and the fractal order, i.e., 𝜆HS1 > 𝜆HS2 > 𝜆HS3 and 𝜆𝑆𝐹𝑆1 > 𝜆𝑆𝐹𝑆2 > 𝜆𝑆𝐹𝑆3 . These results reveal that the hierarchical principle improves energy absorption.
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However, the force-displacement curves of the 2nd order (HS2, FS2) and 3rd order structures (HS3, FS3) are fairly similar, indicating that they have near load bearing
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capacity and energy absorption. This is consistent with the similarity between the folding wavelengths of the 2nd order and the 3rd order structures. Therefore, it be concluded that hierarchical organization improves crashworthiness of thin-walled
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structures, with the 2nd order hierarchical structures being the optimum considering the
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gain versus geometrical complexity.
Fig. 10 Impacting force-displacement of different hierarchical structures: (a) HS structures, (b) FS structures.
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Fig. 11 Deformations and section views of different structures: (a) HS structures, (b) FS structures.
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It is important to note that FS structures offer a dramatic enhancement in SEA
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compared to the HS counterparts, as seen in Fig. 9. The percentages of increase in SEA for different mass are presented in Table. 3, indicating the amount of increase ranging
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between 59.92% and 96.24%. Therefore, the fractal geometry offers huge potential in improving the energy absorption capability of thin-walled structures.
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Table 3 Increasing rate of SEA for FS structures and corresponding HS structures. ΔSEA (FS1 vs HS1)
ΔSEA (FS2 vs HS2)
ΔSEA (FS3 vs HS3)
M1=0.1348kg
74.98%
89.54%
96.24%
M2=0.1685kg
72.88%
75.51%
79.32%
M3=0.2022kg
71.12%
72.86%
76.07%
M4=0.2359kg
59.92%
62.39%
67.94%
*∆𝑆𝐸𝐴 =
(𝑆𝐸𝐴𝑆𝐹𝑆 −𝑆𝐸𝐴𝐻𝑆 ) 𝑆𝐸𝐴𝐻𝑆
× 100%
ACCEPTED MANUSCRIPT To further illustrate the improvement offered by the fractal design over the simple hierarchical counterpart, Fig.12 presents the force-displacement curves for FS and HS structures with the same mass of 0.202 kg. It is clear that at the same hierarchical order, a fractal structure offers much higher load bearing capacity and more energy absorption than the corresponding simple hierarchical structure. Furthermore, the 2nd and 3rd order
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fractal hierarchical structures show more progressive folding lobes than the corresponding simple hierarchical structures. From the deformation shapes shown in
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Fig. 12, strain concentration occurs mainly at the corners of structures. In the case of fractal structures, the strain also concentrates at the corners of sub-hexagon structures
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thus the sub-hexagons trigger more progressive deformations, contributing to better
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energy absorption performance of the fractal structures.
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Fig. 12 Force-displacement curves and deformation modes of hierarchical and fractal structures: (a) 1st structures, (b) 2nd structures, (c) 3rd structures.
The PCF values listed in Table. 2 also confirm that an increase in mass causes the PCF to increase. The reason is that higher mass means thicker walls, increasing the
ACCEPTED MANUSCRIPT stiffness of the thin-walled structures. The values of CLE for different structures are plotted in Fig.13. It is clear that the CLE values for both structures increase with the hierarchical order. More importantly, the CLE values of fractal structures are much greater than that of corresponding simple hierarchical structures, indicating that fractal
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organization improves crush force uniformity, which is desirable for energy absorbers.
Fig. 13 CLE of all structures with different mass.
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Overall, the 2nd fractal hierarchical structures (FS2) has higher energy absorption than other structures examined in this paper. Therefore, FS2 is chosen as a candidate in following parametric design.
4 Parametric design of FS2 The crashworthiness behavior of the FS2 highly depends on the geometric and fractal parameters. Therefore, a parametric design is necessary to explore the influence on crashworthiness under dynamic loading.
ACCEPTED MANUSCRIPT 4. 1 The effect of fractal configuration The location of the sub-hexagons, i.e., the fractal configuration, may influence the energy absorption characteristics of the fractal structures. Fig. 14 presents three fractal configurations of sub-hexagon in vertex of 2nd order FS, which are referred as inner sub-hexagon, central sub-hexagon, and outer sub-hexagon, respectively. Applying
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these three configurations to the three hierarchical geometries gives nine types of fractal designs listed in Table 4. Dynamic impacting simulations of the nine fractal structures
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are performed to investigate their crushing behaviors and to identify the optimum
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candidate for further detailed analysis. The side length b of sub-hexagon and the linking distance d2 shown in Fig. 2(b) are set as 5 mm for all nine variants, and total mass is
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kept constant.
Fig. 14 Different fractal configurations for FS2.
Table 4 Fractal structures with different fractal configurations.
Cross section
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Fractal type
Name
Fractal type
Cross section
Name
Fractal type
Cross section
Name
HOHI
HCHI
HIHI
HOHC
HCHC
HIHC
HOHO
HCHO
HIHO
The results of SEA, PCF and CLE of the nine fractal structures are summarized in
ACCEPTED MANUSCRIPT Table 5. It can be seen that the fractal configuration has an important effect on the energy absorption, with the HIHC design having the highest SEA, more than 33.2% than the HOHI design that yields the minimum energy absorption. The HIHC design features inner sub-hexagons in the first layer and central sub-hexagons in the second layer. This particular design produces both the maximum SEA (29.21 kJ/kg) and the
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maximum CLE, while the PCF is close to other configurations. Therefore, HIHC is the optimal fractal configuration among the nine options, and thus will be further evaluated
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in following to quantify the effects of other geometrical parameters, such as the
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thickness of the cell wall.
Table 5 SEA, PCF and CLE of different fractal structures. PCF (kN)
SEA
PCF
(kJ/kg)
(kN)
27.84
74.42
0.85
21.94
70.42
0.70
27.2
73.63
0.84
29.21
75.04
0.88
CLE
Cross
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section
Deformation
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Cross section
73.99
0.73
27.43
75.77
0.82
24.71
70.37
0.79
24.03
75.16
0.72
70.16
0.77
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23.87
4.2 The effect of geometrical parameters Based on the finding presented in Section 4.1 that the fractal structures HIHC is the best design for crashworthiness, further parametric analysis is performed to characterize the effects of the cell walls on crashworthiness of HIHC, such as the side length b of sub-hexagon, the linking distance d2 shown in Fig. 2(b). To avoid geometric interference, the range of parameter b and d are set as 4 mm-7 mm.
ACCEPTED MANUSCRIPT Fig. 15(a) depicts the 3D surface of SEA of HIHC with different geometrical parameters (b and d2). It is clear that increasing the value of d2 only yields a slight increase in SEA, indicating that SEA is not sensitive to the linking distance d2. However, the side length b of the sub-hexagons shows a strong effect on SEA: increasing b causes SEA to rise first and then fall, reaching the maximum value at b=6 mm. This is further
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illustrated in Fig. 16. At a given crushing displacement, the force is highest at b=6 mm.
(a)
(b)
Fig. 15 SEA and PCF of HIHC with different geometrical parameters: (a) SEA, (b) PCF.
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b=4mm
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b=6mm
b=7mm
80 70
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Impacting Force (kN)
90
60 50 40 30
20 0
10
20
30
40
50
60
70
80
Displacement （ mm)
90
100 110
Fig. 16 Force-displacement curve of HIHC with d2=6 mm.
In addition, PCF values shown in Fig. 15(b) also display a similar phenomenon with SEA. It is seen that PCF is not sensitive for d2, however, PCF gradually increases with the b increasing. Therefore, it can be believed that geometrical parameter b plays
ACCEPTED MANUSCRIPT an important role in the crashworthiness of HIHC.
6 Conclusions This paper presents a comprehensive investigation of the crashworthiness of simple hierarchical and fractal hierarchical structures using experimental tests and
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computational modelling techniques. Several conclusions are drawn from the results: 1. Bionic designs of simple hierarchical and fractal hierarchical configurations
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offer significant improvement in energy absorption capabilities over the
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hexagonal structure (HS1).
2. The energy absorption of the fractal hierarchical design is better than the simple
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hierarchical design, the 2nd order fractal hierarchical structure (FS2) has the optimum crashworthiness considering the gain versus geometrical complexity.
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3. Both fractal configuration and geometry have important effect on the energy absorption of the 2nd order fractal structure, in which the length of sub-hexagon
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Acknowledgments
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has strong sensitivity on the energy absorption.
This work is supported by The National Natural Science Foundation of China (51675190), Program for New Century Excellent Talents in Fujian Province University,
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Fujian Province Natural Science Foundation of China (2015J01204), Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of
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Huaqiao University (ZQN-PY202).
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Graphical Abstract
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Highlights
Thin-walled structures with hierarchical and fractal cross section present great
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potential to improve energy absorption. The high order fractal designs outperform the corresponding simple hierarchical
Fractal configurations and geometry have remarkable influence on the energy
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absorption of fractal structure.
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designs with the same mass.