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Effect of cell-wall angle on the in-plane crushing behaviour of hexagonal honeycombs
Materials and Design 46 (2013) 511–523
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Effect of cell-wall angle on the in-plane crushing behaviour of hexagonal honeycombs Lingling Hu a,⇑, Fanfan You a, Tongxi Yu b a b
Department of Applied Mechanics & Engineering, School of Engineering, Sun Yat-sen University, Guangzhou 510275, PR China Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 28 July 2012 Accepted 27 October 2012 Available online 16 November 2012 Keywords: Honeycomb Crushing strength Cell-wall angle In-plane Impact
a b s t r a c t The dynamic crushing behaviors in the x- and the y-directions of hexagonal honeycombs with various cell-wall angles are explored by both experiments and numerical simulations. Several deformation modes are identified based on the shape of the localization band forming by the cells’ collapse. The respective influence of the cell-wall angle, the crushing velocity and the honeycomb’s relative density on the honeycomb’s mechanical properties is studied. It is shown that these influencing factors affect the honeycomb’s x-directional crushing strength by altering the deformation mode of the honeycomb, while both the y-directional crushing strength and the average crushing strength are dominated by the honeycomb’s density. With retaining the honeycombs’ relative density as a constant, the honeycomb with the cell-wall angle of about 45° exhibits the optimal crushing strength and energy absorption capacity under the y-directional crushing, while the average crushing strength in the x- and the y-directions decreases with the cell-wall angle. Significant orthotropic anisotropy is revealed in the honeycombs with the cell-wall angle greater than 30°, especially under low-velocity compression. The honeycomb with the cell-wall angle of about 25° possesses transversely isotropic mechanical properties. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Cellular material has attracted a great deal of attentions owing to their outstanding properties in efficient energy absorption, high relative stiffness and strength, heat insulation and so on. They have been widely used in lightweight structures and in energy absorption devices, especially under dynamic crushing conditions. One of the notable mechanical characteristics of the cellular material under crushing is that the crushing strain increases without obvious increase in the stress for certain period of time, which is called ‘‘plateau stage’’. The energy absorption of cellular material is dominated by the stress and the length of this stage. The relative density has been recognized as one of the dominant factors in dictating the mechanical properties of cellular material [1,2]. Except the relative density, other factors, such as the crushing velocity, the loading manners, the cell structures and so on, also have important effects on it. Both numerical [3,4] and theoretical analyses [5,6] show that the plateau stress of cellular material under dynamic crushing increases with the impact velocity by following a square law. Honeycombs display much server inhomogeneous deformation under uniaxial compression than that under biaxial compression, which was quantitatively revealed by an ⇑ Corresponding author. Tel.: +86 20 8411 0293. E-mail addresses: [email protected], [email protected] (L. Hu). 0261-3069/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.matdes.2012.10.050
inhomogeneity index defined by Hu et al. [7]. Chung and Waas [8] and Papka and Kyriakides [9] performed biaxial in-plane compression on honeycombs, showing that the crushing strength and the deformation modes of honeycombs were related to the particular ratio between the components of the applied displacements or forces. The effects of the cell size, the cell-wall thickness and the panel thickness on mean crushing strength, energy absorption capacity and folds wavelength of aluminum honeycombs were investigated in Ref. [10]. Liu and Zhang [11] and Qiu et al. [12] studied the influence of the cell micro-topology on the in-plane dynamic crushing of honeycombs by finite element method (FEM). Cell shapes such as hexagonal, rhombus, square, triangular and Kagome were investigated. Honeycombs with regular hexagonal cells are widely used in industry. Extensive experimental [13,14], analytical [5,15] and numerical [3,4] studies have been conducted on their mechanical behaviors. By simply changing the internal cell-wall angle of hexagonal cells, new cell configuration can be obtained, which is easy to implement based on the existing production process. The outof-plane mechanical properties of hexagonal honeycombs with changed cell-wall angles were studied in Ref. [16]. The results show that changing the cell structures by this simple method can enhance the honeycomb’s out-of-plane crushing strength to 1.5 times comparing to the regular hexagonal honeycomb. In some applications, such as using a honeycomb block as an energy
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absorption layer in aircraft against bird or debris collision, the crushing could take place along any direction of the honeycomb. Hence, the effect of changing the cell-wall angle on the in-plane dynamic behaviors of the honeycomb also needs to be known in addition to its out-of-plane behaviors. The regular hexagonal honeycomb is usually regarded to possess the same mechanical properties, such as the Young’s modulus and the plastic collapse strength, in the x- and the y-directions [1]. However, experiments revealed that the regular hexagonal honeycombs are orthotropic in the crushing strength and the energy absorption [7,17]. The orthotropic characteristics of honeycombs are related to the cells’ configuration, and a transversely isotropic behavior is expected by changing the cell-wall angle of the hexagonal honeycomb. Based on the ideas of changing the cell shape by simply changing the cell-wall angle as mentioned above, the objective of the present paper is to examine the effect of the cell shape (or the cell-wall angle) on the in-plane crushing behaviors of hexagonal honeycombs by employing both experiments and numerical method. The crushing strength of honeycombs in both the x- and ydirections is discussed in terms of the impact velocity, the cell-wall angles and the relative density of honeycombs; and it is comprehensively evaluated by defining a crushing strength ratio and the average crushing strength. Finally, a transversely isotropic honeycomb is discovered based on the numerical results. 2. Experiments 2.1. Honeycomb specimens The specimens of the regular hexagonal honeycomb used in this work were cut from a piece of commercial 5052-aluminum honeycomb, as shown in Fig. 1a. The thickness and the length of the cell walls parallel to the y-direction were measured to be h1 = 0.083 mm and l1 = 2.04 mm, respectively; while the sizes for the cell walls unparallel to the y-direction were h2 = 0.046 mm and l2 = 2.53 mm. The cell-wall angle was b = 35.5°, defined as the angle of the honeycomb’s bevel edge to the x-direction, as shown in Fig. 1a. The density of the honeycomb was 72.11 kg/m3. To obtain the honeycomb with a different cell-wall angle from the regular hexagonal one, the cells of a part of the specimens were carefully pre-compressed along the x-direction. Before compression, each cell was filled in by a toothpick at its center, which was perpendicularly inserted into a sponge underneath the specimen, as shown in Fig. 2. The sponge has a similar size to the specimen along the x-direction. Then the specimen and the sponge were carefully compressed together along the x-direction by two rigid planes until the cells being condensed with the filled-in toothpicks. Then, after holding it for a half minute under this condensed status, the force and the toothpicks were removed. Thus the honeycomb with amended cell-configuration was achieved, as shown in Fig. 1b. The cell-wall angle of the pre-compressed honeycomb was measured to be 51.2°, and the density was 85.21 kg/ m3.
specimen is placed on a fixed rigid base at one end and crushed by a steel hammer with an initial impact velocity of 7.8 m/s at the other end. The impact velocity retains almost a constant during the whole crushing process by setting the hammer heavy enough. Sufficient number of cells (more than 11 11 cells) are contained in the honeycomb specimens so that they can represent the global properties of the honeycomb. The deformation process of the specimens was recorded by a high-speed camera, which is demonstrated in Figs. 3 and 4. It is shown that at the early time of the deformation process, the specimen exhibited uniformly transverse expansion with all the cells compressed simultaneously as shown in the second pictures of each figure. Soon after that, the compressed deformation localized at some cells, forming an obvious deformation zone (or the deformation bands) within the specimens, see the third pictures in Figs. 3 and 4. Then with further crushing of the specimen, the cells near the deformation zone collapsed gradually. The initiation of the deformation zone depended on the distribution of the cells’ imperfection. According to the wave trapping theory, once the crushing velocity is higher than a critical value, localization bands would initiate at the loading end or the supporting end of the honeycomb block, despite the initial imperfections distribute among the honeycomb [18]. Based on the theory, the critical velocity is estimated as 6.4– 7.6 m/s for the honeycombs studied in our experiments. It should be noticed that the wave trapping theory comes from the onedimensional wave theory. For the 2-dimensional honeycomb specimens, this critical velocity will become higher resulted by the wave dispersion. The stress–strain curves of the honeycombs are shown in Fig. 5. It indicates that the honeycombs crushed along the y-direction have a higher plateau stage compared with the ones crushed along the x-direction. By averaging the stress in the plateau stage of the stress–strain curve, the crushing strength of the honeycomb, rc, is defined. Table 1 summarized the crushing strength of the honeycombs in the experiments. Both Fig. 5 and Table 1 indicate that the pre-compressed honeycomb exhibits a lower x-directional crushing strength but a higher y-directional crushing strength compared with the regular hexagonal honeycomb. The ratio of the crushing strength between the y- and the x- directions (rcy/rcx) shown in Table 1 denotes that the pre-compressed honeycomb with a larger cell-wall angle displays more significant orthotropic anisotropy than the regular hexagonal one. It is noted that the pre-compressed honeycomb exhibits lower mechanical properties in the x-direction than the regular hexagonal one, although the density of the former is larger than that of the latter, implying that the cell configuration, rather than the relative density of the honeycomb, is the dominant factor in dictating the honeycombs’ x-directional mechanical properties under the conditions of our experiments.
3. Numerical simulation 3.1. Numerical model
2.2. Experimental results Both the regular hexagonal honeycombs and the pre-compressed honeycombs were dynamically crushed along both the xand y-directions, respectively, in the Instron Dynatup 9250 testing system. In our experiments, like most of the ones in this field [6– 9,14,16,17], did not use testing standard. For scientific research on cellular materials including honeycombs, people usually design their own experiments to meet the purpose of verifying theoretical/numerical models. In the dynamic tests, the honeycomb
On account of the limit in the specimens with amended cellwall angle and in the crushing velocity in tests, a finite element model is built using ANSYS–LSDYNA to further understand the mechanical behaviors of hexagonal honeycombs with various cell-wall angles under dynamic crushing. By changing the cell-wall angle, five types of honeycombs are obtained with b being equal to 15°, 30°, 45°, 60° and 75°, respectively, as shown in Fig. 6. As the present work is focused on the influence of the cell-wall angle, the un-equivalence in both the thickness and the length of the cell
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h2
l2
β
h1
l1
y x
(a) Regular hexagonal honeycomb
(b) Pre-compressed honeycomb Fig. 1. Honeycomb specimens in the experiments.
Two groups of honeycombs are studied in the present paper: a group with the same density and a group with the same cell-wall. The relative density is one of the most important factors in affecting the mechanical properties of cellular material. Thus to examine the influence of the cell-wall angle on the honeycomb’s mechanical behaviors, the honeycombs in the first group possess the same rel ative density qr ¼ qq ¼ 0:1; where q⁄ and qs are the density of the s honeycomb and the base material, respectively. Then the cell-wall thickness h of honeycombs with various cell shapes can be calculated according to [1]:
q 3 h : ¼ qs 2 cos bð1 þ sin bÞ l
Fig. 2. A honeycomb with toothsticks inside before pre-compression.
ledges is not considered in the numerical simulations, i.e., all the cells are equilateral hexagon with equal cell-wall thickness. The edge length of the cells l is 4 mm,and the out-of-plane width (along the z-direction) of the honeycomb is b = 0.5 mm.
ð1Þ
In the production process, a more direct method is to change only the honeycomb’s cell-wall angle while remaining the thickness h and the length l of the cell walls unchanged. Thus honeycombs in the second group retain the cell-wall thickness h as a constant of 0.346 mm, and then according to Eq. (1), the relative density of honeycombs with various cell-wall angles is listed in Table 2. In simulations the hexagonal honeycomb block consists of 21 cells in the x-direction and 13 cells in the y-direction, which is placed on a fixed rigid base at one end and crushed along either the x- or the y-direction by a rigid plate with a constant crushing velocity V at the other end, as shown in Fig. 7. The cell-wall material is assumed to be elastic, perfectly plastic with E = 68 GPa, rys ¼ 130 MPa, qs ¼ 2700 kg=m3 and t = 0.3, where E, rys and t are the Young’s modulus, yield stress and Poisson ratio of the base material, respectively. The material of the rigid plates is chosen to be steel with Young’s modulus 210 GPa and density 7800 kg/m3.
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(i)
(ii)
(iii)
(iv)
(v) (a) Regular hexagonal honeycomb
(vi)
(i)
(ii)
(iv)
(iii)
(v) (b) Pre-compressed honeycomb
(vi)
Fig. 3. Deformation process of honeycomb specimens in the x-directional crushing.
Each cell-wall is meshed into 16 shell elements with one layer elements along the out-of-plane direction (the z-direction). The outof-plane displacement of the nodes within the x–y plane (i.e. z = 0) is constrained so as to prevent the specimen from out-ofplane bulking. A single self-contact is defined on the honeycomb model, and a surface-to-surface contact is applied between the honeycomb specimen and the two rigid plates respectively without considering the contact friction.
3.2. X-directional crushing 3.2.1. Deformation mode Honeycombs with various cell-wall angles exhibit distinct deformation modes under x-directional crushing, which are classified into the low-velocity deformation mode (LVDM) and the highvelocity deformation mode (HVDM). In the low-velocity deformation mode (LVDM), the initial localization bands occurred from
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(i)
(ii)
(iv)
(v) (a) Regular hexagonal honeycomb
(iii)
(ii)
(i)
(iv)
(iii)
(v) (b) Pre-compressed honeycomb
(vi)
Fig. 4. Deformation process of honeycomb specimens in the y-directional crushing.
the loading edge produce an ‘‘X’’ shape along the row parallel to the symmetric axis of the honeycomb’s geometry, and then it is soon compressed as an inversed ‘‘V’’-shaped band, as shown in Fig. 8a (LVDMI), where t denotes the crushing time. Or an inversed ‘‘V’’-shaped localization band is directly formed within the honeycomb at the beginning of the deformation, as shown in Fig. 8b (LVDM II). Afterwards, the cells collapse layer by layer along the
inversed ‘‘V’’-shaped band. With the increase of the displacement, a second localized ‘‘V’’-shaped band is developed from the supporting end and ‘‘grows’’ towards the existing one which seems to have stopped developing (Fig. 8a and b). With more layers of cells gradually collapsing along the ‘‘V’’-shaped bands, the two localization bands finally intersect with each other to form a rhombus at the central region of the specimen. After that, localization takes place
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L. Hu et al. / Materials and Design 46 (2013) 511–523 Table 2 The relative densities of honeycombs with the same cell-wall thickness.
0.4 Pre-compressed Y Regular Y Regular X Pre-compressed X
Stress (MPa)
0.3
b (°) Relative density
15 0.1067
30 0.1
45 0.1074
60 0.1391
75 0.255
0.2
Loading plate
0.1
0
0
0.2
0.4
0.6
0.8
1
V
Strain Fig. 5. Stress–strain curves of honeycombs in the dynamic experiments.
Honeycomb block Table 1 Crushing strength of honeycombs in dynamic experiments (unit: kPa).
rcx rcy rcy/rcx
Regular
Pre-compressed
31.40 53.42 1.70
29.09 70.47 2.42
Supporting plate
within the central rhombus until the specimen is completely crushed. Under high crushing velocity, for example, 100 m/s (Fig. 8c), a localized crushing band is initiated at the loading edge perpendicular to the impact direction and continues to propagate layer by layer to the supporting end. This deformation mode is similar to the propagation of a shock wave [5]. Here we call this deformation mode as the high-velocity deformation mode I (HVDM I). Another high-velocity deformation mode, i.e., HVDM II, is shown in Fig. 8d. At the early stage of the impact, a ‘‘V’’-shaped deformation zone forms from the stricken end. Then it gradually turns into a transverse localization band perpendicular to the impact direction similar to HVDM I and continues to propagate layer by layer to the fixed edge. Based on this convention, the deformation mode of the honeycombs involved in the two groups are summarized in Fig. 9a and b, respectively. It is shown in the figures that the deformation mode of honeycombs depends on both the cell-wall angle and the crushing velocity. Under the low velocity crushing, the honeycombs display the low-velocity deformation mode. With the dynamic enhancement due to the crushing velocity, the high-velocity deformation mode is exhibited in the honeycombs. Similar effects can be achieved by enlarging the cell-wall angle instead of increasing the crushing velocity, which makes the honeycombs’ deformation mode varying from LVDM to HVDM. 3.2.2. X-directional crushing strength 3.2.2.1. Honeycombs with the same density. The energy-absorption capacity of the honeycombs is decided by both the crushing
(a) β=15o
(b) β=30o
(c) β=45
o
Fig. 7. Numerical model in simulations.
strength and the densification strain. The experiments show that the cell-wall angle has little effect on the densification strain, as shown in Fig. 5, which can also be confirmed by the numerical results. Thus our interest is mainly focused on the crushing strength of the honeycombs. When the honeycombs with various cell-wall angles have the same relative density of 0.1, the variety of honeycombs’ crushing strength with the cell-wall angle under the four levels of impact velocities of 10 m/s, 60 m/s, 100 m/s and 150 m/s is listed in Table 3 and demonstrated in Fig. 10. It is shown that the crushing strength of all the honeycombs involved is enhanced with the increases of impact velocity on account of the inertial effect. Similar results were reported in literatures [2,4,5]. Since the regular hexagonal honeycomb with b = 30° is most popular in the industry applications, here it is regarded as the standard shape and the normalized crushing strength of honeycombs rcN is defined as:
rcN ¼
rc rcð30Þ
ð2Þ
where rc is the crushing strength of honeycombs with various cellwall angles, and rc(30) is the crushing strength of the regular hexagonal honeycombs with b = 30°. Fig. 11 exhibits the dependence of the normalized crushing strength of honeycombs on the cell-wall angle.
(d) β=60
Fig. 6. Honeycombs with various cell shapes.
o
o
(e) β=75
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(i) t=2.72ms
(ii) t=6.08ms
(iii) t=10.08ms
(iv) t=12.16ms
(a) Low-velocity deformation mode I (LVDM I)
(i) t=1.885ms
(ii) t=4.64ms
(iii) t=7.975ms
(iv) t=11.89ms
(b) Low-velocity deformation mode II (LVDM II)
(i) t=0.35ms
(ii) t=0.767ms
(iii) t=1.19ms
(c) High-velocity deformation mode I (HVDM I)
(i) t=0.564ms
(ii) t=0.735ms
(iii) t=1.274ms
(iv) t=2.009ms
(d) High-velocity deformation mode II (HVDM II) Fig. 8. Deformation modes of honeycombs.
Both Figs. 10 and 11 indicate that under low crushing velocity such as 10 m/s, when the honeycomb exhibits the low-velocity deformation mode as shown in Fig. 9a, the crushing strength of honeycombs in the x-direction monotonously decreases with the increase of the cell-wall angle although retaining their relative density as a constant. The crushing strength of the honeycomb with the cell-wall angle of 75° is diminished to only 10% of that
of a regular honeycomb. However, this influence of the cell-wall angle is weakened sharply with the increase of impact velocity. It is shown from Figs. 10 and 11 that under the higher-velocity impact, once the cell-wall angle is larger than a certain degree, which is 45° for 60 m/s, 30° for 100 m/s, and 15° for 150 m/s, the change of the honeycombs’ crushing strength is very little. While the cellwall angle is smaller than the threshold degree, the honeycombs’
L. Hu et al. / Materials and Design 46 (2013) 511–523
160
V (m/s)
120
LVDM HVDM II
80
HVDM I
40
Crushing strength (MPa)
518
8 6
150m/s 100m/s 60m/s
4
10m/s
2 0
0
0
15
30
45
60
75
β (°)
90
β (°)
Fig. 10. Dependence of the crushing strength on the cell-wall angle of honeycombs under x-directional crushing (qr = 0.1).
(a) Honeycombs with the same density
Normalized crushing strength
120
V (m/s)
100 LVDM
80
HVDM II
60 HVDM I
40 20 0
0
15
30
45
60
75
90
β (°)
Fig. 9. Deformation mode map in terms of cell-wall angle and impact velocity under x-directional impact.
Table 3 Crushing strength of honeycombs in dynamic simulations (Unit: MPa). b (°)
15
30
45
60
75
Group with qr = 0.1 rcx 10 m/s 60 m/s 100 m/s 150 m/s
0.63 1.55 3.58 6.86
0.52 1.27 3.06 6.69
0.32 1.12 2.94 6.54
0.16 1.05 2.88 6.54
0.05 1.08 3.01 6.97
0.48 1.27 3.16
0.60 1.42 3.24
0.66 1.51 3.27
0.61 1.48 3.12
0.35 1.19 2.96
0.76 1.72 3.66
0.52 1.27 3.06
0.38 1.21 3.19
0.31 1.48 4.02
0.29 2.83 7.74
0.56 1.43 3.36
0.60 1.42 3.24
0.72 1.62 3.60
1.14 2.36 4.62
2.42 5.23 9.43
10 m/s 60 m/s 100 m/s
Group with h = 0.346 mm 10 m/s 60 m/s 100 m/s
rcx
rcy
10 m/s 60 m/s 100 m/s
1.2 1 0.8 0.6
150m/s
0.4
100m/s
0.2
60m/s 10m/s
0
β (°)
(b) Honeycombs with the same cell-wall
rcy
1.4
crushing strength decreases with the increase of the cell-wall angle, however the decreasing extent is much smaller than that in the case of 10 m/s. By referring the deformation mode map of the honeycombs shown in Fig. 9a, it is found that the threshold values of the cell-wall angle just lie on the boundary between HVDM I and the HVDM II. This indicates that the crushing strength of the honeycombs still depends on the geometric configuration of the cells under HVDM II in a law similar to that under LVDM, but the dependence is weakened with the increase of impact velocity. Once HVDM I occurs in the honeycomb, the influence of cell-wall angle on the honeycombs’ crushing strength will no longer be obvious. 3.2.2.2. Honeycombs with the same cell-wall. Figs. 12 and 13 respectively exhibit the crushing strength and the normalized crushing
Fig. 11. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under x-directional crushing (qr = 0.1).
strength against the cell-wall angle for the honeycombs in the group with the same cell-wall according to the values in Table 3. In Fig. 13, the dashed curve denotes the normalized density of honeycombs, which is defined as the ratio of the honeycomb’s density to the density of the regular hexagonal honeycomb. This curve reaches the lowest point at b = 30°, and then increases rapidly with the cell-wall angle. Under a low velocity crushing of 10 m/s, all the honeycombs exhibit the low-velocity deformation mode, as shown in Fig. 9b, and it is shown in both Figs. 12 and 13 that the honeycombs’ crushing strength decreases with the cell-wall angle, although their relative density increases with it when b > 30°, as shown in Table 2, again demonstrating that the geometric configuration of the cells is the dominant factor in dictating the honeycombs’ x-directional crushing strength under LVDM, even exceeding the effect of the relative density, which is identical with the experimental results. On the other hand, on account of the increase of the honeycombs’ relative density, the decreasing extent of the curve of 10 m/s shown in Fig. 13 is smaller than that in the group with the same density as shown in Fig. 11, which displays the contribution of the relative density to the honeycombs’ crushing strength, although it is not the dominant factor under LVDM. The crushing strength of honeycombs in the cases of 60 m/s decreases monotonously when b < 45° as dominated by the cell structure, while it increases rapidly when b > 45° as dominated by the relative density, as shown in Fig. 13. By referring to the deformation mode map shown in Fig. 9b, it is found that b = 45° is just the intermediate point between HVDM I and HVDM II. Under the impact of 100 m/s, the intermediate point between HVDM I and HVDM II is also b = 45° (see Fig. 9b), however the curve of the honeycombs’ crushing strength vs. the cell-wall angle is identical with that vs. the normalized density as shown in Fig. 13. Therefore, under HVDM I the relative density is the dominant factor affecting on the honeycombs’ crushing strength, while the influence of the cell-wall angle is very little. Under LVDM, the
L. Hu et al. / Materials and Design 46 (2013) 511–523
regarded as the mixture of progressive collapse mode and global collapse mode, so it is called as the mixed mode.
Crushing strength (MPa)
10 100m/s
8
60m/s 10m/s
6 4 2 0
β (°)
Normalized crushing strength
Fig. 12. Dependence of the crushing strength on the cell-wall angle of honeycombs under x-directional crushing (h = 0.346 mm, l = 0.4 mm).
3 2.5 2
519
100m/s 60m/s 10m/s Normalized density
1.5 1 0.5
3.3.2. y-Directional crushing strength 3.3.2.1. Honeycombs with the same density. Under y-directional crushing with the impact velocity of 10 m/s, 60 m/s and 100 m/s, respectively, the crushing strength of the honeycombs with the relative density of 0.1 is listed in Table 3 and also shown in Fig. 15. The dependence of the normalized crushing strength defined by Eq. (2) on the cell-wall angle is shown in Fig. 16. It is shown in both Figs. 15 and 16 that the y-directional crushing strength of the honeycombs increases with the enlargement of cell-wall angle until it reaches the maximal value at the points with b = 45°. Then it decreases with the increase of the cell-wall angle. This phenomenon becomes more notable under low-velocity impact. Under the impact with the velocity of 10 m/s, the crushing strength of the honeycomb with cell-wall angle of 75° is only 53% of that of the 45° case, while under high-velocity impact with the velocity of 100 m/s, the variation of the crushing strength with the cell-wall angle is within 10%. The higher crushing strength means that the honeycomb possesses a higher stress in the plateau stage of the stress–strain curve, so it can absorb more energy under impact. Thus, the honeycomb with the cell-wall angle of about 45° exhibits the optimal crushing strength and energy absorption capacity under the y-directional crushing, especially under low-velocity impact.
0
β (°) Fig. 13. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under x-directional crushing (h = 0.346 mm, l = 0.4 mm).
inertia effect is negligible, and the crushing strength of the honeycomb is dominated by the geometric configuration of the cells. Under HVDM II, the honeycombs’ crushing strength is affected by both the cell’s configuration and the honeycombs’ relative density. The higher the impact velocity, the more obvious the effect of the honeycomb’s relative density, and it gradually becomes the dominant factor. On the contrary, the lower the impact velocity, the more obvious the effect of the cell’s geometric configuration. 3.3. y-Directional crushing 3.3.1. Deformation mode With the variety of the impact velocity and the honeycombs’ cell-wall angle, three types of deformation modes are observed in the honeycombs crushed along the y-direction: the global collapse mode, the progressive collapse mode, and the mixed mode. Under the impact with the velocity of 10 m/s, honeycombs with b = 15° and b = 30° exhibit the global collapse mode, as shown in Fig. 14a, where e is the crushing strain of the honeycomb along the impact direction. Cells’ collapse initiates near both the impact and supporting ends, and soon spreads all over the honeycomb. Then all the cells within the honeycomb collapse together although their deformations are inhomogeneous. In the cases of 60 m/s and 100 m/s impact, all the honeycombs with various cell-wall angles, except the honeycombs with b = 75°, show the progressive collapse mode, as shown in Fig. 14b. The collapse of cells starts from the loading edge and propagates row by row to the supporting end while no more crush bands appear inside the honeycomb block, which is similar to the high-velocity deformation mode under xdirectional crushing. In the other cases, although the transverse localization band forms at the loading edge and propagates to the supporting end, meanwhile obvious expansion appears in the cells beyond the localization band, as shown in Fig. 14c, which is
3.3.2.2. Honeycombs with the same cell-wall. The crushing strength and the normalized crushing strength of the honeycombs in the group with the same cell-wall against the cell-wall angle are plotted in Figs. 17 and 18, respectively, which are very different from the cases in the group with the same density, as shown in Figs. 15 and 16. The curves almost reach the lowest points at b = 30° and then increase rapidly with the cell-wall angle under three impact velocities of 10 m/s, 60 m/s and 100 m/s, which is very similar to the variation of the honeycombs’ relative density. Under highvelocity impact such as 100 m/s, the curve of the normalized crushing strength almost coincides with that of the normalized density, as shown in Fig. 18. By comparing Figs. 17 and 18 with Figs. 15 and 16, it is found that for the honeycombs with the same cell-wall, the y-directional crushing strength is dominated by the honeycomb’s density rather than the cell-wall angle under both high-velocity and low-velocity impacts. In fact, the influence of the honeycomb’s density on the crushing strength is more serious under low-velocity crushing, as confirmed by the fact that the relative crushing strength of honeycombs with b > 30° under 10 m/s and 60 m/s crushing is higher than that under 100 m/s crushing as a result of the increase of density, as shown in Fig. 18. 3.4. Comprehensive evaluation 3.4.1. Crushing strength ratio Honeycombs usually exhibit different mechanical properties in the x-direction and in the y-direction. To reveal the orthotropic characteristics of the honeycombs, the crushing strength ratio is defined as the ratio of the crushing strength in the y-direction to that in the x-direction. Fig. 19 shows the dependence of the crushing strength ratio on the cell-wall angle of the honeycombs. It is shown that the crushing strength ratio increases with the cell-wall angle both in the group of the same density and in the group of the same cell-wall. Especially under low-velocity impact, significant orthotropic anisotropy is displayed in the honeycombs with cellwall angle larger than 30°, which is consistent with the finding in the tests. The y-directional crushing strength of the honeycomb with b = 75° reaches even 8 times of that in the x-direction under impact of 10 m/s. However, a smaller crushing strength ratio is
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Crushing strength (MPa)
4 3
100m/s 60m/s 10m/s
2 1 0
β (°) Fig. 15. Dependence of the crushing strength on the cell-wall angle of honeycombs under y-directional crushing (qr = 0.1).
Normalized crushing strength
Fig. 14. Deformation modes of honeycombs under y-directional crushing.
1.2 1 0.8 0.6 0.4 0.2
100m/s 60m/s 10m/s
0
β (°) Fig. 16. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under y-directional crushing (qr = 0.1).
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8 100m/s
8
Crushing strength ratio
Crushing strength (MPa)
10 60m/s 10m/s
6 4 2 0
Y/X-100m/s
6
Y/X-60m/s Y/X-10m/s
4
2
0
0
15
30
45
60
75
90
β (°)
β (°)
(a) Honeycombs with the same density
Fig. 17. Dependence of the crushing strength on the cell-wall angle of honeycombs under y-directional crushing (h = 0.346 mm, l = 0.4 mm).
Crushing strength ratio
Normalized crushing strength
10
4.5 4 3.5 3
100m/s 60m/s 10m/s Normalized density
2.5 2 1.5
Y/X-100m/s
8 Y/X-60m/s Y/X-10m/s
6 4 2
1 0
0.5 0
0
15
30
45
60
75
90
β (°) β (°)
Fig. 18. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under y-directional crushing (h = 0.346 mm, l = 0.4 mm).
displayed with the increase of the impact velocity, although the honeycomb’s crushing strength is enhanced with the impact velocity. Under impact of 100 m/s, the difference of honeycombs’ crushing strength between in the x- and in the y-directions is within 10% for the group of the same density and within 20% for the group of the same cell-wall. It is noted that the crushing strength ratio reaches 1 at the point of the cell-wall angle being about 25° for all the curves in Fig. 19, implying that the honeycomb with b 25° possesses the same crushing strength in the x- and the y-directions. It can be observed more clearly by comparing the curves of the honeycombs’ crushing strength in these two directions, as shown in Fig. 20, in which the curves representing the impacts from the two directions intersect at the points of b 25° under all the three impact velocities both in the group with the same density (Fig. 20a) and in the group with the same cell-wall (Fig. 20b). When the cell-wall angle is smaller than 25°, the honeycomb’s crushing strength in the x-direction is larger than that in the y-direction, while once the cell-wall angle is greater than 25°, the y-directional crushing strength will exceed the x-directional one. 3.4.2. Average crushing strength The honeycomb’s crushing strengths in the x- and y-directions vary with the cell-wall angle in different ways as discussed in the last sections. The average of the crushing strengths in the two directions is compared to give a comprehensive evaluation, as shown in Fig. 21, in which the average crushing strength of honeycombs is normalized by that of the regular honeycomb with b = 30°. It is shown in Fig. 21a that the average crushing strength of the honeycombs in the group of the same density decreases with the increase of cell-wall angle. Under the impact with the velocity of 10 m/s, the average crushing strength of the honeycomb with
(b) Honeycombs with the same cell-wall Fig. 19. Dependence of the crushing strength ratio on the cell-wall angle.
b = 75° is only 36% of that of the regular honeycomb. With the increase of the impact velocity, the honeycombs’ crushing strength is enhanced, while the decreasing tendency of the average crushing strength with the cell-wall angel is weakened due to the inertia effect. The variation of the average crushing strength with the cellwall angle is within 10% under impact velocity of 100 m/s. For the honeycombs with the same cell-wall, as shown in Fig. 21b, the variation of the average crushing strength with the cell-wall angle is slightly affected by the impact velocity compared with that of the group with the same density, as shown in Fig. 21a, although the honeycombs’ crushing strengths in both the x- and the y-directions are notably enhanced by the increase of the impact velocity, as shown in Figs. 12 and 15. It is shown in Fig. 21b that the average crushing strength varies with the cell-wall angle in a way similar to the varying of the honeycomb’s density, i.e., the average crushing strength of honeycombs decreases with the cell-wall angle and takes the lowest value at the points of b = 30°, and then increases with the cell-wall angle. The curve of the average crushing strength under the impact with the velocity of 100 m/s almost coincides with that of the normalized density. Hence, for the honeycombs with the same cell-wall, the average crushing strength is dominated by the honeycombs’ density and is almost not affected by the cell-wall angle within the range of the impact velocity concerned in the present paper.
4. Discussion The relative density has been recognized as one of the dominant factors in dictating the mechanical properties of the cellular material under both the quasi-static [1,2] and the dynamic crushing [5,6,12]. A similar result is obtained in the present paper. The dominant status of the relative density is especially irreplaceable under the high-velocity impact, which is due to the inertia effect of the
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4 3 Y-100m/s X-100m/s
2
Y-60m/s X-60m/s Y-10m/s
1
X-10m/s
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
β (°)
(a) Honeycombs with the same density Crushing strength (MPa)
10 8
Y-100m/s X-100m/s
6
Y-60m/s X-60m/s
4
X-10m/s
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
β (°)
(b) Honeycombs with the same cell-wall Fig. 20. Dependence of the crushing strength on the cell-wall angle of honeycombs crushed along the x-direction and the y-direction, respectively.
Normalized crushing strength
1.2 1 0.8 0.6 0.4 0.2
100m/s 60m/s 10m/s
0
β (°)
(a) Honeycombs with the same density
3 2.5
The mechanical properties of honeycombs are affected by several factors, such as the cells’ configuration, the crushing velocity and the honeycomb’s relative density. The latter two are related to the inertia effect. These influencing factors are discussed in the present paper by studying the crushing behaviors of the hexagonal honeycombs with various cell-wall angles. It is shown that both the honeycombs’ relative density and the geometric configuration of cells have important influence on the mechanical properties of honeycombs. The respective weights of the two factors depend on the crushing velocity and the loading manners. On the whole, the influence of the cells’ configuration is weakened with the increase of the crushing velocity, while that of the honeycomb’s relative density is strengthened. For the hexagonal honeycomb crushed along the y-direction, the crushing strength and energy absorption capacity achieve their respective maximum under both high-velocity and low-velocity impacts when the honeycombs’ cell-wall angle is equal to about 45°. Moreover, a transversely isotropic honeycomb with b 25° is discovered based on the numerical results. All of those throw a light on the optimal design of the cell’s configuration for the honeycombs to achieve superior mechanical properties to meet the needs of industry applications. Acknowledgements The authors would like to thank the support from the National Natural Science Foundation of China under Grant No. 11172335 and No. 10802100. The support from the Fundamental Research Funds for the Central Universities No. 3161263 is also gratefully acknowledged.
3.5
Normalized crushing strength
5. Conclusions
Y-10m/s
2 0
notable under the low-velocity crushing compared with that in the cases of high-velocity impact. Besides the honeycomb’s relative density, the cells’ configuration is also revealed to be a non-negligible factor, which plays even more important roles than that of the relative density under some certain conditions (such as the x-directional compression under the low-velocity impact). Similarly, the significant dependence of the honeycombs’ properties, such as the Young’s modulus and the plastic collapse strength, on the cells’ configuration was also found under the quasi-static compression [1]. Although the regular hexagonal honeycomb is usually regarded to possess the same mechanical properties in the x- and the ydirections [1], experiments indeed revealed that the regular hexagonal honeycombs are orthotropic in the crushing strength and the energy absorption [7,17]. Based on the present numerical results, a transversely isotropic hexagonal honeycomb with b 25° is discovered rather than the regular hexagonal one.
100m/s 60m/s 10m/s Normalized density
2
Reference
1.5 1 0.5 0
β (°)
(b) Honeycombs with the same cell-wall Fig. 21. Dependence of the average of crushing strength on the cell-wall angle.
cell structure [5,6]. Moreover, in some cases, such as the y-directional impact, the influence of the relative density is even more
[1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. 2nd ed. Cambridge University Press; 1997. [2] Chuang CH, Huang JS. Theoretical expressions for describing the stiffness and strength of regular hexagonal honeycombs with Plateau border. Mater Des 2003;24:263–72. [3] Ruan D, Lu G, Wang B, Yu TX. In-plane dynamic crushing of honeycombs – finite element study. Int J Impact Eng 2003;28:161–82. [4] Zou Z, Reid SR, Tan PJ, Li S, Harrigan JJ. Dynamic crushing of honeycombs and features of shock fronts. Int J Impact Eng 2009;36:165–76. [5] Hu LL, Yu TX. Dynamic crushing strength of hexagonal honeycombs. Int J Impact Eng 2010;37:467–74. [6] Tan PJ, Reid SR, Harrigan JJ, Zou Z, Li S. Dynamic compressive strength properties of aluminium foams. Part II – ‘Shock’ theory and comparison with experimental data and numerical models. J Mech Phys Solids 2005;53: 2206–30.
L. Hu et al. / Materials and Design 46 (2013) 511–523 [7] Hu LL, Yu TX, Gao ZY, Huang XQ. The inhomogeneous deformation of polycarbonate circular honeycombs under in-plane compression. Int J Mech Sci 2008;7:1224–36. [8] Chung J, Waas AM. Compressive response of circular cell polycarbonate honeycombs under inplane biaxial static and dynamic loading. Part I: Experiments. Int J Impact Eng 2002;27:729–54. [9] Papka SD, Kyriakides S. Biaxial crushing of honeycombs. Part I: Experiments. Int J Solids Struct 1999;35:4367–96. [10] Nia AA, Sadeghi MZ. The effects of foam filling on compressive response of hexagonal cell aluminum honeycombs under axial loading-experimental study. Mater Des 2010;31:1216–30. [11] Liu Y, Zhang X-C. The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs. Int J Impact Eng 2009;36:98–109. [12] Qiu XM, Zhang J, Yu TX. Collapse of periodic planar lattices under uniaxial compression, Part II: dynamic crushing based on finite element simulation. Int J Impact Eng 2009;36:1231–41.
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[13] Hong ST, Pan J, Tyan T, Prasad P. Dynamic crush behaviors of aluminium honeycomb specimens under compression dominant inclined loads. Int J Plasticity 2008;24:89–117. [14] Khan MK, Baig T, Mirza S. Experimental investigation of in-plane and out-of-plane crushing of aluminum honeycomb. Mater Sci Eng A 2012;539: 135–42. [15] Okumura D, Ohno N, Noguchi H. Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression. Int J Solids Struct 2002;39:3487–503. [16] Yamashita M, Gotoh M. Impact behavior of honeycomb structures with various cell specifications – numerical simulation and experiment. Int J Impact Eng 2005;32:618–30. [17] Chung J, Waas AM. Compressive response of honeycombs under in-plane uniaxial static and dynamic loading. Part I: Experiments. AIAA J 2002;40: 966–73. [18] Honig A, Strong WJ. In-plane dynamic crushing of honeycomb. Part I. Crush band initiation and wave trapping [J]. Int J Mech Sci 2002;44:1665–96.
Contents lists available at SciVerse ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Effect of cell-wall angle on the in-plane crushing behaviour of hexagonal honeycombs Lingling Hu a,⇑, Fanfan You a, Tongxi Yu b a b
Department of Applied Mechanics & Engineering, School of Engineering, Sun Yat-sen University, Guangzhou 510275, PR China Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 28 July 2012 Accepted 27 October 2012 Available online 16 November 2012 Keywords: Honeycomb Crushing strength Cell-wall angle In-plane Impact
a b s t r a c t The dynamic crushing behaviors in the x- and the y-directions of hexagonal honeycombs with various cell-wall angles are explored by both experiments and numerical simulations. Several deformation modes are identified based on the shape of the localization band forming by the cells’ collapse. The respective influence of the cell-wall angle, the crushing velocity and the honeycomb’s relative density on the honeycomb’s mechanical properties is studied. It is shown that these influencing factors affect the honeycomb’s x-directional crushing strength by altering the deformation mode of the honeycomb, while both the y-directional crushing strength and the average crushing strength are dominated by the honeycomb’s density. With retaining the honeycombs’ relative density as a constant, the honeycomb with the cell-wall angle of about 45° exhibits the optimal crushing strength and energy absorption capacity under the y-directional crushing, while the average crushing strength in the x- and the y-directions decreases with the cell-wall angle. Significant orthotropic anisotropy is revealed in the honeycombs with the cell-wall angle greater than 30°, especially under low-velocity compression. The honeycomb with the cell-wall angle of about 25° possesses transversely isotropic mechanical properties. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Cellular material has attracted a great deal of attentions owing to their outstanding properties in efficient energy absorption, high relative stiffness and strength, heat insulation and so on. They have been widely used in lightweight structures and in energy absorption devices, especially under dynamic crushing conditions. One of the notable mechanical characteristics of the cellular material under crushing is that the crushing strain increases without obvious increase in the stress for certain period of time, which is called ‘‘plateau stage’’. The energy absorption of cellular material is dominated by the stress and the length of this stage. The relative density has been recognized as one of the dominant factors in dictating the mechanical properties of cellular material [1,2]. Except the relative density, other factors, such as the crushing velocity, the loading manners, the cell structures and so on, also have important effects on it. Both numerical [3,4] and theoretical analyses [5,6] show that the plateau stress of cellular material under dynamic crushing increases with the impact velocity by following a square law. Honeycombs display much server inhomogeneous deformation under uniaxial compression than that under biaxial compression, which was quantitatively revealed by an ⇑ Corresponding author. Tel.: +86 20 8411 0293. E-mail addresses: [email protected], [email protected] (L. Hu). 0261-3069/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.matdes.2012.10.050
inhomogeneity index defined by Hu et al. [7]. Chung and Waas [8] and Papka and Kyriakides [9] performed biaxial in-plane compression on honeycombs, showing that the crushing strength and the deformation modes of honeycombs were related to the particular ratio between the components of the applied displacements or forces. The effects of the cell size, the cell-wall thickness and the panel thickness on mean crushing strength, energy absorption capacity and folds wavelength of aluminum honeycombs were investigated in Ref. [10]. Liu and Zhang [11] and Qiu et al. [12] studied the influence of the cell micro-topology on the in-plane dynamic crushing of honeycombs by finite element method (FEM). Cell shapes such as hexagonal, rhombus, square, triangular and Kagome were investigated. Honeycombs with regular hexagonal cells are widely used in industry. Extensive experimental [13,14], analytical [5,15] and numerical [3,4] studies have been conducted on their mechanical behaviors. By simply changing the internal cell-wall angle of hexagonal cells, new cell configuration can be obtained, which is easy to implement based on the existing production process. The outof-plane mechanical properties of hexagonal honeycombs with changed cell-wall angles were studied in Ref. [16]. The results show that changing the cell structures by this simple method can enhance the honeycomb’s out-of-plane crushing strength to 1.5 times comparing to the regular hexagonal honeycomb. In some applications, such as using a honeycomb block as an energy
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absorption layer in aircraft against bird or debris collision, the crushing could take place along any direction of the honeycomb. Hence, the effect of changing the cell-wall angle on the in-plane dynamic behaviors of the honeycomb also needs to be known in addition to its out-of-plane behaviors. The regular hexagonal honeycomb is usually regarded to possess the same mechanical properties, such as the Young’s modulus and the plastic collapse strength, in the x- and the y-directions [1]. However, experiments revealed that the regular hexagonal honeycombs are orthotropic in the crushing strength and the energy absorption [7,17]. The orthotropic characteristics of honeycombs are related to the cells’ configuration, and a transversely isotropic behavior is expected by changing the cell-wall angle of the hexagonal honeycomb. Based on the ideas of changing the cell shape by simply changing the cell-wall angle as mentioned above, the objective of the present paper is to examine the effect of the cell shape (or the cell-wall angle) on the in-plane crushing behaviors of hexagonal honeycombs by employing both experiments and numerical method. The crushing strength of honeycombs in both the x- and ydirections is discussed in terms of the impact velocity, the cell-wall angles and the relative density of honeycombs; and it is comprehensively evaluated by defining a crushing strength ratio and the average crushing strength. Finally, a transversely isotropic honeycomb is discovered based on the numerical results. 2. Experiments 2.1. Honeycomb specimens The specimens of the regular hexagonal honeycomb used in this work were cut from a piece of commercial 5052-aluminum honeycomb, as shown in Fig. 1a. The thickness and the length of the cell walls parallel to the y-direction were measured to be h1 = 0.083 mm and l1 = 2.04 mm, respectively; while the sizes for the cell walls unparallel to the y-direction were h2 = 0.046 mm and l2 = 2.53 mm. The cell-wall angle was b = 35.5°, defined as the angle of the honeycomb’s bevel edge to the x-direction, as shown in Fig. 1a. The density of the honeycomb was 72.11 kg/m3. To obtain the honeycomb with a different cell-wall angle from the regular hexagonal one, the cells of a part of the specimens were carefully pre-compressed along the x-direction. Before compression, each cell was filled in by a toothpick at its center, which was perpendicularly inserted into a sponge underneath the specimen, as shown in Fig. 2. The sponge has a similar size to the specimen along the x-direction. Then the specimen and the sponge were carefully compressed together along the x-direction by two rigid planes until the cells being condensed with the filled-in toothpicks. Then, after holding it for a half minute under this condensed status, the force and the toothpicks were removed. Thus the honeycomb with amended cell-configuration was achieved, as shown in Fig. 1b. The cell-wall angle of the pre-compressed honeycomb was measured to be 51.2°, and the density was 85.21 kg/ m3.
specimen is placed on a fixed rigid base at one end and crushed by a steel hammer with an initial impact velocity of 7.8 m/s at the other end. The impact velocity retains almost a constant during the whole crushing process by setting the hammer heavy enough. Sufficient number of cells (more than 11 11 cells) are contained in the honeycomb specimens so that they can represent the global properties of the honeycomb. The deformation process of the specimens was recorded by a high-speed camera, which is demonstrated in Figs. 3 and 4. It is shown that at the early time of the deformation process, the specimen exhibited uniformly transverse expansion with all the cells compressed simultaneously as shown in the second pictures of each figure. Soon after that, the compressed deformation localized at some cells, forming an obvious deformation zone (or the deformation bands) within the specimens, see the third pictures in Figs. 3 and 4. Then with further crushing of the specimen, the cells near the deformation zone collapsed gradually. The initiation of the deformation zone depended on the distribution of the cells’ imperfection. According to the wave trapping theory, once the crushing velocity is higher than a critical value, localization bands would initiate at the loading end or the supporting end of the honeycomb block, despite the initial imperfections distribute among the honeycomb [18]. Based on the theory, the critical velocity is estimated as 6.4– 7.6 m/s for the honeycombs studied in our experiments. It should be noticed that the wave trapping theory comes from the onedimensional wave theory. For the 2-dimensional honeycomb specimens, this critical velocity will become higher resulted by the wave dispersion. The stress–strain curves of the honeycombs are shown in Fig. 5. It indicates that the honeycombs crushed along the y-direction have a higher plateau stage compared with the ones crushed along the x-direction. By averaging the stress in the plateau stage of the stress–strain curve, the crushing strength of the honeycomb, rc, is defined. Table 1 summarized the crushing strength of the honeycombs in the experiments. Both Fig. 5 and Table 1 indicate that the pre-compressed honeycomb exhibits a lower x-directional crushing strength but a higher y-directional crushing strength compared with the regular hexagonal honeycomb. The ratio of the crushing strength between the y- and the x- directions (rcy/rcx) shown in Table 1 denotes that the pre-compressed honeycomb with a larger cell-wall angle displays more significant orthotropic anisotropy than the regular hexagonal one. It is noted that the pre-compressed honeycomb exhibits lower mechanical properties in the x-direction than the regular hexagonal one, although the density of the former is larger than that of the latter, implying that the cell configuration, rather than the relative density of the honeycomb, is the dominant factor in dictating the honeycombs’ x-directional mechanical properties under the conditions of our experiments.
3. Numerical simulation 3.1. Numerical model
2.2. Experimental results Both the regular hexagonal honeycombs and the pre-compressed honeycombs were dynamically crushed along both the xand y-directions, respectively, in the Instron Dynatup 9250 testing system. In our experiments, like most of the ones in this field [6– 9,14,16,17], did not use testing standard. For scientific research on cellular materials including honeycombs, people usually design their own experiments to meet the purpose of verifying theoretical/numerical models. In the dynamic tests, the honeycomb
On account of the limit in the specimens with amended cellwall angle and in the crushing velocity in tests, a finite element model is built using ANSYS–LSDYNA to further understand the mechanical behaviors of hexagonal honeycombs with various cell-wall angles under dynamic crushing. By changing the cell-wall angle, five types of honeycombs are obtained with b being equal to 15°, 30°, 45°, 60° and 75°, respectively, as shown in Fig. 6. As the present work is focused on the influence of the cell-wall angle, the un-equivalence in both the thickness and the length of the cell
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h2
l2
β
h1
l1
y x
(a) Regular hexagonal honeycomb
(b) Pre-compressed honeycomb Fig. 1. Honeycomb specimens in the experiments.
Two groups of honeycombs are studied in the present paper: a group with the same density and a group with the same cell-wall. The relative density is one of the most important factors in affecting the mechanical properties of cellular material. Thus to examine the influence of the cell-wall angle on the honeycomb’s mechanical behaviors, the honeycombs in the first group possess the same rel ative density qr ¼ qq ¼ 0:1; where q⁄ and qs are the density of the s honeycomb and the base material, respectively. Then the cell-wall thickness h of honeycombs with various cell shapes can be calculated according to [1]:
q 3 h : ¼ qs 2 cos bð1 þ sin bÞ l
Fig. 2. A honeycomb with toothsticks inside before pre-compression.
ledges is not considered in the numerical simulations, i.e., all the cells are equilateral hexagon with equal cell-wall thickness. The edge length of the cells l is 4 mm,and the out-of-plane width (along the z-direction) of the honeycomb is b = 0.5 mm.
ð1Þ
In the production process, a more direct method is to change only the honeycomb’s cell-wall angle while remaining the thickness h and the length l of the cell walls unchanged. Thus honeycombs in the second group retain the cell-wall thickness h as a constant of 0.346 mm, and then according to Eq. (1), the relative density of honeycombs with various cell-wall angles is listed in Table 2. In simulations the hexagonal honeycomb block consists of 21 cells in the x-direction and 13 cells in the y-direction, which is placed on a fixed rigid base at one end and crushed along either the x- or the y-direction by a rigid plate with a constant crushing velocity V at the other end, as shown in Fig. 7. The cell-wall material is assumed to be elastic, perfectly plastic with E = 68 GPa, rys ¼ 130 MPa, qs ¼ 2700 kg=m3 and t = 0.3, where E, rys and t are the Young’s modulus, yield stress and Poisson ratio of the base material, respectively. The material of the rigid plates is chosen to be steel with Young’s modulus 210 GPa and density 7800 kg/m3.
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(i)
(ii)
(iii)
(iv)
(v) (a) Regular hexagonal honeycomb
(vi)
(i)
(ii)
(iv)
(iii)
(v) (b) Pre-compressed honeycomb
(vi)
Fig. 3. Deformation process of honeycomb specimens in the x-directional crushing.
Each cell-wall is meshed into 16 shell elements with one layer elements along the out-of-plane direction (the z-direction). The outof-plane displacement of the nodes within the x–y plane (i.e. z = 0) is constrained so as to prevent the specimen from out-ofplane bulking. A single self-contact is defined on the honeycomb model, and a surface-to-surface contact is applied between the honeycomb specimen and the two rigid plates respectively without considering the contact friction.
3.2. X-directional crushing 3.2.1. Deformation mode Honeycombs with various cell-wall angles exhibit distinct deformation modes under x-directional crushing, which are classified into the low-velocity deformation mode (LVDM) and the highvelocity deformation mode (HVDM). In the low-velocity deformation mode (LVDM), the initial localization bands occurred from
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(i)
(ii)
(iv)
(v) (a) Regular hexagonal honeycomb
(iii)
(ii)
(i)
(iv)
(iii)
(v) (b) Pre-compressed honeycomb
(vi)
Fig. 4. Deformation process of honeycomb specimens in the y-directional crushing.
the loading edge produce an ‘‘X’’ shape along the row parallel to the symmetric axis of the honeycomb’s geometry, and then it is soon compressed as an inversed ‘‘V’’-shaped band, as shown in Fig. 8a (LVDMI), where t denotes the crushing time. Or an inversed ‘‘V’’-shaped localization band is directly formed within the honeycomb at the beginning of the deformation, as shown in Fig. 8b (LVDM II). Afterwards, the cells collapse layer by layer along the
inversed ‘‘V’’-shaped band. With the increase of the displacement, a second localized ‘‘V’’-shaped band is developed from the supporting end and ‘‘grows’’ towards the existing one which seems to have stopped developing (Fig. 8a and b). With more layers of cells gradually collapsing along the ‘‘V’’-shaped bands, the two localization bands finally intersect with each other to form a rhombus at the central region of the specimen. After that, localization takes place
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L. Hu et al. / Materials and Design 46 (2013) 511–523 Table 2 The relative densities of honeycombs with the same cell-wall thickness.
0.4 Pre-compressed Y Regular Y Regular X Pre-compressed X
Stress (MPa)
0.3
b (°) Relative density
15 0.1067
30 0.1
45 0.1074
60 0.1391
75 0.255
0.2
Loading plate
0.1
0
0
0.2
0.4
0.6
0.8
1
V
Strain Fig. 5. Stress–strain curves of honeycombs in the dynamic experiments.
Honeycomb block Table 1 Crushing strength of honeycombs in dynamic experiments (unit: kPa).
rcx rcy rcy/rcx
Regular
Pre-compressed
31.40 53.42 1.70
29.09 70.47 2.42
Supporting plate
within the central rhombus until the specimen is completely crushed. Under high crushing velocity, for example, 100 m/s (Fig. 8c), a localized crushing band is initiated at the loading edge perpendicular to the impact direction and continues to propagate layer by layer to the supporting end. This deformation mode is similar to the propagation of a shock wave [5]. Here we call this deformation mode as the high-velocity deformation mode I (HVDM I). Another high-velocity deformation mode, i.e., HVDM II, is shown in Fig. 8d. At the early stage of the impact, a ‘‘V’’-shaped deformation zone forms from the stricken end. Then it gradually turns into a transverse localization band perpendicular to the impact direction similar to HVDM I and continues to propagate layer by layer to the fixed edge. Based on this convention, the deformation mode of the honeycombs involved in the two groups are summarized in Fig. 9a and b, respectively. It is shown in the figures that the deformation mode of honeycombs depends on both the cell-wall angle and the crushing velocity. Under the low velocity crushing, the honeycombs display the low-velocity deformation mode. With the dynamic enhancement due to the crushing velocity, the high-velocity deformation mode is exhibited in the honeycombs. Similar effects can be achieved by enlarging the cell-wall angle instead of increasing the crushing velocity, which makes the honeycombs’ deformation mode varying from LVDM to HVDM. 3.2.2. X-directional crushing strength 3.2.2.1. Honeycombs with the same density. The energy-absorption capacity of the honeycombs is decided by both the crushing
(a) β=15o
(b) β=30o
(c) β=45
o
Fig. 7. Numerical model in simulations.
strength and the densification strain. The experiments show that the cell-wall angle has little effect on the densification strain, as shown in Fig. 5, which can also be confirmed by the numerical results. Thus our interest is mainly focused on the crushing strength of the honeycombs. When the honeycombs with various cell-wall angles have the same relative density of 0.1, the variety of honeycombs’ crushing strength with the cell-wall angle under the four levels of impact velocities of 10 m/s, 60 m/s, 100 m/s and 150 m/s is listed in Table 3 and demonstrated in Fig. 10. It is shown that the crushing strength of all the honeycombs involved is enhanced with the increases of impact velocity on account of the inertial effect. Similar results were reported in literatures [2,4,5]. Since the regular hexagonal honeycomb with b = 30° is most popular in the industry applications, here it is regarded as the standard shape and the normalized crushing strength of honeycombs rcN is defined as:
rcN ¼
rc rcð30Þ
ð2Þ
where rc is the crushing strength of honeycombs with various cellwall angles, and rc(30) is the crushing strength of the regular hexagonal honeycombs with b = 30°. Fig. 11 exhibits the dependence of the normalized crushing strength of honeycombs on the cell-wall angle.
(d) β=60
Fig. 6. Honeycombs with various cell shapes.
o
o
(e) β=75
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L. Hu et al. / Materials and Design 46 (2013) 511–523
(i) t=2.72ms
(ii) t=6.08ms
(iii) t=10.08ms
(iv) t=12.16ms
(a) Low-velocity deformation mode I (LVDM I)
(i) t=1.885ms
(ii) t=4.64ms
(iii) t=7.975ms
(iv) t=11.89ms
(b) Low-velocity deformation mode II (LVDM II)
(i) t=0.35ms
(ii) t=0.767ms
(iii) t=1.19ms
(c) High-velocity deformation mode I (HVDM I)
(i) t=0.564ms
(ii) t=0.735ms
(iii) t=1.274ms
(iv) t=2.009ms
(d) High-velocity deformation mode II (HVDM II) Fig. 8. Deformation modes of honeycombs.
Both Figs. 10 and 11 indicate that under low crushing velocity such as 10 m/s, when the honeycomb exhibits the low-velocity deformation mode as shown in Fig. 9a, the crushing strength of honeycombs in the x-direction monotonously decreases with the increase of the cell-wall angle although retaining their relative density as a constant. The crushing strength of the honeycomb with the cell-wall angle of 75° is diminished to only 10% of that
of a regular honeycomb. However, this influence of the cell-wall angle is weakened sharply with the increase of impact velocity. It is shown from Figs. 10 and 11 that under the higher-velocity impact, once the cell-wall angle is larger than a certain degree, which is 45° for 60 m/s, 30° for 100 m/s, and 15° for 150 m/s, the change of the honeycombs’ crushing strength is very little. While the cellwall angle is smaller than the threshold degree, the honeycombs’
L. Hu et al. / Materials and Design 46 (2013) 511–523
160
V (m/s)
120
LVDM HVDM II
80
HVDM I
40
Crushing strength (MPa)
518
8 6
150m/s 100m/s 60m/s
4
10m/s
2 0
0
0
15
30
45
60
75
β (°)
90
β (°)
Fig. 10. Dependence of the crushing strength on the cell-wall angle of honeycombs under x-directional crushing (qr = 0.1).
(a) Honeycombs with the same density
Normalized crushing strength
120
V (m/s)
100 LVDM
80
HVDM II
60 HVDM I
40 20 0
0
15
30
45
60
75
90
β (°)
Fig. 9. Deformation mode map in terms of cell-wall angle and impact velocity under x-directional impact.
Table 3 Crushing strength of honeycombs in dynamic simulations (Unit: MPa). b (°)
15
30
45
60
75
Group with qr = 0.1 rcx 10 m/s 60 m/s 100 m/s 150 m/s
0.63 1.55 3.58 6.86
0.52 1.27 3.06 6.69
0.32 1.12 2.94 6.54
0.16 1.05 2.88 6.54
0.05 1.08 3.01 6.97
0.48 1.27 3.16
0.60 1.42 3.24
0.66 1.51 3.27
0.61 1.48 3.12
0.35 1.19 2.96
0.76 1.72 3.66
0.52 1.27 3.06
0.38 1.21 3.19
0.31 1.48 4.02
0.29 2.83 7.74
0.56 1.43 3.36
0.60 1.42 3.24
0.72 1.62 3.60
1.14 2.36 4.62
2.42 5.23 9.43
10 m/s 60 m/s 100 m/s
Group with h = 0.346 mm 10 m/s 60 m/s 100 m/s
rcx
rcy
10 m/s 60 m/s 100 m/s
1.2 1 0.8 0.6
150m/s
0.4
100m/s
0.2
60m/s 10m/s
0
β (°)
(b) Honeycombs with the same cell-wall
rcy
1.4
crushing strength decreases with the increase of the cell-wall angle, however the decreasing extent is much smaller than that in the case of 10 m/s. By referring the deformation mode map of the honeycombs shown in Fig. 9a, it is found that the threshold values of the cell-wall angle just lie on the boundary between HVDM I and the HVDM II. This indicates that the crushing strength of the honeycombs still depends on the geometric configuration of the cells under HVDM II in a law similar to that under LVDM, but the dependence is weakened with the increase of impact velocity. Once HVDM I occurs in the honeycomb, the influence of cell-wall angle on the honeycombs’ crushing strength will no longer be obvious. 3.2.2.2. Honeycombs with the same cell-wall. Figs. 12 and 13 respectively exhibit the crushing strength and the normalized crushing
Fig. 11. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under x-directional crushing (qr = 0.1).
strength against the cell-wall angle for the honeycombs in the group with the same cell-wall according to the values in Table 3. In Fig. 13, the dashed curve denotes the normalized density of honeycombs, which is defined as the ratio of the honeycomb’s density to the density of the regular hexagonal honeycomb. This curve reaches the lowest point at b = 30°, and then increases rapidly with the cell-wall angle. Under a low velocity crushing of 10 m/s, all the honeycombs exhibit the low-velocity deformation mode, as shown in Fig. 9b, and it is shown in both Figs. 12 and 13 that the honeycombs’ crushing strength decreases with the cell-wall angle, although their relative density increases with it when b > 30°, as shown in Table 2, again demonstrating that the geometric configuration of the cells is the dominant factor in dictating the honeycombs’ x-directional crushing strength under LVDM, even exceeding the effect of the relative density, which is identical with the experimental results. On the other hand, on account of the increase of the honeycombs’ relative density, the decreasing extent of the curve of 10 m/s shown in Fig. 13 is smaller than that in the group with the same density as shown in Fig. 11, which displays the contribution of the relative density to the honeycombs’ crushing strength, although it is not the dominant factor under LVDM. The crushing strength of honeycombs in the cases of 60 m/s decreases monotonously when b < 45° as dominated by the cell structure, while it increases rapidly when b > 45° as dominated by the relative density, as shown in Fig. 13. By referring to the deformation mode map shown in Fig. 9b, it is found that b = 45° is just the intermediate point between HVDM I and HVDM II. Under the impact of 100 m/s, the intermediate point between HVDM I and HVDM II is also b = 45° (see Fig. 9b), however the curve of the honeycombs’ crushing strength vs. the cell-wall angle is identical with that vs. the normalized density as shown in Fig. 13. Therefore, under HVDM I the relative density is the dominant factor affecting on the honeycombs’ crushing strength, while the influence of the cell-wall angle is very little. Under LVDM, the
L. Hu et al. / Materials and Design 46 (2013) 511–523
regarded as the mixture of progressive collapse mode and global collapse mode, so it is called as the mixed mode.
Crushing strength (MPa)
10 100m/s
8
60m/s 10m/s
6 4 2 0
β (°)
Normalized crushing strength
Fig. 12. Dependence of the crushing strength on the cell-wall angle of honeycombs under x-directional crushing (h = 0.346 mm, l = 0.4 mm).
3 2.5 2
519
100m/s 60m/s 10m/s Normalized density
1.5 1 0.5
3.3.2. y-Directional crushing strength 3.3.2.1. Honeycombs with the same density. Under y-directional crushing with the impact velocity of 10 m/s, 60 m/s and 100 m/s, respectively, the crushing strength of the honeycombs with the relative density of 0.1 is listed in Table 3 and also shown in Fig. 15. The dependence of the normalized crushing strength defined by Eq. (2) on the cell-wall angle is shown in Fig. 16. It is shown in both Figs. 15 and 16 that the y-directional crushing strength of the honeycombs increases with the enlargement of cell-wall angle until it reaches the maximal value at the points with b = 45°. Then it decreases with the increase of the cell-wall angle. This phenomenon becomes more notable under low-velocity impact. Under the impact with the velocity of 10 m/s, the crushing strength of the honeycomb with cell-wall angle of 75° is only 53% of that of the 45° case, while under high-velocity impact with the velocity of 100 m/s, the variation of the crushing strength with the cell-wall angle is within 10%. The higher crushing strength means that the honeycomb possesses a higher stress in the plateau stage of the stress–strain curve, so it can absorb more energy under impact. Thus, the honeycomb with the cell-wall angle of about 45° exhibits the optimal crushing strength and energy absorption capacity under the y-directional crushing, especially under low-velocity impact.
0
β (°) Fig. 13. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under x-directional crushing (h = 0.346 mm, l = 0.4 mm).
inertia effect is negligible, and the crushing strength of the honeycomb is dominated by the geometric configuration of the cells. Under HVDM II, the honeycombs’ crushing strength is affected by both the cell’s configuration and the honeycombs’ relative density. The higher the impact velocity, the more obvious the effect of the honeycomb’s relative density, and it gradually becomes the dominant factor. On the contrary, the lower the impact velocity, the more obvious the effect of the cell’s geometric configuration. 3.3. y-Directional crushing 3.3.1. Deformation mode With the variety of the impact velocity and the honeycombs’ cell-wall angle, three types of deformation modes are observed in the honeycombs crushed along the y-direction: the global collapse mode, the progressive collapse mode, and the mixed mode. Under the impact with the velocity of 10 m/s, honeycombs with b = 15° and b = 30° exhibit the global collapse mode, as shown in Fig. 14a, where e is the crushing strain of the honeycomb along the impact direction. Cells’ collapse initiates near both the impact and supporting ends, and soon spreads all over the honeycomb. Then all the cells within the honeycomb collapse together although their deformations are inhomogeneous. In the cases of 60 m/s and 100 m/s impact, all the honeycombs with various cell-wall angles, except the honeycombs with b = 75°, show the progressive collapse mode, as shown in Fig. 14b. The collapse of cells starts from the loading edge and propagates row by row to the supporting end while no more crush bands appear inside the honeycomb block, which is similar to the high-velocity deformation mode under xdirectional crushing. In the other cases, although the transverse localization band forms at the loading edge and propagates to the supporting end, meanwhile obvious expansion appears in the cells beyond the localization band, as shown in Fig. 14c, which is
3.3.2.2. Honeycombs with the same cell-wall. The crushing strength and the normalized crushing strength of the honeycombs in the group with the same cell-wall against the cell-wall angle are plotted in Figs. 17 and 18, respectively, which are very different from the cases in the group with the same density, as shown in Figs. 15 and 16. The curves almost reach the lowest points at b = 30° and then increase rapidly with the cell-wall angle under three impact velocities of 10 m/s, 60 m/s and 100 m/s, which is very similar to the variation of the honeycombs’ relative density. Under highvelocity impact such as 100 m/s, the curve of the normalized crushing strength almost coincides with that of the normalized density, as shown in Fig. 18. By comparing Figs. 17 and 18 with Figs. 15 and 16, it is found that for the honeycombs with the same cell-wall, the y-directional crushing strength is dominated by the honeycomb’s density rather than the cell-wall angle under both high-velocity and low-velocity impacts. In fact, the influence of the honeycomb’s density on the crushing strength is more serious under low-velocity crushing, as confirmed by the fact that the relative crushing strength of honeycombs with b > 30° under 10 m/s and 60 m/s crushing is higher than that under 100 m/s crushing as a result of the increase of density, as shown in Fig. 18. 3.4. Comprehensive evaluation 3.4.1. Crushing strength ratio Honeycombs usually exhibit different mechanical properties in the x-direction and in the y-direction. To reveal the orthotropic characteristics of the honeycombs, the crushing strength ratio is defined as the ratio of the crushing strength in the y-direction to that in the x-direction. Fig. 19 shows the dependence of the crushing strength ratio on the cell-wall angle of the honeycombs. It is shown that the crushing strength ratio increases with the cell-wall angle both in the group of the same density and in the group of the same cell-wall. Especially under low-velocity impact, significant orthotropic anisotropy is displayed in the honeycombs with cellwall angle larger than 30°, which is consistent with the finding in the tests. The y-directional crushing strength of the honeycomb with b = 75° reaches even 8 times of that in the x-direction under impact of 10 m/s. However, a smaller crushing strength ratio is
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Crushing strength (MPa)
4 3
100m/s 60m/s 10m/s
2 1 0
β (°) Fig. 15. Dependence of the crushing strength on the cell-wall angle of honeycombs under y-directional crushing (qr = 0.1).
Normalized crushing strength
Fig. 14. Deformation modes of honeycombs under y-directional crushing.
1.2 1 0.8 0.6 0.4 0.2
100m/s 60m/s 10m/s
0
β (°) Fig. 16. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under y-directional crushing (qr = 0.1).
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8 100m/s
8
Crushing strength ratio
Crushing strength (MPa)
10 60m/s 10m/s
6 4 2 0
Y/X-100m/s
6
Y/X-60m/s Y/X-10m/s
4
2
0
0
15
30
45
60
75
90
β (°)
β (°)
(a) Honeycombs with the same density
Fig. 17. Dependence of the crushing strength on the cell-wall angle of honeycombs under y-directional crushing (h = 0.346 mm, l = 0.4 mm).
Crushing strength ratio
Normalized crushing strength
10
4.5 4 3.5 3
100m/s 60m/s 10m/s Normalized density
2.5 2 1.5
Y/X-100m/s
8 Y/X-60m/s Y/X-10m/s
6 4 2
1 0
0.5 0
0
15
30
45
60
75
90
β (°) β (°)
Fig. 18. Dependence of the normalized crushing strength on the cell-wall angle of honeycombs under y-directional crushing (h = 0.346 mm, l = 0.4 mm).
displayed with the increase of the impact velocity, although the honeycomb’s crushing strength is enhanced with the impact velocity. Under impact of 100 m/s, the difference of honeycombs’ crushing strength between in the x- and in the y-directions is within 10% for the group of the same density and within 20% for the group of the same cell-wall. It is noted that the crushing strength ratio reaches 1 at the point of the cell-wall angle being about 25° for all the curves in Fig. 19, implying that the honeycomb with b 25° possesses the same crushing strength in the x- and the y-directions. It can be observed more clearly by comparing the curves of the honeycombs’ crushing strength in these two directions, as shown in Fig. 20, in which the curves representing the impacts from the two directions intersect at the points of b 25° under all the three impact velocities both in the group with the same density (Fig. 20a) and in the group with the same cell-wall (Fig. 20b). When the cell-wall angle is smaller than 25°, the honeycomb’s crushing strength in the x-direction is larger than that in the y-direction, while once the cell-wall angle is greater than 25°, the y-directional crushing strength will exceed the x-directional one. 3.4.2. Average crushing strength The honeycomb’s crushing strengths in the x- and y-directions vary with the cell-wall angle in different ways as discussed in the last sections. The average of the crushing strengths in the two directions is compared to give a comprehensive evaluation, as shown in Fig. 21, in which the average crushing strength of honeycombs is normalized by that of the regular honeycomb with b = 30°. It is shown in Fig. 21a that the average crushing strength of the honeycombs in the group of the same density decreases with the increase of cell-wall angle. Under the impact with the velocity of 10 m/s, the average crushing strength of the honeycomb with
(b) Honeycombs with the same cell-wall Fig. 19. Dependence of the crushing strength ratio on the cell-wall angle.
b = 75° is only 36% of that of the regular honeycomb. With the increase of the impact velocity, the honeycombs’ crushing strength is enhanced, while the decreasing tendency of the average crushing strength with the cell-wall angel is weakened due to the inertia effect. The variation of the average crushing strength with the cellwall angle is within 10% under impact velocity of 100 m/s. For the honeycombs with the same cell-wall, as shown in Fig. 21b, the variation of the average crushing strength with the cell-wall angle is slightly affected by the impact velocity compared with that of the group with the same density, as shown in Fig. 21a, although the honeycombs’ crushing strengths in both the x- and the y-directions are notably enhanced by the increase of the impact velocity, as shown in Figs. 12 and 15. It is shown in Fig. 21b that the average crushing strength varies with the cell-wall angle in a way similar to the varying of the honeycomb’s density, i.e., the average crushing strength of honeycombs decreases with the cell-wall angle and takes the lowest value at the points of b = 30°, and then increases with the cell-wall angle. The curve of the average crushing strength under the impact with the velocity of 100 m/s almost coincides with that of the normalized density. Hence, for the honeycombs with the same cell-wall, the average crushing strength is dominated by the honeycombs’ density and is almost not affected by the cell-wall angle within the range of the impact velocity concerned in the present paper.
4. Discussion The relative density has been recognized as one of the dominant factors in dictating the mechanical properties of the cellular material under both the quasi-static [1,2] and the dynamic crushing [5,6,12]. A similar result is obtained in the present paper. The dominant status of the relative density is especially irreplaceable under the high-velocity impact, which is due to the inertia effect of the
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Crushing strength (MPa)
4 3 Y-100m/s X-100m/s
2
Y-60m/s X-60m/s Y-10m/s
1
X-10m/s
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
β (°)
(a) Honeycombs with the same density Crushing strength (MPa)
10 8
Y-100m/s X-100m/s
6
Y-60m/s X-60m/s
4
X-10m/s
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
β (°)
(b) Honeycombs with the same cell-wall Fig. 20. Dependence of the crushing strength on the cell-wall angle of honeycombs crushed along the x-direction and the y-direction, respectively.
Normalized crushing strength
1.2 1 0.8 0.6 0.4 0.2
100m/s 60m/s 10m/s
0
β (°)
(a) Honeycombs with the same density
3 2.5
The mechanical properties of honeycombs are affected by several factors, such as the cells’ configuration, the crushing velocity and the honeycomb’s relative density. The latter two are related to the inertia effect. These influencing factors are discussed in the present paper by studying the crushing behaviors of the hexagonal honeycombs with various cell-wall angles. It is shown that both the honeycombs’ relative density and the geometric configuration of cells have important influence on the mechanical properties of honeycombs. The respective weights of the two factors depend on the crushing velocity and the loading manners. On the whole, the influence of the cells’ configuration is weakened with the increase of the crushing velocity, while that of the honeycomb’s relative density is strengthened. For the hexagonal honeycomb crushed along the y-direction, the crushing strength and energy absorption capacity achieve their respective maximum under both high-velocity and low-velocity impacts when the honeycombs’ cell-wall angle is equal to about 45°. Moreover, a transversely isotropic honeycomb with b 25° is discovered based on the numerical results. All of those throw a light on the optimal design of the cell’s configuration for the honeycombs to achieve superior mechanical properties to meet the needs of industry applications. Acknowledgements The authors would like to thank the support from the National Natural Science Foundation of China under Grant No. 11172335 and No. 10802100. The support from the Fundamental Research Funds for the Central Universities No. 3161263 is also gratefully acknowledged.
3.5
Normalized crushing strength
5. Conclusions
Y-10m/s
2 0
notable under the low-velocity crushing compared with that in the cases of high-velocity impact. Besides the honeycomb’s relative density, the cells’ configuration is also revealed to be a non-negligible factor, which plays even more important roles than that of the relative density under some certain conditions (such as the x-directional compression under the low-velocity impact). Similarly, the significant dependence of the honeycombs’ properties, such as the Young’s modulus and the plastic collapse strength, on the cells’ configuration was also found under the quasi-static compression [1]. Although the regular hexagonal honeycomb is usually regarded to possess the same mechanical properties in the x- and the ydirections [1], experiments indeed revealed that the regular hexagonal honeycombs are orthotropic in the crushing strength and the energy absorption [7,17]. Based on the present numerical results, a transversely isotropic hexagonal honeycomb with b 25° is discovered rather than the regular hexagonal one.
100m/s 60m/s 10m/s Normalized density
2
Reference
1.5 1 0.5 0
β (°)
(b) Honeycombs with the same cell-wall Fig. 21. Dependence of the average of crushing strength on the cell-wall angle.
cell structure [5,6]. Moreover, in some cases, such as the y-directional impact, the influence of the relative density is even more
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