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Strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs: Experiment and theory
Composite Structures 132 (2015) 644–651
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs: Experiment and theory Yong Tao a, Mingji Chen b,⇑, Haosen Chen c,d,⇑, Yongmao Pei a, Daining Fang a a
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China National Center for Nanoscience and Technology, Beijing 100190, China c Beijing Institute of Technology, Beijing 100081, China d Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China b
a r t i c l e
i n f o
Article history: Available online 9 June 2015 Keywords: Metallic honeycomb Dynamic compression Strain rate effect Inertia effect Plateau stress
a b s t r a c t Many studies reveal that the dynamic compressive strength of metallic honeycombs is higher than the quasi-static one, but the reasons for that are still debatable. This paper aims to study the strain rate effect of parent materials on the out-of-plane dynamic compressive behavior of metallic honeycombs. Quasi-static and dynamic tests on aluminum honeycombs were performed with universal testing machine and Split Hopkinson Pressure Bar, respectively. The velocity values of dynamic tests were from about 6 to 19 m/s. The present and existing measures of plateau stress are evaluated by both the rate-independent (R-I) and rate-dependent (R-D) shock theories. It is shown that the R-D shock theory proposed in our previous study provides more accurate predictions at low, medium and high impact velocities. Based on the R-D shock theory, the influences of strain rate effect are analyzed quantitatively and the change tendencies of measured plateau stresses with impact velocities are explained reasonably. The analysis indicates that the strain rate effect has a large contribution to the dynamic enhancement of metallic honeycombs in a wide velocity range. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Metallic honeycombs are widely used in the packaging, transport and aircraft industries due to their lightweight, high specific stiffness and strength and outstanding energy absorption performance. In these engineering applications, dynamic loadings are often encountered. Under dynamic loading, the response of honeycombs is significantly different from that under static loading [1– 13], and honeycomb structures designed with static properties are usually too conservative and wasteful. Much effort has been made on the in-plane dynamic compressive properties of honeycombs [14–19]. However, honeycombs are generally used as core materials of sandwich plates [20], and the out-of-plane stiffness and strength are much higher than those with respect to the in-plane directions [21]. Therefore, it is necessary and important to study the out-of-plane dynamic compressive behavior of metallic honeycombs.
⇑ Corresponding authors at: Tel./fax: +86 010 82545661 (M. Chen). Tel./fax: +86 010 62753795 (H. Chen). E-mail addresses: [email protected] (M. Chen), [email protected]. edu.cn (H. Chen). http://dx.doi.org/10.1016/j.compstruct.2015.06.015 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
As for the out-of-plane compressive behavior of metallic honeycombs, McFarland [22] provided a semi-empirical expression to predict the plateau stress of hexagonal honeycombs. Since that pioneering study, Wierzbicki [23] developed a more reasonable model as the correct cell wall deformation patterns were identified. Zhang and Ashby [24] proposed a formula for the plateau stress of hexagonal honeycombs with uniform cell wall thickness. Systematic and detailed studies of honeycombs were published in the book of Gibson and Ashby [21]. In the numerical and experimental fields, Aktay et al. [25] simulated the compressive behavior of honeycomb with a meso-structure model and a homogenized model and compared the numerical results with the experimental data. Wilbert et al. [26] presented a comprehensive study of the compressive response of hexagonal honeycombs through experiments and simulations. More recently, Zhang et al. [27] performed simulations and experiments to evaluate the influences of cell number and central angle on the out-of-plane compressive resistance of aluminum honeycombs. Beside the quasi-static investigations mentioned above, the out-of-plane dynamic compression of honeycombs was also frequently studied, and the compressive strength of honeycombs under dynamic loading has been shown to be higher than that under quasi-static loading [1–13]. The inertia effect has been
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regarded as the main reason for the dynamic enhancement, and a rate-independent (R-I) shock theory considering the inertia effect was proposed to predict the dynamic compressive plateau stress [6,28]. This model has been widely used to evaluate the dynamic enhancement of cellular materials [17,15,29–34]. However, such a model do not account for the strain rate effect of parent materials from which cellular materials are made. More dynamic finite element studies can be found in literature, but most of them adopted a rate independent elastic-perfectly plastic or bilinear constitutive law [8,11,35–37]. Recently, Tao et al. [38] studied the strain rate effect on the behavior of metallic honeycombs through numerical analysis and a rate-dependent (R-D) shock theory was proposed. As the strain rate effect of parent materials is considered to be small and often neglected in the current studies, its influences on the dynamic enhancement are still unclear and the quantitative analysis is not available. Besides, it is observed that the change tendencies of measured plateau stresses are different in the low [12], medium [3] and high [6] velocity ranges, and that also lacks a convincing theoretical explanation. Therefore, more studies need to be done to further understand the out-of-plane dynamic behavior of metallic honeycombs. In this paper, the strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs is studied through experimental and theoretical investigations. Firstly, quasi-static and dynamic tests on aluminum honeycombs were performed by using the universal testing machine and Split Hopkinson Pressure Bar (SHPB) system, respectively. Secondly, the experimental results are presented and analyzed, and the present and existing measured plateau stresses are evaluated by the R-I and R-D shock theories at low, medium and high impact velocities. Finally, based on the R-D shock theory proposed in our previous study [38], the influences of inertia and strain rate effects are analyzed, and the change tendencies of measured plateau stresses with impact velocities are also explained. 2. Experimental procedures of quasi-static and dynamic tests 2.1. Specimen specification Typical honeycomb specimens contain 19 complete cells used in this study are shown in Fig. 1. The honeycomb specimens were made of Al3003-H18 with density of 2730 kg/m3 and elastic modulus of 68.9 GPa. The honeycomb is regarded as an orthotropic material, and the orthotropic directions can be denoted as the L, W, and T directions [39], as shown in Fig. 2(a). The T direction, also called the out-of-plane direction, is paralleled with the axis of the cell wall. The other two directions are referred to in-plane (L and W) directions. The L direction is called the ribbon direction, and the cell wall thickness in the L direction is double that in the width or W direction, which is transverse to the L direction [40].
Fig. 1. Typical hexagonal aluminum honeycomb specimens used in this study.
Specimens employed in tests were designed as cylindrical columns with diameter of 20 mm and height of 14 mm. The cell wall thickness t is 0.06 mm and the cell wall length l is 2 mm. All specimens were carefully cut from a big square honeycomb panel using wire electric discharge machining (WEDM). It should be noted that honeycomb specimens with accurate dimensions and little damage can be obtained by taking the advantages of a WEDM method [41,42]. Detailed geometrical configuration of a honeycomb specimen is shown in Fig. 2. 2.2. Experimental arrangement of quasi-static tests The quasi-static tests were carried out with an electronic universal testing machine (Fig. 3(a)) at a loading velocity of 0.5 mm/min based on both the Chinese test method [43] and the ASTM standards [44–46]. In this machine, the lower platen was stationary, whereas the upper platen was movable towards the lower one. The plane surface of the upper platen was smooth enough to avoid the influences of friction, but the surface of lower platen had circle-shaped notches. It must be noted that the platen-specimen interface friction should be avoided for compression tests [45,46]. Therefore, an iron cylinder with polished upper surface was put on the lower platen. Then the honeycomb specimen was placed on the polished surface in order to minimize friction. Three specimens were tested for this case and mean value of the data was used as the result. 2.3. Experimental arrangement of dynamic tests The dynamic tests were performed on a SHPB system, which was often used as an experimental setup to study the dynamic behaviors of materials [9–11,31]. The SHPB setup used in this study is illustrated in Fig. 3(b), and its schematic diagram is shown in Fig. 4. It can be seen that a typical SHPB setup is composed of input and output bars with a honeycomb specimen sandwiched between them. A striker bar launched by a gas gun impacts the free end of the input bar and generates a compressive longitudinal incident wave ei(t). When the incident wave arrives at the input bar-specimen interface, part of it is reflected and develops a reflected wave er(t) in the input bar, while the rest passes through the specimen and develops the transmitted wave et(t) in the output bar. Based on the strain measured by two gauges, the forces and velocities can be calculated as follows [9–11]:
v input ¼ C 0 ðei ðtÞ er ðtÞÞ v output ¼ C 0 et ðtÞ
F input ¼ Sbar Eðei ðtÞ þ er ðtÞÞ F output ¼ Sbar Eet ðtÞ
ð1Þ
where vinput, vouput, Finput, Foutput are velocities and forces at the bars-specimen interfaces, Sbar, E and C0 are the cross sectional area, elastic modulus and the longitudinal wave velocity of the bars, respectively, ei(t), er(t), et(t) are the strain signals at the bars-specimen interfaces. A range of impact velocities was covered from 6 to 19 m/s by varying the air pressure in the gas gun. It should be noticed that the density of aluminum honeycomb is small, so its impedance is low and the induced strain in an ordinary pressure bar (like steel bar) is too small to measure accurately [9]. In order to get an accurate measurement, the low impedance PMMA bars with diameter of 30 mm were used to match the impedance of aluminum honeycomb specimen. The striker bar had a length of 290 mm, and both the input and output bars were 2000 mm long. The surfaces at the bars-specimen interfaces were lubricated to reduce the influence of friction. Three and five specimens were tested for dynamic tests with velocities of 19.01 and 8.15 m/s, respectively. Six tests were performed for the velocities of 9.68 and 14.62 m/s, respectively. For dynamic tests with velocities of 6.04, 11.16 and 13.52 m/s, four specimens were tested for each case.
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Fig. 2. Geometric dimensions of a hexagonal honeycomb specimen: (a) sketch, (b) circle cross section and (c) unit cell.
Fig. 3. Experimental systems: (a) electronic universal testing machine for quasi-static tests and (b) SHPB setup for dynamic tests.
Fig. 4. Schematic diagram of SHPB setup.
2.4. Parameters definition and calculation for the compression of honeycombs The typical stress–strain curve of honeycomb under out-of-plane compression displays three regions, that is, the initial linear elastic region followed by the plateau region, and finally the rapidly rising densification region [21], as shown in Fig. 5. Nominal stress and strain are defined as:
F A
r¼ ; e¼
d H
ð2Þ
where F is the compressive force, A is the initial cross-sectional area of honeycomb, d is the compression displacement and H is the initial height of the honeycomb structure. Here, the average value of the stresses between the yield strain and densification strain is used as the plateau stress, which can be calculated by the following relation:
Fig. 5. Typical stress–strain curve of honeycomb under out-of-plane compression.
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rpl ¼
1 eD ey
Z
eD
rðeÞde
ð3Þ
ey
where ey is the yield strain, which is the strain corresponding to the initial peak stress rpk, and eD is the densification strain, which can be defined through [29]:
d de
1 rðeÞ
Z
e 0
rðeÞde
¼0
ð4Þ
e¼ eD
It should be noted that the results of the aforementioned integral formulas are equal to the area under the nominal stress–strain curve, which can be calculated by using the function ‘‘Integrate’’ in Origin software. 3. Experimental results and analysis 3.1. Repeatability of the experimental stress–strain curves Figs. 6(a) and 6(b) show the stress–strain curves for three independent quasi-static and dynamic tests on the aluminum honeycombs, respectively. Although the measured data of peak stress has slight dispersity, the stresses in the plateau region are of good repeatability, which means the quasi-static and dynamic test results are reliable. It should be noticed that the compression reached in dynamic test is quite limited and the densification region is absent due to the limited length of impulse in the SHPB
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system [9,11]. Therefore, for dynamic tests, the average value of the stress between the yield strain and the maximum strain is used as the plateau stress. 3.2. Effect of impact velocity The mean values of both the quasi-static and dynamic experimental results are listed in Table 1. Rpl denotes the plateau stress ratio, which is the ratio between dynamic and quasi-static plateau stresses. As shown in Table 1, when compared with the value under quasi-static loading, the plateau stresses are raised by about 38–57% in a velocity range from about 6–19 m/s, indicating that the impact velocity has a great influence on the out-of-plane plateau stress. The present experimental results of dynamic compressive plateau stress enhancement for honeycombs are quite similar with previous researches. For example, Goldsmith and Sackman [1] observed an increase of about 20–50% in their dynamic tests by using a rounded projectile with impact velocities up to 35 m/s. Wu and Jiang [3] performed dynamic tests at velocities up to about 28 m/s and found that the dynamic plateau stress was about 74% higher than the quasi-static value. Baker et al. [4], Xu et al. [5], Harrigan et al. [6], Yamashita and Gotoh [8], Zhao and Gray [9], Zhao et al. [10], Hou et al. [11], Alavi Nia and Sadeghi [12], and Wang et al. [13] have also found a certain dynamic plateau stress enhancement for metallic honeycombs in their experiments. 3.3. Deformation patterns under out-of-plane quasi-static and dynamic loadings Some pictures were also taken during quasi-static loading, as shown in Fig. 7(a). For out-of-plane quasi-static compression, the deformation patterns of aluminum honeycomb specimens were characterized by localized and progressive buckling. The progressive buckling was started from the layers near the interface between the lower platen and the honeycomb specimen, and then induced the deformation of adjacent layers to form folds. In fact, the onset of this progressive buckling can occur anywhere in the specimen where the weakest zone located [6,47]. For dynamic compression, a typical progressive buckling deformation process was also observed, as shown in Fig. 7(b). In dynamic compression, most of the aluminum honeycomb specimens began to buckle from the layers adjacent to the input or output bar-specimen interface, then the plastic buckling progressed in nearby layers. After the plastic buckling progressed in nearby layers at a certain stage, the honeycomb layers near the other end of the honeycomb specimen started to buckle. Then the plastic buckling of both ends of the honeycomb continued and finished until the densification was achieved. The deformation patterns were similar with the quasi-static case, but the start position of buckling may be different under dynamic compressive loading. Figs. 7(c) and 7(d) show the final deformation patterns of honeycomb specimens under quasi-static and dynamic compression, respectively. The final deformation patterns under dynamic compression were also similar to that under quasi-static compression. In both cases, clear folds of cell wall caused by plastic buckling could be observed. The present results agree with the observations reported by Xu et al. [5] and Harrigan et al. [6] in the low and high velocity ranges, respectively. 4. Comparison and discussion 4.1. A brief overview of the R-I and R-D shock theories
Fig. 6. Repeatability of the stress–strain curves under (a) quasi-static (v = 0.5 mm/ min) and (b) dynamic (v = 19 m/s) loadings.
The R-I shock theory was first proposed by Reid and Peng [28] in order to explain the dynamic enhancement of wood specimens
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Table 1 The measurement results from Alavi Nia et al. [12], present experiments and Harrigan et al. [6], respectively. t = 0.0508 mm, l = 1.833 mm
v (m/s)
4.16 10 4.300 1.000
rpl (MPa) Rpl
5
0.1 5.000 1.163
t = 0.0508 mm, l = 2.748 mm 0.2 5.200 1.209
4.16 105 1.860 1.000
0.1 2.500 1.344
0.2 2.600 1.398
8.15 4.171 1.419
9.68 4.168 1.418
11.16 4.610 1.568
13.52 4.540 1.544
14.62 4.386 1.492
19.01 4.351 1.480
35.76 1.002 1.891
59.67 1.229 2.319
82.80 1.905 3.594
91.74 1.906 3.596
96.99 2.218 4.185
132.46 2.097 3.957
t = 0.06 mm, l = 2.00 mm
v (m/s)
8.33 106 2.940 1.000
rpl (MPa) Rpl
6.04 4.047 1.377
t = 0.049 mm, l = 3.67 mm
v (m/s)
1.67 105 0.530 1.000
rpl (MPa) Rpl
32.85 0.961 1.813
Fig. 7. Typical deformation patterns of aluminum honeycomb specimens subjected to out-of-plane compression: (a) quasi-static and (b) dynamic deformation processes and the final deformation patterns under (c) quasi-static and (d) dynamic loadings.
under dynamic compression. Afterwards, it was used to predict the dynamic strength of hexagonal honeycombs [6]. Since then, the model has been widely employed to study the dynamic enhancement of cellular materials [17,15,29–34]. The dynamic plateau stress of honeycomb predicted by the R-I shock theory can be expressed in the following form [6,28]:
rdpl ¼ rspl þ
qv eD
2
ð5Þ
respectively. q represents the density of honeycomb and v is the impact velocity. The quasi-static plateau stress in Eq. (5) can be calculated as follows [23]:
53 t l
rdpl ¼ rspl 1 þ
where rspl and rdpl are the quasi-static and dynamic plateau stresses,
rspl ¼ 6:63rsy
In order to incorporate the strain rate effect of parent materials into the theoretical model, the R-D shock theory was proposed by introducing the Cowper–Symonds relation into the R-I shock theory [38]. According to the R-D shock theory, the formula of dynamic compressive plateau stress can be written as [38]:
ð6Þ
where rsy denotes the yield stress of parent material. As can be seen in Eq. (5), the R-I shock theory has taken into account the inertia effect, but the strain rate effect of parent materials which has been regarded as a possible cause [3,11,48] for dynamic enhancement of metallic honeycombs was neglected.
v
1P
2:24lD
þ
q v 2 eD
ð7Þ
where D and P are the strain rate material constants, e.g., for Al5050 and Al5052, D = 6500 s1 and P = 4 [49], and for Al3003, D = 2.7 105 s1 and P = 8 [50]. 4.2. Comparison between theory and experiment The dynamic experiments in the present study were performed with impact velocities varied from about 6 to 19 m/s, with associated nominal strain rates from about 429 to 1357 s1. Theoretical predictions given by the R-I and R-D shock theories have also been compared with the present experimental data, as shown in Fig. 8(a). It can be seen that the R-D shock theory follows the experimental data well, whereas the R-I shock theory gives much lower predictions. Both the measured plateau stresses and
Y. Tao et al. / Composite Structures 132 (2015) 644–651
predictions of the R-D shock theory are almost proportional to the impact velocity in the studied velocity range. Wu and Jiang [3] also fitted their experimental results into linear relation in a similar velocity range. The present experimental results are in the medium velocity range. Fig. 8(b) also provides some low and high impact velocity data obtained from existing experiments in order to make a comparison between the theoretical and experimental results in a wide velocity range. By using an Instron 8503 apparatus, Alavi Nia and Sadeghi [12] performed experimental investigation on the effect of impact velocity on the out-of-plane compressive behavior of bare and foam-filled Al5052-H39 honeycombs in a low velocity range. Fig. 8(b) shows their experimental data at impact velocities from 4.16 105 to 0.2 m/s, together with the plateau stresses predicted by the R-I and R-D shock theories. The aluminum honeycomb specimens are square panels with thickness of 19.05 mm, which means the associated nominal strain rates vary from 2.2 103 to 10.5 s1. Their geometrical parameters and experimental results are listed in Table 1. As shown in Fig. 8(b), there is a reasonable agreement between the results of R-D shock theory and experiment, whereas the R-I shock theory underestimates the measured plateau stresses. The experimental results show that an increase in plateau stress is much more pronounced at low impact velocity. For example, increasing the impact velocity from quasi-static (4.16 105 m/s) to 0.1 m/s increases the plateau stress of 16.3% for honeycombs
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with cell wall length of 1.833 mm. However, changing the impact velocity from 0.1 to 0.2 m/s increases the plateau stress only by 4.6%. The two values for honeycombs with cell wall length of 2.748 mm are 34.4% and 5.4%, respectively. The R-D shock theory can successfully predict this change tendency while the R-I shock theory fails to do so. An investigation concerning dynamic out-of-plane compression of Al5050 honeycombs in the high velocity range was reported by Harrigan et al. [6]. Their aluminum honeycomb specimens were cylindrical columns with diameter of 45 mm and height of 30 mm, and the high velocity impact tests were performed with an impact testing rig developed by the authors themselves. The experimental results for backed, initially uncrushed specimens are used to compare with theoretical predictions, as shown in Fig. 8(b). Corresponding geometrical parameters and experimental results are listed in Table 1. It can be seen that the dynamic plateau stresses are related to the impact velocities by conic curve at high impact velocity. The predictions of both the R-I and R-D shock theories show similar tendency with the experimental results, but the predicted values of R-I shock theory are still smaller than the experimental data. Fig. 9 shows a comparison between theoretical predictions and experimental results at low, medium and high impact velocities. Experimental data from Wu and Jiang [3], Xu et al. [5], and Wang et al. [13] are also included in this figure. In Fig. 9, the solid lines are predicted by the R-D shock theory and the dash lines represent the predictions of the R-I shock theory. Lines and symbols in the same color are plotted with corresponding geometrical and material parameters in a certain literature. As the geometrical parameters and materials are different in different literatures, the corresponding theoretical curves are different and cannot collapse into a single curve. It can be seen that the plateau stresses predicted by the R-D shock theory correspond better to the experiment results over the whole range of impact velocities studied, whereas the R-I shock theory always underestimates the measured plateau stresses. Moreover, the change tendencies between the plateau stress and the impact velocity are distinct in different velocity ranges. The reason for that will be discussed in Section 4.3. 4.3. Theoretical analysis of experimental results To study the influences of the strain rate effect, the predicted values of R-D shock theory with different strain rate parameters under different impact velocities is plotted in Fig. 10. For
Fig. 8. Comparison between theoretical predictions and experimental results in the (a) medium and (b) low and high velocity ranges.
Fig. 9. Comparison between theoretical predictions and experimental results over a wide velocity range.
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comparison purpose, the quasi-static plateau stress [23] and the predictions of R-I shock theory [6,28] are also plotted. It can be seen from Fig. 10 that the dynamic plateau stress predicted by R-I shock theory increases with the square of impact velocity. While the plateau stress predicted by the R-D shock theory exhibits three regions, namely, the initial rapidly increasing region at low impact velocity followed by the approximately linear region at medium impact velocity, and finally the conic curve region at high impact velocity. In addition, the dynamic plateau stress of honeycomb made of a relatively stronger strain rate dependent (D = 6500 s1, P = 4) parent material is always larger than honeycomb made of a relatively weaker strain rate dependent (D = 270,000 s1, P = 8) parent material. In order to further study the influences of the strain rate effect on the dynamic enhancement of honeycombs, the R-D shock theory as expressed in Eq. (7) is divided into three parts, namely, static term, strain rate term and inertia term, as given in Eqs. (8)–(10), respectively. s rd1 Static term pl ¼ rpl
s rd2 pl ¼ rpl
rd3 pl ¼
v
1P
Strain rate term
2:24lD
q v 2 eD
ð8Þ
Inertia term
ð9Þ
ð10Þ
Therefore, the influences of different effects on the dynamic compressive plateau stress can be study quantitatively. Furthermore, in order to identify the dominant factor in different velocity ranges, a critical velocity can be defined by equaling the strain rate term and the inertia term. The critical velocity can be expressed as:
v cr ¼
1 2:24lD
1 2P1
rspl eD 2P1 q P
ð11Þ
where all of terms in the equation are as defined earlier. It should be noted that the critical velocity is dependent not only on the parent material parameters of honeycomb, but also on the geometric dimensions. Fig. 11(a) shows the stress values of the inertia and strain rate terms under different impact velocities. It can be seen that the curves can be divided into two regions according to the critical velocity. In region I, the strain rate term always larger than the inertia term. Meanwhile, the strain rate term increases sharply at the
Fig. 11. Comparison of the (a) stress values and (b) proportions of inertia and strain rate terms under different impact velocities.
beginning. As the static term is kept unchanged under different impact velocities, it can be conclude that the rapidly increase of plateau stress at low impact velocity is mainly caused by the strain rate effect of parent materials. It can also find that the strain rate term relates to the impact velocity by a convex curve and the inertia term relates to it by a concave curve. This may result in a linear relationship between the plateau stress and the impact velocity. In region II, the inertia term becomes dominant, thus the plateau stress increases with the square of impact velocity. The rapidly increase of plateau stress in this region mainly due to the inertia effect. In the whole velocity range studied, both the strain rate term and the inertia term increase with the increasing impact velocity. Moreover, the strain rate term is higher than the inertia term when the impact velocity is less than the critical velocity. Therefore, the strain rate effect should not be ignored in region I. Although the inertia effect becomes dominant at high impact velocity, the strain rate term still takes a large proportion in the plateau stress, as shown in Fig. 11(b). These results imply that the strain rate effect of parent materials plays an important role on the dynamic plateau stress of honeycombs in a wide velocity range. 5. Conclusions
Fig. 10. Theoretical predictions of quasi-static and dynamic plateau stresses under different impact velocities.
In this paper, the strain rate effect of parent materials on the out-of-plane dynamic compression behavior of metallic honeycombs is studied. Through quasi-static and dynamic out-of-plane
Y. Tao et al. / Composite Structures 132 (2015) 644–651
compression tests on aluminum honeycombs, the dynamic plateau stress enhancement of about 38–57% was observed in a velocity range from about 6–19 m/s, and similar deformation patterns were also observed during quasi-static and dynamic compressive tests. Comparisons show that the R-D shock theory corresponds well with the present and existing experimental results in a wide velocity range while the R-I shock theory always underestimates the experimental results. Based on the R-D shock theory, the plateau stress is divided into three terms, namely static term, inertia term and strain rate term, and the influences of the inertia and strain rate effects are studied quantitatively. It is demonstrated that the strain rate effect of parent materials plays an important role on the dynamic enhancement of metallic honeycombs over a wide velocity range. Besides, reasonable explanations for the change tendencies of measured plateau stresses with impact velocities are provided by calculating and analyzing the contribution of inertia and strain rate effects. Acknowledgements The authors are grateful for the support provided by the National Natural Science Foundation of China (11227801, 11232001) and the National Basic Research Program of China (2011CB610303, G2010CB832701, 2011CB606105). References [1] Goldsmith W, Sackman JL. An experimental study of energy absorption in impact on sandwich plates. Int J Impact Eng 1992;12(2):241–62. [2] Goldsmith W, Louie DL. Axial perforation of aluminum honeycombs by projectiles. Int J Solids Struct 1995;32(8):1017–46. [3] Wu E, Jang WS. Axial crush of metallic honeycombs. Int J Impact Eng 1997;19(5):439–56. [4] Baker WE, Togami TC, Weydert JC. Static and dynamic properties of highdensity metal honeycombs. Int J Impact Eng 1998;21(3):149–63. [5] Xu SQ, Beynon JH, Ruan D, Lu GX. Experimental study of the out-of-plane dynamic compression of hexagonal honeycombs. Compos Struct 2012;94(8):2326–36. [6] Harrigan JJ, Reid SR, Peng C. Inertia effects in impact energy absorbing materials and structures. Int J Impact Eng 1999;22(9):955–79. [7] Zhou Q, Mayer RR. Characterization of aluminum honeycomb material failure in large deformation compression, shear, and tearing. J Eng Mater Technol 2002;124(1):412–20. [8] Yamashita M, Gotoh M. Impact behavior of honeycomb structures with various cell specifications—numerical simulation and experiment. Int J Impact Eng 2005;32(1):618–30. [9] Zhao H, Gary G. Crushing behaviour of aluminium honeycombs under impact loading. Int J Impact Eng 1998;21(10):827–36. [10] Zhao H, Elnasri I, Abdennadher S. An experimental study on the behaviour under impact loading of metallic cellular materials. Int J Mech Sci 2005;47(4):757–74. [11] Hou B, Zhao H, Pattofatto S, Liu JG, Li YL. Inertia effects on the progressive crushing of aluminium honeycombs under impact loading. Int J Solids Struct 2012;49(19):2754–62. [12] Alavi Nia A, Sadeghi MZ. An experimental investigation on the effect of strain rate on the behaviour of bare and foam-filled aluminium honeycombs. Mater Des 2013;52(1):748–56. [13] Wang ZG, Tian HQ, Lu ZJ, Zhou W. High-speed axial impact of aluminum honeycomb – experiments and simulations. Composite Part B 2014;56(1):1–8. [14] Chung J, Waas AM. Compressive response of honeycombs under in-plane uniaxial static and dynamic loading, part 1: experiments. AIAA J 2002;40(5):966–73. [15] Ruan D, Lu G, Wang B, Yu TX. In-plane dynamic crushing of honeycombs—a finite element study. Int J Impact Eng 2003;28(2):161–82. [16] Sun DQ, Zhang WH, Zhao YC, Li GZ, Xing YQ, Gong GF. In-plane crushing and energy absorption performance of multi-layer regularly arranged circular honeycombs. Compos Struct 2013;96(1):726–35. [17] 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(10):1231–41. [18] Shen CJ, Lu G, Yu TX. Dynamic behavior of graded honeycombs–a finite element study. Compos Struct 2013;98(1):282–93.
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[19] Hu LL, You FF, Yu TX. Analyses on the dynamic strength of honeycombs under the y-directional crushing. Mater Des 2014;53(1):293–301. [20] Carlsson LA, Kardomateas GA. Structural and failure mechanics of sandwich composites. Springer; 2011. [21] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge: Cambridge University Press; 1997. [22] McFarland RK. Hexagonal cell structures under post-buckling axial load. AIAA J 1963;1(6):1380–5. [23] Wierzbicki T. Crushing analysis of metal honeycombs. Int J Impact Eng 1983;1(2):157–74. [24] Zhang J, Ashby MF. The out-of-plane properties of honeycombs. Int J Mech Sci 1992;34(6):475–89. [25] Aktay L, Johnson AF, Kröplin BH. Numerical modelling of honeycomb core crush behaviour. Eng Fract Mech 2008;75(9):2616–30. [26] Wilbert A, Jang WY, Kyriakides S, Floccari JF. Buckling and progressive crushing of laterally loaded honeycomb. Int J Solids Struct 2011;48(5):803–16. [27] Zhang X, Zhang H, Wen ZZ. Experimental and numerical studies on the crush resistance of aluminum honeycombs with various cell configurations. Int J Impact Eng 2014;66(1):48–59. [28] Reid SR, Peng C. Dynamic uniaxial crushing of wood. Int J Impact Eng 1997;19(5):531–70. [29] Tan PJ, Reid SR, Harrigan JJ, Zou Z, Li S. Dynamic compressive strength properties of aluminium foams. Part I—experimental data and observations. J Mech Phys Solids 2005;53(1):2174–205. [30] 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(10):2206–30. [31] Elnasri I, Pattofatto S, Zhao H, Tsitsiris H, Hild F, Girard Y. Shock enhancement of cellular structures under impact loading: part I experiments. J Mech Phys Solids 2007;55(12):2652–71. [32] Pattofatto S, Elnasri I, Zhao H, Tsitsiris H, Hild F, Girard Y. Shock enhancement of cellular structures under impact loading: part II analysis. J Mech Phys Solids 2007;55(12):2672–86. [33] 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(1):165–76. [34] Tan PJ, Reid SR, Harrigan JJ. On the dynamic mechanical properties of open-cell metal foams – a re-assessment of the ‘simple-shock theory’. Int J Solids Struct 2012;49(19–20):2744–53. [35] Sun GY, Li GY, Stone M, Li Q. A two-stage multi-fidelity optimization procedure for honeycomb-type cellular materials. Compos Mater Sci 2010;49(3):500–11. [36] Sun DQ, Zhang WH, Wei YB. Mean out-of-plane dynamic plateau stresses of hexagonal honeycomb cores under impact loadings. Compos Struct 2010;92(11):2609–21. [37] Partovi Meran A, Toprak T, Mug˘an A. Numerical and experimental study of crashworthiness parameters of honeycomb structures. Thin Wall Struct 2014;78(1):87–94. [38] Tao Y, Chen MJ, Pei YM, Fang DN. Strain rate effect on mechanical behavior of metallic honeycombs under out-of-plane dynamic compression. ASME J Appl Mech 2015;82(2):021007. [39] Hexcel Composites. HexWeb™ honeycomb attributes and properties. A comprehensive guide to standard Hexcel honeycomb materials, configurations, and mechanical. Copyright 1999. [40] Campbell FC. Manufacturing processes for advanced composites. Elsevier; 2003. [41] Yamada H, Yamaguchi S, Yamamoto N, Kato T. Cutting speed of electric discharge machining for SiC ingot. Mater Sci Forum 2012;717:861–4. [42] Klocke F, Lung D, Thomaidis D, Antonoglou G. Using ultra thin electrodes to produce micro-parts with wire-EDM. J Mater Process Tech 2004;149(1):579–84. [43] GB/T 1453-2005. Test method for flatwise compression properties of sandwich constructions or cores. Beijing: Standards Press of China; 2005 (in Chinese). [44] ASTM C365/C365M-11a. Standard test method for flatwise compressive properties of sandwich cores. ASM International; 2011. [45] ASTM E9-09. Standard test methods of compression testing of metallic materials at room temperature. ASM International; 2009. [46] Kuhn H, Medlin Dana. ASM handbook volume 8: mechanical testing and evaluation. ASM International; 2000. p. 226. [47] Zheng ZJ, Wang CF, Yu JL, Reid SR, Harrigan JJ. Dynamic stress–strain states for metal foams using a 3D cellular model. J Mech Phys Solids 2014;72(1):93–114. [48] Liu YD, Yu JL, Zheng ZJ, Li JR. A numerical study on the rate sensitivity of cellular metals. Int J Solids Struct 2009;46(22):3988–98. [49] Bodner SR, Symonds PS. Experimental and theoretical investigation of the plastic deformation of cantilever beams subjected to impulsive loading. ASME J Appl Mech 1962;29(4):719–28. [50] Bodner SR, Speirs WG. Dynamic plasticity experiments on aluminium cantilever beams at elevated temperature. J Mech Phys Solids 1963;11(2):65–77.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs: Experiment and theory Yong Tao a, Mingji Chen b,⇑, Haosen Chen c,d,⇑, Yongmao Pei a, Daining Fang a a
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China National Center for Nanoscience and Technology, Beijing 100190, China c Beijing Institute of Technology, Beijing 100081, China d Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China b
a r t i c l e
i n f o
Article history: Available online 9 June 2015 Keywords: Metallic honeycomb Dynamic compression Strain rate effect Inertia effect Plateau stress
a b s t r a c t Many studies reveal that the dynamic compressive strength of metallic honeycombs is higher than the quasi-static one, but the reasons for that are still debatable. This paper aims to study the strain rate effect of parent materials on the out-of-plane dynamic compressive behavior of metallic honeycombs. Quasi-static and dynamic tests on aluminum honeycombs were performed with universal testing machine and Split Hopkinson Pressure Bar, respectively. The velocity values of dynamic tests were from about 6 to 19 m/s. The present and existing measures of plateau stress are evaluated by both the rate-independent (R-I) and rate-dependent (R-D) shock theories. It is shown that the R-D shock theory proposed in our previous study provides more accurate predictions at low, medium and high impact velocities. Based on the R-D shock theory, the influences of strain rate effect are analyzed quantitatively and the change tendencies of measured plateau stresses with impact velocities are explained reasonably. The analysis indicates that the strain rate effect has a large contribution to the dynamic enhancement of metallic honeycombs in a wide velocity range. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Metallic honeycombs are widely used in the packaging, transport and aircraft industries due to their lightweight, high specific stiffness and strength and outstanding energy absorption performance. In these engineering applications, dynamic loadings are often encountered. Under dynamic loading, the response of honeycombs is significantly different from that under static loading [1– 13], and honeycomb structures designed with static properties are usually too conservative and wasteful. Much effort has been made on the in-plane dynamic compressive properties of honeycombs [14–19]. However, honeycombs are generally used as core materials of sandwich plates [20], and the out-of-plane stiffness and strength are much higher than those with respect to the in-plane directions [21]. Therefore, it is necessary and important to study the out-of-plane dynamic compressive behavior of metallic honeycombs.
⇑ Corresponding authors at: Tel./fax: +86 010 82545661 (M. Chen). Tel./fax: +86 010 62753795 (H. Chen). E-mail addresses: [email protected] (M. Chen), [email protected]. edu.cn (H. Chen). http://dx.doi.org/10.1016/j.compstruct.2015.06.015 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
As for the out-of-plane compressive behavior of metallic honeycombs, McFarland [22] provided a semi-empirical expression to predict the plateau stress of hexagonal honeycombs. Since that pioneering study, Wierzbicki [23] developed a more reasonable model as the correct cell wall deformation patterns were identified. Zhang and Ashby [24] proposed a formula for the plateau stress of hexagonal honeycombs with uniform cell wall thickness. Systematic and detailed studies of honeycombs were published in the book of Gibson and Ashby [21]. In the numerical and experimental fields, Aktay et al. [25] simulated the compressive behavior of honeycomb with a meso-structure model and a homogenized model and compared the numerical results with the experimental data. Wilbert et al. [26] presented a comprehensive study of the compressive response of hexagonal honeycombs through experiments and simulations. More recently, Zhang et al. [27] performed simulations and experiments to evaluate the influences of cell number and central angle on the out-of-plane compressive resistance of aluminum honeycombs. Beside the quasi-static investigations mentioned above, the out-of-plane dynamic compression of honeycombs was also frequently studied, and the compressive strength of honeycombs under dynamic loading has been shown to be higher than that under quasi-static loading [1–13]. The inertia effect has been
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regarded as the main reason for the dynamic enhancement, and a rate-independent (R-I) shock theory considering the inertia effect was proposed to predict the dynamic compressive plateau stress [6,28]. This model has been widely used to evaluate the dynamic enhancement of cellular materials [17,15,29–34]. However, such a model do not account for the strain rate effect of parent materials from which cellular materials are made. More dynamic finite element studies can be found in literature, but most of them adopted a rate independent elastic-perfectly plastic or bilinear constitutive law [8,11,35–37]. Recently, Tao et al. [38] studied the strain rate effect on the behavior of metallic honeycombs through numerical analysis and a rate-dependent (R-D) shock theory was proposed. As the strain rate effect of parent materials is considered to be small and often neglected in the current studies, its influences on the dynamic enhancement are still unclear and the quantitative analysis is not available. Besides, it is observed that the change tendencies of measured plateau stresses are different in the low [12], medium [3] and high [6] velocity ranges, and that also lacks a convincing theoretical explanation. Therefore, more studies need to be done to further understand the out-of-plane dynamic behavior of metallic honeycombs. In this paper, the strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs is studied through experimental and theoretical investigations. Firstly, quasi-static and dynamic tests on aluminum honeycombs were performed by using the universal testing machine and Split Hopkinson Pressure Bar (SHPB) system, respectively. Secondly, the experimental results are presented and analyzed, and the present and existing measured plateau stresses are evaluated by the R-I and R-D shock theories at low, medium and high impact velocities. Finally, based on the R-D shock theory proposed in our previous study [38], the influences of inertia and strain rate effects are analyzed, and the change tendencies of measured plateau stresses with impact velocities are also explained. 2. Experimental procedures of quasi-static and dynamic tests 2.1. Specimen specification Typical honeycomb specimens contain 19 complete cells used in this study are shown in Fig. 1. The honeycomb specimens were made of Al3003-H18 with density of 2730 kg/m3 and elastic modulus of 68.9 GPa. The honeycomb is regarded as an orthotropic material, and the orthotropic directions can be denoted as the L, W, and T directions [39], as shown in Fig. 2(a). The T direction, also called the out-of-plane direction, is paralleled with the axis of the cell wall. The other two directions are referred to in-plane (L and W) directions. The L direction is called the ribbon direction, and the cell wall thickness in the L direction is double that in the width or W direction, which is transverse to the L direction [40].
Fig. 1. Typical hexagonal aluminum honeycomb specimens used in this study.
Specimens employed in tests were designed as cylindrical columns with diameter of 20 mm and height of 14 mm. The cell wall thickness t is 0.06 mm and the cell wall length l is 2 mm. All specimens were carefully cut from a big square honeycomb panel using wire electric discharge machining (WEDM). It should be noted that honeycomb specimens with accurate dimensions and little damage can be obtained by taking the advantages of a WEDM method [41,42]. Detailed geometrical configuration of a honeycomb specimen is shown in Fig. 2. 2.2. Experimental arrangement of quasi-static tests The quasi-static tests were carried out with an electronic universal testing machine (Fig. 3(a)) at a loading velocity of 0.5 mm/min based on both the Chinese test method [43] and the ASTM standards [44–46]. In this machine, the lower platen was stationary, whereas the upper platen was movable towards the lower one. The plane surface of the upper platen was smooth enough to avoid the influences of friction, but the surface of lower platen had circle-shaped notches. It must be noted that the platen-specimen interface friction should be avoided for compression tests [45,46]. Therefore, an iron cylinder with polished upper surface was put on the lower platen. Then the honeycomb specimen was placed on the polished surface in order to minimize friction. Three specimens were tested for this case and mean value of the data was used as the result. 2.3. Experimental arrangement of dynamic tests The dynamic tests were performed on a SHPB system, which was often used as an experimental setup to study the dynamic behaviors of materials [9–11,31]. The SHPB setup used in this study is illustrated in Fig. 3(b), and its schematic diagram is shown in Fig. 4. It can be seen that a typical SHPB setup is composed of input and output bars with a honeycomb specimen sandwiched between them. A striker bar launched by a gas gun impacts the free end of the input bar and generates a compressive longitudinal incident wave ei(t). When the incident wave arrives at the input bar-specimen interface, part of it is reflected and develops a reflected wave er(t) in the input bar, while the rest passes through the specimen and develops the transmitted wave et(t) in the output bar. Based on the strain measured by two gauges, the forces and velocities can be calculated as follows [9–11]:
v input ¼ C 0 ðei ðtÞ er ðtÞÞ v output ¼ C 0 et ðtÞ
F input ¼ Sbar Eðei ðtÞ þ er ðtÞÞ F output ¼ Sbar Eet ðtÞ
ð1Þ
where vinput, vouput, Finput, Foutput are velocities and forces at the bars-specimen interfaces, Sbar, E and C0 are the cross sectional area, elastic modulus and the longitudinal wave velocity of the bars, respectively, ei(t), er(t), et(t) are the strain signals at the bars-specimen interfaces. A range of impact velocities was covered from 6 to 19 m/s by varying the air pressure in the gas gun. It should be noticed that the density of aluminum honeycomb is small, so its impedance is low and the induced strain in an ordinary pressure bar (like steel bar) is too small to measure accurately [9]. In order to get an accurate measurement, the low impedance PMMA bars with diameter of 30 mm were used to match the impedance of aluminum honeycomb specimen. The striker bar had a length of 290 mm, and both the input and output bars were 2000 mm long. The surfaces at the bars-specimen interfaces were lubricated to reduce the influence of friction. Three and five specimens were tested for dynamic tests with velocities of 19.01 and 8.15 m/s, respectively. Six tests were performed for the velocities of 9.68 and 14.62 m/s, respectively. For dynamic tests with velocities of 6.04, 11.16 and 13.52 m/s, four specimens were tested for each case.
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Fig. 2. Geometric dimensions of a hexagonal honeycomb specimen: (a) sketch, (b) circle cross section and (c) unit cell.
Fig. 3. Experimental systems: (a) electronic universal testing machine for quasi-static tests and (b) SHPB setup for dynamic tests.
Fig. 4. Schematic diagram of SHPB setup.
2.4. Parameters definition and calculation for the compression of honeycombs The typical stress–strain curve of honeycomb under out-of-plane compression displays three regions, that is, the initial linear elastic region followed by the plateau region, and finally the rapidly rising densification region [21], as shown in Fig. 5. Nominal stress and strain are defined as:
F A
r¼ ; e¼
d H
ð2Þ
where F is the compressive force, A is the initial cross-sectional area of honeycomb, d is the compression displacement and H is the initial height of the honeycomb structure. Here, the average value of the stresses between the yield strain and densification strain is used as the plateau stress, which can be calculated by the following relation:
Fig. 5. Typical stress–strain curve of honeycomb under out-of-plane compression.
Y. Tao et al. / Composite Structures 132 (2015) 644–651
rpl ¼
1 eD ey
Z
eD
rðeÞde
ð3Þ
ey
where ey is the yield strain, which is the strain corresponding to the initial peak stress rpk, and eD is the densification strain, which can be defined through [29]:
d de
1 rðeÞ
Z
e 0
rðeÞde
¼0
ð4Þ
e¼ eD
It should be noted that the results of the aforementioned integral formulas are equal to the area under the nominal stress–strain curve, which can be calculated by using the function ‘‘Integrate’’ in Origin software. 3. Experimental results and analysis 3.1. Repeatability of the experimental stress–strain curves Figs. 6(a) and 6(b) show the stress–strain curves for three independent quasi-static and dynamic tests on the aluminum honeycombs, respectively. Although the measured data of peak stress has slight dispersity, the stresses in the plateau region are of good repeatability, which means the quasi-static and dynamic test results are reliable. It should be noticed that the compression reached in dynamic test is quite limited and the densification region is absent due to the limited length of impulse in the SHPB
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system [9,11]. Therefore, for dynamic tests, the average value of the stress between the yield strain and the maximum strain is used as the plateau stress. 3.2. Effect of impact velocity The mean values of both the quasi-static and dynamic experimental results are listed in Table 1. Rpl denotes the plateau stress ratio, which is the ratio between dynamic and quasi-static plateau stresses. As shown in Table 1, when compared with the value under quasi-static loading, the plateau stresses are raised by about 38–57% in a velocity range from about 6–19 m/s, indicating that the impact velocity has a great influence on the out-of-plane plateau stress. The present experimental results of dynamic compressive plateau stress enhancement for honeycombs are quite similar with previous researches. For example, Goldsmith and Sackman [1] observed an increase of about 20–50% in their dynamic tests by using a rounded projectile with impact velocities up to 35 m/s. Wu and Jiang [3] performed dynamic tests at velocities up to about 28 m/s and found that the dynamic plateau stress was about 74% higher than the quasi-static value. Baker et al. [4], Xu et al. [5], Harrigan et al. [6], Yamashita and Gotoh [8], Zhao and Gray [9], Zhao et al. [10], Hou et al. [11], Alavi Nia and Sadeghi [12], and Wang et al. [13] have also found a certain dynamic plateau stress enhancement for metallic honeycombs in their experiments. 3.3. Deformation patterns under out-of-plane quasi-static and dynamic loadings Some pictures were also taken during quasi-static loading, as shown in Fig. 7(a). For out-of-plane quasi-static compression, the deformation patterns of aluminum honeycomb specimens were characterized by localized and progressive buckling. The progressive buckling was started from the layers near the interface between the lower platen and the honeycomb specimen, and then induced the deformation of adjacent layers to form folds. In fact, the onset of this progressive buckling can occur anywhere in the specimen where the weakest zone located [6,47]. For dynamic compression, a typical progressive buckling deformation process was also observed, as shown in Fig. 7(b). In dynamic compression, most of the aluminum honeycomb specimens began to buckle from the layers adjacent to the input or output bar-specimen interface, then the plastic buckling progressed in nearby layers. After the plastic buckling progressed in nearby layers at a certain stage, the honeycomb layers near the other end of the honeycomb specimen started to buckle. Then the plastic buckling of both ends of the honeycomb continued and finished until the densification was achieved. The deformation patterns were similar with the quasi-static case, but the start position of buckling may be different under dynamic compressive loading. Figs. 7(c) and 7(d) show the final deformation patterns of honeycomb specimens under quasi-static and dynamic compression, respectively. The final deformation patterns under dynamic compression were also similar to that under quasi-static compression. In both cases, clear folds of cell wall caused by plastic buckling could be observed. The present results agree with the observations reported by Xu et al. [5] and Harrigan et al. [6] in the low and high velocity ranges, respectively. 4. Comparison and discussion 4.1. A brief overview of the R-I and R-D shock theories
Fig. 6. Repeatability of the stress–strain curves under (a) quasi-static (v = 0.5 mm/ min) and (b) dynamic (v = 19 m/s) loadings.
The R-I shock theory was first proposed by Reid and Peng [28] in order to explain the dynamic enhancement of wood specimens
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Table 1 The measurement results from Alavi Nia et al. [12], present experiments and Harrigan et al. [6], respectively. t = 0.0508 mm, l = 1.833 mm
v (m/s)
4.16 10 4.300 1.000
rpl (MPa) Rpl
5
0.1 5.000 1.163
t = 0.0508 mm, l = 2.748 mm 0.2 5.200 1.209
4.16 105 1.860 1.000
0.1 2.500 1.344
0.2 2.600 1.398
8.15 4.171 1.419
9.68 4.168 1.418
11.16 4.610 1.568
13.52 4.540 1.544
14.62 4.386 1.492
19.01 4.351 1.480
35.76 1.002 1.891
59.67 1.229 2.319
82.80 1.905 3.594
91.74 1.906 3.596
96.99 2.218 4.185
132.46 2.097 3.957
t = 0.06 mm, l = 2.00 mm
v (m/s)
8.33 106 2.940 1.000
rpl (MPa) Rpl
6.04 4.047 1.377
t = 0.049 mm, l = 3.67 mm
v (m/s)
1.67 105 0.530 1.000
rpl (MPa) Rpl
32.85 0.961 1.813
Fig. 7. Typical deformation patterns of aluminum honeycomb specimens subjected to out-of-plane compression: (a) quasi-static and (b) dynamic deformation processes and the final deformation patterns under (c) quasi-static and (d) dynamic loadings.
under dynamic compression. Afterwards, it was used to predict the dynamic strength of hexagonal honeycombs [6]. Since then, the model has been widely employed to study the dynamic enhancement of cellular materials [17,15,29–34]. The dynamic plateau stress of honeycomb predicted by the R-I shock theory can be expressed in the following form [6,28]:
rdpl ¼ rspl þ
qv eD
2
ð5Þ
respectively. q represents the density of honeycomb and v is the impact velocity. The quasi-static plateau stress in Eq. (5) can be calculated as follows [23]:
53 t l
rdpl ¼ rspl 1 þ
where rspl and rdpl are the quasi-static and dynamic plateau stresses,
rspl ¼ 6:63rsy
In order to incorporate the strain rate effect of parent materials into the theoretical model, the R-D shock theory was proposed by introducing the Cowper–Symonds relation into the R-I shock theory [38]. According to the R-D shock theory, the formula of dynamic compressive plateau stress can be written as [38]:
ð6Þ
where rsy denotes the yield stress of parent material. As can be seen in Eq. (5), the R-I shock theory has taken into account the inertia effect, but the strain rate effect of parent materials which has been regarded as a possible cause [3,11,48] for dynamic enhancement of metallic honeycombs was neglected.
v
1P
2:24lD
þ
q v 2 eD
ð7Þ
where D and P are the strain rate material constants, e.g., for Al5050 and Al5052, D = 6500 s1 and P = 4 [49], and for Al3003, D = 2.7 105 s1 and P = 8 [50]. 4.2. Comparison between theory and experiment The dynamic experiments in the present study were performed with impact velocities varied from about 6 to 19 m/s, with associated nominal strain rates from about 429 to 1357 s1. Theoretical predictions given by the R-I and R-D shock theories have also been compared with the present experimental data, as shown in Fig. 8(a). It can be seen that the R-D shock theory follows the experimental data well, whereas the R-I shock theory gives much lower predictions. Both the measured plateau stresses and
Y. Tao et al. / Composite Structures 132 (2015) 644–651
predictions of the R-D shock theory are almost proportional to the impact velocity in the studied velocity range. Wu and Jiang [3] also fitted their experimental results into linear relation in a similar velocity range. The present experimental results are in the medium velocity range. Fig. 8(b) also provides some low and high impact velocity data obtained from existing experiments in order to make a comparison between the theoretical and experimental results in a wide velocity range. By using an Instron 8503 apparatus, Alavi Nia and Sadeghi [12] performed experimental investigation on the effect of impact velocity on the out-of-plane compressive behavior of bare and foam-filled Al5052-H39 honeycombs in a low velocity range. Fig. 8(b) shows their experimental data at impact velocities from 4.16 105 to 0.2 m/s, together with the plateau stresses predicted by the R-I and R-D shock theories. The aluminum honeycomb specimens are square panels with thickness of 19.05 mm, which means the associated nominal strain rates vary from 2.2 103 to 10.5 s1. Their geometrical parameters and experimental results are listed in Table 1. As shown in Fig. 8(b), there is a reasonable agreement between the results of R-D shock theory and experiment, whereas the R-I shock theory underestimates the measured plateau stresses. The experimental results show that an increase in plateau stress is much more pronounced at low impact velocity. For example, increasing the impact velocity from quasi-static (4.16 105 m/s) to 0.1 m/s increases the plateau stress of 16.3% for honeycombs
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with cell wall length of 1.833 mm. However, changing the impact velocity from 0.1 to 0.2 m/s increases the plateau stress only by 4.6%. The two values for honeycombs with cell wall length of 2.748 mm are 34.4% and 5.4%, respectively. The R-D shock theory can successfully predict this change tendency while the R-I shock theory fails to do so. An investigation concerning dynamic out-of-plane compression of Al5050 honeycombs in the high velocity range was reported by Harrigan et al. [6]. Their aluminum honeycomb specimens were cylindrical columns with diameter of 45 mm and height of 30 mm, and the high velocity impact tests were performed with an impact testing rig developed by the authors themselves. The experimental results for backed, initially uncrushed specimens are used to compare with theoretical predictions, as shown in Fig. 8(b). Corresponding geometrical parameters and experimental results are listed in Table 1. It can be seen that the dynamic plateau stresses are related to the impact velocities by conic curve at high impact velocity. The predictions of both the R-I and R-D shock theories show similar tendency with the experimental results, but the predicted values of R-I shock theory are still smaller than the experimental data. Fig. 9 shows a comparison between theoretical predictions and experimental results at low, medium and high impact velocities. Experimental data from Wu and Jiang [3], Xu et al. [5], and Wang et al. [13] are also included in this figure. In Fig. 9, the solid lines are predicted by the R-D shock theory and the dash lines represent the predictions of the R-I shock theory. Lines and symbols in the same color are plotted with corresponding geometrical and material parameters in a certain literature. As the geometrical parameters and materials are different in different literatures, the corresponding theoretical curves are different and cannot collapse into a single curve. It can be seen that the plateau stresses predicted by the R-D shock theory correspond better to the experiment results over the whole range of impact velocities studied, whereas the R-I shock theory always underestimates the measured plateau stresses. Moreover, the change tendencies between the plateau stress and the impact velocity are distinct in different velocity ranges. The reason for that will be discussed in Section 4.3. 4.3. Theoretical analysis of experimental results To study the influences of the strain rate effect, the predicted values of R-D shock theory with different strain rate parameters under different impact velocities is plotted in Fig. 10. For
Fig. 8. Comparison between theoretical predictions and experimental results in the (a) medium and (b) low and high velocity ranges.
Fig. 9. Comparison between theoretical predictions and experimental results over a wide velocity range.
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comparison purpose, the quasi-static plateau stress [23] and the predictions of R-I shock theory [6,28] are also plotted. It can be seen from Fig. 10 that the dynamic plateau stress predicted by R-I shock theory increases with the square of impact velocity. While the plateau stress predicted by the R-D shock theory exhibits three regions, namely, the initial rapidly increasing region at low impact velocity followed by the approximately linear region at medium impact velocity, and finally the conic curve region at high impact velocity. In addition, the dynamic plateau stress of honeycomb made of a relatively stronger strain rate dependent (D = 6500 s1, P = 4) parent material is always larger than honeycomb made of a relatively weaker strain rate dependent (D = 270,000 s1, P = 8) parent material. In order to further study the influences of the strain rate effect on the dynamic enhancement of honeycombs, the R-D shock theory as expressed in Eq. (7) is divided into three parts, namely, static term, strain rate term and inertia term, as given in Eqs. (8)–(10), respectively. s rd1 Static term pl ¼ rpl
s rd2 pl ¼ rpl
rd3 pl ¼
v
1P
Strain rate term
2:24lD
q v 2 eD
ð8Þ
Inertia term
ð9Þ
ð10Þ
Therefore, the influences of different effects on the dynamic compressive plateau stress can be study quantitatively. Furthermore, in order to identify the dominant factor in different velocity ranges, a critical velocity can be defined by equaling the strain rate term and the inertia term. The critical velocity can be expressed as:
v cr ¼
1 2:24lD
1 2P1
rspl eD 2P1 q P
ð11Þ
where all of terms in the equation are as defined earlier. It should be noted that the critical velocity is dependent not only on the parent material parameters of honeycomb, but also on the geometric dimensions. Fig. 11(a) shows the stress values of the inertia and strain rate terms under different impact velocities. It can be seen that the curves can be divided into two regions according to the critical velocity. In region I, the strain rate term always larger than the inertia term. Meanwhile, the strain rate term increases sharply at the
Fig. 11. Comparison of the (a) stress values and (b) proportions of inertia and strain rate terms under different impact velocities.
beginning. As the static term is kept unchanged under different impact velocities, it can be conclude that the rapidly increase of plateau stress at low impact velocity is mainly caused by the strain rate effect of parent materials. It can also find that the strain rate term relates to the impact velocity by a convex curve and the inertia term relates to it by a concave curve. This may result in a linear relationship between the plateau stress and the impact velocity. In region II, the inertia term becomes dominant, thus the plateau stress increases with the square of impact velocity. The rapidly increase of plateau stress in this region mainly due to the inertia effect. In the whole velocity range studied, both the strain rate term and the inertia term increase with the increasing impact velocity. Moreover, the strain rate term is higher than the inertia term when the impact velocity is less than the critical velocity. Therefore, the strain rate effect should not be ignored in region I. Although the inertia effect becomes dominant at high impact velocity, the strain rate term still takes a large proportion in the plateau stress, as shown in Fig. 11(b). These results imply that the strain rate effect of parent materials plays an important role on the dynamic plateau stress of honeycombs in a wide velocity range. 5. Conclusions
Fig. 10. Theoretical predictions of quasi-static and dynamic plateau stresses under different impact velocities.
In this paper, the strain rate effect of parent materials on the out-of-plane dynamic compression behavior of metallic honeycombs is studied. Through quasi-static and dynamic out-of-plane
Y. Tao et al. / Composite Structures 132 (2015) 644–651
compression tests on aluminum honeycombs, the dynamic plateau stress enhancement of about 38–57% was observed in a velocity range from about 6–19 m/s, and similar deformation patterns were also observed during quasi-static and dynamic compressive tests. Comparisons show that the R-D shock theory corresponds well with the present and existing experimental results in a wide velocity range while the R-I shock theory always underestimates the experimental results. Based on the R-D shock theory, the plateau stress is divided into three terms, namely static term, inertia term and strain rate term, and the influences of the inertia and strain rate effects are studied quantitatively. It is demonstrated that the strain rate effect of parent materials plays an important role on the dynamic enhancement of metallic honeycombs over a wide velocity range. Besides, reasonable explanations for the change tendencies of measured plateau stresses with impact velocities are provided by calculating and analyzing the contribution of inertia and strain rate effects. Acknowledgements The authors are grateful for the support provided by the National Natural Science Foundation of China (11227801, 11232001) and the National Basic Research Program of China (2011CB610303, G2010CB832701, 2011CB606105). References [1] Goldsmith W, Sackman JL. An experimental study of energy absorption in impact on sandwich plates. Int J Impact Eng 1992;12(2):241–62. [2] Goldsmith W, Louie DL. Axial perforation of aluminum honeycombs by projectiles. Int J Solids Struct 1995;32(8):1017–46. [3] Wu E, Jang WS. Axial crush of metallic honeycombs. Int J Impact Eng 1997;19(5):439–56. [4] Baker WE, Togami TC, Weydert JC. Static and dynamic properties of highdensity metal honeycombs. Int J Impact Eng 1998;21(3):149–63. [5] Xu SQ, Beynon JH, Ruan D, Lu GX. Experimental study of the out-of-plane dynamic compression of hexagonal honeycombs. Compos Struct 2012;94(8):2326–36. [6] Harrigan JJ, Reid SR, Peng C. Inertia effects in impact energy absorbing materials and structures. Int J Impact Eng 1999;22(9):955–79. [7] Zhou Q, Mayer RR. Characterization of aluminum honeycomb material failure in large deformation compression, shear, and tearing. J Eng Mater Technol 2002;124(1):412–20. [8] Yamashita M, Gotoh M. Impact behavior of honeycomb structures with various cell specifications—numerical simulation and experiment. Int J Impact Eng 2005;32(1):618–30. [9] Zhao H, Gary G. Crushing behaviour of aluminium honeycombs under impact loading. Int J Impact Eng 1998;21(10):827–36. [10] Zhao H, Elnasri I, Abdennadher S. An experimental study on the behaviour under impact loading of metallic cellular materials. Int J Mech Sci 2005;47(4):757–74. [11] Hou B, Zhao H, Pattofatto S, Liu JG, Li YL. Inertia effects on the progressive crushing of aluminium honeycombs under impact loading. Int J Solids Struct 2012;49(19):2754–62. [12] Alavi Nia A, Sadeghi MZ. An experimental investigation on the effect of strain rate on the behaviour of bare and foam-filled aluminium honeycombs. Mater Des 2013;52(1):748–56. [13] Wang ZG, Tian HQ, Lu ZJ, Zhou W. High-speed axial impact of aluminum honeycomb – experiments and simulations. Composite Part B 2014;56(1):1–8. [14] Chung J, Waas AM. Compressive response of honeycombs under in-plane uniaxial static and dynamic loading, part 1: experiments. AIAA J 2002;40(5):966–73. [15] Ruan D, Lu G, Wang B, Yu TX. In-plane dynamic crushing of honeycombs—a finite element study. Int J Impact Eng 2003;28(2):161–82. [16] Sun DQ, Zhang WH, Zhao YC, Li GZ, Xing YQ, Gong GF. In-plane crushing and energy absorption performance of multi-layer regularly arranged circular honeycombs. Compos Struct 2013;96(1):726–35. [17] 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(10):1231–41. [18] Shen CJ, Lu G, Yu TX. Dynamic behavior of graded honeycombs–a finite element study. Compos Struct 2013;98(1):282–93.
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