Inertia effects on the progressive crushing of aluminium honeycombs under impact loading

International Journal of Solids and Structures 49 (2012) 2754–2762

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Inertia effects on the progressive crushing of aluminium honeycombs under impact loading B. Hou a, H. Zhao b,⇑, S. Pattofatto b, J.G. Liu b, Y.L. Li a a b

School of Aeronautics, Northwestern Polytechnical University, 710072 Xi’an, China Laboratoire de Mécanique et Technologie, ENS-Cachan/CNRS-UMR8535/Université Paris 6, 61 avenue du président Wilson, 94235 Cachan cedex, France

a r t i c l e

i n f o

Article history: Available online 17 May 2012 Keywords: Honeycombs Impact loading Lateral inertia Kolsky’s bar Numerical simulation

a b s t r a c t This paper presents the test results under quasi-static and impact loadings for a series of aluminum honeycombs (3003 and 5052 alloys) of different cell sizes, showing significantly different enhancements of the crushing pressure between 3003 honeycombs and the 5052 ones. A comprehensive numerical investigation with rate insensitive constitutive laws is also performed to model the experimental results for different cell size/wall thickness/base material, which suggests that honeycomb crushing pressure enhancement under impact loading is mostly due to a structural effect. Such simulated tests provide detailed local information such as stress and strain fields (in the cell wall) during the whole crushing process of honeycombs. A larger strain (in the cell wall) under impact loading than for the quasi-static case before each successive folding of honeycombs is observed, because of the lateral inertia effect. Thus, differences of the ratios of the stress increase due to strain hardening over the yield stress between 3003 and 5052 alloys lead to the different enhancements of crushing pressure. This result illustrates that the lateral inertia effect in the successive folding of honeycombs is the main factor responsible for the enhancement of the crushing pressure under impact loading. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Aluminium honeycombs are widely used in railway, automotive and aircraft industries because of their excellent physical and mechanical properties such as an interesting strength/weight ratio and an outstanding capability in absorbing energy. Mechanical behavior for small strain under quasi-static loadings such as the elastic behavior and failure strength are well investigated for structural applications (Gibson and Ashby, 1988). Elastic and fracture models for out-of-plane crushing (Zhang and Ashby, 1992a) and in-plane crushing (Zhang and Ashby, 1992b), as well as for transverse shearing (Shi and Tong, 1995), have been developed. For the case of larger strain, theoretical, experimental and numerical studies have also been reported. Theoretical models can predict the crushing pressure of honeycombs from its geometrical parameters and wall material behavior such as the out-of-plane crushing pressure (Wierzbicki, 1983), the in-plane crushing pressure (Klintworth and Stronge, 1988), and multi-axial collapse envelope (Mohr and Doyoyo, 2004a). Other related topics such as fracture detection using elastic waves (Thwaites and Clark, 1995), negative Poisson’s ratio honeycombs (Prall and Lakes, 1997), and foam-filled honeycombs (Wu et al., 1995), have also been reported in the open literatures. ⇑ Corresponding author. Tel.: +33 1 47 40 20 39; fax: +33 1 47 40 22 40. E-mail address: [email protected] (H. Zhao). 0020-7683/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijsolstr.2012.05.005

For the energy absorption applications such as protective design for accidental collisions of high speed vehicles or the bird strike of aircrafts, the out-of-plane behavior for large strains (up to 80%) under impact loading is desired. Experimental results show that the crushing pressure of honeycombs under impact loading is higher than that under quasi-static loading. For example, (Goldsmith and Sackman, 1992; Goldsmith and Louie, 1995) have reported some experimental works on out-of-plane crushing and on the ballistic perforation of honeycombs. They have fired a rigid projectile to a target made of honeycomb and have shown that the mean crushing pressures sometimes increase up to 50% compared to the static results. Wu and Jiang (1997), Baker et al. (1998), Harrigan and Reid (1999), Zhao and Gary (1998), and Zhao et al. (2005) have also found the similar phenomenon for metallic honeycombs with an enhancement ranging from 10% to 50%. As the aluminium alloy is hardly rate sensitive in the range of moderate impact speed, there exists no plausible explanation of this enhancement. Indeed, the possible effect due to the eventual shock front (Reid and Peng, 1997; Pattofatto et al., 2007), air trapped in the cell (Gibson and Ashby, 1988) cannot be applied here. Besides, since the testing methods used in previous works are rather different from each other, reasonable doubt also exists on the validity of those experimental results. Enhancement or not, how much, why, are still the open questions. This paper describes an experimental and numerical study of the out-of-plane compressive behavior for a series of aluminium

B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762

honeycombs of different size (relative densities varying from 1.78% to 4.72%) and different base material (3003 and 5052 alloys) under moderate impact velocities. Tests in a range of impact speed from 10 m/s to 28 m/s were performed using large diameter polymer Split Hopkinson bars. The accuracy of polymeric SHPB systems is known to be adequate to obtain reliable testing results on such soft materials. The testing results confirmed that there does exist an enhancement from 10% to 60% for aluminum 5052 and 3003 honeycombs. A numerical model of the studied honeycombs (3003 and 5052 alloys) was performed afterward using ABAQUS commercial codes. Similar enhancements were found with numerical results (>40% for 3003 honeycombs and <20% for 5052 ones). Finally, on the basis of numerical models, a comprehensive numerical study of successive crushing process was performed in order to understand the reason of this enhancement as well as its dependence on the base materials. It shows that the lateral inertia in the successive folding of thin-wall tube structures can explain such observed enhancement of out-of-plane crushing pressure. 2. Experimental impact rate sensitivity of 5052 and 3003 aluminium honeycombs 2.1. Experimental methods and procedures Honeycombs of various wall thickness and cell size made of 5052 or 3003 alloys were tested under axial compression in the out-of-plane direction. The quasi-static experiments were performed using a universal tension/compression testing machine. The dynamic experiments were conducted on a SHPB (Split Hopkinson Pressure Bar) apparatus (Hopkinson, 1914; Kolsky, 1949), commonly used as an experimental technique to study constitutive laws of materials at high strain rates. A typical SHPB set-up is composed of long input and output bars with a short specimen placed between them. A projectile launched by a gas gun strikes the free end of the input bar and develops a compressive longitudinal incident wave ei(t). Once this wave reaches the bar-specimen interface, part of it er(t) is reflected, whereas the other part goes through the specimen and develops the transmitted wave et(t) in the output bar. As the stress and particle velocity of a longitudinal stress wave can be calculated from the strain measured by gauges, and shifted at any other place, the transmitted wave can be shifted to the output bar-specimen interface to obtain the output force and velocity, whereas the input force and velocity can be determined via incident and reflected waves shifted to the input bar-specimen interface. The forces and particle velocities can be then calculated as follows (Eq. (1)):

F input ðtÞ ¼ SB E ðei ðtÞ þ er ðtÞÞ V input ðtÞ ¼ C 0 ðei ðtÞ  er ðtÞÞ V output ðtÞ ¼ C 0

et ðtÞ

F output ðtÞ ¼ SB E et ðtÞ

ð1Þ

where F input ; F output ; V input ; V output are forces and particle velocities at the interfaces, SB, E and C0 are respectively the cross sectional area, Young’s modulus and the longitudinal wave speed in the pressure bars. ei ðtÞ; er ðtÞ; et ðtÞ are the strain signals at the bar-specimen interface. The standard SHPB analysis (Hopkinson, 1914; Kolsky, 1949) provides an average nominal stress–strain curve, dividing the displacement and force respectively by the initial length and the cross-sectional area of the specimen (see Zhao and Gary, 1996). For the out-of-plane crushing tests of honeycombs, mechanical fields are not uniform (even under quasi-static loading). Therefore, Hopkinson bars here are considered as only a loading and measuring system which can give accurately the force and displacement time histories on the specimen faces without considering the deforming characteristics (uniform or not) of the sandwiched

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specimen. Instead of average stress and strain in a common SHPB test, we use only the pressure p(t) as a function of the crush D(t) to give an overall idea of the behavior of the honeycombs. They are defined as follows (Eq. (2)):

pðtÞ ¼ ðF input ðtÞ þ F output ðtÞÞ=2Ss Z t DðtÞ ¼ ðV output ðsÞ  V input ðsÞÞds

ð2Þ

0

where Ss is the apparent area of the specimen face. It is worthwhile to notice that impact tests on such soft cellular materials using a SHPB have two major difficulties. One is the large scatter due to the small cell/sample ratio. To overcome this difficulty, a large diameter pressure bar is necessary to host a larger specimen. Another is the weak signal due to the weak strength of honeycombs, which leads to a low signal/noise ratio. Large diameter, soft, but polymeric pressure bars are used to overcome these difficulties. In practice, two Hopkinson bar systems were used: a 60 mm PA6.6 Hopkinson bar system with input and output bars of 3 m (in LMT-Cachan), and a 30 mm diameter PMMA bar system with 2 m input and output bars (in the Laboratory of Dynamics and Strength, NWPU). The projectiles were made of same materials as pressure bars and their lengths are respectively 1.2 m and 0.5 m (less than half input bar length to avoid superimposition of the tail of incident impulse in viscoelastic bars, see Zhao et al., 1997). Thus, the impulse is not long enough to reach the densification point in dynamic test. Moreover, soft polymeric bars are viscoelastic materials, and the wave dispersion effect increases greatly with the diameter of the bars. Consequently, as the three waves in Eq. (1) are not measured at bar-specimen interfaces to avoid their superimposition, they have to be shifted from the position of the strain gauges to the specimen faces. Kolsky’s original SHPB analysis on the basis of a one-dimensional wave propagation theory is no longer valid here. The shifting along pressure bars is performed in our experiments with Pochhammer and Chree’s harmonic wave propagation theory in an infinite cylindrical bar (Davies, 1948; Follansbee and Franz, 1983), extended to viscoelastic bars (Zhao and Gary, 1995). Detailed data processing procedure can be found in Zhao et al. (1997). 2.2. Honeycomb specimens and testing results Firstly, four 3003 alloy honeycombs referenced by the single wall thickness h, and the minimum cell diameter S (Fig. 1) were studied. Samples were designed as columns of 30 mm high, with hexagonal cross section containing a maximum number of cells in a circle of 30 mm diameter (Fig. 2). Table 1 provides a summary of honeycomb geometry parameters and the relative densities (see Gibson and Ashby, 1988) of four types of Al3003 honeycombs. Quasi-static tests with loading speed of 0.03 mm/s and SHPB test between 26 and 28 m/s were performed in the out-of-plane direction (T direction in Fig. 2). For the quasi-static tests, the pressure (measured force divided by the nominal cross-sectional area)

Fig. 1. Geometry of the unit cell.

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B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762 4.0

Dynamic Quasi-static

3.5 3.0

Pressure (MPa)

o

No 4 (Al3003)

2.5 2.0 1.5

Fig. 2. Hexagonal honeycomb specimens. 1.0

10

0.5

9

0.0 -3

8

0

3

6

9

12

15

18

21

24

27

30

Pressure (MPa)a

Crush (mm) 7 o

No 3 (Al3003)

Fig. 4. Dynamic enhancement of honeycomb No. 4 pressure/crush curve.

6 5

Table 1a Summary of 3003 honeycomb parameters.

4 3

Honeycomb number

Material

h/S (mm/mm)

Relative density (%)

2

1 2 3 4

Al3003 Al3003 Al3003 Al3003

0.05/5.2 0.06/4.33 0.06/3.46 0.04/3.46

2.57 3.70 4.61 3.08

1 0

-2

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

Crush (mm) Fig. 3a. Reproducibility of quasi-static experiments on honeycomb No. 3.

11 10

Table 1b Summary of the parameters and experimental results of 3003 honeycombs. Honeycomb number

Material

h/S (mm)

Relative density (%)

pquasi-static (MPa)

pimpact (MPa)

c

1

Al3003

2.57

1.24

1.96

58.1

2

Al3003

3.70

2.79

4.06

45.5

3

Al3003

4.61

4.00

5.51

37.8

4

Al3003

0.05/ 5.2 0.06/ 4.33 0.06/ 3.46 0.04/ 3.46

3.08

1.20

1.75

45.8

9 8

o

Pressure (MPa)

No 3 (Al3003) 7

(%)

6 5 4 3 2 1 0 -1

0

1

2

3

4

5

6

7

Crush (mm) Fig. 3b. Reproducibility of impact experiments on honeycomb No. 3.

can be depicted with respect to the measured crush displacement. For example, Fig. 3a shows the pressure/crushing displacement curves for the three independent quasi-static tests on the specimen No. 3 (h = 0.06 mm/S = 3.46 mm). There is a scatter on the initial buckling peak force and the locking strain, but the average plateau level is very stable. The average value of the pressure between the initial peak and locking strain is used as the quasi-static pressure pquasi-static. For the SHPB tests, the pressure and the crush are calculated using Eq. (2). Fig. 3b illustrates the three repeated tests under

impact loading for the same kind of specimens No. 3 (Table 1). There exist larger oscillations probably due to impact testing imperfection. However, the scatter on the average plateau level is small. It is noticed that the crush reached in impact test is quite limited (around 6 mm) because of limited length of impulse in the Hopkinson bars system, which gives also an enlarged impression of this oscillations. A direct comparison between dynamic and quasi-static pressure/crush curves is shown in Fig. 4 for the case of specimen No. 4 (Table 1a). Quasi-static specimens undergo much larger crush compared to the dynamic ones. Oscillations under impact loading in the plateau stage in such figure are not significantly higher than the quasi-static case. As in the case of quasi-static loading, an average plateau value of the pressure pdynamic can be derived. The impact enhancement ratio c is defined as follows (Eq. (3)):

c ¼ ðpdynamic  pquasistatic Þ=pquasistatic

ð3Þ

Table 1b gives the summary of testing results of all the four 3003 honeycombs. From those testing results, it is clear that the strength of 3003 alloy honeycomb under out-of-plane compression exhibits an important impact enhancement around 40%.

B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762 Table 2 Summary of the parameters and experimental results of 5052 honeycombs. Honeycomb number

Material

5

Al5052

6

Al5052

7

Al5052

pimpact (MPa)

c

1.7

2

17.6

2.76

3

3.5

16.7

4.72

5.1

5.7

11.8

h/S (mm/ mm)

Relative density (%)

0.076/ 9.52 0.076/ 6.35 0.076/ 4.76

1.78

pquasistatic

(%)

(MPa)

Fig. 5. Scheme of honeycomb cross section and the unit cell-model.

Secondly, three Alcore 5052 honeycombs (Duracore) were also studied. Samples were designed as columns of 50 mm high with hexagonal cross section containing a maximum number of cells in a circle of 60 mm diameter. For the honeycomb of largest cells, there are still 5 cells at least in one edge of the hexagon. Quasi-static and SHPB tests (1014 m/s) were performed in the out-of-plane direction. Table 2 gives the characteristics of different tested honeycombs (density, cell size and wall thickness) as well as the quasistatic and dynamic mean crushing pressure results (Zhao et al., 2005). It is found that the enhancement for 5052 alloy honeycombs is rather small (<20%), compared to that of 3003 alloy honeycombs (>40%), noting that this is a general trend obtained from different wall thickness/cell size ratios in the same range of relative density (3 for 5052 and 4 for 3003). 2.3. Numerical simulation of honeycombs The rate sensitivity of bulk aluminium alloy is known to be small (<10%, see Duffy et al., 1971). Recent testing results on thin

(a)

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aluminium sheet metals (6065 T5 in compression (Zhao, 1997), 2024 T3 under shear loadings (Zhao et al., 2007)) reveal also a limited rate sensitivity (<10%). Therefore, the significant impact enhancement (>40%) observed in 3003 honeycombs cannot be directly derived from the rate sensitivity of the cell wall base materials. There must be a structural reason responsible for this enhancement. For this purpose, numerical simulation of the above honeycomb testing results is developed. Since our study focuses on the behavior of honeycombs, the modelling of the whole testing environment is not necessary. Thus, only honeycomb structures were modeled here and the loading environment was modeled by two rigid planes moving at the prescribed velocities corresponding to the average value of those measured in the experiments. Commercial FEM code of ABAQUS/ Explicit was employed for this simulated work. In order to reduce the calculation cost with a complete honeycomb model with the same geometry as the hexagonal honeycomb specimen (Fig. 2) honeycomb specimen was simplified into a unit cell consisting of three conjoint half walls in Y-shape (Fig. 5) because of its periodicity (Mohr and Doyoyo, 2004b; Hou et al., 2011; Wilbert et al., 2011). The simplified models work with symmetric boundary conditions applied on the three non-intersecting edges of each cell wall. It is noticed that the leg of this Y-shape cell-model is a thick wall in a real honeycomb typically made of two single-thickness thin walls which are bonded together. In this model, we ignore the rare delamination of the bonded interfaces and consider the strength of the adhesive bond as infinite. Thus, the simulations are carried out for a monolithic structure, where the thick walls are represented by the same shell elements but with a double thickness value. The model is meshed with 4-node doubly curved thick shell elements with a reduced integration, active stiffness hourglass control (S4R) and 5 integration points through the cell wall thickness. In order to determine the appropriate element size, a convergence study was performed among different element sizes. The element size is finally chosen to be 0.1 mm. The numerical specimen is placed between two rigid planes moving at constant velocities, which take the mean value of real tests (i.e. 27 m/s for 3003 alloy honeycomb and 15 m/s for 5052 alloy honeycomb). In this model, general contact with frictionless tangential behavior is defined for the whole model excluding the contact pairs of rigid planes and tested honeycomb specimen, which are redefined by surface-to-surface rough contact to make sure that no slippage occurs. As the real honeycomb is always far from perfect, it includes all kinds of imperfections which affect the initial peak value, but have little influence on the crush behavior at a large strain. These imperfections are due to various reasons, like irregular cell geometry, uneven or pre-buckled cell walls, wall thickness variation etc. Here in

(b)

Fig. 6. Two honeycomb cell-model with different cell-size (with initial imperfection).

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B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762

this work, we generated the imperfections by preloading the perfect specimen uniaxially by 0.1 mm. Then, the obtained displacement of nodes are introduced geometrically into the actual model used for further calculation. The value of 0.1 mm is chosen to make sure that the simulated initial peak force is the same as the one from experimental curve at uniaxial compression. Fig. 6 illustrates two numerical models of different cell sizes as well as details of introduced imperfection. It shows that the imperfections introduced by preloading is very small. Quasi-static simulations are almost impossible to achieve with ABAQUS/Standard which uses Newton’s method (or quasi Newton’s method) as a numerical technique due to the complex nonlinear effects, e.g. the geometrical and material nonlinearity, the complex contact conditions as well as the local instability during crush. An alternative is to use also ABAQUS/Explicit for quasi-static problems. However, the explicit integration scheme of dynamic simulation codes usually leads to very small time step which in our simulation is around ten nanoseconds for the chosen element size. Thus, with the loading velocity of 0.1 mm/s, the computational duration for the quasi-static simulation (e.g. 130 s) will be too large. To overcome this difficulty, automatic mass scaling technique was employed to increase the time increment to 10 ls. The quasi-static loading conditions are guaranteed by ensuring the ratio of the kinetic energy to the strain energy as a small value (of the order of 104) with the chosen time increment. It is known that such mass scaling might introduce an artificial strength enhancement when the introduced imperfection is small (Langseth et al., 1999). In our simulation, the imperfection obtained by 0.1 mm axial crushing seems to be well-adapted because the different prescribed time-step do not generate significant scatter (Hou, 2011). A bilinear elastic-plastic material model until 20% strain and perfect plastic afterwards was employed to describe the cell wall material of the aluminum honeycombs. For 5052 H38 honeycomb, a yield stress of 290 MPa and a small hardening was usually admitted in many previous works (Papka and Kyriakides, 1994). For 3003 honeycombs, parameters were identified initially with the tensile testing results of 1 mm thick dog bone type 3003-O sheet specimen using universal testing machine and Split Hopkinson tensile bar test (Fig. 7). One can see that its small rate sensitivity is confirmed. However, the hardening or thermal treatment of our 3003 honeycomb is unknown. The testing results of 3003-O give only a reference of the real base material behavior of 3003 honeycomb. The model parameters of the base material such as yield stress and

Table 3 Bilinear material parameters. Material

Al3003 Al5052

Hardening modulus Et (MPa)

2700 2700

70 70

0.35 0.35

70 290

1150 500

3.0

2.5

400

Pressure (MPa)

Al3003 Static Exp. Al3003 Dyn. Exp. Al3003 Moel Al5052 Model

500

Stress (MPa)

Yield stress rs (MPa)

hardening modulus (Table 3) were finally determined by fitting one quasi-static calculation result (i.e. No. 1 (Al3003) and No. 6 (Al5052)) to the experimental one for two base materials respectively. These bilinear curves are also illustrated in Fig. 7. Fig. 8 shows a comparison between experimental and simulated pressure/crush curves for honeycomb No. 1 (Al3003) under quasistatic loading. It shows that the cell-model exhibits significant fluctuations at the plateau stage, which is probably due to the application of excessive symmetric boundary constraints. Actually, it is well known that the crushing behavior of honeycombs under out-of-plane compression is regulated by the successive folding procedure of honeycomb cell walls. With the symmetric boundary condition on three non-intersecting edges, the cell-model is actually equivalent to a honeycomb specimen consisting of repeated cells with identical deforming procedure, which results in a strictly simultaneous collapse of all the honeycomb cells. Thus, in the pressure/crush curve, each fluctuation represents one fold formation of the cell wall in honeycomb structure. For the large size model, the neighboring cells interact with each other while forming the folds and reach their local peak value at different instants, which makes the macroscopic resulting curves smoother (Hou et al., 2011; Wilbert et al., 2011). However, the mean crushing pressure is hardly affected. We finally use this cell-model for the subsequent calculations. Fig. 9 shows the comparison of experimental and numerical pressure/crush curves for the honeycomb No. 1 (Al3003) under quasi-static and impact loadings. Even though the numerical pressure enhancement is smaller than the experimental one, which could be due to base material rate sensitivities or other effects not taken into account in this numerical model, the basic trend is preserved. Table 4 gives the simulated average plateau values of crushing pressure under quasi-static and impact loading for all the tested 3003 and 5052 honeycombs. These numerical results suggest also a significant enhancement under impact for 3003 honeycombs and a small enhancement for 5052 honeycombs.

600

500MPa 300

Poisson’s ratio m

m3)

Young’s modulus E (GPa)

Density

q (kg/

290MPa 1150MPa

Cell-model Experiment

2.0

1.5

1.0

200

Quasi-static 0.5 o

100

No 1 (Al3003)

70MPa 0.0

0 −0.05

-1

0

0.05

0.1

0.15

0.2

0.25

Strain Fig. 7. Prescribed constitutive relation of 5052 and 3003 alloys.

0.3

0

1

2

3

4

5

6

7

8

9

Crush (mm) Fig. 8. Comparison between numerical and experimental results for Al3003 cellmodel under quasi-static loading.

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b=1.833mm

3.5

Static Exp. Static Cal. Dynamic Exp. Dynamic Cal.

3.0 2.5

Pressure (MPa)

h=152 m

v

L=0.5mm

1

2.0

3

1.5

2

1.0 0.5 o

0.0

No 1 (Al3003)

Fig. 11. Scheme of double-plate model.

-0.5 -1

0

1

2

3

4

5

6

7

8

9

Crush (mm)

2.4. Lateral inertia effect on the successive folding mechanism of honeycombs

Fig. 9. Comparison between the calculating and experimental results.

Table 4 Simulated pressure enhancements of 3003 and 5052 honeycombs. Honeycomb number

Material

h/S (mm/ mm)

pquasi-static (MPa)

pimpact (MPa)

c (%)

1 2 3 4 5 6 7

Al3003 Al3003 Al3003 Al3003 Al5052 Al5052 Al5052

0.05/5.2 0.06/4.33 0.06/3.46 0.04/3.46 0.076/9.52 0.076/6.35 0.076/4.76

1.24 2.27 3.61 1.80 1.55 2.99 4.62

1.83 3.57 4.92 2.49 1.66 3.58 5.39

48 57.1 36.3 38.6 7.38 19.8 16.7

70 Al3003 h/S=0.06/4.33

Dynamic enhancement (%)

60

Experiment Calculation Al3003 h/S=0.04/3.46

50 Al3003 h/S=0.05/5.2

40

Al3003 h/S=0.06/3.46

30

Al5052 h/S=0.076/6.35

20 Al5052 h/S=0.076/9.52

10

Al5052 h/S=0.076/4.72 0 0

1

2

3

4

5

6

7

8

Specimen Number Fig. 10. Summary of the calculated and experimental dynamic enhancement for both 3003 and 5052 alloys.

Fig. 10 depicts comparison of the simulated and experimental enhancement ratios for all the 5052 and 3003 honeycombs. The numerical models follow well the experiments. It shows that the numerical enhancement is around 40% for Al3003 honeycombs and less than 20% for 5052 ones. Thus, this enhancement has its origin in the structural response and this structural response should be related to the constitutive relationship of two materials because the geometry and loading conditions are similar in numerical models for 3003 and 5052 honeycombs.

2.4.1. Lateral inertia effect in a simple double-plate model The early theoretical work on the lateral inertia effect was reported by Budiansky and Hutchinson (1964). Gary (1983) showed experimentally that the buckling of a column under compressive impact occurs at a larger plastic strain. Calladine and English (1984) identified velocity-sensitive type II structure and Tam and Calladine (1991) revealed that there exists, under dynamic loading, an initial phase where the compression is dominant before a second phase of bending. More sophisticated models were also reported (Karagiozova and Jones, 1995; Su et al., 1995). It leads to the fact that the buckling of an elastic-plastic column under compressive impact occurs at a larger strain (than under quasi-static loading) because of necessary transverse acceleration. If the elastic-plastic column has a strain-hardening behavior, the buckling peak force will also increase as shown in the numerical work of Webb et al. (2001). In order to illustrate this lateral inertia effect without long analytical formulas and to quantitatively evaluate the magnitude of potential augmentation of dynamic buckling force due to inertia effect in the honeycombs, a double-plate numerical model with dimensions comparable to honeycomb is built using ABAQUS. The scheme of this model is shown in Fig. 11 (like most previous works cited above), which is composed of two plates connected with an angle (for the initial imperfection). The size of the model is in the same order with honeycomb cell walls with the plate thickness h = 152 lm, plate width b = 1.833 mm and height of one plate L = 0.5 mm, d is the maximum deviation of plates from the vertical line, which represents the magnitude of initial imperfection of this model. In this study, the initial imperfection employed is fixed to 3.2 lm (much more exaggerated in Fig. 11) in order to avoid the undesirable elastic buckle before the plastic collapse. The doubleplate model is sandwiched between two parallel rigid walls (one fixed and another moving at prescribed velocity). The loading velocity is 0.1 mm/s for quasi-static case and 15 m/s for dynamic loading. A surface-to-surface rough contact is defined at the interfaces of double-plate model and rigid walls to make sure that no slippage occurs. Such simulation permits to obtain the force and displacement time histories, which can be converted to nominal stress and strain by being divided by the plate cross sectional area and the initial distance between rigid walls. Fig. 12 shows the calculated quasi-static and impact nominal stress-strain curves for both 3003 and 5052 alloys, compared with the prescribed constitutive relations. It can be found that the collapse point (at which the curve begin to decrease rapidly) of the quasi-static curve coincides with the yield point. However, under impact loading, the collapse cannot take place at yield point and

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5052 material 5052 dynamic 5052 static 3003 material 3003 dynamic 3003 static

400

Stress (MPa)

350 300 250 200 150 100 50 0 -0.025

0.000

0.025

0.050

0.075

0.100

0.125

0.150

Strain Fig. 12. Force divided by cross section area vs. displacement divided by length.

the double-plate model is further compressed. Therefore, the model undergoes larger plastic strains in axial direction under impact loading before the collapse occurs. As the strain hardening curve is different, the stress increase due to this larger strain is different and especially the ratio of this stress increase over the yield stress is different. From constitutive relation shown in Fig. 7, for the 3003 alloy, with a yield stress of 70 MPa and a hardening modulus of 1150 MPa, 5% of strain enhancement induces more than 80% stress increase. For the 5052 alloy, with a yield stress of 290 MPa and a hardening modulus of 500 MPa, 5% of strain enhancement induces only less than 9% of rise in stress. This is considered to be the reason for the pressure enhancement difference between 5052 and 3003 honeycombs.

Fig. 14. Formation of the second fold of honeycomb cell-model.

of this corner part determines the successive peak load. For the honeycomb structure, this corner part corresponds to the region near the intersectional line. In our Y-shape cell-model (as shown in Fig. 13(b) the deformed profile), the analysis should be then focused on the central intersectional line of three cell walls. Fig. 13(a) shows the pressure/crush curve of a cell-model for honeycomb No. 1 (Al3003) taken as an example. Large fluctuation with each wave representing one fold formation is observed. We consider the formation of the second fold to illustrate the

2.5. Lateral inertia effect in the honeycomb cell-models Such a lateral inertia effect under impact loading exists also in the successive folding of tube-like hollow structures (Langseth and Hopperstad, 1996; Zhao and Abdennadher, 2004; Karagiozova and Alves, 2008). It has been found that in the successive crushing process of square tube, the corner region (intersection of two flat plates) supports most of the external loadings and the buckling

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Fig. 13. Deformation profile (a) and pressure/crush curve (b) of honeycomb cell-model.

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successive folding system of honeycomb under out-of-plane compression. The deformed profile of cell-model at point B in Fig. 13(a) is shown in Fig. 13(b). The deformation sequence of the basic cellmodel is shown in Fig. 14 (only the thick wall is displayed for sake of illustration’s clarity). The formation of the second fold begins from point A in Fig. 13(a). At this moment, the first fold has completely collapsed and the material of the second fold begins to support loading (Fig. 14(a)). The continuous axial deformation of the second fold enables the carrying capacity of the cell-model to increase gradually (Segment a in Fig. 13(a) and the deformation image in Fig. 14(b)). During this process, the intersectional line (as shown in Fig. 14(a)) and its adjacent region remains straight, while the plate region has been buckled. The peak load of the second fold is reached (Point B in Fig. 13(a)) when the intersectional line and its adjacent region begin to buckle (as show in Fig. 14(c)). After this peak point, the overall carrying capacity decreases dramatically (segment b in Fig. 13(a)) and the corresponding deformation of the cell-model is characterized with the bending of intersection region (as shown in Fig. 14(d)). When the carrying capacity of the cell-model reaches the trough C in Fig. 13(a), the third fold initiates

and will repeat the above-mentioned process, i.e. the C-c-D-d process in Fig. 13(a). The successive folding is controlled by the successive buckling of region near the central intersectional line. It is important to notice that the intersectional line remain rather straight (Fig. 14(c)) before its buckling. Thus, the initial imperfection is small enough for each single successive fold so that the lateral inertia effect like in the double-plate model will apply. Figs. 15a and 15b depicts the equivalent strain profiles (still considering that the strain here is the calculated wall material equivalent strain) along the central line of one fold (see Fig. 14(d)) in both the double thickness wall and the single thickness wall just before the buckling (at the successive force peaks, points B, D in Fig. 13(a)). For honeycomb No. 1 (Al3003) model (Fig. 15a) as well as No. 6 (Al5052) model (Fig. 15b), one can see that the strain reached before buckling is higher under impact loading (27 m/s for Al3003 and 15 m/s for Al5052) than the case under quasi-static loading. The closer is the position to the intersectional line; the larger is the increase in strain. The lateral inertia effect is then clearly seen in the successive buckling of the central intersectional line of the Y-shape cell-model.

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Fig. 15a. Strain distribution in the central line of a fold under quasi-static and dynamic loading, honeycomb No. 1 (Al3003).

Fig. 16a. Stress distribution in the central line of a fold under quasi-static and dynamic loading, honeycomb No. 1 (Al3003).

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Distance from the intersection line (mm) Fig. 16b. Stress distribution in the central line of a fold under quasi-static and dynamic loading, honeycomb No. 6 (Al5052).

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The stress profiles just before the buckling are also shown in Figs. 16a and 16b. On the one hand, there is a larger difference of stress levels between impact and quasi-static loadings for 3003 model which is due to the important strain hardening behavior of prescribed 3003 constitutive law, especially the ratio between the stress increase due to strain hardening and the yield stress. On the other hand, because of prescribed flat strain hardening behavior for 5052 model, the stress level differences for 5052 are less significant even though a noticeable strain difference due to lateral inertia is observed. 3. Conclusion Quasi-static and impact tests with large diameter polymer SHPB were performed on 7 kinds of aluminium (5052 or 3003) honeycomb with a relative density varying from 1.78% to 4.72%. The results show that the enhancement under impact loading of the crushing pressure depends on the base material. In fact, 4 different 3003 honeycombs have an enhancement larger than 40% while the 3 types of 5052 honeycombs exhibit an enhancement less than 20%. Abaqus models using a Y-shape cell-model with rate insensitive constitutive law were performed to simulate the studied sets of honeycombs under quasi-static and impact loadings. This result confirmed numerically this difference of enhancement. Finally, strain and stress profiles in single Y-shape cell during the successive folding (especially at the force peaks) were analysed. The strain reached before collapse (successive force peak) under impact loading is bigger than that under quasi-static loading. This is considered to be due to the lateral inertia effect of type II structure, well studied in the past. As the 3003 alloy has a bigger ratio between the stress increase due to strain hardening and the yield stress than that of 5052 alloy, the crushing pressure enhancement under impact loading of 3003 honeycombs is therefore bigger than 5052 honeycombs. Acknowledgement The authors would like to thank 111 project of China (Contract No.1307050) for funding the cooperation between NPU and LMT. B. Hou and Y. L. Li would also like to thank the supports of the National Science Foundation of China (Contract Nos. 10932008 and 11072202). References Baker, W.E., Togami, T.C., Weydert, J.C., 1998. Static and dynamic properties of highdensity metal honeycombs. Int. J. Impact Engng. 21, 149–163. Budiansky, B., Hutchinson, J. W., 1964. Dynamic buckling of imperfection sensitive structures. In: Proceedings of 11th international congress of Applied Mechanics. Springer Verlag, Munich. Calladine, C.R., English, R.W., 1984. Strain-rate and inertia effects in the collapse of two types of energy-absorbing structures. Int. J. Mech. Sci. 26 (11–12), 689–701. Davies, R.M., 1948. A critical study of Hopkinson pressure bar. Phil. Trans. Roy. Soc. A240, 375–457. Duffy, J., Campbell, J.D., Hawley, R.H., 1971. On the use of a Torsional Split Hopkinson Bar the study rate effects in 11000 Aluminum. J. Appl. Mech. 38, 83– 91. Follansbee, P.S., Franz, C., 1983. Wave propagation in the split Hopkinson pressure bar. J. Engng, Mater. Tech. 105, 61–66. Gary, G., 1983. Dynamic buckling of an elastoplastic column. Int. J. Impact Engng. 2, 357–375. Gibson, L.J., Ashby, M.F., 1988. Cellular Solids. Pergamon Press, Oxford. Goldsmith, W., Sackman, J.L., 1992. An experimental study of energy absorption in impact on sandwich plates. Int. J. Impact Engng. 12, 241–262. Goldsmith, W., Louie, D.L., 1995. Axial perforation of aluminium honeycombs by projectiles. Int. J. Solids Struct. 32, 1017–1046.

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