Microstructure and mechanical properties of ALPORAS closed-cell aluminium foam

Materials Characterization 107 (2015) 228–238

Contents lists available at ScienceDirect

Materials Characterization journal homepage: www.elsevier.com/locate/matchar

Microstructure and mechanical properties of ALPORAS closed-cell aluminium foam Wen-Yea Jang ⁎, Wen-Yen Hsieh, Ching-Chien Miao, Yu-Chang Yen Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 21 April 2015 Received in revised form 9 July 2015 Accepted 10 July 2015 Available online 17 July 2015 Keywords: Closed-cell foam Microstructure Mechanical properties Compressive response

a b s t r a c t In recent years, closed-cell foam has drawn increasing attention in applications ranging from energy absorption devices to serving as the core material of light-weight structures. Although closed-cell foams are critical in many applications, the microstructural characteristics and mechanical properties of these foams are not fully understood. In this paper, we performed a comprehensive study on the commercially available ALPORAS closed-cell aluminium foam by means of experiment. First, inspection of the foam microstructure was systematically performed. Based on this inspection, the geometric features of the foam were characterised. Crushing experiments were subsequently conducted on specimens under various conditions to examine the effect of each of the testing parameters such as the specimen size and geometry as well as the loading direction, on the mechanical response of closed-cell foam. We found that the cells were irregular polyhedra with approximately 14 faces and that each face had approximately 5 sides. The cell wall was thinnest in the middle section, became increasingly thicker towards the edges, and eventually formed circular fillets as it intersected with neighbouring walls. The foam was geometrically and mechanically anisotropic. The factors that most influenced the mechanical properties were the loading direction and relative density. In this paper, we also compare the experimental results on the foam mechanical properties with existing equations from the literature. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Because of their complex microstructures, closed-cell foams have unique physical and mechanical properties, making them niche candidates in a variety of engineering applications. For instance, because of their high strength-to-weight and modulus-to-weight ratios, closed-cell polymethacrylimide foams are used in the aerospace industry as the core material in helicopter fuselage panels and satellite launch vehicles [1,2] to reduce weight, thereby saving fuel. Because of their ability to undergo large deformation with relatively constant reaction force, closed-cell aluminium foams are used in energy absorption devices in various types of vehicle [3]. In addition to toughness, corrosion resistance, and biological compatibility (biocompatible with the human body), titanium-based foams have much lower stiffness (compared with solid titanium metal) that is more similar to that of human cancellous bones. Titanium foams are widely used to replace solid metal for skeletal repair and as bone implants [4] because they can alleviate the ‘stress shielding’ phenomenon. Although the aforementioned applications concern the mechanical properties of foams, foam mechanical properties depend heavily on (1) the cellular microstructure geometry and (2) constitutive behaviour of the material from which the foam is made, also referred to as the ‘base material’ [5,6]. In applications that exploit foam mechanical properties, ⁎ Corresponding author. E-mail address: [email protected] (W.-Y. Jang).

http://dx.doi.org/10.1016/j.matchar.2015.07.012 1044-5803/© 2015 Elsevier Inc. All rights reserved.

the relationship between the cellular microstructure and foam mechanical properties must be established [7,8]. The mechanical behaviours of open-cell foams have been extensively investigated in the past few decades (e.g., Gibson and Ashby [6]). By contrast, literature on the mechanical properties of closed-cell foams, unlike their open-cell counterparts, is limited. For example, Brunke el al. [9] performed characterisation on the cellular microstructure of several closed-cell aluminium foams through microtomography. Furthermore, Sugimura et al. [10] performed an experimental study on several types of commercially available closed-cell aluminium foams in which the effect of microstructural imperfections (such as curves and wiggles in the cell walls) on the stiffness and strength was investigated. Additionally, Simone and Gibson [11] tested 2 types of closed-cell aluminium foams made using different liquid-state production methods and proposed (empirical) equations describing the mechanical parameters. Moreover, Bastawros et al. [12] studied the evolution of deformation mechanisms of ALPORAS closed-cell aluminium foam through a digital image correlation procedure. In addition, Ramamurty and Paul [13] examined the dependence of relative density on Young's modulus and the compressive strength of ALPORAS closed-cell aluminium foam. Furthermore, Santosa and Wierzbicki [14] developed a numerical model based on truncated cube geometry to study the deformation mechanism as well as crushing resistance of closed-cell foam. Meguid et al. [15] employed a representative unit cell model to study the crush behaviour of closed-cell metallic foams and successfully captured

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

150 mm

2 3

1 (G)

600 mm

TOP

50 mm

MID

50 mm

BOT

100 mm

50 mm

600 mm

Fig. 1. Coordinate system and region definitions of the as-received foam block. The direction G is marked by the supplier and is presumed to be the gravity direction during the manufacturing process. The TOP, MID, and BOT regions are cut from the top, middle, and bottom sections of the original block.

deformation localisation patterns similar to those observed in other experiments. Additionally, Youssef et al. [16] generated realistic numerical models of closed-cell polyurethane foam using the X-ray tomography

229

technique. Chen et al. [17] conducted micromechanical modelling of closed-cell M310 polymeric foams in which the effect of cell size and wall thickness distribution on foam stiffness was investigated. Although studies dedicated to experimental and numerical issues have been published continuously, a comprehensive understanding of the optimal use of closed-cell foams is still under development. In the present study the characterisation of the foam cellular microstructure was first conducted. The measured microstructural geometry certainly offers useful information that lays a solid foundation on the development of representative numerical models. A series of systematically designed crushing experiments were subsequently conducted in order to access the effect of various testing conditions on the foam mechanical properties. The experimental findings can undoubtedly provide helpful guidance on the design and use of closed-cell foams. 2. Microstructure characterisation of ALPORAS closed-cell aluminium foam The foam investigated in this study was ALPORAS closed-cell aluminium foam manufactured using a batch casting process developed by Shinko Wire of Japan (the detailed production process can be found in Miyoshi et al. [18]). The as-received foam block had dimensions of 600 mm × 600 mm × 150 mm. Fig. 1 shows a photograph of the entire foam block from which the cellular microstructure can clearly be seen. For convenience, we define the coordinate system shown in Fig. 1 and mark the following 3 regions: the top region (TOP), middle region (MID), and bottom region (BOT).

5mm Fig. 2. Cellular microstructure in various regions: (a) TOP, (b) MID, and (c) BOT regions. The yellow arrows mark the ‘collapse-like’ bands, and the red arrows indicate the missing or broken cell walls. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1 Geometric parameters of ALPORAS closed-cell foam analyzed. Region

h1 (mm)

h1|min − max(mm)

σ h1 =h1

h2 (mm)

h2|min − max(mm)

σ h 2 =h2

λa

ℓ (mm)

σ ℓ =ℓ

MID BOT

4.977 5.120

3.028–8.554 3.023–8.990

0.22820 0.19389

4.157 3.876

2.633–6.915 2.420–6.095

0.20533 0.15724

1.203 1.321

1.630 1.436

0.16941 0.25221

a

λ≡h1/h2.

230

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

(a)

(b) Fig. 3. Cells extracted from various regions of the foam block: (a) MID and (b) BOT regions.

Fig. 2 shows the appearance of the cellular microstructure of foam specimens originating from 3 different regions of the block. In the TOP region, shown in Fig. 2(a), the size and shape of each cell were

different from one another. In other words, the cells tended to have arbitrary sizes and shapes. In some cases, the cellular nature was not obvious compared with cells from the other 2 regions.

50 N

N

1600 1400

40

1200

30

1000 800

20

600 400

10

200

0 6

8

10

12

14 16 18 20 22 Number of faces per cell

Fig. 4. Plot of frequency versus number of faces per cell.

0 2

3

4

5

6

7

8

9

10

Number of edges per face Fig. 5. Plot of frequency versus number of edges per face.

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

231

3 1

(a) Fig. 6. Schematic for calculating the aspect ratio. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

Furthermore, ‘collapse-like’ bands, as marked by yellow arrows in Fig. 2(a), could be observed, which likely resulted from the manufacturing procedures. Because of the abnormality of the cellular microstructure, some of the statistical data presented later in this paper (e.g., data in Tables 1 and 3) do not include the measurements obtained for the TOP region. Similar photos for the MID and BOT regions are shown respectively in Fig. 2(b) and (c). These 2 figures show that the cellular nature in these 2 regions was more evident, and the cell size seemed more uniform. Additionally, foam specimens originating from all 3 regions had microstructural defects and imperfections such as missing and broken cell walls, as indicated in Fig. 2(a)–(c) by red arrows. The number of cells with missing or broken cell walls was estimated in the order of 1 in 10. Measurements of cell dimension, the number of faces per cell, the number of edges per face, and other cellular morphological statistics were then performed. The geometric features of the cellular microstructure were first carefully characterised and recorded. For this purpose, cubical specimens were removed from the foam block. Typically, each specimen was a 50 mm cube, rendering approximately 15 cells in each of the 3 orthogonal directions mentioned previously. Unless otherwise stated, specimens used in this study had this size and shape. The effect of specimen size on the foam mechanical properties was then addressed. The specimens were cut from the foam block by using an electrical discharge machining (EDM) technique, which minimised damage and distortion of the microstructure. To precisely quantify the cellular geometry of the microstructure, 300 individual cells were manually extracted from the three regions of the foam block in a random manner. Two such cells along with their skeletal outlines, obtained by connecting the centres of neighbouring nodes with straight lines, are shown in Fig. 3. The cell shown in Fig. 3(a) had 5 hexagons, 4 pentagons, 4 rectangles, and 1 triangle, totalling 14 faces, and the average number of faces was 4.9. The cell shown in Fig. 3(b) had 1 hexagon, 7 pentagons, 4 rectangles, and 1 triangle, totalling 13 faces, and the average number of faces was 4.7. The cells were irregular polyhedra with, in most cases, nearly straight cell edges. Flat and curved and corrugated

(b)

Area = A 1

2

(c) r 2

t 1

(d) Fig. 7. Schematic for measuring the fillet radius and thickness. (a) A typical cell wall, (b) construction of fillet circles and radii, (c) the length of ℓ1 and ℓ2, and (d) an idealised cell wall with measured r, t, ℓ1, and ℓ2. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

faces were present. With all cell membranes removed, the skeletal outline formed by joining the cell edges indicated that cells extracted from ALPORAS closed-cell foam resembled those extracted from open-cell aluminium foams (e.g., ERG open-cell aluminium foams reported in Jang et al. [8]).

Table 2 Cell aspect ratio. Region

K 12

K 13

K 23

K12|min − max

K13|min − max

K23|min − max

TOP MID BOT

0.891 1.082 1.144

0.856 1.099 1.130

0.991 1.004 0.995

0.802–1.098 1.020–1.145 1.031–1.235

0.755–1.038 1.020–1.182 1.014–1.248

0.844–1.203 0.965–1.069 0.956–1.044

2.1. Cell dimensions Based on measurements performed on 300 cells (MID: 100 cells, BOT: 200 cells), statistical measurements, including average cell size and cell edge length for cells from various regions are reported in

232

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

1000

Table 3 Cell wall measurements.

Average Range STDEV/AVG

ALPORAS Aluminum Foams BOT_Exp.1

Fillet radius r (mm)

Cell wall length ℓ1 (mm)

Cell wall thickness t (mm)

0.8568 0.3646–2.1152 0.35046

1.5584 0.6050–3.8680 0.370

0.1825 0.0748–0.50 0.5039

*

(psi) 800

6

~ = 12.33%

600

4 3

400

2

Table 4 Relative densities of specimens originating from different regions. 

Region TOP MID BOT

200

ρ =ρs (%)

ρ*/ρs|min − max (%)

8.67 8.48 11.66

7.64–10.18 7.77–9.87 10.60–12.38

Table 1. In the BOT region, the average dimension in the ‘semimajor axis’ direction (the h1 direction) was approximately 5.1 mm, and in the other 2 orthogonal directions (the h2 and h3 directions; the lengths in these 2 directions were found to be similar and thus only h2 measurements are shown), this value was approximately 3.9 mm, resulting in an anisotropy of λ = 1.32 (λ≡h1/h2). A similar observation was made in the MID region (average values of h1 and h2 were respectively 4.98 and 4.16 mm and λ = 1.20). Such an observation indicated that instead of a presumably spherical shape, the cells tended to have the shape of a prolate ellipsoid. It should be noted that since h1 always depicts the longest dimension of each extracted cell, its direction does not correspond to any of the D1, D2, and D3 directions defined in Fig. 1. A subsection titled ‘cell aspect ratio’ is dedicated to more effectively describing the cell dimensions in the 3 directions. The value of polydispersity (defined as σ h 1 =h1 , where σ h 1 and h1 denote respectively the standard deviation and the average of h1) was approximately 0.2 for the MID and BOT regions. This implied that the cell size of ALPORAS foam varied and thus might not be appropriately regarded as ‘monodispersed’.

Fig. 4 shows the number of faces (n) versus the frequency (N). The number of faces per polyhedron (cell), n, ranged from 6 to 22. The average value n and standard deviation σn were 13.68 and 3.86 respectively, based on measurements performed on 300 cells. The data percentage that fell within one standard deviation (n
σ n ) was approximately 64, whereas the value within 2 standard deviations (n
2σ n ) was approximately 95. The distribution was similar to a Gaussian distribution.

BOT_L03_D1 BOT_L04_D2 BOT_L02_D3

0

(psi) 800

s

6

0.3

0.4

0.5

0 0.7

0.6

Fig. 9. Comparison of compressive responses in various loading directions for foam specimens from the BOT region.

2.3. Number of edges per face Fig. 5 shows the number of edges per face (n) versus the frequency (N). The number of edges per face, n, was found to be between 3 and 9. The average n and standard deviation σn were 4.60 (close to the value reported in Gong et al. [7] for open-cell polymeric foams) and 0.32 respectively. The data percentage that fell within one standard deviation (n
σ n ) was approximately 66, whereas the value within 2 standard deviations ( n
2σ n ) was approximately 97. Similar to the distribution in Fig. 4, the distribution in Fig. 5 was similar to a Gaussian distribution as well. 2.4. Cell aspect ratio Another critical issue was to examine whether the cellular microstructure was anisotropic geometrically. Unlike open-cell foams, the existence of cell membranes made this task difficult. To quantify the cell anisotropy, a measure referred to as cell aspect ratio Kij is defined as (DeHoff and Rhines [19]; Mu et al. [20]) di dj

; i; j ¼ 1; 2; 3 ði≠jÞ

ð1Þ

where di and d j are the average length of the cells projected on the test plane in the Di and Dj directions, respectively. The method used to determine the cell aspect ratio Kij is demonstrated schematically in Fig. 6, which shows a square cutting face (side length = 20 mm) on the D1 − D3 plane. To begin, equally spaced (spacing = 4 mm) horizontal and vertical test lines (dashed lines)

(MPa)

(psi) 400

*

s

3

~ 9.04% =

5

p

600

0.2

ALPORAS Aluminum Foams TOP_Exp.1

= 12.34%

I

0.1

II

1

/h

ALPORAS Aluminum Foams *

(MPa)

2

III

4 3

400

200 1

I

2

*

E

200 0

0

Ki j ¼

2.2. Number of faces per cell

1000

(MPa)

5

s

1 0

0.1

TOP_L04_D1 TOP_L02_D2 TOP_L01_D3

p

0.2

0.3

0.4

0.5

0.6 /h

0 0.7

Fig. 8. Compressive displacement-stress response of ALPORAS closed-cell foam.

0

0

0.1

0.2

0.3

0.4

0.5

0.6

/h

0 0.7

Fig. 10. Comparison of compressive responses in various loading directions for foam specimens from the TOP region.

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

233

Table 5 Experimental results of foam specimens from different regions (a) BOT, (b) MID,(c) TOP. Specimen # (a) Exp. 1 BOT_L03_D1 BOT_L04_D2 BOT_L02_D3 Exp. 2 BOT_L09_D1 BOT_L12_D2 BOT_L10_D3 Exp. 3 BOT_L05_D1 BOT_L08_D2 BOT_L06_D3 Exp. 4 BOT_R01_D1 BOT_R02_D2 BOT_R03_D3 Exp. 5 BOT_L04_D1 BOT_L05_D2 BOT_L06_D3 Exp. 6 BOT_L07_D1 BOT_L08_D2 BOT_L09_D3 (b) Exp. 1 MID_L09_D1 MID_L10_D2 MID_L11_D3 Exp. 2 MID_L02_D1 MID_L01_D2 MID_L04_D3 Exp. 3 MID_L05_D1 MID_L06_D2 MID_L07_D3 Exp. 4 MID_R01_D1 MID_R02_D2 MID_R03_D3 Exp. 5 MID_L04_D1 MID_L05_D2 MID_L06_D3 Exp. 6 MID_L07_D1 MID_L08_D2 MID_L09_D3 (c) Exp. 1 TOP_L04_D1 TOP_L02_D2 TOP_L01_D3 Exp. 2 TOP_L12_D1 TOP_L11_D2 TOP_L09_D3 Exp. 3 TOP_L08_D1 TOP_L06_D2 TOP_L05_D3 Exp. 4 TOP_R06_D1 TOP_R04_D2 TOP_R05_D3 Exp. 5 TOP_L09_D1 TOP_L07_D2 TOP_L08_D3 Exp. 6 TOP_L03_D1 TOP_L01_D2 TOP_L02_D3

ρ*/ρs(%)

E*/Es(%)

σI (MPa)

σ P (MPa)

ΔεP(%)

12.34 12.28 12.36

1.603 1.213 1.143

3.32 2.60 2.58

3.56 2.82 2.79

49 45 45

11.27 11.05 10.61

1.095 1.107 1.202

2.41 2.37 2.47

2.66 2.51 2.50

48 48 49

11.78 11.62 11.60

1.523 1.160 1.127

3.17 2.47 2.54

3.38 2.87 2.89

49 47 48

12.35 12.36 12.38

1.515 1.197 1.258

3.38 2.70 2.68

3.61 2.87 2.89

49 46 46

11.91 11.75 11.75

1.494 1.147 1.160

3.07 2.55 2.57

3.34 2.57 2.71

49 48 48

10.83 10.85 10.65

1.293 1.111 1.050

2.53 2.40 2.38

2.73 2.47 2.48

47 50 50

9.87 9.75 9.26

1.071 0.847 0.848

2.05 1.77 1.61

2.10 1.85 1.76

53 50 49

7.82 7.83 7.88

0.895 0.662 0.656

1.72 1.38 1.29

1.60 1.38 1.41

60 56 52

8.39 8.45 8.40

0.934 0.586 0.791

1.81 1.69 1.63

1.77 1.66 1.68

58 56 54

7.82 7.77 7.80

0.888 0.787 0.723

1.75 1.62 1.42

1.60 1.50 1.46

62 61 59

8.35 8.28 8.38

0.918 0.888 0.877

1.82 1.75 1.72

1.78 1.70 1.72

58 58 58

9.21 8.79 8.74

0.938 0.870 0.810

1.90 1.76 1.68

1.95 1.76 1.76

55 55 53

9.18 9.10 8.83

0.368 0.777 0.831

1.11 1.90 1.56

1.21 1.74 1.63

46 62 55

8.01 7.95 7.88

0.500 0.730 0.567

1.25 1.55 1.30

1.32 1.44 1.30

55 61 58

8.75 8.60 8.68

0.359 0.767 0.803

1.02 1.66 1.71

1.10 1.55 1.65

46 61 60

9.03 8.97 9.29

0.431 0.819 0.858

1.14 1.76 1.94

1.22 1.65 1.78

48 60 60

7.85 8.21 7.65

0.511 0.682 0.566

1.23 1.51 1.22

1.27 1.47 1.27

53 59 56

9.53 10.18 10.13

0.306 0.843 1.012

1.09 1.94 2.28

1.19 1.94 2.05

44 57 60

234

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

20mm cube

35mm cube

50mm cube

65mm cube

Fig. 11. Foam specimens of various sizes.

were drawn, as shown in Fig. 6. The first horizontal test line (bold blue dashed line on the top) crossed 3 cells, and the length of the test line was 20 mm. The quantity di (defined as the ratio between the test line length and the cells being crossed) was then d1= 20 mm/3 = 6.67 mm, which physically represented the cell length projected to the cutting face in the horizontal (D1) direction corresponding to the particular test line. A similar procedure was applied to the first vertical test line (bold green dashed line on the left). With the same test length of 20 mm, 5 cells being crossed yielded d3 = 20 mm/5 = 4 mm. Repeating the same procedure for all horizontal test lines, the average value for d1 andd1 were found to be 4.76. Similarly,d3 was found to be 4.35. Moreover,

Fig. 7(a) were taken for 101 different cell walls. On each of the 4 corners, for example, Corner 1 takes 3 points (the 3 orange dots on the cell wall edge) to determine a circle (the fillet circle) and its radius (the fillet radius r1). The fillet radius r of this particular cell wall is then taken as the average of the 4 radii corresponding to the 4 corners: r¼

4 1X r: 4 i¼1 i

ð2Þ

The length ℓ2 is found by subtracting a length d from the cell wall length ℓ1 (the cell wall length reported in Table 1). The length d

the ratio of d1 to d3 yielded the aspect ratio K13 for the cutting plane K 13 ¼ d1 =d3 ¼ 4:76=4:35 ¼ 1:09. When a Kij value is greater than 1, the cell length in the Di direction is longer than that in the Dj direction, and vice versa. For each of the specimens prepared, 3 of 6 faces were measured using the described method. The results are reported in Table 2. Based on the information in Table 2, in the MID and BOT regions, K12 and K13 N 1, whereas K23 ≈ 1, indicating that the cells were longest in the D1 direction and have similar dimensions in the other 2 directions. However, in the TOP region, cells were shortest in the D1 direction and again had similar dimensions in the other 2 directions (K12, K13 b 1; K23 ≈ 1). Although it has been considered geometrically isotropic in many studies (e.g., Simone and Gibson [21]; Deshpande and Fleck [22]; Ramamurty and Paul [13]), ALPORAS foam investigated in the present study was found to be anisotropic. Furthermore, geometric anisotropy has been shown to result in mechanical anisotropy (e.g., Huber and Gibson [23]; Gong et al. [7]; Jang et al. [8]). This was addressed when we examined the experimental data.

600

4

ALPORAS Aluminum Foams *

(psi)

~ = 8.75%

(MPa)

s

3

400

2 200 65mm 50mm 35mm 20mm

0

0

0.1

0.2

0.3

0.4

0.5

1

0 0.7

0.6 /h

(a)

2.5. Cell wall geometry

E

Close examination of ALPORAS closed-cell foam revealed that the cell wall thickness was not constant. A stereomicroscopic image of a typical wall cross section is shown in Fig. 7(a). The thickness varied in the longitudinal direction. It appeared thicker near 2 ends (junctions of the neighbouring cell walls) and became thinner and more uniform around the middle portion. This trend of cell wall thickness variation resembles a similar observation in open-cell foam. In the case of open-cell foam (e.g., Gong et al. [7]; Jang et al. [8]), the cross-sectional areas of struts tapered such that the area was minimal at the mid-span of the strut but increased as the junctions (nodes) with other struts on either end were approached. This particular geometric shape can be approximately represented by the schematic shown in Fig. 7(b), a rectangular region (the portion where the cell wall thickness is constant) along with 4 circular fillets at 2 ends. To quantify the length of the rectangular region ℓ2, its thickness t, and the fillet radius r, micrographs similar to the one shown in

E (%)

* 1

1.8

1.2 65mm 50mm 35mm 20mm

0.6 8

9

10

11 *

/

12 (%) s

(b) Fig. 12. The effect of specimen size on the (a) complete response and (b) relative Young's modulus.

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

depends on the angle between the present cell wall and the intersected neighbouring cell wall. For example, if the angle is 120°, d ¼ r cos ð120 =2Þ ¼

r : 2

ð3Þ

Therefore, ℓ2 ¼ ℓ1 −2d ¼ ℓ1 −r:

ð4Þ

Next, the rectangular region was extracted, as shown in Fig. 7(c). We measured the area of this region A by using the software ImageJ and obtained the thickness t of this region as t¼

A : ℓ2

ð5Þ

An approximated representation of this cell wall is shown in Fig. 7(d). Following this procedure, 101 cell walls were measured. The average values of the length ℓ1, thickness t, and fillet radius r are reported in Table 3. Such an idealisation not only helps to visualise the geometrical shape of the cell walls but also offers an option in the construction of numerical models. In most existing numerical studies (e.g., Santosa and Wierzbicki [14]; Meguid et al. [15]; Nammi et al. [24]) on foam mechanics, the foam cell wall thickness is generally assumed to be constant. According to observations made in the present study, such an assumption might not be appropriate. 2.6. Relative density distribution The relative density, denoted as ρ*/ρs (where ρ∗ is the density of the foam and ρs is the density of the base material), is one of the most crucial parameters responsible for foam mechanical properties. The as-received foam block was first weighted and the dimensions were measured in order to determine the relative density. The relative density of the foam

235

block studied was found to be 8.9%. Relative densities for 21 specimens originating from the 3 regions are reported in Table 4. Specimens extracted from various regions of the block tended to exhibit variations in relative density. The foam specimens obtained from the TOP and MID regions tended to have lower relative densities (b 8.9%, with average values of 8.67% and 8.48% respectively), whereas relative densities of specimens from the BOT region were much higher (average of 11.66%, much greater than 8.9%). Because of this variation, the results of this study are reported with the relative density for all specimens tested. In brief, ALPORAS foam exhibited polydispersity and was anisotropic geometrically. Cells were irregular polyhedra with approximately 14 faces. The cell size was approximately 5 mm in the ‘semimajor axis’ direction and approximately 4 mm in the other 2 orthogonal directions. The ‘semimajor axis’ direction was not parallel to any of the D1, D2, and D3 directions defined herein. The number of edges per face was approximately 5. After the microstructural characterisation, the specimens were subsequently compressed to study the mechanical response of the closedcell foam under uniaxial compressive loadings.

3. Crushing experiments Crushing experiments were conducted on foam specimens under various testing conditions to determine the effect of each of the conditions on the foam mechanical properties. Foam specimens were again removed from the foam block, using the aforementioned EDM technique. The relative density of each foam specimen was determined, in a similar manner to the as-received foam block, and recorded prior to each experiment. The foam specimen was then crushed between 2 rigid plates under displacement control, using an MTS 810 universal material testing machine. For the response to be quasistatic, all experiments were conducted at the 

same displacement rate of δ =H ¼ 7:5  10−4 s−1 .

39.5mm 35mm 53.2mm 35mm

53.2mm

35mm

Fig. 13. Foam specimens of various shapes: (a) various cross sections and (b) the same height.

236

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

3.1. Typical compression response Fig. 8 shows a typical crushing response of a foam specimen (ρ*/ ρs = 12.36 %, from the BOT region) under compression. The response consisted of 3 regimes, i.e., the linearly elastic, plateau, and densification regimes, which are the main characteristics of most cellular solids. At the beginning of the crushing test, the response traced a linear trajectory. The deformation in the cellular microstructure was relatively small and uniform, whereas most portion of the material remained elastic. As the stress increased, the deformation became more pronounced. The combination of geometric and material nonlinearities gradually softened the structure and the response eventually reached a local load maximum. The stress corresponding to this load maximum, referred to as the initiation stress σI, was 482 psi at δ/H = 4.78 % in our experiment. Because the response was approximately linear, and most portion of the base material (aluminium in the case of ALPORAS foam) remained elastic, this stage of the experiment was usually referred to as the linearly elastic regime and is marked as Regime I in Fig. 8. After σI, the response dropped slightly and subsequently transformed into an extensive stress plateau. In this regime, marked as Regime II in Fig. 8, the deformation started to localise, whereas the stress remained essentially unchanged. The deformation localisation can also be observed in the compression of many other cellular solids such as honeycombs (e.g., Papka and Kyriakides [25,26]; Jang and Kyriakides [26]), woods (Da Silva and Kyriakides [27]), and open-cell foams (Gong et al. [7]; Jang et al. [8]). The extensive stress plateau is responsible for the excellent energy absorption capability and has now been exploited in various foam applications ranging from energy absorbers to collision protection devices (e.g., Banhart [3]). At an ‘average strain’ of approximately 55%, the response started to trace a second stiff branch. This regime was referred to as the densification regime and is marked as Regime III in Fig. 8. Four parameters concerning foam mechanical properties are defined in Fig. 8, the elastic modulus E* (or the elastic modulus), the initiation stress σI (the stress corresponds to the local load maximum of the compressive response), the plateau stress σ P (the average stress at the plateau regime of the compressive response), and the extent of the plateau regime ΔεP, which is defined as the ‘average strain’ δ/h between σI and 1.4σI (Jang et al. [8]). Although the response in Regime I was approximately linear, in our experiments, we performed an unloading right after σI and the Young's modulus E* was measured as the slope of the unloading section of the response as shown in Fig. 8. The mechanical response for foams under compressive loads can be clearly understood once the 4 parameters are determined. The goal of this part of our study was to determine the effects of each of the testing conditions such as the relative densities and the loading directions on the foam mechanical properties, the 4 parametersfE ; σ I ; σ P ; Δε P g, through a series of crushing experiments.

Experiments conducted on the MID region yielded virtually the same conclusion, and the results for this region are thus not reported here. Similar experiments were conducted for foam specimens from the TOP region. Fig. 10 shows a comparison for the compressive response of 3 specimens with similar densities, ρ*/ρs ≅ 9.04 %, all from the TOP region, loaded again in the 3 directions. The D1 response was the lowest among the 3 cases, whereas the responses in the D2 and D3 cases were similar to each other. Additionally, the stress drop after the initiation stress σI was least prominent in the D1 response. Another observation that can be made from Fig. 10 was that the responses from the TOP region usually exhibited more zigzag type of noises while the ones from other two regions (e.g., the response shown in Fig. 9) were much smoother. This can probably be attributed to the presence of microstructual abnormality (such as the ‘collapse-like’ bands) mentioned in Section 2. Table 5 summarises the 4 mechanical parameters fE ; σ I ; σ P ; Δε P g obtained from each of the experiments. For the specimens from the BOT and MID regions, shown in Table 5(a) and (b), the parameters E*, σI, and σ P were highest in the D1 direction, whereas these parameters were smaller and similar in the other 2 directions. However, exceptions to these outcomes were found. For instance, in experiment set #2, shown in Table 5(a), the parameters E*, σI, and σ P for the 3 tests seemed similar. However, for specimens from the TOP region, shown in Table 5(c), the parameters E*, σI, and σ P were lowest in the D1 direction, and these parameters were greater and similar in the other 2 directions. For ΔεP, no clear trend could be observed. As mentioned in the previous section that for the BOT and MID regions, the cells were longest in the D1 direction and shorter and similar in the other 2 directions. For the TOP region, however, the cells were shortest in the D1 direction and longer and similar in the other 2 directions. Therefore, loading along the direction with the greatest dimension yields the highest compressive response (as well as the parameters E*, σI, and σ P ), and vice versa.

3.3. Specimen size Specimen size is another critical factor concerning foam applications. Many researchers have argued that cellular solids should encompass certain numbers of cells in each direction to yield a ‘typical response’ during mechanical testing (e.g., Ashby et al. [2]; Papka and Kyriakides [25]). For specimens of large size, the response resembles that of specimens of infinite size (and is thus referred to as a ‘typical response’). To investigate the effect of specimen size on the foam mechanical response, specimens of 4 different sizes (i.e., cubes with 20, 35, 50, and 65 mm side length, as shown in Fig. 11), all taken from the BOT region, were tested. A 65 mm3 specimen comprised 18 cells in each of the 3 principal directions. For the cases of 50, 35, and 20 mm3, this number was reduced to approximately 15, 10, and 7 respectively. Fig. 12(a) shows the results of a comparison for compressive responses of specimens of different sizes (ρ*/ 800

3.2. Loading direction As mentioned in the previous section, the foam cellular microstructure exhibited geometric anisotropy. Because mechanical anisotropy is critical for mechanical properties in foam applications, examining whether this geometric anisotropy causes mechanical anisotropy is crucial. For this purpose, crushing experiments in different directions (i.e., the D1, D2, and D3 directions defined previously) were performed. Fig. 9 shows a comparison of the compressive responses of 3 specimens with similar relative densities, ρ*/ρs ≅ 12.33 %, all from the region BOT, loaded in the directions D1, D2, and D3. Generally, the 3 responses were similar. However, the D1 case was unique. Among the 3 cases, the D1 specimen not only had the highest overall response but also yielded the most prominent stress drop immediately following the initiation stress σI. The responses of the D2 and D3 specimens were similar, and in both cases the stress drop was not as pronounced as the stress drop in the D1 case.

ALPORAS Aluminum Foams *

(psi)

5

~ 9.36% =

(MPa)

s

600

4 3

400

2 200 Rect. Tri. Circ.

0

0

0.1

0.2

0.3

0.4

0.5

1 0 0.7

0.6 /h

Fig. 14. The effect of specimen shape on the overall response.

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

ρs ≅ 8.75 %). Although responses for the 3 larger specimens were similar, the smallest specimen (i.e., the 20 mm3) exhibited evident scattering. Moreover, although the relative density remained similar among the specimens, some specimens yielded high responses (even higher than those of the larger specimens), whereas others yielded low responses (lower than those of the larger specimens); the reproducibility of test results was poor. The elastic moduli E* of specimens of various sizes are shown in Fig. 12(b). The results for parameters σI and σ P exhibited the same trend as the elastic moduli E* and therefore are not shown. For specimens equal to or larger than 35 mm3, the mechanical parameters seemed ‘converged’. For specimens of the smallest size, 20 mm3, even though the results were not always smaller, they were much more scattered. Therefore, in foam applications, the foam should have a sufficiently large size (e.g., ≥10 cells in one direction) for the more convergent mechanical properties.

E*1 3.6 E (%)

3 2.4 1.8 1.2

BOT MID TOP Eq. (8) Eq. (10)

0.6 0 7

/

s

13 (%)

5 600

(MPa)

4 3

400 BOT MID TOP Eq. (9) Eq. (11) Eq. (12)

200

2 1 0

8

9

10

11

12

13 *

/

s

(%)

(b) p

600

4

(psi) 500

p

3 (MPa) 400 300

2 BOT

200

MID TOP

4. Comparisons and discussion

1

100 7

The literature contains equations describing mechanical properties of closed-cell foams. These equations can be evaluated by comparing predictions (obtained from the equations) and experimental results. For example, Gibson and Ashby [6] derived equations for the Young's modulus E* and yield strength σpl⁎ (the definition of σpl⁎ is the same as the initiation stress σI defined in the present study) of a foam using dimensional analysis as

¼ 0:3ϕ

12

0

The mechanical parameters E*, σI, and σ P for all our experiments are plotted against the relative density ρ*/ρs in Fig. 15(a)–(c) respectively. Fig. 15 shows the results for the specimens compressed in the D1 direction, demonstrating that specimens with higher relative densities tended to have greater values of E*, σI, and σ P , and vice versa. This was also true for specimens originating in the BOT and MID regions. For the specimen from the TOP region, no obvious trend could be observed. This is likely attributable to the fact that the cellular microstructure in this region was highly ‘abnormal’ because of the existence of the collapse-like bands and zones. Although not shown here, for specimens compressed in the other 2 directions, similar observations were made, and the same conclusion was drawn. Included in Fig. 15 for comparison are (empirical) equations quoted from the literature. Comments on the comparison are provided in the next section.

σ ys

11

(psi)

7

  3=2   ρ ρ þ ð1−ϕÞ ρs ρs

10

(a)

3.5. Relative density

3=2

9

800

Foams can be tailored into many shapes according to their applications. Examining whether a specimen's shape affects its mechanical properties is critical. For this purpose, specimens from the BOT region with different shapes were prepared and tested. In addition to the cubes described previously, the shapes considered included circular and triangular cylinders, as shown in Fig. 13. Fig. 14 shows the results for the mechanical parameters of specimens of various shapes. Based on these results, if a specimen is large enough, the effect of specimen shape is not significant.

σ pl

8

*

3.4. Specimen shape

  2   E ρ ρ ¼ ϕ2 þ ð1−ϕÞ Es ρs ρs

237

ð6Þ

ð7Þ

where Es, ρs, and σys are the Young's modulus, density, and yield stress for the material from which the foam is made (also referred to as the ‘base material’) respectively. The term ϕ is the solid fraction that is contained in the cell edges (ρ*/ρs b ϕ b 1), and the remaining (1 − ϕ) occupies the cell faces.

8

9

10

(c)

11

12 * /

s

13 (%)

Fig. 15. Measured mechanical parameters: (a) Young's modulus, (b) initiation stress σI, and (c) plateau stress σ P.

In another study, Simone and Gibson [21] proposed similar equations based on FEM simulations (Es = 70 GPa, σys = 150 MPa):   2   E ρ ρ ¼ 0:3163 þ 0:3188 Es ρs ρs

σ pl σ ys

¼ 0:4445

  2   ρ ρ : þ 0:3346 ρs ρs

ð8Þ

ð9Þ

However no comparisons with experimental data were performed in their study.

238

W.-Y. Jang et al. / Materials Characterization 107 (2015) 228–238

To compare the measured results of E* and σpl⁎, Benouali et al. [28] adopted Eqs. (6) and (7) and argued that ϕ was 0.75. On this basis, the foregoing equations become   2   E ρ ρ ¼ 0:5625 þ 0:25 Es ρs ρs σ pl σ ys

  3=2   ρ ρ : ¼ 0:195 þ 0:25 ρs ρs

ð10Þ

ð11Þ

The density and Young's modulus of the solid aluminium alloys were assumed to be the reference value for aluminium (2.7 g/cm3 and 70 GPa), and the cell wall yield strength was assumed to be 130 MPa (given by the manufacturers). They then compared the predicted E* and σpl⁎ with experimental results from the investigated closed-cell foams. The equations were found to overestimate both values by a factor of 2–10. To remedy this overestimation, Idris et al. [29] incorporated experimental data and modified Eq. (11) as σ pl σ ys

¼ 0:44

  6:95=2   ρ ρ : þ 0:1835 ρs ρs

ð12Þ

Eq. (12) was found to agree with the experimental results of ALPORAS foam presented. Results from Eqs. (8)–(12) were plotted along with the experimental results obtained in the present study. Fig. 15(a) shows experimental results for E1⁎/Es along with the results calculated from Eqs. (8) and (10). Both equations were found to highly overestimate the Young's modulus. Furthermore, if data for E2⁎/Es were included in the figure, the discrepancy became larger. Fig. 15(b) shows a comparison of the initiation stress between the experimental results and results calculated from Eqs. (9), (11), and (12). Eqs. (9) and (11) overestimated σpl⁎ by a large margin. Although Eq. (12) adequately described the experimental results of σI, the predicted results seemed closer to the experimental results in the D1 direction than in the D2 direction (this is not shown in the figure, but the numbers can be found in Table 5). In summary, existing equations are simple and easy to apply. They offer a convenient method of identifying the correct trend of the mechanical properties and gaining preliminary physical insight. However, these equations should be used with caution because of their limitations. First, these equations typically contain only the relative density and fraction parameter variables. For anisotropic foams such as ALPORAS foam investigated in the present study, these equations seem oversimplified because crucial parameters such as the loading directions are not included. Additionally, most existing equations offer only simple mechanical parameters (such as E* and σI). If more detailed information (e.g., the deformation patterns and complete crushing responses) regarding foam crushing is relevant, then these equations are not sufficient. Models based on realistic microstructural geometric features are necessary. 5. Conclusions In the present study, ALPORAS closed-cell aluminium foam was investigated. The microstructure of ALPORAS foam was first characterised. The measured microstructural geometric features can offer valuable information in the development of models for analytic and numerical studies. The cell sizes in ALPORAS foam were found to vary. The polydispersity was approximately 0.2. Furthermore, the cells were anisotropic. In the BOT and MID regions, cells were longest in the D1 direction and similar in the other 2 directions. In the TOP region, cells were shortest in the D1 direction and similar in the other 2 directions. Cells were irregular polyhedrons with about 14 faces and 5 sides per face. Cell wall thickness varied such that cell walls were thinnest around the midspan and thickest near the nodes.

A series of experiments concerning various testing conditions was subsequently conducted to determine the effect of each of the testing conditions on the mechanical properties of ALPORAS foam. The compressive response comprised 3 regimes: the linearly elastic, plateau, and densification regimes, which are the main characteristics of most cellular solids. The effect of the loading direction was also investigated. Loading in the major-axis direction yielded a higher response, greater E*, σI, and σ P , and vice versa. Regarding specimen size, if the specimen length was equal to or greater than 35 mm (containing approximately 10 cells in one direction), the response was ‘converged’. For the 20 mm3 specimens, the results of the overall response and mechanical parameters were scattered. Additionally, if the specimen was sufficiently large, the effect of the specimen shape was not substantial. Relative density was the most critical factor influencing the mechanical properties. Generally, the overall response and mechanical parameters E*, σI, and σ P increased with the relative density if other testing conditions were the same, but some exceptions were noted (e.g., specimens from the TOP region). It should be noted that the aforementioned conclusions may not be applicable to foams manufactured using other methods as they can have microstructures that are distinctly different from those investigated in the present study. Finally the experimental results were compared with equations in the literature. Although the equations are valuable because of their simplicity, they are typically oversimplified. Theoretical models should consider more factors such as the anisotropy and thickness variation to account for the discrepancy between current theoretical values and experimental data. Acknowledgements The work reported was supported in part by the Ministry of Science and Technology through grants 99-2218-E-009-027- and 100-2221-E009-030-. The authors wish to thank Dr. Keng-hui Lin for her help in multiple ways in the course of the present study. The literature on our subject is sufficiently vast that a more complete citation of related references is impractical. The authors have chosen to cite only papers that they found useful in their work. References [1] H.F. Seibert, Reinf. Plast. 50 (2006) 44. [2] M.F. Ashby, A. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley, Metal Foams: A Design Guide, Butterworth-Heinemann, 2000. [3] J. Banhart, Int. J. Veh. Des. 37 (2005) 114. [4] T. Imwinkelried, J. Biomed. Mater. Res. A 81 (2007) 964. [5] A.G. Dement'ev, O.G. Tarakanov, Polym. Mech. 6 (1970) 519. [6] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, 2 edition Cambridge University Press, 1997. [7] L. Gong, S. Kyriakides, W.Y. Jang, Int. J. Solids Struct. 42 (2005) 1355. [8] W.-Y. Jang, A.M. Kraynik, S. Kyriakides, Int. J. Solids Struct. 45 (2008) 1845. [9] O. Brunke, S. Odenbach, F. Beckmann, Structural characterization of aluminium foams by means of microcomputed tomography, in: U. Bonse(Ed.), Developments in X-ray Tomography IV, 5535 2004, p. 453. [10] Y. Sugimura, J. Meyer, M.Y. He, H. Bart-Smith, J. Grenstedt, A.G. Evans, Acta Mater. 45 (1997) 5245. [11] A.E. Simone, L.J. Gibson, Acta Mater. 46 (1998) 3109. [12] A.-F. Bastawros, H. Bart-Smith, A.G. Evans, J. Mech. Phys. Solids 48 (2000) 301. [13] U. Ramamurty, A. Paul, Acta Mater. 52 (2004) 869. [14] S. Santosa, T. Wierzbicki, J. Mech. Phys. Solids 46 (1998) 645. [15] S.A. Meguid, S.S. Cheon, N. El-Abbasi, Finite Elem. Anal. Des. 38 (2002) 631. [16] S. Youssef, E. Maire, R. Gaertner, Acta Mater. 53 (2005) 719. [17] Y. Chen, R. Das, M. Battley, Int. J. Solids Struct. 52 (2015) 150. [18] T. Miyoshi, M. Itoh, S. Akiyama, A. Kitahara, Adv. Eng. Mater. 2 (2000) 179. [19] R.T. DeHoff, F.N. Rhines, Quantitative Microscopy, McGraw-Hill Education, 1968. [20] Y. Mu, G. Yao, H. Luo, Mater. Des. 31 (2010) 1567. [21] A.E. Simone, L.J. Gibson, Acta Mater. 46 (1998) 2139. [22] V.S. Deshpande, N.A. Fleck, J. Mech. Phys. Solids 48 (2000) 1253. [23] A.T. Huber, L.J. Gibson, J. Mater. Sci. 23 (1988) 3031. [24] S.K. Nammi, P. Myler, G. Edwards, Mater. Des. 31 (2010) 712. [25] S.D. Papka, S. Kyriakides, Int. J. Solids Struct. 35 (1998) 239. [26] W.-Y. Jang, S. Kyriakides, Int. J. Mech. Sci. 91 (2015) 81–90. [27] A. Da Silva, S. Kyriakides, Int. J. Solids Struct. 44 (2007) 8685. [28] A.H. Benouali, L. Froyen, T. Dillard, S. Forest, F. N'guyen, J. Mater. Sci. 40 (2005) 5801. [29] M.I. Idris, T. Vodenitcharova, M. Hoffman, Mater. Sci. Eng. A 517 (2009) 37.