Triaxial compression of aluminium foams

COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 67 (2007) 1218–1234 www.elsevier.com/locate/compscitech

Triaxial compression of aluminium foams D. Ruan a, G. Lu a

a,*

, L.S. Ong b, B. Wang

c

Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Hawthorn, Vic. 3122, Australia b School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798 c Department of Mechanical, Aerospace and Civil Engineering, University of Manchester, UK Received 18 November 2005; received in revised form 1 May 2006; accepted 1 May 2006 Available online 22 June 2006

Abstract Triaxial compressive tests have been conducted on CYMAT closed-cell aluminium foams of five different relative densities to investigate their initial failure surfaces under multiaxial loading. Quasi-static uniaxial compressive and tensile tests have also been performed to obtain their uniaxial strength. The experimentally measured yield surfaces are compared with various published phenomenological yield surface models. Reasonable agreement has been observed when suitable Poisson’s ratios are employed. Triaxial compressive tests have been carried out on foams with nominal relative density of 17% at various axial loading velocities to study the effect of strain rate on the initial failure surface. The results showed that the initial yield stresses of CYMAT closed-cell aluminium foams are not sensitive to the axial strain rate ranged from 104 to 10+1 s1.  2006 Elsevier Ltd. All rights reserved. Keywords: A. Aluminium foam; B. Mechanical properties; B. Strength; B. Stress/strain curves; C. Failure criterion

1. Introduction Aluminium foams have the potential for use in lightweight structural components and in energy absorption. They may be subjected to multiaxial loads in these applications and thus a criterion to determine the failure of aluminium foams is essential. The overall properties of foams are governed by the response of cell structures. Foams may fail at the cell level by elastic buckling, plastic collapse and brittle fracture, depending on the properties of the parent materials of which foams are made. Zhang [1] has pointed out that plastic collapse is the dominating mechanism for most practical metal foams under multiaxial loading. Aluminium foams may fail by plastic collapse and the initial failure of aluminium foams can be determined by probing a yield surface. Yield surface is a convex envelope in stress space, within which the material remains elastic and on which plastic deformation may take place [2]. *

Corresponding author. Tel.: +61 3 9214 8669; fax: +61 3 9214 8264. E-mail address: [email protected] (G. Lu).

0266-3538/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.05.005

Several yield criteria have been proposed for metal foams. Gibson et al. [3] used dimensional arguments to analyse an ideal foam and gave the yield surface as !2   re q rm   þ 0:81 ¼1 ð1Þ rpl qs rpl where re, rm and rpl are the von Mises equivalent stress, mean stress and the uniaxial plastic collapse strength. This theoretical prediction neglected the effect of imperfection on the mechanical properties and overestimated the hydrostatic strengths. As von Mises criterion for engineering alloys (metals) and Drucker–Prager criterion for soils, similarly Miller [2] proposed a yield criterion for foams and other types of materials exhibiting plastic compressibility and different yield points in tension and compression. It is given by f ¼ re  cp þ

a0 2 p  d 0 rc 6 0 d 0 rc

ð2Þ

where p is the hydrostatic compressive stress, rc is the uniaxial compressive strength, and the constants c, a 0 and d0

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

are related to b defined as the ratio of the uniaxial compressive strength (rc) to tensile strength (rt), and the plastic Poisson’s ratio mpl. They are given by rc b¼ rt 6b2  12b þ 6 þ 9ðb2  1Þ=ð1 þ mpl Þ c¼ 2ðb þ 1Þ2 45 þ 24c  4c2 þ 4mpl ð2 þ mpl Þð9 þ 6c  c2 Þ a0 ¼ 2 16ð1 þ mpl Þ ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 c c 2 4a0 1 þ 1 d0 ¼ þ 2 3 3 9 It is to be noted that, when c = 0, a 0 = 0.81(q*/qs) and d0 = 1, Miller yield surface retracts the model proposed by Gibson et al. [3]. Deshpande and Fleck [4] have proposed a phenomenological yield surface for metal foams, which is a self-similar model, given by ^Y 60 U¼r ^2 ¼ r

1 1 þ ða=3Þ

2

 2  re þ a2 r2m ;

 1=2 0:5  mpl a¼3 1 þ mpl

ð3Þ ð4Þ

where, Y is the uniaxial yield strength, Y = (rc + rt)/2. A more complicated differential hardening model has also been put forward, which can give a higher level of accuracy. Gioux et al. [5] and Miller [2] suggested that both Miller and Deshpand and Fleck yield criteria can be deduced from Gibson et al. [3] by accounting for cell wall curvature. This gives a morphological related explanation for the above two phenomenological criteria. Hanssen et al. [6] used numerical simulation method to valid these current constitutive models for aluminium foams. Doyoyo and Wierzbicki [7] presented a phenomenological yield surface model for foams, based on the measured biaxial yield stress data for Alporas and Hydro closed-cell aluminium foams. The uniaxial strength asymmetry and anisotropy of foams were considered in this model. Although plastic Poisson’s ratio was not included in this model, another independent parameter, a shape factor, was introduced. The value of this shape factor was determined by fitting the experimental data to the yield surface model. Huang [8] extended a simple approach, which was originally proposed for shape memory alloys, to the failure surfaces of both polymer and aluminium foams under multiaxial loads. Straight lines in some particular stress planes were used to give an approximation of the failure surface. Three selected tests will be sufficient to estimate the complete failure surface of a foam. The recommended testing stress states are r1 = r2 = r3 > 0, r1 = r2 = r3 < 0 and r1 = r2 = r3 (or r1 = r2, r3 = 0). However, in practice, it is difficult to carry out a hydrostatic tension test (r1 = r2 = r3 > 0) and there has been no report on that test so far. Blazy et al. [9] proposed a statistic failure surface model to predict accurately the mean failure stress and the disper-

1219

sion or standard deviation of aluminium foams under multiaxial loading. This model accounted for the size effect and the scatter observed in the test results. A variety of basic tests (tensile or compression and torsion tests) need to be performed to obtain the values of relevant parameters in the yield surface model. Limited experimental data are available to verify these criteria. Triantafillou et al. [10] first reported the triaxial tests on an open-cell aluminium foam. The applied radial load was tensile, and the axial load was either tensile or compressive. Their data are located in the first and fourth quadrants in the stress space. Fleck and co-workers [4,11] performed the axisymmetric compressive tests on three types of aluminium foams: Alporas foams (closed-cell, with relative densities of 8.4% and 16%), Duocel foam (open-cell, with a relative density of 7%) and Alulight foams (closed-cell, with relative densities ranged from 9% to 30%). Their experimental data shows that their self-similar model can predict the stress–strain response with a reasonable accuracy and that the differential hardening model can give a higher level of accuracy for proportional loading paths. Gioux et al. [5] carried out experimental work on lowdensity (<10%) open-cell (Duocel) and closed-cell (Alporas) aluminium foams subjected to biaxial and triaxial loading to verify the above three theoretical criteria for initial failure. Their experimental data obtained in biaxial tests agreed well with the Miller and Deshpande and Fleck criteria. But there were some difference between the triaxial experimental data and the yield criteria (Figs. 5a and b in [5]). The difference was attributed to the density variation and local imperfections between the specimens. Moreover, Gioux et al. [5] used their measured mpl = 0.024 when plotting Miller yield surfaces for 8% Alporas foam, but a different value was used for the same foam when plotting Deshpande and Fleck yield surface; the best fitted parameter, a  1.4 [4], corresponding to mpl = 0.23, was used. Limited experimental data available and these discrepancies in Poisson’s ratio triggered us to conduct more experimental work on the yield surfaces of foams with various densities, ranging from 5% to 20%. Three types of tests, uniaxial compressive, uniaxial tensile and triaxial compressive tests, were performed at a nominal axial strain rate of 104 s1. 2. Materials and testing procedure CYMAT closed-cell aluminium foams with nominal relative densities of 5%, 10%, 15%, 17% and 20% were used in the experiment. Foam panels supplied by CYMAT Corporation were made of aluminium with a Young’s modulus of 93 GPa and upper yield strength of 310 MPa in compression. The average cell sizes of these foams are 10 mm, 6 mm, 4 mm, 3.5 mm and 3 mm for 5%, 10%, 15%, 17% and 20%, respectively (data from CYMAT Corporation). The specimen size is critical for foams subjected to compressive, tensile and shear loads [12–15]. Andrews et al.

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[16] pointed out that macroscopically representative measurement of compressive strength could be obtained by using closed-cell Alporas foam specimens with at least five cells along the axial direction. We here assume that this rule can be applied to CYMAT closed-cell foams. All dimensions of the specimens in all tests are at least five times larger than the cell size in all directions. The actual density of each specimen was measured by using an electronic balance and an electronic vernier.

Table 1 Dimensions of aluminium foam specimens Nominal relative density of aluminium foama (%)

Dimensions of aluminium foam specimens (mm)

5 10 15 17 20

150 · 150 · 50 90 · 90 · 50 70 · 70 · 25 60 · 60 · 25 50 · 50 · 25

2.1. Quasi-static compressive tests

a The aluminium foams were supplied as sheets. Nominal densities are the densities of these sheets provided by the supplier. The actual density of each specimen was measured before the test by using a electronic balance.

Compressive tests of CYMAT foams with nominal relative densities of 5%, 10%, 15%, 17% and 20% at various nominal strain rates ranged from 103 to 10+1 s1 were

conducted in our previous study [17,18]. One additional set of compressive tests at a nominal strain rate of 104 s1 was performed here. The dimensions of specimens

Fig. 1. Dogbone specimens used in the quasi-static uniaxial tensile tests: (a) photograph of two specimens, (b) dimensions of a dogbone specimen used for foams with relative density of 5% and 10%, (c) dimensions of a dogbone specimen used for foams with relative density of 15%, 17% and 20%.

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

(Table 1) and testing procedure are exactly the same as in our previous study [17]. 2.2. Quasi-static tensile tests Uniaxial tensile strength is required in order to construct the theoretical yield surfaces of foams, and so uniaxial tensile

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tests were performed quasi-statically with dogbone specimens (shown in Fig. 1(a)) for all the five types of CYMAT foams. The nominal relative densities of these foams are 5%, 10%, 15%, 17% and 20%, respectively. Because foams with relative density of 5% and 10% have a large cell size, large dogbone specimens were used and their dimensions are shown in Fig. 1(b). The thickness of the specimen is

Fig. 2. Photographs of the tensile test setting: (a) MTS, jig and a specimen; (b) jig – front view; (c) jig – side view.

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D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

50 mm and the effective gauge length is 100 mm. Foams with relative density of 15%, 17% and 20% have a relatively small cell size, hence small dogbone specimens were used with all dimensions half that of large specimens (Fig. 1(c)). Quasi-static uniaxial tensile tests were carried out using MTS with specially made grips, since the standard MTS grips are not large enough for our dogbone specimens (Fig. 2). The surfaces of the grips are knurled to increase friction. Suitable forces should be applied to tighten up the four bolts to avoid slip as well as pre-crushing the end of the specimen. The cross-head speed was set to 0.01 mm/s and 0.005 mm/s, for large and small dogbone specimens, respectively, corresponding to a nominal strain rate of 104 s1. 2.3. Triaxial compressive tests Cylindrical specimens, shown in Fig. 3, were cut from foam panels. Since the largest average cell size of those foams is 10 mm, a device that can accommodate specimens with a diameter of 55 mm was selected to satisfy the dimension requirement for all the foams. Therefore the diameter of all specimens is 55 mm. The height of the specimen is 50 mm for foams with relative densities of 5% and 10%, and 25 mm for remaining relative densities. A high-pressure triaxial test system was used to measure the axisymmetric compressive stress–strain curves and to

Fig. 3. A cylindrical foam specimen used in the triaxial test.

probe the yield surfaces of foams. The test system consists of an MTS, an ELE-Hoek Cell (model EL70-1710) and a modified ENERPAC pump (model P80), as shown in Fig. 4. A sketch of the ELE-Hoek Cell is also shown in

Fig. 4. Triaxial testing equipment.

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

Fig. 5. It comprises a steel body and two steel end caps, which are screwed to the body of the cell when in use. The body incorporated two self-sealing couplings: one for connecting to the hydraulic pressure system (the pump), the other for de-airing the cell chamber. The Hoek Cell we used is available to accommodate specimens with a diameter of 55 mm. An ENERPAC P80 pump was modified by attaching a relief valve so that an accurate constant (self-compensating) pressure could be maintained. In each test, a specimen was placed at the centre of the Hoek Cell and the pump was used to apply the desired confining pressure. The MTS was then used to apply axial compressive load. Fig. 6 shows the stress condition of a typical aluminium foam specimen. Teflon spray was used to lubricate the contact surfaces between the specimens and platens to reduce friction. Axial load–displacement curve was recorded by a computer. Axial stress was calculated as the ratio of the axial load to the cross-sectional area and axial strain was obtained from the ratio of displacement to the original length of the specimen. The axial stress–strain curve could be easily obtained from the axial load–displacement curve. By using the same convention in Gioux et al. [5], the yield stress was defined as the upper yield point if one existed; otherwise, it was defined as the stress at which the slopes of the linear and plateau region intercepted each other. Multiple tests were conducted on different specimens at various confining pressure. Each specimen produced a test point to plot the yield surface in the third quadrant in the stress space of axial stress and confining pressure. Triaxial tests at different axial loading rates were performed only on foam with a nominal relative density of 17%. For these tests, confining pressure was set first and then the axial load was applied at cross-head speeds of

1223

Fig. 6. (a) Triaxial loading of an aluminium foam specimen, (b) the stress state of an element within the specimen.

0.0025 mm/s, 0.025 mm/s, 0.25 mm/s, 2.5 mm/s, 25 mm/s and 250 mm/s, corresponding to the nominal axial strain rates in the range of 104–10+1 s1. 3. Results 3.1. Quasi-static uniaxial compressive response

Fig. 5. Sketch of ELE-Hoek Cell.

The axial stress versus strain curves for the tested foam specimens are similar; they all have an initial linear elastic region, an yield point, a plateau region where the stress increases slowly as the cells deform plastically, and a densification region where the stress increases rapidly. Fig. 7 shows the stress versus strain curves of CYMAT specimens at a nominal strain rate of 104 s1. Our previous study [17,18] indicated that the compressive strength of CYMAT foams is less sensitive to the strain rate and the density of foam specimen has a greater effect on the strength. The uniaxial compressive strength for each type of foam, as listed in Table 1, was taken as the average value of yield strength from [17] and current study.

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D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234 20 18

ρ/ρs=14.70%

16

ρ/ρs=17.86%

Stress (MPa)

14 12

ρ/ρs=20.82%

10 8

ρ/ρs=11.90%

6 4

ρ/ρs=7.56%

2 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nominal strain Fig. 7. The quasi-static uniaxial compressive stress vs. stain curves of CYMAT aluminium foams at a strain rate of 104 s1.

3.2. Quasi-static uniaxial tensile response Because conventional extensometers available are not large enough for the dogbone foam specimens and only

the maximum tensile strength was of interest, no extensometer was used in the tensile tests. Load and displacement of the whole specimen were recorded by a computer. The stress was defined as the ratio of the load to the waisted cross-sectional area and the strain was defined as the ratio of the displacement to the original length of the waisted section. Fig. 8(a) is a typical curve of stress and strain for a foam specimen. Deformation before failure is small. The tensile strength of a foam was defined as the maximum value on the stress–strain curve. The measured tensile strength of the foams are listed in Table 2. They are slightly higher than the corresponding uniaxial compressive strength. Similar conclusion has been drawn by Sugimura et al. [19] and Gioux et al. [5]. Fig. 8(b) is a photograph of the fractured specimen corresponding to Fig. 8(a) after the test; a main crack occurred and propagated through the specimen, which caused the failure in the mid-region of the specimen. No necking and no localized band were observed in tension, which was very common in compression.

Fig. 8. (a) Stress vs. strain curve of a 5% CYMAT aluminium foam specimen subjected to uniaxial tensile load, the nominal strain rate is 104 s1, (b) a photograph of the fractured specimen corresponding to (a) after the test.

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234 Table 2 Uniaxial compressive and tensile strength of CYMAT aluminium foams Relative density of foams Compressive strength rc (MPa) Tensile strength rt (MPa)

5%

10%

15%

17%

20%

0.44

0.97

2.34

3.77

4.14

0.73

1.70

3.19

4.77

5.20

1225

rpl is the uniaxial strength and was taken as the average of the uniaxial compressive strength rc and tensile strength rt, i.e. rpl ¼ ðrc þ rt Þ=2. Hence, the above equation can be re-written as !2    2 r1  r2 rc q r1 þ 2r2 rc    þ 0:09 ¼1 rc rpl qs rc rpl Gibson et al. yield surfaces for the foams are

3.3. Triaxial compressive response and initial failure surfaces Fig. 9 is a typical axial stress–strain curve of CYMAT aluminium foam specimen while the confining pressure, rp, is kept constant. Its shape is very similar to that under uniaxial compression, with a linear elastic region, a yield stress, a ‘‘plateau’’ region and a densification region. Testing a foam specimen at a given confining pressure gives a stress–strain curve with a defined yield stress. Changing the confining pressure will vary the stress–strain curve and the yield stress. Yield stresses and the corresponding confining pressures form the yield surfaces in the stress space. The diamond points in Fig. 10(a)–(e) are the experimental data for foams with nominal relative densities of 5%, 10%, 15%, 17% and 20%, respectively. These experimental data are normalised by uniaxial compressive strengths listed in Table 2 for each individual foam. The theoretical yield criterion proposed by Gibson et al. [3] can be determined by using the uniaxial compressive and tensile strengths obtained in Sections 3.1 and 3.2. The criterion given by Gibson et al. is !2   re q rm   þ 0:81 ¼1 rpl qs rpl

12

Axial stress σ1 (MPa)

10

ð5Þ

10% foam :

ð6Þ

15% foam : 17% foam : 20% foam :

yield stress

densification

10% foam :

6

15% foam : 4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Axial strain ε1 Fig. 9. A typical experimental axial stress–strain curves of CYMAT aluminium foam specimens under triaxial loading (q/qs = 11.92% and rp = 2 MPa).

ð9Þ

þ 0:014ðr1 þ 2r2 Þ =rC  1:27rC ¼ 0 jr1  r2 j  0:28ðr1 þ 2r2 Þ

2

ð10Þ

þ 0:003ðr1 þ 2r2 Þ2 =rC  1:28rC ¼ 0

ð11Þ

jr1  r2 j  0:16ðr1 þ 2r2 Þ 2

17% foam :

jr1  r2 j  0:12ðr1 þ 2r2 Þ

20% foam :

þ 0:016ðr1 þ 2r2 Þ =rC  1:14rC ¼ 0 jr1  r2 j  0:12ðr1 þ 2r2 Þ

linear elastic region 0 0.0

ð8Þ

jr1  r2 j  0:25ðr1 þ 2r2 Þ

þ 0:004ðr1 þ 2r2 Þ =rC  1:16rC ¼ 0

plateau region 2

ð7Þ

where r1 is the axial stress and r2 is the radial stress. One additional parameter, plastic Poisson’s ratio, should be measured in order to determine the yield criteria proposed by Miller [2] and Deshpande and Fleck [4], respectively. Efforts have been made to measure the value of the plastic Poisson’s ratio for the tested foams, but no reasonable result has been achieved. The values of the plastic Poisson’s ratio used here are assumed to be 0.17 for foams with relative density of 5% and 10%, 0.29 for foams with density of 15% and 0.3 for foams with density of 17% and 20%, respectively (the reason will be stated in Section 4). The relevant constants in the yield criteria of Miller [2] and Deshpande and Fleck [4] are listed in Tables 3 and 4. The yield surfaces are Eqs. (10)–(19)). Miller [2] yield surfaces for the CYMAT foams reduces to 5% foam :

8

   2 r1 r2 r1 2r2  þ ¼1 þ 0:0026 rc rc rc rc    2 r1 r2 r1 2r2 0:73  þ ¼1 þ 0:0048 rc rc rc rc    2 r1 r2 r1 2r2 0:85  þ ¼1 þ 0:0097 rc rc rc rc    2 r1 r2 r1 2r2 0:88  þ ¼1 þ 0:012 rc rc rc rc    2 r1 r2 r1 2r2 0:89  þ ¼1 þ 0:014 rc rc rc rc

5% foam : 0:76

ð12Þ

2

ð13Þ

2

ð14Þ

þ 0:018ðr1 þ 2r2 Þ =rC  1:14rC ¼ 0

Deshpande and Fleck [4] proposed yield surfaces for the CYMAT foams are expressed as

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D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

σ1 / σC 6

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-6

-4

4

2

2

-2

6

4

σ2 / σC

-2

-4

(a)

-6

σ1 / σC 4

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

2

-2

-4

2

4

σ2 / σC

-2

-4

(b)

σ1 / σC 4

triaxial tets data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-4

2

-2

2

4

σ2 / σC

-2

(c)

-4

Fig. 10. Yield surfaces for triaxial tests of CYMAT aluminium foams with nominal relative density of (a) 5%; (b) 10%; (c) 15%; (d) 17%; (e) 20%.

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

1

/

-3

1227

C

-1

-2

0

-4 -1

triaxial test data (strain rate=10 s ) -3 -1 triaxial test data (strain rate=10 s ) -2 -1 triaxial test data (strain rate=10 s ) -1 -1 triaxial test data (strain rate=10 s ) 0 -1 triaxial test data (strain rate=10 s ) +1 -1 triaxial test data (strain rate=10 s ) uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-1

2

/

C

-2

-3

(d)

σ1 / σC 3

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

1

-1

-2

-3

2

1

2

σ2 / σC

3

-1

-2

(e)

-3

Fig. 10 (continued)

Table 3 Values of the parameters in Miller yield surface Nominal foam density (%)

b = rc/rt

c

a0

d0

5 10 15 17 20

0.61 0.57 0.73 0.79 0.80

0.76 0.83 0.47 0.36 0.36

0.16 0.037 0.039 0.17 0.18

1.27 1.28 1.16 1.14 1.14

5% foam :

2

5 10 15 17 20

0.17 0.17 0.29 0.30 0.30

1.59 1.59 1.21 1.18 1.18

1.32rC 1.38rC 1.18rC 1.13rC 1.13rC

2

0:78ðr1  r2 Þ þ 0:22ðr1 þ 2r2 Þ  1:75r2C ¼ 0 ð15Þ

10% foam :

Table 4 Values of the parameters in Deshpande and Fleck yield surface  1=2 t 0:5m Nominal foam density mpl Y ¼ rc þr 2 a ¼ 3 1þmplpl

2

2

0:78ðr1  r2 Þ þ 0:22ðr1 þ 2r2 Þ  1:89r2C ¼ 0 ð16Þ 2

2

15% foam :

0:86ðr1  r2 Þ þ 0:14ðr1 þ 2r2 Þ  1:40r2C ¼ 0 ð17Þ

17% foam :

0:87ðr1  r2 Þ2 þ 0:13ðr1 þ 2r2 Þ2  1:28r2C ¼ 0 ð18Þ

20% foam :

2

2

0:87ðr1  r2 Þ þ 0:13ðr1 þ 2r2 Þ  1:27r2C ¼ 0 ð19Þ

The solid, dotted and dashed lines in Fig. 10(a)–(e) are the yield criteria proposed by Gibson et al. [3], Miller [2] and Deshpande and Fleck [4], respectively. Gibson et al. criterion overestimates experimental yield stresses. Fig. 10(d) also shows the experimental data of foam specimens with a nominal relative density of 17% crushed at different axial loading rates or strain rates. No significant

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D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

influence exists of strain rate on the yield stress of aluminium foams in the strain rate range of 104–10+1 s1. 4. Discussions 4.1. Effect of friction As a possible refinement, the friction between the specimen and rubber jacket may be taken into account with respect to the experimental data used in Fig. 10(a)–(e). Teflon spray was used to lubricate the specimens and the platens; hence, transverse friction between the contact surfaces was regarded as insignificant. During a typical test, foam specimens were crushed by several millimetres in the axial direction. Hence longitudinal friction may exist between the specimen and rubber jacket along the axial direction. However, no information was available on the value of the frictional coefficient, l. A simple test

was conducted to estimate this value. A foam specimen was placed into the rubber jacket (Fig. 11(a)) and then one end of the rubber jacket was slowly raised until the foam specimen just started to slip. By measuring the angle between the jacket axis and horizontal line, h, the frictional coefficient was determined to be, approximately, l = tan h = 0.4. Referring to Fig. 6(a), frictional force acts in the axial (r1) direction. Note that when this friction is present, the axial stress, r1, varies along the length (Fig. 11(b)), as follows: r1  A þ Frictional force ¼ r1 tested  A

ð20Þ

where r1 tested is the measured axial stress and A is the cross-section area of a foam specimen of radius R. Taking A = pR2 and frictional force = lrp · 2pRx, r1  pR2 þ lrp  2pRx ¼ r1 tested  pR2

Fig. 11. (a) Sketch of the estimation of the frictional coefficient between the specimen and rubber jacket, (b) sketch of the frictional force.

ð21Þ

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

Hence r1 ¼ r1 tested  lrp 

2x R

(55/2) = 1.8 and the average difference in r1 is approximately 20%. While for foams with a relative density of 15%, 17% and 20%, l/R = 25/(55/2) = 0.9 and the average difference in r1 is approximately 15%. The experimentally determined yield surface is then re-plotted as a function of r1 adjusted and r2(=rp) in Fig. 12(a)–(e). In this case, agreement between theories and experiments is better than for the unadjusted case as shown in Fig. 10(a)–(e).

ð22Þ

where x varies from zero to the total length, l. rp is the confining pressure. An average value for the last term in the above equation (Eq. (22)) is lrp  2l=2 . Hence the axial R stress r1 may be adjusted to allow for the frictional effect, as follows: r1 adjusted ¼ r1 tested 

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4.2. Effect of plastic Poisson’s ratio

l lrp R

ð23Þ Particular attention has been paid to measurement of the plastic Poisson’s ratio, mpl, but in spite of this, no consistent result was obtained. Deshpande and Fleck [4] considered the plastic Poisson’s ratio instead of the elastic Poisson’s ratio in their criterion because the elastic Poisson’s ratio of aluminium foams appears to be close to zero

The difference between r1 adjusted and r1 tested is dependent upon rp and l/R, for a given l. In the test series different values of rp were used. Hence the average value of difference between r1 adjusted and r1 tested was calculated. For foams with a relative density of 5% and 10%, l/R = 50/

σ1 / σC 6

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-6

-4

4

2

-2

2

4

σ2 / σC

6

-2

-4

(a)

-6

σ1 / σC 4

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-4

2

-2

2

4

σ2 / σC

-2

(b)

-4

Fig. 12. Yield surfaces for triaxial tests of CYMAT aluminium foams with nominal relative density of (a) 5%; (b) 10%; (c) 15%; (d) 17%; (e) 20%. Experimental data were calibrated by taking friction into account.

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σ1 / σC 4

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-4

2

-2

4

2

σ2 / σC

-2

(c)

-4 σ1 / σC -1

-2

-3 -4 -1

triaxial test data (strain rate=10 s ) -3 -1 triaxial test data (strain rate=10 s ) -2 -1 triaxial test data (strain rate=10 s ) -1 -1 triaxial test data (strain rate=10 s ) 0 -1 triaxial test data (strain rate=10 s ) +1 -1 triaxial test data (strain rate=10 s ) uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

-1

σ2 / σC -2

-3

(d)

σ1 / σC 3

triaxial test data uniaxial test data Gibson et al. criterion Miller criterion Deshpande & Fleck criterion

2

1

-3

-2

-1

1 -1

-2

(e)

-3

Fig. 12 (continued)

2

3

σ2 / σC

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

and is difficult to measure from the experiments. They suggested that measurement of mpl is best performed by compressing a specimen to a uniaxial strain of 20–30%, using suitably lubricated loading platens. However, one particular feature of aluminium foam is localisation of deformation, i.e. the deformation is not evenly distributed within the specimen when crushed. Some region deforms more than other regions. Certain portion may not even start to deform even when the whole specimen is compressed to 20–30%. The value of mpl depends largely on the region or layer measured. Measuring mpl at different regions or layers and taking an average may help. But the data are too scattered. Theoretically speaking, it is possible to plot the variation of plastic Poisson’s ratio with foam relative density for various foams and fit a curve through them. Then an appropriate value of plastic Poisson’s ratio for the considered CYMAT foams is determined for a given density. However, experimental data of plastic Poisson’s ratio seem meagre [20]. Moreover, the available data are too scattered and the difference in the measured values could vary up to 10 times [20]. The plastic Poisson’s ratio of Alporas foam with relative density of 8% was 0.33 as reported by Motz and Pippan [21], while it was 0.024 by Gioux et al. [5]. Therefore it would be difficult to fit the experimental data of plastic Poisson’s ratio to a curve and then independently select a value for our density. It appears that the value of the plastic Poisson’s ratio of metal foam or aluminum foam is difficult to ascertain. However, both the yield surfaces of Miller and Deshpande and Fleck are very sensitive to the value of this plastic Poisson’s ratio mpl. For the same type of foams, say foams with a relative density of 5%, Deshpande and Fleck’s yield surface expands when a large value of mpl is employed (as

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shown in Fig. 13). Similarly, Miller’s yield surface increases with mpl (Fig. 14). But a large value of mpl may result in a saddle-shaped Miller’s yield surface (Fig. 14), which is contradictory with the requirement that the yield surface be convex rather than concave in stress space. Obviously, the values of mpl cannot be too large in order to obtain a reasonable Miller’s yield surface. Based on the above observation, values of the plastic Poisson’s ratio mpl were assumed to be, respectively, 0.17 for 5% and 10% foams, 0.29 for 15% foam and 0.3 for 17% and 20% foams in order to best fit their surfaces to our test data. Table 5 lists the parameters in Deshpande and Fleck [4] yield surface of foam with relative density of 5% when different values of plastic Poisson’s ratio were employed. The corresponding equations for the yield surfaces in Fig. 13 are given as follows: For mpl ¼ 0:30; 0:87ðr1  r2 Þ2 þ 0:13ðr1 þ 2r2 Þ  1:75r2C ¼ 0 ð24Þ For mpl ¼ 0:17; Eq: ð15Þ For mpl ¼ 0:10; 0:73ðr1  r2 Þ2 þ 0:27ðr1 þ 2r2 Þ2  1:75r2C ¼ 0 ð25Þ

Table 6 lists the parameters in Miller yield surfaces of foam with relative density of 5% when different values of plastic Poisson’s ratio were employed. The corresponding equations for the yield surfaces in Fig. 14 are For mpl ¼ 0:30; jr1  r2 j  0:22ðr1 þ 2r2 Þ 2

 0:034ðr1 þ 2r2 Þ =rC  1:19rC ¼ 0 For mpl ¼ 0:17; Eq: ð10Þ For mpl ¼ 0:10; jr1  r2 j  0:27ðr1 þ 2r2 Þ þ 0:045ðr1 þ 2r2 Þ2 =rC  1:32rC ¼ 0

σ1 / σc 2

γpl=0.30 γpl=0.17

1

γpl=0.10

-2

-1

ð26Þ

0

1

2

σ2 / σc

-1

-2 Fig. 13. Effect of plastic Poisson’s ratio mpl on Deshpande and Fleck yield surfaces.

ð27Þ

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σ1 / σc

6

γpl=0.30 γpl=0.17 γpl=0.10

4

0

2 -4

-2

2

4

6

σ2 / σc

-2

-4 Fig. 14. Effect of plastic Poisson’s ratio mpl on Miller yield surfaces.

Table 5 Parameters in Deshpande and Fleck yield surface of foam with relative density of 5% when different values of the plastic Poisson’s ratio were employed  1=2 t 0:5m mpl Y ¼ rc þr 2 a ¼ 3 1þmplpl 0.30 0.17 0.10

1.18 1.59 1.81

1.32rC

and Waas [12], using two different test methods. In order to compare those data with the Deshpande and Fleck [4] yield surfaces, we have re-plotted them in the equivalent stress and mean stress space (Fig. 15(a) and (b)). Deshpande and Fleck yield surfaces in terms of equivalent stress and mean stress for 10% foam and 17% foam are  2  2 re rm þ 2:5 ¼ 2:4 rc rc

Table 6 Parameters in Miller yield surfaces of foam with relative density of 5% when different values of plastic Poisson’s ratio were employed b = rc/rt

mpl

c

a0

d0

0.6069

0.30 0.17 0.10

0.67 0.76 0.82

0.37 0.16 0.53

1.19 1.27 1.32

Gioux et al. [5] used their measured mpl = 0.024 when plotting Miller yield surface. But when plotting Deshpande and Fleck yield surface for the same foam, a  1.4 was used [4], which corresponds to mpl = 0.23. Theoretically, plastic Poisson’s ratio mpl should be a constant value for a foam of particular relative density. A practical method of how to measure the plastic Poisson’s ratio mpl used in the equations of Miller’s [2] and Deshpande and Fleck’s [4] yield criteria should be investigated and specified clearly. Alternatively other parameters may be proposed in place of the plastic Poisson’s ratio in those criteria. 4.3. Comparison with shear strength Shear strength of nominally the same CYMAT aluminium foams has been reported by Hou et al. [22] and Rakow

and  2  2 re rm þ 1:4 ¼ 1:5 rc rc

ð28Þ

ð29Þ

pffiffiffi For pure shear of stress s, re ¼ 3s, rm = 0. As obtained in the present study, the uniaxial compressive strength used to normalise the shear strength is 0.97 MPa for 10% foam and 3.77 MPa for 17% foam. No uniaxial compressive strength was reported in [12], and measured compressive strength data, 0.97 MPa, was adopted to normalise the results of shear strength from [12]. Good agreement has been found for the 10% foam (Fig. 15(a)), but the shear strength is slightly higher (by 20%) for 17% foam (Fig. 15(b)). 4.4. A final remark The brittle nature of CYMAT foam cell material (aluminium alloy AlSi8Mg with SiC particles) would have a primary effect on the failure mechanism of cell walls. Hence for foams it would govern the initial value of the yield stress and the overall post yield (hardening) behaviour. Its effect on the initial yielding surface normalised with respect to the uniaxial compressive strength under present discussion may be less significant.

D. Ruan et al. / Composites Science and Technology 67 (2007) 1218–1234

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Fig. 15. Triaxial compressive data, shear strength data and Deshpande and Fleck yield surface in the equivalent stress and mean stress space for CYMAT foams: (a) 10%; (b) 17%.

5. Conclusions Three different types of tests, i.e. uniaxial compression, uniaxial tensile and triaxial compression, were conducted at a nominal axial strain rate of 104 s1 in order to construct the initial yield surfaces for CYMAT aluminium foams. Five types of closed-cell foams with nominal relative densities of 5%, 10%, 15%, 17% and 20%, respectively, were employed. Uniaxial compressive tests were conducted to obtain the uniaxial compressive strength for each foam. Quasi-static uniaxial tensile tests were carried out with dogbone foam specimens to investigate the tensile strength of CYMAT aluminium foams. Aluminium foams exhibit less ductility in tension than compression. The tensile strength of these

foams was found slightly higher than the corresponding compressive strength. Triaxial compressive tests were conducted on cylindrical foam specimens by using MTS and a Hoek Cell. Initial yield stresses were measured for a range of axisymmetric compressive stress states to construct the initial yield surfaces. It has been found that Gibson et al. criterion overestimates experimental yield stresses. The experimental data are consistent with both Miller and Deshpande and Fleck criteria when suitable plastic Poisson’s ratios are employed. An accurate independent measurement of the plastic Poisson’s ratio is essential. Triaxial compressive tests of aluminium foam with nominal relative density of 17% at various nominal axial strain rates (from 104 to 10+1 s1) indicate that the yield stresses

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of CYMAT closed-cell aluminium foams are not sensitive to the axial strain rate. Acknowledgement The authors thank the Australia Research Council for the financial support. References [1] Zhang J. The mechanics of foams and honeycombs. PhD thesis, University of Cambridge, Engineering Department, 1989. [2] Miller RE. A continuum plasticity model for the constitutive and indentation behaviour of foamed metals. Int J Mech Sci 2000;42:729–54. [3] Gibson LJ, Ahsby MF, Zhang J, Triantafillou TC. Failure surfaces of cellular materials under multiaxial loads – I. Modelling. Int J Mech Sci 1989;31:635–63. [4] Deshpande VS, Fleck NA. Isotropic constitutive models for metallic foams. J Mech Phys Solids 2000;48:1253–83. [5] Gioux G, McCormack TM, Gibson LJ. Failure of aluminium foams under multiaxial loads. Int J Mech Sci 2000;42:1097–117. [6] Hanssen AG, Hopperstad OS, Langseth M, Ilstad H. Validation of constitutive models applicable to aluminium foams. Int J Mech Sci 2002;44:359–406. [7] Doyoyo M, Wierzbicki T. Experimental studies on the yield behaviour of ductile and brittle aluminium foams. Int J Plast 2003;19:1195–214. [8] Huang WM. A simple approach to estimate failure surface of polymer and aluminium foams under multiaxial loads. Int J Mech Sci 2003;45:1531–40. [9] Blazy J-S, Marie-Louise A, Forest S, Chastel Y, Pineau A, Awade A, et al. Deformation and fracture of aluminium foams under proportional and non proportional multi-axial loading: statistical analysis and size effect. Int J Mech Sci 2004;46:217–44.

[10] Triantafillou TC, Zhang J, Shercliff TL, Gibson LJ, Ashby MF. Failure surfaces for cellular materials under multiaxial loads – II. Comparison of models with experiment. Int J Mech Sci 1989;31:665–78. [11] Sridhar I, Fleck NA. The multiaxial yield behaviour of an aluminium alloy foam. J Mater Sci 2005;40:4005–8. [12] Rakow JF, Waas AM. Size effects and the shear response of aluminum foam. Mech Mater 2005;37:69–82. [13] Rakow JF, Waas AM. Size effects in metal foam cores for sandwich structures. AIAA J 2004;42:1331–7. [14] Bazˇant ZP, Zhou Y, Zi G, Daniel IM. Size effect and asymptotic matching analysis of fracture of closed-cell polymeric foam. Int J Solids Struct 2003;40:7197–217. [15] Bazˇant ZP, Zhou Y, Nova’k D, Daniel IM. Size effect in fracture of sandwich structure components: foam and laminate. American Society of Mechanical Engineers, Applied Mechanics Division, 2001, AMD 248, p. 19–30. [16] Andrews EW, Gioux G, Onck P, Gibson LJ. Size effects in ductile cellular solids. Part II: experimental results. Int J Mech Sci 2001;43:701–13. [17] Ruan D, Lu G, Chen FL, Siores E. Compressive behaviour of aluminium foams at low and medium strain rates. Compos Struct 2002;57:331–6. [18] Lu G, Yu TX. Energy absorption of structures and materials. Cambridge, England: Woodhead Publishing Limited; 2003. [19] Sugimura Y, Meyer J, He MY, Bart-Smith H, Grenstedt J, Evans AG. On the mechanical performance of closed cell Al alloy foams. Acta Mater 1997;45:5245–59. [20] Wicklein M, Thoma K. Numerical investigations of the elastic and plastic behaviour of an open-cell aluminium foam. Mat Sci Eng A – Struct 2005;397:391–9. [21] Motz C, Pippan R. Deformation behaviour of closed-cell aluminium foams in tension. Acta Mater 2001;49:2463–70. [22] Hou W, Shen J, Lu G, Ong LS. Strength and energy absorption of aluminium foam under quasi-static shear loading. Key Eng Mater 2006;312:269–74.