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The influence of cell-size distribution on the plastic deformation in metal foams
Scripta Materialia 50 (2004) 295–300 www.actamat-journals.com
The influence of cell-size distribution on the plastic deformation in metal foams P. Kenesei *, Cs. K ad ar, Zs. Rajkovits, J. Lendvai Department of General Physics, E€otv€os University, P azm any P. s et any 1/A, Budapest H-1117, Hungary Received 9 April 2003; received in revised form 9 September 2003; accepted 18 September 2003
Abstract Analytical and numerical descriptions are developed and compared with results of compression tests for the increasing of the plateau stress of metal foams. Both uniform and non-uniform cell-size distributions are investigated. The energy absorption properties were also studied. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Compression test; Foams; Modelling
1. Introduction In the past few years there has been a remarkable increase of interest in metal foams, especially in aluminum foams with low relative density. The reason for this is the wide range of applicability: from sound and heat insulation, porous electrodes, catalytic surfaces to energy absorption, vibration reducers, high strength aircraft wing panels, etc. [1–3]. Currently there is a high interest in using Al-based foams for lightweight structural components in energy absorption systems for protection from impacts. Therefore it is important to investigate the compression properties of these foams. The stress–strain curves of metal foams can be usually characterized by a long plateau regime, which is constant in the case of a perfect cellular material. A model was given by Gibson and Ashby to determine the plateau stress of a perfect foam in which all cells are of the same shape and size [1]. According to this, during compression the plateau stress of a metal foam is considered to be a constant. If the foam is not perfect, then the plateau stress rises with increasing stress, due to the size distribution of the pores. In order to predict the energy absorption properties of metal foams, it is important to describe this kind of changes in plateau stress more precisely. *
Corresponding author. Tel.: +36-1-2090555/6458; fax: +3613722811. E-mail address: [email protected] (P. Kenesei).
The aim of this paper is to describe the plateau stress and the energy absorption as a function of the strain during compression. We found that an analytical formula could not be obtained if we used experimentally determined cell-size distributions for our calculations. By using simplifying assumption for the cell-size distribution (CSD), for example taking the CSD as a Gaussian function, the solution remained numerical. An analytical solution could be obtained if an open cell foam with uniform CSD was considered. Although this seemed to be a very rough approximation for closed-cell foams, it was found to satisfactorily describe the rising plateau stress.
2. Experimental Alporas metal foams manufactured by the Shinko Wire process were investigated. In this procedure 1–2% of calcium is stirred into the molten aluminum to increase the viscosity. To obtain low relative density foams 1–3% of TiH2 is added to the melt. The titanium hydride decomposes in the molten aluminum, releasing hydrogen gas to form bubbles in the metal. Adjusting the overpressure, temperature, time, the volume fraction of calcium and titanium hydride foams with different relative density and cell-size can be obtained [4]. In this paper foams with different relative densities ranging from 0.06–0.17 were studied.
1359-6462/$ - see front matter 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2003.09.046
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Table 1 The data of the investigated metal foams Name Relative density (%) Min. cell-size, l1 (mm) Max. cell-size, l2 (mm) Average wall thickness, tw (mm) Average edge thickness, te (mm)
Ap0.8
Ap1.1
Ap2.08
6 0.6 5.4 0.1 0.2
9 0.8 4.2 0.1 0.3
17 0.6 3.6 0.2 0.4
Compression tests were performed on three types of foams with different relative densities: q(Ap0.8) ¼ 6%, q(Ap1.1) ¼ 9% and q(Ap2.08) ¼ 17% (Table 1). An MTS hydraulic machine was used for quasi-static compression tests at 103 s1 strain rate. The dimensions of the rectangular samples were 30 · 30 · 25 mm3 . The displacement was determined from the crosshead displacement. The relative density of the specimens was calculated by measuring the weight and the volume of each specimen. The cell-size distribution and other structural parameters of the foams, like the approximate average cell wall and edge thickness were obtained by analyzing digitalized pictures taken from two dimensional slices of the foams (Table 1).
3. Results 3.1. Plateau stress In the model given by Gibson and Ashby [1] open-cell foams are considered as a cubic array of identical members of length l and square cross-section of side t, so that the cells meet at the midpoint of the struts of the adjoining cell. The plateau stress of a foam ðrpl Þ is given in this model as t 3 rpl ¼ Ary;s ; ð1Þ l where ry;s is the yield stress of the material of which the foam is made, A is a constant and the densification strain ðeD Þ eD ¼ 1 Dqrel ;
ð2Þ
where qrel is the relative density of the foam and D is a constant ðD 1:4Þ. It is also known that in cellular materials the deformation takes place in deformation bands. The first deformation band appears in the layer of the lowest density [5]. Let us suppose that cells in a deformation band are all of the same size. Consequently, first the cells with the largest cell-size ðl2 Þ and last the cells with the smallest cell-size ðl1 Þ will collapse during compression. This process controls the rising of the plateau stress with increasing strain.
In the case of foams the cross-sectional deformation can be neglected and the strain e is connected to the cellsize l, the size of the cell just about to collapse through the following equation R l2 03 0 0 R l2 l f ðl Þ dl l 3Dt2 l0 f ðl0 Þ dl0 ; ð3Þ eðlÞ ¼ l R l2 l03 f ðl0 Þ dl0 l1 where f ðlÞ is the cell-size distribution function (CSDF). The second term in the numerator of Eq. (3) takes into account the volume of the totally compressed foam. The simplest possible CSDF can be obtained supposing that the cell-size varies from l1 to l2 continuously and it is uniformly distributed between l1 and l2 , so the distribution function is f ðlÞ ¼
1 ; l2 l1
ð4Þ
where l2 > l1 . Substituting this distribution function into Eq. (3) eðlÞ ¼
ðl2 l22 Þðl2 þ l22 6Dt2 Þ l41 l42
ð5Þ
is obtained. In order to determine the plateau stress as a function of the relative density and strain, l has to be specified as a function of the strain from Eq. (5). Since the geometrical constraint for the cell edge thickness t due to the uniform distribution is rffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 l2 qrel t¼ ; ð6Þ 3 using Eq. (1) the following connection is obtained Ary;s 1 rpl ðeÞ ¼ pffiffiffi ffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 D þ D2 þ k2 2Dk þ ð1k4 Þe 3=2 qrel q2 k 2 q2 rel
ð7Þ
rel
which depends on four parameters: Ary;s , D, k ¼ l2 =l1 and the relative density. However, D, k and qrel can be directly measured consequently only Ary;s should be fitted. It is worth to note that in the case of e ¼ 0 using Eq. (6), Eq. (1) is recovered for the largest cell-size 3 Ary;s 1 t rpl ðe ¼ 0Þ ¼ pffiffiffi 3=2 ¼ Ary;s : ð8Þ l2 3 3 k qrel
In the case of k ¼ 1, when the foam consists of cells of the same cell-size the Gibson–Ashby model is recovered, substituting k ¼ 1 in Eq. (7) a constant (strain independent) plateau stress is obtained. Using a more realistic than uniform CSDF, for example Gaussian distribution, the expression for the plateau stress is no longer analytical, but numerical evaluation can be performed.
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3.2. Energy absorption The energy absorption capacity is defined as the energy necessary to deform a given specimen to a strain e. So the absorbed energy per unit volume ðWvol Þ is the area under the stress–strain curve up to a certain strain, namely Z e Wvol ðeÞ ¼ rðe0 Þ de0 : ð9Þ 0
Using Eq. (7), Eq. (9) can be expressed as 4Ak 2 q2 ry;s Wvol ðeÞ ¼ pffiffiffi rel 3 3ðk 4 1Þ
! Dqrel þ k 2D þ wðeÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; kqrel D þ wðeÞ
ð10Þ Fig. 1. The stress–strain curve of Alporas foams of different densities. The continuous curves in the figure are the functions calculated according to Eq. (7).
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 eÞk 4 þ e 2Dk 3 qrel þ D2 k 2 q2rel wðeÞ ¼ : kqrel
ð11Þ
The expression of the absorbed energy per unit volume contains the same parameters as Eq. (7) for the plateau stress. In the k ¼ 1 limit case the energy absorption capacity for ideal foam can be obtained from Eq. (10), which is a linear function of the strain.
4. Discussion Compressive tests were performed on several closedcell aluminum foams. The compressive stress–strain curves and the stress–strain curves predicted from Eq. (7) are shown in Fig. 1. If the foam is made from liquid (as Alporas foams investigated here), surface tension draws the material from the faces of cells into the cell edges, so the mechanical properties are determined predominantly by the edges [1]. Although in all calculations uniform CSDF (Eq. (4)) was assumed (see Fig. 1), the calculated curves are in good agreement with the experiments except the end of the plateau region ðe > 0:5Þ. The best fit was obtained for the foam Ap0.8. There are two possible reasons for this: i(i) among the foams investigated, the time available for drainage before solidification was the largest in the case of Ap0.8, so surface tension could draw the material into cell edges causing very fragile cell walls, and (ii) CSD is the most uniform in Ap0.8 among the three different foams. With this simple model not only the stress–strain curves can be described in reasonable agreement with the measurements, but the calculated function of energy absorption capacity vs. strain as well. Fig. 2 shows the measured energy absorption and the calculated energy
Fig. 2. The energy absorption capacity as a function of strain. The continuous curves in the figure are the functions calculated according to Eq. (9).
absorption function on the basis of Eq. (10) for the different foams. As it can be seen in Fig. 1, although the calculated curves exhibit a gradually increasing part at large strain, they fail at the end of the plateau region. To describe better the compression behavior of Alporas foams, two further corrections must be considered. i(i) First, we should use a more realistic CSDF (like e.g. Gaussian distribution) or the experimentally measured CSD. As mentioned before, except using the simplest CSDF (uniform distribution), we could not find analytical expressions for the plateau stress. (ii) Second, we have to take into account that the compression tests were done on closed-cell foams, but the calculations were made for open-cell foams. (As a first approximation, we have neglected the effect of the cell walls.)
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Fig. 3. The experimentally measured cell-size distribution of the different Alporas foams and the fitted Gaussian curves.
In order to take the effect of the walls into consideration, the following expression for the plateau stress of closed-cell foams [1] was used for the calculations instead of Eq. (1) t 3 t rpl e w ¼ B1 þ B2 ; ð12Þ ry;s l l
Fig. 5. The stress–strain curve of Alporas foams of different densities. The continuous curves in the figure are the numerically calculated stress–strain curves using the experimentally observed interpolated CSDFs.
where B1 and B2 are constants, and te and tw are the thickness of the cell edges and walls, respectively. For substituting the uniform CSDF with a more realistic one, the CSD was measured on approximately 150 pores (Fig. 3) and an interpolation formula of the measured distribution was applied in Eq. (3) for f ðlÞ. Since in the case of Ap2.08 Gaussian function is more realistic than the interpolated CSDF, Gaussian functions were also fitted to the measured CSDs and the effect of f ðlÞ was investigated. Following the procedures above, numerical expressions were obtained for both rðeÞ and Wvol ðeÞ with only two fitting parameters: B1 and B2 . The fitted curves are shown in Figs. 4–6. The uniform CSDF (Fig. 1) and the
Gaussian CSDF give nearly the same stress–strain curves. The plateau stress is practically the same for 0:1 < e < 0:5, while the stress is slightly lower in the quasi-elastic part in the case of Gaussian CSDF. Comparing these two CSDs the main difference occurs at the end of the plateau stage: the fitted curves based on the Gaussian CSD exhibit gradually increasing stress. However, these curves show a marked deviation from the measured stress–strain curves. By using the measured CSDF the stress in the plateau region is very similar to the simplest assumptions (the uniform CSDF). Since the low cell-size region of the CSD determines the stress at the end of the plateau, a better description for the end of plateau can be given with a better determination of CSD of small cells, with measured CSD. The effect of the CSDF and of the fitting parameters B1 and B2 on the plateau stress can be investigated separately in this model. The effect of the CSD can be investigated by a family of Gaussian distribution functions, where the variance ðrÞ of the Gaussian CSDF is
Fig. 4. The stress–strain curves of Alporas foams of different densities. The continuous curves in the figure are the numerically calculated stress–strain curves using Gaussian CSDFs.
Fig. 6. The energy absorption capacity as a function of strain. The continuous curves in the figure are the numerically calculated curves using the experimentally measured interpolated CSDFs.
P. Kenesei et al. / Scripta Materialia 50 (2004) 295–300
Fig. 7. The effect of the ratio of the maximum and minimum cell-size on the stress–strain curve. The stress–strain curves are evaluated according to Eq. (7) for k ¼ 1, 2, 3 and 6 using D ¼ 1:4 and qrel ¼ 0:1 as constants.
changing but the mean value (m ¼ 2:6 mm) and the boundary of the domain of variability (l1 ¼ 0:6 mm, l2 ¼ 5:4 mm) are fixed (Fig. 8 lower part). The two limit cases r ! 0 and r ! 1 recover the CSDF for the Gibson–Ashby arrangement and the uniform CSDF, respectively. In the upper part of Fig. 8 the effect of the thickness of edges compared to the thickness of cell walls was investigated by varying B1 while B2 was fixed. The plateau stress was found to be independent of B1 =B2 if the foam can be considered as closed-cell foam ðB1 =B2 < 1Þ. Further increase in B1 =B2 results in a rising plateau with higher slope.
299
It should be emphasized however, that the cell-size distribution does not influence strongly the character of the stress–strain curve in the plateau region. Consequently, to predict the behavior of foams and to analyze the dependence on the different parameters like the relative density of the foam ðqrel Þ and the ratio of the maximum and minimum cell-size ðkÞ the results of the analytical description, Eqs. (7) and (10) can be used. By examining the effect of these parameters it can be stated that the shape of the stress–strain curve is extremely sensitive to k: changes in k may modify the increment of the stress during the plateau region and the onset of the densification as well. As it can be seen in Fig. 7 for k ¼ 1 the stress–strain curve is constant. By increasing k the densification strain eD first decreases, then above about k ¼ 2 increases and the slope of the plateau stress decreases, in addition to this the initial value of the plateau stress decreases significantly as k increases. The parameters qrel and D influence also the shape of the curves; and these effects are not negligible. Obviously, if D decreases eD increases, as follows from Eq. (2). To investigate the influence of qrel , rpl resulting from Eq. (7) can be compared to the constant plateau stress, rGA pl from the Gibson–Ashby model [1]: 3=2
0 rGA ð13Þ pl ¼ A ry;s qrel ; p ffiffi ffi where A0 ¼ A=ð3 3Þ. On the left-hand side of Fig. 9 rGA pl and rpl are shown as a function of qrel for gradually increasing strains. The curves rpl and rGA pl intersect at a given strain. Drawing these intersections two regions
Fig. 8. The effect of CSDF and B1 =B2 on the stress–strain curve. A family of Gaussian distribution function is used in Eq. (3) for f ðlÞ. The variance ðrÞ of a Gaussian CSDF was changed but the mean value and the boundary of the domain of variability were fixed (lower). A family of stress–strain curves with changing B1 =B2 , B2 was fixed (upper).
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Fig. 9. The effect of qrel on the plateau stress. The plateau stress rpl compared to the rGA pl of the Gibson–Ashby model for different strains (left). The regions where the rpl smaller than rGA pl and contrary (right).
can be separated in which the stress calculated by Eq. (7) is smaller or larger than rGA pl (right-hand side of Fig. 9).
5. Conclusions The changes in the plateau stress of metal foams during compression were described with a simple model, using the Gibson–Ashby cubic arrangement. Supposing an open cell structure and uniform, continuous CSDF, an analytical expression can be derived for the stress and the energy absorption capacity as a function of strain. By using more realistic CSDFs or closed-cell structures only numerical methods can be applied for the evaluation. The results for the plateau stress and the energy absorption capacity show that these parameters are not very sensitive to the cell-size distribution. To draw conclusions concerning the influence of quantities like the relative density ðqrel Þ, the ratio of the maximum and the minimum cell-size ðkÞ and the densification factor ðDÞ one can use the analytical formulae of Eqs. (7) and (10). It is worth noting that the deformation behavior of metal foams can be satisfactorily predicted up to the densification strain (approximately e ¼ 0:5), by this kind
of description. In these calculations the work hardening of the alloy, which might have a slight effect on the plateau regime was neglected. We also would like to emphasize that although the model was found not to depend very sensitively on the precise form of the CSDF, it is not possible to get the increment of the plateau stress without taking the cell-size distribution into account.
Acknowledgements The authors thank to N. Babcsan for providing samples. This work was supported by the Hungarian Scientific Research Fund (OTKA) under grant T 043247.
References [1] Gibson LJ, Ashby MF. Cellular solids––structure and properties. Cambridge: Cambridge University Press; 1997. [2] Banhart J, Baumeister J. J Mater Sci 1998;33:1431. [3] Andrews E, Sanders W, Gibson LJ. Mater Sci Eng A 1999;270:113. [4] Ashby MF, Evans A, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG. Metal foams––a design guide. Oxford: ButterworthHeinemann; 2000. [5] Weaire D, Fortes MA. Adv Phys 1994;43:685.
The influence of cell-size distribution on the plastic deformation in metal foams P. Kenesei *, Cs. K ad ar, Zs. Rajkovits, J. Lendvai Department of General Physics, E€otv€os University, P azm any P. s et any 1/A, Budapest H-1117, Hungary Received 9 April 2003; received in revised form 9 September 2003; accepted 18 September 2003
Abstract Analytical and numerical descriptions are developed and compared with results of compression tests for the increasing of the plateau stress of metal foams. Both uniform and non-uniform cell-size distributions are investigated. The energy absorption properties were also studied. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Compression test; Foams; Modelling
1. Introduction In the past few years there has been a remarkable increase of interest in metal foams, especially in aluminum foams with low relative density. The reason for this is the wide range of applicability: from sound and heat insulation, porous electrodes, catalytic surfaces to energy absorption, vibration reducers, high strength aircraft wing panels, etc. [1–3]. Currently there is a high interest in using Al-based foams for lightweight structural components in energy absorption systems for protection from impacts. Therefore it is important to investigate the compression properties of these foams. The stress–strain curves of metal foams can be usually characterized by a long plateau regime, which is constant in the case of a perfect cellular material. A model was given by Gibson and Ashby to determine the plateau stress of a perfect foam in which all cells are of the same shape and size [1]. According to this, during compression the plateau stress of a metal foam is considered to be a constant. If the foam is not perfect, then the plateau stress rises with increasing stress, due to the size distribution of the pores. In order to predict the energy absorption properties of metal foams, it is important to describe this kind of changes in plateau stress more precisely. *
Corresponding author. Tel.: +36-1-2090555/6458; fax: +3613722811. E-mail address: [email protected] (P. Kenesei).
The aim of this paper is to describe the plateau stress and the energy absorption as a function of the strain during compression. We found that an analytical formula could not be obtained if we used experimentally determined cell-size distributions for our calculations. By using simplifying assumption for the cell-size distribution (CSD), for example taking the CSD as a Gaussian function, the solution remained numerical. An analytical solution could be obtained if an open cell foam with uniform CSD was considered. Although this seemed to be a very rough approximation for closed-cell foams, it was found to satisfactorily describe the rising plateau stress.
2. Experimental Alporas metal foams manufactured by the Shinko Wire process were investigated. In this procedure 1–2% of calcium is stirred into the molten aluminum to increase the viscosity. To obtain low relative density foams 1–3% of TiH2 is added to the melt. The titanium hydride decomposes in the molten aluminum, releasing hydrogen gas to form bubbles in the metal. Adjusting the overpressure, temperature, time, the volume fraction of calcium and titanium hydride foams with different relative density and cell-size can be obtained [4]. In this paper foams with different relative densities ranging from 0.06–0.17 were studied.
1359-6462/$ - see front matter 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2003.09.046
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Table 1 The data of the investigated metal foams Name Relative density (%) Min. cell-size, l1 (mm) Max. cell-size, l2 (mm) Average wall thickness, tw (mm) Average edge thickness, te (mm)
Ap0.8
Ap1.1
Ap2.08
6 0.6 5.4 0.1 0.2
9 0.8 4.2 0.1 0.3
17 0.6 3.6 0.2 0.4
Compression tests were performed on three types of foams with different relative densities: q(Ap0.8) ¼ 6%, q(Ap1.1) ¼ 9% and q(Ap2.08) ¼ 17% (Table 1). An MTS hydraulic machine was used for quasi-static compression tests at 103 s1 strain rate. The dimensions of the rectangular samples were 30 · 30 · 25 mm3 . The displacement was determined from the crosshead displacement. The relative density of the specimens was calculated by measuring the weight and the volume of each specimen. The cell-size distribution and other structural parameters of the foams, like the approximate average cell wall and edge thickness were obtained by analyzing digitalized pictures taken from two dimensional slices of the foams (Table 1).
3. Results 3.1. Plateau stress In the model given by Gibson and Ashby [1] open-cell foams are considered as a cubic array of identical members of length l and square cross-section of side t, so that the cells meet at the midpoint of the struts of the adjoining cell. The plateau stress of a foam ðrpl Þ is given in this model as t 3 rpl ¼ Ary;s ; ð1Þ l where ry;s is the yield stress of the material of which the foam is made, A is a constant and the densification strain ðeD Þ eD ¼ 1 Dqrel ;
ð2Þ
where qrel is the relative density of the foam and D is a constant ðD 1:4Þ. It is also known that in cellular materials the deformation takes place in deformation bands. The first deformation band appears in the layer of the lowest density [5]. Let us suppose that cells in a deformation band are all of the same size. Consequently, first the cells with the largest cell-size ðl2 Þ and last the cells with the smallest cell-size ðl1 Þ will collapse during compression. This process controls the rising of the plateau stress with increasing strain.
In the case of foams the cross-sectional deformation can be neglected and the strain e is connected to the cellsize l, the size of the cell just about to collapse through the following equation R l2 03 0 0 R l2 l f ðl Þ dl l 3Dt2 l0 f ðl0 Þ dl0 ; ð3Þ eðlÞ ¼ l R l2 l03 f ðl0 Þ dl0 l1 where f ðlÞ is the cell-size distribution function (CSDF). The second term in the numerator of Eq. (3) takes into account the volume of the totally compressed foam. The simplest possible CSDF can be obtained supposing that the cell-size varies from l1 to l2 continuously and it is uniformly distributed between l1 and l2 , so the distribution function is f ðlÞ ¼
1 ; l2 l1
ð4Þ
where l2 > l1 . Substituting this distribution function into Eq. (3) eðlÞ ¼
ðl2 l22 Þðl2 þ l22 6Dt2 Þ l41 l42
ð5Þ
is obtained. In order to determine the plateau stress as a function of the relative density and strain, l has to be specified as a function of the strain from Eq. (5). Since the geometrical constraint for the cell edge thickness t due to the uniform distribution is rffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 l2 qrel t¼ ; ð6Þ 3 using Eq. (1) the following connection is obtained Ary;s 1 rpl ðeÞ ¼ pffiffiffi ffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 D þ D2 þ k2 2Dk þ ð1k4 Þe 3=2 qrel q2 k 2 q2 rel
ð7Þ
rel
which depends on four parameters: Ary;s , D, k ¼ l2 =l1 and the relative density. However, D, k and qrel can be directly measured consequently only Ary;s should be fitted. It is worth to note that in the case of e ¼ 0 using Eq. (6), Eq. (1) is recovered for the largest cell-size 3 Ary;s 1 t rpl ðe ¼ 0Þ ¼ pffiffiffi 3=2 ¼ Ary;s : ð8Þ l2 3 3 k qrel
In the case of k ¼ 1, when the foam consists of cells of the same cell-size the Gibson–Ashby model is recovered, substituting k ¼ 1 in Eq. (7) a constant (strain independent) plateau stress is obtained. Using a more realistic than uniform CSDF, for example Gaussian distribution, the expression for the plateau stress is no longer analytical, but numerical evaluation can be performed.
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3.2. Energy absorption The energy absorption capacity is defined as the energy necessary to deform a given specimen to a strain e. So the absorbed energy per unit volume ðWvol Þ is the area under the stress–strain curve up to a certain strain, namely Z e Wvol ðeÞ ¼ rðe0 Þ de0 : ð9Þ 0
Using Eq. (7), Eq. (9) can be expressed as 4Ak 2 q2 ry;s Wvol ðeÞ ¼ pffiffiffi rel 3 3ðk 4 1Þ
! Dqrel þ k 2D þ wðeÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; kqrel D þ wðeÞ
ð10Þ Fig. 1. The stress–strain curve of Alporas foams of different densities. The continuous curves in the figure are the functions calculated according to Eq. (7).
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 eÞk 4 þ e 2Dk 3 qrel þ D2 k 2 q2rel wðeÞ ¼ : kqrel
ð11Þ
The expression of the absorbed energy per unit volume contains the same parameters as Eq. (7) for the plateau stress. In the k ¼ 1 limit case the energy absorption capacity for ideal foam can be obtained from Eq. (10), which is a linear function of the strain.
4. Discussion Compressive tests were performed on several closedcell aluminum foams. The compressive stress–strain curves and the stress–strain curves predicted from Eq. (7) are shown in Fig. 1. If the foam is made from liquid (as Alporas foams investigated here), surface tension draws the material from the faces of cells into the cell edges, so the mechanical properties are determined predominantly by the edges [1]. Although in all calculations uniform CSDF (Eq. (4)) was assumed (see Fig. 1), the calculated curves are in good agreement with the experiments except the end of the plateau region ðe > 0:5Þ. The best fit was obtained for the foam Ap0.8. There are two possible reasons for this: i(i) among the foams investigated, the time available for drainage before solidification was the largest in the case of Ap0.8, so surface tension could draw the material into cell edges causing very fragile cell walls, and (ii) CSD is the most uniform in Ap0.8 among the three different foams. With this simple model not only the stress–strain curves can be described in reasonable agreement with the measurements, but the calculated function of energy absorption capacity vs. strain as well. Fig. 2 shows the measured energy absorption and the calculated energy
Fig. 2. The energy absorption capacity as a function of strain. The continuous curves in the figure are the functions calculated according to Eq. (9).
absorption function on the basis of Eq. (10) for the different foams. As it can be seen in Fig. 1, although the calculated curves exhibit a gradually increasing part at large strain, they fail at the end of the plateau region. To describe better the compression behavior of Alporas foams, two further corrections must be considered. i(i) First, we should use a more realistic CSDF (like e.g. Gaussian distribution) or the experimentally measured CSD. As mentioned before, except using the simplest CSDF (uniform distribution), we could not find analytical expressions for the plateau stress. (ii) Second, we have to take into account that the compression tests were done on closed-cell foams, but the calculations were made for open-cell foams. (As a first approximation, we have neglected the effect of the cell walls.)
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Fig. 3. The experimentally measured cell-size distribution of the different Alporas foams and the fitted Gaussian curves.
In order to take the effect of the walls into consideration, the following expression for the plateau stress of closed-cell foams [1] was used for the calculations instead of Eq. (1) t 3 t rpl e w ¼ B1 þ B2 ; ð12Þ ry;s l l
Fig. 5. The stress–strain curve of Alporas foams of different densities. The continuous curves in the figure are the numerically calculated stress–strain curves using the experimentally observed interpolated CSDFs.
where B1 and B2 are constants, and te and tw are the thickness of the cell edges and walls, respectively. For substituting the uniform CSDF with a more realistic one, the CSD was measured on approximately 150 pores (Fig. 3) and an interpolation formula of the measured distribution was applied in Eq. (3) for f ðlÞ. Since in the case of Ap2.08 Gaussian function is more realistic than the interpolated CSDF, Gaussian functions were also fitted to the measured CSDs and the effect of f ðlÞ was investigated. Following the procedures above, numerical expressions were obtained for both rðeÞ and Wvol ðeÞ with only two fitting parameters: B1 and B2 . The fitted curves are shown in Figs. 4–6. The uniform CSDF (Fig. 1) and the
Gaussian CSDF give nearly the same stress–strain curves. The plateau stress is practically the same for 0:1 < e < 0:5, while the stress is slightly lower in the quasi-elastic part in the case of Gaussian CSDF. Comparing these two CSDs the main difference occurs at the end of the plateau stage: the fitted curves based on the Gaussian CSD exhibit gradually increasing stress. However, these curves show a marked deviation from the measured stress–strain curves. By using the measured CSDF the stress in the plateau region is very similar to the simplest assumptions (the uniform CSDF). Since the low cell-size region of the CSD determines the stress at the end of the plateau, a better description for the end of plateau can be given with a better determination of CSD of small cells, with measured CSD. The effect of the CSDF and of the fitting parameters B1 and B2 on the plateau stress can be investigated separately in this model. The effect of the CSD can be investigated by a family of Gaussian distribution functions, where the variance ðrÞ of the Gaussian CSDF is
Fig. 4. The stress–strain curves of Alporas foams of different densities. The continuous curves in the figure are the numerically calculated stress–strain curves using Gaussian CSDFs.
Fig. 6. The energy absorption capacity as a function of strain. The continuous curves in the figure are the numerically calculated curves using the experimentally measured interpolated CSDFs.
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Fig. 7. The effect of the ratio of the maximum and minimum cell-size on the stress–strain curve. The stress–strain curves are evaluated according to Eq. (7) for k ¼ 1, 2, 3 and 6 using D ¼ 1:4 and qrel ¼ 0:1 as constants.
changing but the mean value (m ¼ 2:6 mm) and the boundary of the domain of variability (l1 ¼ 0:6 mm, l2 ¼ 5:4 mm) are fixed (Fig. 8 lower part). The two limit cases r ! 0 and r ! 1 recover the CSDF for the Gibson–Ashby arrangement and the uniform CSDF, respectively. In the upper part of Fig. 8 the effect of the thickness of edges compared to the thickness of cell walls was investigated by varying B1 while B2 was fixed. The plateau stress was found to be independent of B1 =B2 if the foam can be considered as closed-cell foam ðB1 =B2 < 1Þ. Further increase in B1 =B2 results in a rising plateau with higher slope.
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It should be emphasized however, that the cell-size distribution does not influence strongly the character of the stress–strain curve in the plateau region. Consequently, to predict the behavior of foams and to analyze the dependence on the different parameters like the relative density of the foam ðqrel Þ and the ratio of the maximum and minimum cell-size ðkÞ the results of the analytical description, Eqs. (7) and (10) can be used. By examining the effect of these parameters it can be stated that the shape of the stress–strain curve is extremely sensitive to k: changes in k may modify the increment of the stress during the plateau region and the onset of the densification as well. As it can be seen in Fig. 7 for k ¼ 1 the stress–strain curve is constant. By increasing k the densification strain eD first decreases, then above about k ¼ 2 increases and the slope of the plateau stress decreases, in addition to this the initial value of the plateau stress decreases significantly as k increases. The parameters qrel and D influence also the shape of the curves; and these effects are not negligible. Obviously, if D decreases eD increases, as follows from Eq. (2). To investigate the influence of qrel , rpl resulting from Eq. (7) can be compared to the constant plateau stress, rGA pl from the Gibson–Ashby model [1]: 3=2
0 rGA ð13Þ pl ¼ A ry;s qrel ; p ffiffi ffi where A0 ¼ A=ð3 3Þ. On the left-hand side of Fig. 9 rGA pl and rpl are shown as a function of qrel for gradually increasing strains. The curves rpl and rGA pl intersect at a given strain. Drawing these intersections two regions
Fig. 8. The effect of CSDF and B1 =B2 on the stress–strain curve. A family of Gaussian distribution function is used in Eq. (3) for f ðlÞ. The variance ðrÞ of a Gaussian CSDF was changed but the mean value and the boundary of the domain of variability were fixed (lower). A family of stress–strain curves with changing B1 =B2 , B2 was fixed (upper).
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Fig. 9. The effect of qrel on the plateau stress. The plateau stress rpl compared to the rGA pl of the Gibson–Ashby model for different strains (left). The regions where the rpl smaller than rGA pl and contrary (right).
can be separated in which the stress calculated by Eq. (7) is smaller or larger than rGA pl (right-hand side of Fig. 9).
5. Conclusions The changes in the plateau stress of metal foams during compression were described with a simple model, using the Gibson–Ashby cubic arrangement. Supposing an open cell structure and uniform, continuous CSDF, an analytical expression can be derived for the stress and the energy absorption capacity as a function of strain. By using more realistic CSDFs or closed-cell structures only numerical methods can be applied for the evaluation. The results for the plateau stress and the energy absorption capacity show that these parameters are not very sensitive to the cell-size distribution. To draw conclusions concerning the influence of quantities like the relative density ðqrel Þ, the ratio of the maximum and the minimum cell-size ðkÞ and the densification factor ðDÞ one can use the analytical formulae of Eqs. (7) and (10). It is worth noting that the deformation behavior of metal foams can be satisfactorily predicted up to the densification strain (approximately e ¼ 0:5), by this kind
of description. In these calculations the work hardening of the alloy, which might have a slight effect on the plateau regime was neglected. We also would like to emphasize that although the model was found not to depend very sensitively on the precise form of the CSDF, it is not possible to get the increment of the plateau stress without taking the cell-size distribution into account.
Acknowledgements The authors thank to N. Babcsan for providing samples. This work was supported by the Hungarian Scientific Research Fund (OTKA) under grant T 043247.
References [1] Gibson LJ, Ashby MF. Cellular solids––structure and properties. Cambridge: Cambridge University Press; 1997. [2] Banhart J, Baumeister J. J Mater Sci 1998;33:1431. [3] Andrews E, Sanders W, Gibson LJ. Mater Sci Eng A 1999;270:113. [4] Ashby MF, Evans A, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG. Metal foams––a design guide. Oxford: ButterworthHeinemann; 2000. [5] Weaire D, Fortes MA. Adv Phys 1994;43:685.