Elastic and electric properties of closed-cell aluminum foams

Materials Science and Engineering A 420 (2006) 87–99

Elastic and electric properties of closed-cell aluminum foams Cross-property connection Igor Sevostianov a,∗ , Jaroslav Kov´acˇ ik b , Frantiˇsek Simanˇc´ık b a

b

Department of Mechanical Engineering, New Mexico State University, Las Cruces, NM 88001, USA Institute of Materials, Machine Mechanics, Slovak Academy of Sciences, Racianska 75, Bratislava 3, SK-831 02, Slovak Republic Received 26 September 2005; accepted 17 January 2006

Abstract Foamed aluminum (AlMg1Si0.6) in the porosity range 0.45–0.85 produced by the powder metallurgy method is analyzed with regard to its elastic and electric properties. Various predictive models for the electrical conductivity and Young’s modulus of closed-cell metal foam are assessed based on the experimental measurements. It is shown that the differential scheme provides the best predictions of the electrical conductivity in the porosity range 0.7–0.85, while Mori–Tanaka’s scheme gives the best results for the Young’s modulus. Comparing the two sets of the experimental data, cross-property coefficient that connects changes in the Young’s modulus and electrical conductivity of a material due to pores was determined. A non-trivial finding is that the best prediction of the cross-property coefficient is obtained in the framework of non-interaction approximation. © 2006 Elsevier B.V. All rights reserved. Keywords: Cross-property connection; Metal foams; Effective properties; Elasticity; Conductivity

1. Introduction Metal foams are highly porous materials with cellular structure. Due to this, they possess an excellent combination of mechanical properties (strength and stiffness) at the low weight, absorb high impact energies regardless of the impact direction, are electrically and thermally conductive, and are highly efficient in electromagnetic shielding and vibration damping. As mentioned by Grenestedt [1] aluminum foams seem to have a potential to greatly outperform the polymer foams (due to mechanical properties) and honeycomb structures (due to environmental properties). Structure and properties of metal foams (and cellular solids, in general) are discussed in detail in books [2,3]. A number of theoretical and experimental papers have been published on macroscopic behavior of the metal foams during last decade. Various constitutive laws have been suggested for the characterization and modeling of the macroscopic properties of the metal foams as functions of porosity p. Most of them, however, contain fitting parameters (see review [4]), which indicates that the derivation of the microstructure-property relationships for open or closed-cell foams is still not an accomplished issue.

∗

Corresponding author. Tel.: +1 505 646 3322; fax: +1 505 646 6111. E-mail address: [email protected] (I. Sevostianov).

0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.01.064

Semi-empirical modeling is usually based on percolation theory [5]. There, the effective property K behaves as a power of 1 − p: K = K0 (1 − p)t

(1.1)

where K0 is the corresponding property of the cell wall material. Experimentally measured properties of the foam can be fitted to (1.1) and the exponent t can be determined (see, for example [6]). The main disadvantage of this approach is that the exponent may be different for different properties and its micromechanical meaning is unclear (and, therefore, cannot be strictly predicted from microstructural parameters like shape and size of the pores). In Section 3, however, we show that for the electrical conductivity exponent t has very clear micromechanical sense and can be evaluated from the foam morphology. For elastic properties, such evaluation can be done approximately (see Section 4). Semi-numerical model was proposed [7,8] in the context of calculation of the Young’s modulus and the yield stress of closedcell foams. In these papers, however, the material is idealized as a perfectly periodic structure and the influence of the perturbation of periodicity is not discussed. Several micromechanical models for the overall elastic moduli of the closed-cell foams were analyzed in [1]. Substantial

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disagreement between the prediction and experimental data has been reported and the possible reasons for this disagreement were indicated. Unfortunately, the paper does not recommend any model as an appropriate one. Concerning electrical or thermal conductivity of the metal foams, substantial experimental results are presented in [6,9]. In these papers formula (1.1) is used to predict effective properties of aluminum foam and exponent t is evaluated from the experimental measurements. The present work is focused on further experimental study and micromechanical modeling of elastic and conductive properties and cross-property connection for closed-cell metal foam (AlMg1Si0.6). For 2D cellular structures, cross-property connection was first examined in [10] where the results are obtained in the form of bounds. We consider three-dimensional structures and show that the cross-property connection proposed in [11,12] work with good accuracy for such materials. It allows one to evaluate the elastic moduli of metal foam from relatively simple measurements of the electrical conductivity. As a by-product, the assessment of the predictive power of various micromechanical models is done for the electrical conductivity and Young’s modulus of AlMg1Si0.6 foam with the porosity range 0.45–0.85. Since none of the models presented in the literature can rigorously account for the pores of irregular shape, we only compared the predictions of those models that are adapted for randomly oriented ellipsoidal inhomogeneities. We approximate the pore shapes by oblate spheroids with an aspect ratio 0.7, a typical value found by image analysis (see Section 2). 2. Experiment 2.1. Specimens preparation There are two basic methods to produce aluminum foams: (A) Melt processing, when a gas is let into molten aluminum or aluminum alloy to blow up the melt and forming the aluminum foam. Another way is to mix a solid foaming agent (TiH2 , for instance) with molten aluminum or aluminum alloy. The foaming agent immediately decomposes and the released gas blows up the melt forming the aluminum foam. Then the obtained cellular structure is immediately cooled down to room temperature. (B) Powder metallurgical processing, using a compacted (extruded, rolled or isostatically pressed) mixture of aluminum powder and powder of a foaming agent almost pore

less. As the mixture is heated above the melting temperature of the aluminum, the gas released from the foaming agent expands the compacted precursor into a cellular solid, which is immediately cooled down to room temperature. The second method provides more flexibility for producing specimens of diverse shape due to the steel moulds using. Foamable precursor of the diameter of 8 mm was hot extruded from the compacted mixture of AlMg1Si0.6 powder and powdered foaming agent (0.4 wt.% TiH2 ). The specimens for testing have been foamed in steel moulds in an electrically heated furnace in the form of plates (140 mm × 140 mm × 8.6 mm) for measuring the electric conductivity and rods (diameter 25 mm, length 300 mm) for measuring the Yong’s modulus. To exactly reveal the inner pore structure, electric discharge machining has been used to cut the test specimens. The density of the foam specimens was determined by a volumetric method (from the weight and geometry) and computed from the photographs of the inner pore structures. (Note that for pores close to spherical, three-dimensional volume fraction of pores in a specimen statistically coincides with the twodimensional porosity of its cross-sections [13].) In the second case, the pores were filled with black resin to achieve high contrast between the pores and the pore walls. The specimens were then scanned with the resolution of 600 dpi. The density was computed in various segments of the images obtained along both axes using the ratio of the wall area to the whole segment area. The results are in good agreement with ones obtained by the volumetric method. Note that the computer analysis provides additional information about microstructure of specimens. The pores in the aluminum foam are essentially of slightly oblate shape and partially closed (Fig. 1). The extraction of microstructural information from the images has been done as follows [14]. We analyzed 13 images of specimens of different porosity. Four square segments 20 mm × 20 mm have been randomly chosen in each image and 2D shape factors for each region were calculated as R=

1
Ai γi2D A

(2.1)

i

where γi2D is the aspect ratio and Ai is the area of ith 2D pore, A is the total area of the pores. Then the average aspect ratio γ of 3D pores has been calculated for each value of porosity according to [14]: γ = 23 R

Fig. 1. Typical microstructure of AlMg1Si0.6 foam.

(2.2)

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Table 1 Measured values of the electrical conductivity of AlMg1Si0.6 foam at different levels of porosity

Fig. 2. Average aspect ratios of pores calculated by expressions (2.1) and (2.2) at various levels of porosity.

The results are presented in Fig. 2. Based on these data, we cannot identify any dependence of the aspect ratio on porosity and the average aspect ratio 0.7 has been chosen as a good approximation for micromechanical modeling in Sections 3 and 4. 2.2. Measurement of electrical conductivity

Porosity (%)

Electric conductivity (×106 S/m)

0.835576 0.822992 0.821637 0.817155 0.813750 0.811985 0.811421 0.811175 0.808165 0.802824 0.789039 0.788417 0.771546 0.769407 0.759225 0.707239 0.638198 0.550741 0.456011 0.000000

2.234 2.577 2.628 2.586 2.579 2.626 2.346 2.603 2.772 2.923 2.961 3.233 3.193 3.503 3.475 4.826 5.394 7.520 10.025 37.600

calculated from the resonant frequencies fn according to   2Lfn 2 E=ρ (2.3) n where n is the order of resonant frequency in harmonic oscillation, ρ the density, and L is the length of the specimen. The measured values of the Young’s modulus are given in Table 2 for various levels of porosity. 3. Electrical conductivity

The electrical conductivity of the flat aluminum foam samples was calculated from the geometry and resistance of the specimens. The resistance measurements were performed by the “four point” method in which four sharp tungsten electrodes are positioned under an optical microscope and are mechanically pressed in the sample surface. All the electrodes should be aligned in one line. The outer two electrodes are current bearing while the inner two electrodes in between are used for the voltage tap over the electrode distance. The measured values of the electric conductivity are presented in Table 1 for various levels of porosity.

We first consider a reference volume V of an infinite threedimensional solid (with the isotropic electrical conductivity k0 ) containing a pore (insulating inhomogeneity). Assuming a linear conduction law (linear relation between the gradient of electric potential U and the electric current vector Q per volume V), the resistivity contribution tensor of a pore HR is defined by the following relation [11]:

2.3. Measurement of the Young’s modulus

U=

Due to effects of clamping and plastic deformation of very thin cell walls at low stress levels, it is not easy to obtain the Young’s modulus of the aluminum foam from the slope of the stress–strain curve. Instead, it is more appropriate to determine it from free vibrations of the specimens. The specimens of cylindrical shape (diameter of 17 mm and length of 300 mm) were vibrated longitudinally using “impact hammer method” [15]. The amplitude of the specimen’s vibration reaches maxima at different resonant frequencies corresponding to harmonic oscillation. The Young’s modulus can be

For the pores of ellipsoidal shape, tensors HR can be expressed in terms of the Eshelby’s tensor for conductivity problem sC as follows:

3.1. Electrical resistivity contribution tensors

1 1 1 Q + U = Q + H R Q k0 k0 V

HR =

V∗ 1 −1 (I − sC ) V k0

(3.1)

(3.2)

where I is the second rank unit tensor. In the case of a spheroidal pore of aspect ratio γ, tensor sC has the following form: sK = f0 (γ)(I − nn) + (1 − 2f0 (γ))nn

(3.3)

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Table 2 Vibrational Young’s modulus of AlMg1Si0.6 foam at different levels of porosity

Then one gets the following relation for HR

Porosity (%)

Young’s modulus (GPa)

HR =

0.851562 0.840270 0.823922 0.822941 0.811623 0.811271 0.808756 0.808605 0.805034 0.797791 0.797288 0.796030 0.795024 0.793767 0.793264 0.783706 0.777369 0.762278 0.759562 0.756293 0.754280 0.745226 0.740699 0.732148 0.729558 0.727370 0.719070 0.708255 0.705640 0.703200 0.703150 0.701892 0.699453 0.698120 0.697692 0.696862 0.695579 0.694372 0.690600 0.686802 0.685544 0.685519 0.681772 0.000000

3.04 4.25 4.18 3.86 3.54 4.03 4.60 3.81 5.12 3.78 3.76 3.81 4.32 3.82 4.21 5.04 5.53 5.80 5.35 5.06 5.76 6.05 6.20 6.28 8.01 6.65 6.42 7.07 7.36 8.79 10.06 9.83 8.39 8.58 8.90 8.77 8.55 10.24 8.80 8.59 11.54 11.18 8.47 70.00

γ 2 (1 − g) 2(γ 2 − 1)

(3.6)

where A1 =

1 , 1 − f0 (γ)

A2 =

1 − 3f0 (γ) 2f0 (γ)[1 − f0 (γ)]

(3.7)

3.2. Non-interaction approximation This approximation is reasonably accurate at low concentration of inhomogeneities (“dilute limit”). If interaction between the inhomogeneities is neglected, each inhomogeneity can be assumed to be subjected to the same remotely applied electric current field. Contribution of the inhomogeneity into the change in the voltage can be treated separately and the total gradient of the electric potential is

1 1
R U = Q+ Q (3.8) H k0 V i

The summation over inhomogeneities may be changed by the integration over orientations if convenient. In particular, in the case of isotropic orientation distribution of spheroidal pores of aspect ratio γ (randomly oriented spheroidal pores):    1 A1 (γ) + A2 (γ) U= 1+p Q (3.9) k0 3 Using the methodology presented in [16], one can introduce the non-interaction resistivity contribution tensor H R NI such that the effective electrical resistivity tensor K−1 is expressed in its terms as follows: K−1 =

1 I + HR NI k0

(3.10)

For randomly distributed spheroidal pores of aspect ratio γ:   1 p A1 (γ) + A2 (γ) HR = η (γ)I = I (3.11) NI NI k0 k0 3 and the effective electrical conductivity of (isotropic) porous material is

where f0 =

V∗ 1 (A1 I + A2 nn) V 0 k0

(3.4)

and the shape factor g is expressed in terms of the aspect ratio γ as follows: ⎧  ⎪ 1 1 − γ2 ⎪ ⎪  arctan , oblate shape(γ < 1) ⎪ ⎨ γ 1 − γ2 γ  g(γ) = ⎪ γ + γ2 − 1 ⎪ ⎪ 1 ⎪  , prolate shape(γ > 1) ln ⎩ γ − γ2 − 1 2γ γ 2 − 1 (3.5)

k=

k0 1 + pηNI (γ)

(3.12)

Parameter ηNI is shape dependent. Fig. 3 illustrates its dependence on the aspect ratio γ. Note, that the non-interaction approximation, besides being rigorous at small concentration of inhomogeneities, is of a fundamental importance since it serves as a basic build block for various commonly used approximate schemes (self-consistent, differential, Mori–Tanaka’s, etc.) that place non-interacting inclusions into some sort of “effective environment” (effective field or effective matrix).

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the volume where “new” inhomogeneities are placed is already occupied by the “old” ones. For isotropic distribution of spheroidal pores, the differential scheme gives the effective electrical conductivity as follows: k = k0 (1 − p)ηNI (γ)

(3.14)

Fig. 3. Dependence of the parameter ηNI entering expression for the effective electrical conductivity on the aspect ratio γ.

Remark. The differential scheme gives micromechanical explanation for the semi-empirical formulas of the percolation theory (1.1) that is often used to describe overall properties of porous materials. Moreover, comparison of formulas (3.14) and (1.1) allows one to extract information about an “average” pore aspect ratio using the fitting parameter that may be obtained experimentally and the curve presented in Fig. 3. For example, the value of critical exponent in (1.1) for electrical conductivity was evaluated as 1.55 and 1.75 [6]. Substitution of these values into differential scheme (3.14) together with Fig. 3 gives the average aspect ratios of the (spheroidal) pores as 0.57 and 0.32, respectively.

3.3. Self consistent (effective media) scheme

3.5. Effective field method

For higher volume concentration of inhomogeneities it becomes unrealistic to consider them as isolated ones and their mutual interactions have to be accounted for. One of the simplest ways to do it is the application of self-consistent scheme that was first employed probably by Clausius [17]. In this approximation, each inhomogeneity is assumed to be surrounded by material possessing the (unknown) overall properties of the composite and subjected to the remotely applied external field [18–22]. For isotropic distribution of spheroidal pores, this method yields the following formula for the effective electrical conductivity:

In this scheme, each particle is treated as isolated one embedded into the matrix material and the effect of interaction is accounted for through the assumption that each particle lies within a certain effective field that differs from the applied macroscopic one. The basic idea of the method has roots in works of Mossotti (see [30], Chapter 11). Below, we consider two variants of the effective field method. Mori–Tanaka scheme. These approach [31] as interpreted in [32] is based on the assumption that the effective field acting on each inhomogeneity is equal to the average over the matrix. Then the macroscopic properties may be calculated from the non-interaction approximation with appropriate change of the remotely applied field. For isotropic distribution of spheroidal pores, the overall electrical conductivity according to the Mori–Tanaka method is

k = k0 (1 − pηNI (γ))

(3.13)

Remark. We note that for isotropically distributed pores (3.11) represents a linearization of formula (3.12). This fact should not confuse a reader—weak points of the self-consistent scheme have been widely discussed in literature. For references, we recommend reviews of Markov [23] and Hashin [24]. 3.4. Differential scheme The differential scheme can be considered as a kind of infinitesimal implementation of the self-consistent idea [23]. It was initiated by Bruggeman [18,25]. More recently it was elaborated in [26–28]. In [29] it was reported that the differential scheme gives the most accurate results on electrical conductivity of metal matrix composites containing insulating inclusions. The differential scheme assumes that the inhomogeneities are incrementally added to the material until the final volume fraction is reached. On each increment, a set of non-interacting inhomogeneities is added to the homogenized material with the properties determined by the previously embedded inhomogeneities. As pointed out in [27], the total concentration of inhomogeneities introduced to the matrix does not coincide with the volume concentration of the dispersed phase since part of

k=

1+

k0 p 1−p ηNI (γ)

(3.15)

Levin–Kanaun scheme represents a more refined variant of the effective field method. The main assumption of this method is that the effective field acting on each inhomogeneity is equal to the average over the whole composite (i.e. over matrix and inhomogeneities). It is free from the disadvantages of the Mori–Tanaka scheme discussed in [33,34]. This method has been proven on a variety of microstructures and, as shown in [35] its results coincide with the ones obtained by variational methods [36]. The method was developed for elastic properties of composites [37–39]. The conductivity problem has been addressed in [40]. For the case under consideration (isotropic distribution of spheroidal pores), the Levin–Kanaun scheme gives the following relation for the effective electrical conductivity: k=

k0 1+

pηNI (γ) 1−pηNI (γ)/ηNI (1)

(3.16)

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Fig. 4. Comparison of the experimentally measured effective electrical conductivities of the aluminum foam with predictions by various approximate micromechanical models.

Note that for spherical pores (γ = 1) ηNI (1) = 1.5 and Mori–Tanaka scheme coincides with the one of Levin–Kanaun (and with the upper Hashin–Shtrikman bound).

and compare the predictions with the experimental data given in Section 2. 4.1. Compliance contribution tensor

3.6. Comparison with the experimental data The results for the electrical conductivity obtained with the five methods discussed above are checked against the experimental data presented in Table 1. The comparison is illustrated in Fig. 4. It is seen that the differential scheme provides the best agreement with the experimental data for the porosity variation from 0.7 to 0.85. This result is in agreement with [28] where several predictive schemes are checked against experimental data for aluminum/alumina particulate composites with inhomogeneities of various shapes. For lower porosity levels, one can see growing divergence of the predicted values from the experimental data. Possible explanations may be related to non-adequate choice of the average pore aspect ratios at these porosity levels or to the non-negligible level of anisotropy (note that the degree of anisotropy of the foam with low porosity is usually higher than those of highly porous material). 4. Young’s modulus All the predictive methods mentioned in the previous section can be rewritten for the elastic properties of the inhomogeneous materials as well. Instead of second rank tensors, now we have to operate with fourth rank ones, and instead of one isotropic constant (effective conductivity) we have to evaluate two isotropic elastic constants. Again, all the predictive schemes are rooted in the non-interaction approximation and compliance contribution tensor. The general tensor equations (without solutions and specifications) related these schemes to non-interaction approximation are given in [16]. In this section we provide the formulas for effective Young’s modulus of a material containing isotropic mixtures of non-spherical pores

Utilizing the same approach as the one for the electrical conductivity problem, the compliance contribution tensor is defined by the following relation for the overall strain εij per volume V [41,12]: 0 εij = Sijkl σkl + Hijkl σkl

(4.1)

where the second term represents the strain change εij due to the presence of the inclusion. H-tensor depends on the inclusion shape and its elastic properties. (S0 is the matrix compliance tensor and σ is the “remotely applied” stress, assumed to be uniform in the absence of the inclusion.) For ellipsoidal pores, H-tensor is related to Eshelby’s tensor s as follows [12]: H=

V∗ 0 −1 [C : (J − s)] V

(4.2)

where C0 = (S0 )−1 , (J)ijkl = 21 (δik δjl + δil δjk ) and the inverse −1 −1 fourth rank tensor is defined as Sijkl Sklmn = Sijkl Sklmn = Jijmn . For a general ellipsoid, components Hijkl are expressed in terms of elliptic integrals. They reduce to elementary functions for the spheroidal shapes. Our analysis requires explicit analytic inversions of fourth rank tensors. Such inversions can be done by representing these tensors in terms of a certain “standard” tensor basis T(1) , . . ., T(6) [42,43] (see Appendix 1): V∗ 1
hk T (k) V G0 6

H=

(4.3)

k=1

(where G0 is the shear modulus of the matrix material) so that finding these tensors reduces to calculation of factors hk . Using the representations for the tensor of elastic stiffness, the Eshelby’s tensor and the unit tensor in terms of this basis

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(Appendix 1) yields the following relations for the coefficients: h1 =

κf0 − f1 , 2(4κ − 1)[2(κf0 − f1 ) − (4κ − 1)f02 ]

h2 =

1 , 2[1 − (2 − κ)f0 − f1 ]

h3 = h4 =

−(2κf0 − f0 + 2f1 ) , 4(4κ − 1)[2(κf0 − f1 ) − (4κ − 1)f02 ]

h5 =

4 , 4[f0 + 4f1 ]

h6 =

4κ − 1 − 6κf0 + 2f0 − 2f 4(4κ − 1)[2(κf0 − f1 ) − (4κ − 1)f02 ]

(4.4)

where κ=

1 , 2(1 − ν0 )

f1 =

κγ 2 4(γ 2 − 1)

2

[(2γ 2 + 1)g − 3]

(4.5)

and the shape factors f0 and g are given by (3.4) and (3.5), respectively. 4.2. Non-interaction approximation In the non-interaction approximation, the effective compliance tensor can be calculated as
S = S0 + (4.6) H (i) = S 0 + H NI i

Again, summation over the inhomogeneities may be replaced by the integration over orientations. In particular, it is convenient for spheroidal pores randomly distributed in isotropic matrix. In this case

   3B1 −B2  1 1 B2 2 H NI = p J − II II + (4.7) G0 3 G0 3 where p is the overall porosity and (II)ijkl = δij δkl ,

(J)ijkl = 21 (δik δjl + δil δjk )

Fig. 5. Shape factors B1 and B2 as functions of the spheroid aspect ratios (for Poisson’s ratio of dense AlMg1Si0.6 ν0 = 0.33).

4.3. Self-consistent scheme In contrast with the conductivity problem, the self-consistent scheme for the effective elastic constants leads to a system of two non-linear algebraic equations. Utilizing Eq. (4.6) we get the following system for the bulk and shear moduli:    2(1 + ν) 3B1 (γ, ν) − B2 (γ, ν) K = K0 1 − p , 1 − 2ν 2 G = G0 [1 − 2pB2 (γ, ν)]

(4.11)

Dividing the first equation by the second one gives a single nonlinear equation for the Poisson’s ratio: 1 + ν0 (1 + ν)[1 − 2pB2 (γ, ν)] =  3B1 (γ,ν)−B2 (γ,ν) 1 − 2ν0 1 − 2ν − p(1 + ν) 2

(4.12)

(4.8)

The coefficients entering expression for HNI can be obtained via integration of (4.3) over all possible orientations: 26h1 + 3h2 + 28h3 + 4h5 + 6h6 , 30 2h1 + 11h2 − 4h3 + 8h5 + 2h6 B2 = (4.9) 30 Shape factors B1 and B2 are functions of the spheroid aspect ratios and the Poisson’s ratio of the matrix and are illustrated in Fig. 5. The effective Young’s modulus can be calculated now in terms of parameter ξ(γ, ν0 ) = 2(1 + ν0 )(B1 + B2 /2) (dependence of ξ on the aspect ratio γ at ν0 = 0.33 is illustrated in Fig. 6). B1 =

E=

E0 1 + pξ(γ, ν0 )

(4.10)

This result serves as a basic block for various micromechanical approximate schemes discussed below.

Fig. 6. Dependence of the shape factor ξ(γ,ν0 ) = 2(1 + ν0 )(B1 + B2 /2) on the aspect ratio γ of pores at ν0 = 0.33.

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When this equation is solved, the effective Young’s modulus can be obtained from E = E0 (1 − pξ(γ, ν))

(4.13)

Note that the self-consistent scheme leads to substantial underestimation of the effective elastic properties. In particular, it predicts vanishing of the elastic response (E = 0) for spherical pores at p = 0.5 [28] and for non-spherical ones—even earlier (Fig. 6 shows that ξ(γ,ν) reaches its minimum equal to 2.0 when pores are spherical and ν0 = 0.33). Due to this well known disadvantage, we are not comparing this method with the experimental data obtained for porosity higher than 0.7. 4.4. Differential scheme As it was discussed in Section 3, the differential scheme may be interpreted as the infinitesimal implementation of the selfconsistent approach. Thus, it also leads to a system of two (differential) equations for elastic constants. Utilizing expression (4.6) the equations for bulk and shear moduli can be obtained as follows:   1 dK 1 2(1 + ν) 3B1 (γ, ν) − B2 (γ, ν) =− , K dp 1 − p 1 − 2ν 2 1 dG 2 =− B2 (γ, ν) G dp 1−p

(4.14)

(with obvious initial conditions K|p=0 = K0 ; G|p=0 = G0 ). A detailed analysis of the system (4.14) for spherical inhomogeneities has been performed in [28]. It was observed that the

system is autonomous and suggested to form a single differential equation for the Poisson’s ratio. In the case of spheroidal pores, such an equation takes the form 1 (1 − 2ν)(1 + ν) dν = dp 1−p 3    2(1 + ν) 3B1 (γ, ν) − B2 (γ, ν) × − + 2B2 (γ, ν) 1 − 2ν 2 (4.15) (with the initial condition ν|p=0 = ν0 ). Solution for (4.15) is illustrated in Fig. 7. Note, that in the contrast with spherical pores [28], ν0 = 0.2 is not exactly a point of attraction anymore (it is significant for the oblate shapes, in particular). When Eq. (4.15) is solved, the effective Young’s modulus can be expressed in its terms as   p  ξ(γ, ν(ρ)) E = E0 exp − dρ (4.16) 1−ρ 0 Fig. 8 shows that for any value of the aspect ratio γ, ξ(γ, ν(ρ)) is approximately constant (with the accuracy better than 1% for γ ≥ 0.1) for the whole range of porosity variation and thus its dependence on the variation of the Poisson’s ratio with porosity is negligible: ξ(γ,ν(ρ)) = ξ(γ,ν0 ) and (4.16) gives E = E0 (1 − p)ξ(γ,ν0 )

(4.17)

Using this approximation, Eq. (1.1) for Young’s modulus can be obtained from the differential scheme and strict micromechanical sense can be attributed to the exponent in this equation.

Fig. 7. Effective Poisson’s ratio of a porous material for various average pore aspect ratios. In the contrast with spherical pores ν0 = 0.2 is not the point of attraction. It is especially substantial for pores of oblate shape.

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Fig. 8. Dependence of the shape factor ξ(γ,ν(ρ)) on porosity for various values of the aspect ratio γ. For the whole range of porosity it is approximately constant (with the accuracy better than 1%). Thus, its dependence on the variation of the Poisson’s ratio with porosity is negligible.

Similarly to the electrical conductivity case discussed in the previous section, the information on the average pore shape can be extracted from the experimental measurement of the exponent and Fig. 6. 4.5. Effective field methods In contrast with the self-consistent and differential schemes, effective field approach applying for porous materials allows one to work with a single equation rather than a system. In particular, Mori–Tanaka and Levin–Kanaun schemes give the following expressions for the effective Young’s modulus: E=

E=

1+

E0 p 1−p ξ(γ, ν0 )

(Mori–Tanaka)

E0 1+

pξ(γ,ν0 ) 1−pξ(γ,ν0 )/ξ(1,ν0 )

(Levin–Kanaun)

(4.18)

(4.19)

4.6. Comparison with the experimental data The values of the Young’s modulus calculated according to the five methods discussed above are checked against the experimental data presented in Table 2. The comparison is illustrated in Fig. 9. For porosity range 0.68–0.85, the measured values of the Young’s modulus lie in between the lines predicted by the effective filed methods and the differential scheme. In this context it seems interesting to compare our results with ones presented in [27] for a sintered glass containing spherical pores (experimental data from [44]), where it is shown that for lower porosity levels (up to 0.5) the differential scheme gives the best agreement with the experimentally measured bulk modulus while the Mori–Tanaka approach works better for higher porosity. Since, the self-consistent scheme gives zero value of the Young’s modulus already at p = 0.5, the corresponding curve is not shown in the figure.

5. Cross-property connection for metal foams Cross-property connection (CPC) recently developed in [11,12] for inhomogeneous materials can be used to evaluate elastic properties of metal foams from the relatively simple measurements of the electrical conductivity. Attempts to reach lower levels of porosity (0.4–0.5) of foams lead to formation of anisotropic porous space. To the best of our knowledge, there are no simple methods to evaluate the full set of anisotropic elastic constants of such materials. CPC technique may provide simple and reasonably accurate method to do it. Another possible application of CPC method is the control over the mechanical properties of isotropic foams. At this time the reproducibility of the macroscopic properties of foams is insufficient that makes such a control desirable. In this section we illustrate the workability of CPC technique for materials with isotropic microstructures. The cross-property connection for such materials, can be written in the form E0 − Eeff k0 − keff = ΨCP Eeff keff

(5.1)

where Ψ CP is a cross-property coefficient. In [45], the crossproperty coefficient for the aluminum foam was derived from the non-interaction approximation of spherical pores: ΨCP =

(1 − ν0 )(9 + 5ν0 ) (7 − 5ν0 )

(5.2)

Now we can construct the coefficients according to the five models discussed in Sections 3 and 4. Non-interaction approximation gives ΨCP =

ξ(γ, ν0 ) η(γ)

(5.3)

The influence of the non-sphericity of pores is taken into account. This effect, however, is important for sufficiently oblate pores only (γ < 0.5).

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Fig. 9. Comparison of the experimentally measured effective Young’s modulus of the aluminum foam with predictions by various approximate micromechanical models.

5.1. Self-consistent scheme ΨCP =

ξ(g, ν(ρ)) η(γ)

(5.4)

Note that this formula cannot be used for metal foams since it utilizes the value of the Poisson’s ratio ν(p) calculated at given porosity. Since the method predicts vanishing of elastic properties at p = 0.5, it leads to wittingly wrong results at the higher levels of porosity. 5.2. Differential scheme Eqs. (3.14) and (4.16) give the following formula for the cross-property coefficient:   p exp 0 ξ(γ,ν(ρ)) 1−ρ dρ − 1 ΨCP = (5.5) (1 − p)−η(γ) − 1

Taking into account the approximation ξ(γ,ν(ρ)) = ξ(γ,ν0 ) (see discussion in Section 4), one can rewrite this expression as ΨCP =

(1 − p)−ξ(γ,ν0 ) − 1 (1 − p)−η(γ) − 1

(5.6)

5.3. Mori–Tanaka method Mori–Tanaka method gives the cross-property coefficient coinciding with one given by non-interaction approximation (since the interaction between the pores in the Mori–Tanaka scheme is accounted for conductive and elastic properties with the same coefficient): ΨCP =

ξ(γ, ν0 ) η(γ)

(5.7)

Fig. 10. Comparison of the experimentally measured cross-property coefficient of the aluminum foam with predictions by various approximate micromechanical models.

I. Sevostianov et al. / Materials Science and Engineering A 420 (2006) 87–99

5.4. Levin–Kanaun method According to formulas (3.16) and (4.19) the cross-property coefficient can be calculated as   NI (γ) ξ(γ, ν0 ) 1−pη ηNI (1)   ΨCP = (5.8) 0) ηNI (γ) 1−pξ(γ,ν ξ(1,ν0 ) To compare the predictions according all these formulas with experimental measurements, we calculated the cross-property coefficient using (5.1) and data presented in Tables 1 and 2 for the same porosity levels. The results of the comparison are presented in Fig. 10. One can see that the experimental values of the cross-property coefficient do not vary significantly (from 1.2 to 1.38 in the considered porosity range). These data are well approximated by non-interaction approximation and Mori–Tanaka scheme. Note that the formula (5.2) derived for the spherical pores gives quite a good agreement with the experimental data. The differential scheme and Levin–Kanaun method give Ψ CP depending on the porosity. An interesting observation is that the differential scheme giving sufficient accuracy in the prediction of both the electrical conductivity and the Young’s modulus leads to significant disagreement with the experimentally obtained cross-property coefficient. The explanation is obvious if we compare Figs. 4 and 9. The differential scheme always slightly overestimates the electrical conductivity and underestimates the Young’s modulus. When the cross-property coefficient is calculated, these two factors are multiplied and the disagreement may become remarkable (Fig. 10).

97

non-interaction approximation for the spherical pores (formula (5.2)). In the light that non-interaction approximation gives qualitatively incorrect prediction for both elastic and conductive properties of the foams (predicted properties are out of the Hashin–Shtrikman bounds), this fact is quite nontrivial: it means that the effects of the pore shape and pores interaction affect both elastic and conductive properties in similar manner. This hypothesis has been first formulated by Bristow [46] regarding microcracked materials. Overall, the cross-property coefficient for closed-cell aluminum foam may be well described by formula (5.2) which neglects pore shapes (all the pores are considered as spherical) and pore interactions. Appendix A. Tensor basis in the space of fourth rank tensors We outline a convenient technique of analytic inversion and multiplication of fourth rank tensors. It is based on expressing tensors in “standard” tensor bases to Kunin [42] and Walpole [43]. In the case of the transversely isotropic elastic symmetry, the following basis is most convenient: (1)

Tijkl = θij θkl , (3)

Tijkl = θij mk ml , (5)

Tijkl =

(2)

Tijkl =

θik θlj + θil θkj − θij θkl , 2

(4)

Tijkl = mi mj θkl ,

θik ml mj + θil mk mj + θjk ml mi + θjl mk mi , 4

(6)

Tijkl = mi mj mk ml

(A.1)

6. Conclusions 1. Young’s modulus and electrical conductivity of closedcell foamed aluminum (AlMg1Si0.6) in the porosity range 0.45–0.85 produced by the powder metallurgy method exhibit a marked similarity in dependence on the pores volume fraction. This similarity can be used to establish the cross-property connection for metal foams which allows one to evaluate elastic constants of the material from the relatively simple electrical conductivity measurements. 2. The electrical conductivity of the closed-cell aluminum foam is well predicted by the differential scheme for randomly oriented spheroidal pores. In the porosity range 0.7–0.85, only slight overestimation is observed (less than 10%). The effective field approaches (Mori–Tanaka and Levin–Kanaun schemes) overestimate the electrical conductivity (the disagreement is up to 50%). The self-consistent scheme underestimates the electrical conductivity considerably. In particular it predicts vanishing of the conductivity at p = 0.62. 3. The Young’s modulus of the closed-cell aluminum foam is satisfactory predicted by the differential scheme (underestimation less than 20%) and by the effective field methods (overestimation less than 20%). Self-consistent scheme gives qualitatively incorrect results. 4. Cross-property coefficient is well predicted by the noninteraction approximation, the effective field methods and the

where θ ij = δij − mi mj and m = m1 e1 + m2 e2 + m3 e3 is a unit vector along the axis of transverse symmetry. These tensors form the closed algebra with respect to the operation of (non-commutative) multiplication (contraction over two indices): (α)

(β)

(T (α) : T (β) )ijkl ≡ Tijpq Tpqkl

(A.2)

The table of multiplication of these tensors has the following form (the column represents the left multipliers):

Then the inverse of any fourth rank tensor X, as well as the product X:Y of two such tensors are readily found in the closed

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• Tensor of elastic stiffness of the isotropic material by Cijkl =  m has components cm Tijkl

form, as soon as the representations in the basis X=

6


Xk T (k) ,

Y=

k=1

6


Yk T (k)

(A.3)

k=1

X−1

(a) inverse tensor defined −1 (Xijmn Xmnkl ) = Jijkl is given by

by

−1 Xijmn Xmnkl

=

(1)

Jijkl =

2X1 (6) T ∆

(A.4)

where ∆ = 2(X1 X6 − X3 X4 ). (b) product of two tensors X:Y (tensor with ijkl components equal to Xijmn Ymnkl ) is X : Y = (2X1 Y1 + X3 Y4 )T

(1)

+(2X1 Y3 + X3 Y6 )T

+ X 2 Y2 T

(3)

+ (2X4 Y1 + X6 Y4 )T (4) (A.5)

is the axis of transverse symmetry, tensors T(1) , . . ., T(6)

If x3 given by (A.1) have the following non-zero components: (1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

(2)

(2)

(4)

(4)

(2)

(2)

T 1212 = T 2121 = T 1221 = T 2112 = T 1111 = T 2222 = 21 ,

(5)

(3)

T 1133 = T 2233 = 1, (5)

T 3311 = T 3322 = 1,

(5)

(5)

(5)

T 1313 = T 2323 = T 1331 = T 2332

(5)

(5)

(5)

= T 3113 = T 3223 = T 3131 = T 3232 = 41 , (6)

T 3333 = 1

(A.6)

General transversely isotropic fourth rank tensor, being represented in this basis Ψijkl =




m ψm Tijkl

has the following components: (Ψ1111 + Ψ1122 ) , ψ2 = 2Ψ1212 , ψ3 = Ψ1133 , 2 (A.7) ψ5 = 4Ψ1313 , ψ6 = Ψ3333 ψ4 = Ψ3311 ,

ψ1 =

Utilizing (A.7) one obtains the following representations: • Tensor of elastic compliances of the isotropic material Sijkl =  m has the following components sm Tijkl s1 =

1−ν , 4G(1 + ν)

s5 =

1 , G

s6 =

s2 =

1 , 2G

1 . 2G(1 + ν)

(A.10)

(2)

1 3 4 6 Jijkl = δij δkl = Tijkl + Tijkl + Tijkl + Tijkl

• Eshelby’s tensor for spheroidal inclusion sijkl = has components 1 f0 + f1 , 2(1 − ν) ν f0 − 2f1 , s3e = 1−ν

s5e = 2(1 − f0 − 4f1 ),

(A.11) 

e Tm sm ijkl

3 − 4ν f 0 + f1 , 2(1 − ν) ν s4e = (1 − 2f0 ) − 2f1 , 1−ν s2e =

s6e = 1 − 2f0 + 4f1

(A.12)

where f0 and f1 are given by (3.4) and (4.5). References

T 1111 = T 2222 = T 1122 = T 2211 = 1,

(3)

(A.9)

1 1 δik δlj + δil δkj 2 5 6 = Tijkl + Tijkl + 2Tijkl + Tijkl 2 2

s1e =

(2)

+ 21 X5 Y5 T (5) + (X6 Y6 + 2X4 Y3 )T (6)

T 1122 = T 2211 = − 21 ,

c5 = 4G,

where λ = 2Gν/(1 − 2ν). • Unit fourth rank tensors are represented in the form

X6 (1) 1 (2) X3 (3) X4 (4) 4 (5) T + T − T − T + T 2∆ X2 ∆ ∆ X5 +

c3 = c4 = λ,

c6 = λ + 2G.

are established. Indeed:

X−1 =

c1 = λ + G,

s3 = s4 = −

ν , 2G(1 + ν) (A.8)

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