On the elastic deformation behavior of nanoporous metal foams

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ScienceDirect Scripta Materialia 69 (2013) 781–784 www.elsevier.com/locate/scriptamat

On the elastic deformation behavior of nanoporous metal foams Giorgio Pia and Francesco Delogu⇑ Dipartimento di Ingegneria Meccanica, Chimica, e dei Materiali, Universita` degli Studi di Cagliari, via Marengo 2, 09123 Cagliari, Italy Received 30 June 2013; revised 26 August 2013; accepted 27 August 2013 Available online 6 September 2013

A refined modeling of the elastic deformation behavior of nanoporous metal foams is discussed. The foam structure is described in terms of relatively massive cubic junctions joining six ligaments of square cross-section. Experimental data are used to roughly estimate the mass agglomerated at junctions and to show that ligaments must be regarded as thick beams. Their bending behavior is described by a hyperbolic shear deformation theory. A good agreement between experimental and theoretical estimates of the Young’s modulus is obtained. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Porous material; Nanostructured material; Metals and alloys; Elastic behavior; Modeling

Nanoporous (NP) metal foams exhibit a disordered porous structure with nanometer-sized ligaments and pores [1,2]. Reduced dimensionality and a high surface area-to-volume ratio give rise to unusual physical and chemical properties that make NP metals promising candidates as structural and functional materials in different fields of advanced technology [3–10]. Although the emergence of novel properties can sometimes be related to size effects [1,2,11–13], the combination of characteristic lengths and structural features also plays a crucial role [1–13]. Therefore, a clear understanding of the relationship between structure and properties is needed to fully exploit the technological potential of NP metals. In this respect, both experimental and theoretical studies face significant challenges. As recently pointed out [14], even obtaining accurate estimate of the relative density, /, is a difficult issue, requiring combined scanning electron microscopy and Rutherford backscattering spectrometry analyses. Thus, it is not surprising that the mechanical properties of NP metals, which are intimately related to these materials’ porosity and connectivity, are still far from being properly understood [10,14–17]. Attempts to rationalize these properties have mostly relied upon the model proposed by Gibson and Ashby to account for the mechanical response of open-cell

⇑ Corresponding author. Tel.: +39 70 675 50 73; fax: +39 70 675 50

macroporous foams [18,19]. Based on the bending behavior of thin beams, the model predicts that the Young’s modulus and yield strength vary with the relative density as /2 and /3/2, respectively [18,19]. Predictions and experimental measurements agree as long as / is <0.1 [18,19]. However, for NP metals / is of the order of 0.3–0.6, and the Young’s modulus and yield strength are unexpectedly large [1,2,5,10,14]. Liu and Antoniou recently suggested that the increase in Young’s modulus can be partly ascribed to the mass agglomeration at ligament junctions [14]. Starting from this result, the present work provides an improved modeling description of the mechanical response of NP metals by taking into account the influence of the ligament thickness. The same simplified structural unit proposed by Liu and Antoniou is used to describe the NP metal foam structure [14]. As schematically shown in Figure 1a, this consists of a cubic node with sides t + 2d long joining six thinner h/2 long semi-ligaments with square cross-section of area t2, where t is the ligament thickness and d measures the mass aggregated at junctions. This unit is enclosed in a cubic volume of side l. As l = t + 2d + h, t, d, and h cannot vary independently of each other. In the case of a bulk metal, the cubic volume Vtot = l3 is fully occupied by the solid, and the corresponding mass is equal to mb = ql3, where q is the bulk density. For a NP that occupies the same macroscopic volume, 3 the mass is equal to mNP ¼ bðt þ 2dÞ þ 3t2 hcq, and the

67; e-mail: [email protected] 1359-6462/$ - see front matter Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2013.08.027

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Figure 1. (a) The elementary unit of the NP metal foam structure with the characteristic lengths. These are given as average values calculated on the basis of literature data [14]. (b) Schematic representation of the bending behavior.

density to qNP ¼ bðt þ 2dÞ3 þ 3t2 hcq=l3 . It follows that the NP metal has relative density q 3 / ¼ NP ¼ ða þ 2bÞ þ 3a2 ð1  a  2bÞ; ð1Þ q where a and b are equal to t/l and d/l, respectively, and h/l has been expressed as 1  a  2b recalling that h = l  t  2d. When a  1 and b = 0, Eq. (1) simplifies to / = a3 + 3a2, and not to a4 as predicted for foams formed by thin beams [18,19]. This is due to the fact that, unlike the present case, the original Gibson–Ashby cubic cell model cannot be periodically extended, so that the density cannot recover its bulk value [18,19]. Instead, the / expression above overlaps with that proposed by Gibson and Ashby to account for the density contribution of cell corners [18,19]. The present model also predicts a specific surface area equal to: 2

S sp ¼

2

12 a  a þ 2b : lq ða þ 2bÞ3 þ 3a2 ð1  a  2bÞ

ð2Þ

When a  1 and b = 0, Eq. (2) reduces to / = 4(qt)1, not dissimilar to the equations obtained by De Hosson and co-workers in their recent work on the specific surface area of NP metals [20]. Once / and a are known, Eq. (1) can be used to calculate b. Such calculations have been performed on the refined experimental / and a values reported by Liu and Antoniou [14]. The results obtained are shown in Table 1. It appears that a is invariably larger than 0.1, with an average value around 0.31. This agrees with previous results on NP Au, for which a ranges approximately between 0.33 and 0.50 [21]. u is also quite large, averaging around 0.55. Furthermore, b values are comparable with a

ones, their average amounting to 0.24. Finally, the plot in Figure 2 indicate a decrease in the mass agglomerated at junctions as the ligament thickness increases, suggesting that a and b are correlated. This evidence indicates that the structure of NP metals is very different to that underlying the Gibson–Ashby model [18,19]. The ligaments are quite thick, and the agglomeration of mass at junctions significantly reduces the portion of the structural unit undergoing bending. The average thickness of nodes and ligaments is equal to about 0.31l and 0.78l, respectively. It should also be noted that ligaments are about 0.22l long on the average, i.e. they are thicker than they are long. As schematically shown in Figure 1b, under these circumstances only ligaments can bend significantly in response to mechanical loads, whereas the nodes remain substantially rigid. In the light of these features, it appears unlikely that the Euler–Bernoulli theory can satisfactorily describe the bending behavior of ligaments in NP metals [22]; in fact, this theory ignores transverse shear deformation effects, which are important for thick beams [22]. Timoshenko’s first-order shear deformation theory is also unsatisfactory, due to the limitations related to the assumption of a constant transverse shear strain distribution [23–25]. Such limitations can be tackled by using case-specific higher-order shear deformation theories [26–32]. The case of a thick, simply supported, uniform isotropic beam with a central concentrated load, commonly invoked in the elastic deformation of foams, can be satisfactorily discussed within the hyperbolic shear deformation theory [33–37]. Here, only a short description of the calculations needed to estimate the maximum transverse deflection of the bent beam is given. Details can be found elsewhere [33–37]. The solid beam occupies the initial volume defined by the Cartesian coordinates 0 6 x 6 h, t/2 6 y 6 t/2 and t/2 6 z 6 t/2. In other words, coherent with previous considerations, only the thinner parts of the NP metal foam are allowed to bend. The axial displacement field can be written as: dwðxÞ uðx; zÞ ¼ z dx      1 1 þ z cosh  t sinh uðxÞ: 2 2

ð3Þ

The first term in Eq. (3) expresses the axial displacement according to the Euler–Bernoulli theory, where w(x) is the transverse deflection of the beam center line. The term in square brackets represents the hyperbolic deflection due to transverse shear deformation, and u(x) is a function associated with the rotation of the beam cross-section at the neutral axis [33–37]. The normal and shear strains are respectively expressed as [33–37]: duðx; zÞ dx    z duðxÞ d 2 wðxÞ 1 ¼ z þ z cosh ;  t sinh 2 t dx dx2

ex ¼

ð4Þ

G. Pia, F. Delogu / Scripta Materialia 69 (2013) 781–784

783

Table 1. The relative density /, the relative ligament thickness a, the measure of mass agglomeration at junctions b, and the relative Young’s modulus Erel. Experimental / and a data, and Erel ones, are taken from Tables 1 and 2 of Ref. [14]. Data refer to NP Au. Data in bold refer to NP Pt. /

a

b

Erel

0.56 0.53 0.61 0.64 0.65 0.64 0.62 0.63 0.69 0.55 0.65 0.62 0.60 0.60 0.65

0.31 0.23 0.30 0.39 0.34 0.41 0.32 0.36 0.39 0.37 0.36 0.39 0.38 0.33 0.38

0.24 0.28 0.26 0.22 0.25 0.20 0.25 0.23 0.23 0.20 0.24 0.21 0.21 0.24 0.23

0.098 0.049 0.094 0.167 0.125 0.183 0.108 0.140 0.168 0.142 0.142 0.165 0.155 0.115 0.159

0.09 0.06 0.16 0.20

0.11 0.10 0.11 0.19 0.17

/

a

b

Erel

0.47 0.32 0.46 0.57 0.56 0.58 0.47 0.43 0.46 0.48 0.44 0.47 0.55 0.48 0.52

0.40 0.33 0.40 0.35 0.36 0.32 0.14 0.13 0.15 0.17 0.19 0.17 0.32 0.32 0.27

0.14 0.12 0.13 0.22 0.21 0.24 0.32 0.31 0.31 0.30 0.28 0.30 0.23 0.22 0.26

0.156 0.084 0.153 0.128 0.135 0.106 0.013 0.009 0.016 0.022 0.028 0.022 0.108 0.097 0.074

0.16

0.28

0.3 0.45 0.33 0.37

where I = t4/12 is the moment of inertia of the beam area, and A, B and C, are equal to cosh(1/2) – 12[cosh (1/2) – 2sinh(1/2)], cosh2(1/2) + 6[sinh(1)  1]  24 cosh (1/2)[cosh(1/2)  2 sinh(1/2)] and cosh2(1/2) + (1/2) [sinh(1) + 1]  4cosh(1/2)sinh(1/2), respectively. The associated boundary conditions are expressed by d3w(x)/ dx3  Ad2u(x)/dx2 = 0, d2w(x)/dx2  Adu(x)/dx = 0 and Ad2w(x)/dx2  Bdu(x)/dx = 0. Eqs. (7) and (8) can be integrated and rearranged to give the single differential equation: Figure 2. The measure of mass agglomeration at junctions, b, as a function of the ligament thickness, a. Data refer to NP Au and Pt [14]. The dotted line is a visual guide.

duðx; zÞ dwðxÞ þ dz dx    z 1 ¼ cosh /ðxÞ:  cosh 2 t

exz ¼

ð5Þ

Correspondingly, the normal and shear stresses are equal to rx = Eex and rxz = Gexz, where E and G represent the Young’s and shear moduli. When the principle of virtual work is applied to a beam subjected to a transverse load q(x) along the beam length, the following expression is obtained: Z x¼h Z z¼t=2 t ðrx þ dex þ rxz dexz Þdxdz x¼0



Z

z¼t=2 x¼h

qðxÞdwðxÞdx x¼0

¼ 0:

ð6Þ

Here, d represents the variational operator. The use of Green’s theorem allows the coupled Euler–Lagrange governing equations to be obtained [33–37]: d 4 wðxÞ d 3 uðxÞ qðxÞ  A ¼ ; EI dx4 dx3 3

A

ð7Þ

d 2 uðxÞ  Gt2 CuðxÞ ¼ AQðxÞ; ð9Þ dx2 R where Q(x) = q(x) dx is the generalized shear force along the beam length [33–37]. Once solved, Eq. (9) provides a general expression for u(x). In turn, this expression can be substituted in Eqs. (7) and (8) to obtain a general expression for the transverse deflection w(x) of the beam center line [33–37]. The general expression for w(x) can be specialized by imposing suitable boundary conditions. For a simply supported uniform isotropic beam with a central concentrated load F at x = h/2, the boundary conditions are dw(x)/dx = u(x) = 0 at x = h/2, and d2w(x)/ 2 dx = du(x)/dx = w(x) = 0 at x = 0, L [33–37]. In addition, the central beam cross-section must remain normal to the beam surface and planar. Under these conditions, the maximum transverse deflection at x = h/2 is equal to:   t 2  Fh3 wðh=2Þ ¼ 1 þ 2:4ð1 þ mÞ ; ð10Þ h 48EI ðB  A2 ÞEI

where m is the Poisson ratio of the material. Following Gibson and Ashby’s approach [18,19], it can be shown that the relative Young’s modulus of NP metal foams is equal to: Erel ¼

Eeff a4 ¼4 2 E ð1  a  2bÞ  1 a2  1 þ 2:4ð1 þ mÞ ; 1  a  2b

ð11Þ

2

d wðxÞ d uðxÞ B þ Gt2 CuðxÞ ¼ 0; 3 dx dx2

ð8Þ

where Eeff is the effective Young’s modulus. The term in square brackets can be regarded as a correction factor

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Figure 3. The relative Young’s modulus of NP metal foams, Erel, as a function of the relative density /. Experimental data ( ) are taken from Table 2 of Ref. [14]. The points enclosed in the ellipse refer to NP Pt. The curve is best-fitted to the modeled data.

for the first term, which corresponds to the effective Young’s modulus predicted by the Gibson–Ashby model [18,19]. Eq. (11) indicates that thick beams exhibit smaller Eeff values than thin beams. The Erel estimates obtained from Eq. (11) are compared in Table 2 with the available experimental data taken from Liu and Antoniou’s work [14]. Theoretical and experimental estimates are also plotted in Figure 3 as a function of the relative density / of the NP metal foams. A relatively good agreement is obtained, with experimental findings and theoretical predictions overlapping in a few cases regarding NP Au. However, the Erel estimates for NP Pt are considerably larger than predicted, which could be related to size effects [14]. Although Eq. (11) does not indicate any trivial relationship between Eeff and /, it should be noted that the data are scattered around the curve Erel = 0.35/2, which is reminiscent of the Gibson–Ashby model predictions. In summary, the elastic deformation behavior of NP metal foams has been described by using a refined modeling approach to the bending behavior of thick ligaments. The model has been applied to a regular foam structure with massive cubic nodes and short ligaments with a square cross-section. Our model predicts a smaller Young’s modulus for NP metals than that predicted by the Gibson–Ashby model. In the case of NP Au foams, the predictions satisfactorily agree with experimental data. Financial support has been given by the University of Cagliari. [1] B.C. Tappan, S.A. Steiner III, E.P. Luther, Angew. Chem. Int. Ed. 49 (2010) 4544. [2] J. Zhang, C.M. Li, Chem. Soc. Rev. 41 (2012) 7016. [3] J. Biener, A.M. Hodge, J.R. Hayes, C.A. Volkert, L.A. Zepeda-Ruiz, A.V. Hamza, F.F. Abraham, Nano Lett. 6 (2006) 2379. [4] A.M. Hodge, J. Biener, J.R. Hayes, P.M. Bythrow, C.A. Volkert, A.V. Hamza, Acta Mater. 55 (2007) 1343. [5] A. Mathur, J. Erlebacher, Appl. Phys. Lett. 90 (2007) 061910.

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