Multiaxial creep of low density open-cell foams

Materials Science and Engineering A 540 (2012) 83–88

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Multiaxial creep of low density open-cell foams Z.G. Fan a,c , C.Q. Chen b,∗ , T.J. Lu a a

School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, PR China Department of Engineering Mechanics, AML & CNMM, Tsinghua University, Beijing 100084, PR China c Institute of Structural Mechanics, CAEP, Mianyang 621900, PR China b

a r t i c l e

i n f o

Article history: Received 30 November 2011 Received in revised form 19 January 2012 Accepted 20 January 2012 Available online 30 January 2012 Keywords: Open-cell foam Voronoi model Phenomenological constitutive model Creep

a b s t r a c t Open-cell foams have wide applications in structural components, energy adsorption, heat transfer, sound insulation, and so on. When their in-service temperature is high, time dependent creep may become significant. To investigate the secondary creep of low density foams under multiaxial loading, threedimensional (3D) finite element (FE) Voronoi models are developed. The effects of relative density, temperature, cell irregularity, and stress state on the uniaxial creep are explored. By taking the mass at strut nodes into account, the creep foam model by Gibson and Ashby (Cellular Solids: Structure and Properties, 2nd ed., Cambridge University Press, Cambridge, UK, 1997) is modified. Obtained results show that the uniaxial secondary foam creep rate predicted by the FE simulations can be well captured by the modified creep model. For multiaxial secondary creep, a phenomenological elastoplastic constitutive model is extended to include the rate effect into the creep response of 3D Voronoi foams. Again, the model predictions agree well with the FE results. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Low density foams made of high thermal conductive materials from the investment casting route (such as aluminum and copper) have excellent heat exchange performance and can be used as heat exchanger in autos, trains, aircrafts and micro-electronics [1–3]. Metallic foams can also be fabricated via the sintering route from a variety of materials including iron-based super alloys with iron-based alloys (e.g., FeCrAlY). The high strength and high melting temperature of this class of materials allow the foams to be used as components in extreme environments such as the acoustic liner for a gas turbine engine [4]. The in-service temperature of these applications can be well above 0.3Tm , where Tm is the melting temperature. Under such circumstances, time-dependent creep deformation in metals is usually deemed to be significant and cannot be neglected [5–7]. It is thus necessary to study the creep responses of foams carrying loads for long period of time and at relatively high temperatures. A number of analytical micromechanics models have been developed for the creep behavior of foams. Under uniaxial loading at environmental temperatures above 0.3Tm , the secondary creep

∗ Corresponding author. Tel.: +86 10 62783488; fax: +86 10 62783488. E-mail address: [email protected] (C.Q. Chen). 0921-5093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2012.01.086

rate of the constituent solid metal of the foam can be modeled using the classical power law equation [2,5]:

 Q  s

n ε˙ U ss = As exp −

RT

= ε˙ 0s

  n s

(1)

0s

where the superscript U represents the uniaxial loading; ε˙ U ss is the uniaxial strain rate; A, ε˙ 0s and 0s are the creep constants; s is the uniaxial stress; n is the stress exponent; Q is the creep activation energy; R is the ideal gas constant and T is the absolute temperature. Based upon a bending dominated cubic model, Gibson and Ashby [5] derived the uniaxial secondary creep rate of low density foams, as: ε˙ U = A

C4 n+2

 C (2n + 1) n 5 n

 Q

−(3n+1)/2  n exp −

RT

(2a)

or ε˙ U C4 = n+2 ε˙ 0s

 C (2n + 1)  n 5 n

0s

−(3n+1)/2

(2b)

where  is the relative density (defined as the volume fraction of the solid material of the foam) and C4 = 0.6 and C5 = 1.7 are the semi-empirical parameters determined by experiments and/or simulations. Eq. (2) is named as the Gibson–Ashby bending (GA-b) model by Boonyongmaneerat and Dunand [8] and will be identified as the GA-b model in the present study.

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Z.G. Fan et al. / Materials Science and Engineering A 540 (2012) 83–88

Hodge and Dunand [9] developed an axial compression dominant foam model and obtained the uniaxial secondary creep rate for low density foams.

  −n

ε˙ U = A

3

 Q

 n exp −

(3)

RT

Taking into account the presence of mass at strut nodes, Boonyongmaneerat and Dunand [8] developed a series of modified expressions to predict the secondary creep of cellular materials over a broad range of relative density. For low density open-cell foams, the modified expression was as follows [8],

⎧ U  k (2n + 1)  n  a − 1 2+n  a + 1 2n−1 k1 ε˙ 2 ⎪ ⎨ = ε˙ 0s

n+2

⎪ ⎩  = (9/4)a + 1 (a + 1)

n

0s

2

2

(4)

3

where a is the ratio of strut length to its thickness, and k1 and k2 are the strut geometric constants. Eq. (4) is a modified version of Eq. (2) and will be labeled here as the GA-b-1 model. In addition to analytical models, finite element (FE) models have been developed to explore the uniaxial creep responses of foams. Andrews and Gibson [10] simulated the secondary creep rate of two-dimensional (2D) random honeycombs under uniaxial stressing. Huang and Gibson [11] developed a three-dimensional (3D) Voronoi model with 27 random seeds and calculated its secondary creep rate. Oppenheimer and Dunand [12] constructed four different 3D regular architectures representing four types of foam model having different deformation mechanisms, and calculated their uniaxial secondary creep rates. Ajdari et al. [13] studied the foam creep behavior using 2D regular honeycomb models as well as 2D Voronoi structures. Existing numerical studies indicated that changes in the microstructure of a foam model affect the creep behavior remarkably [10–12] and foam imperfections (e.g., missing cell walls) have a dramatic effect on foam creep behavior [11,13]. Apart from modeling, a number of experimental studies are also available for open cell Duocel foam [6], Ni–Al, Ni–Cr–Al and Ni–Cr foams [14,15], as well as Al–Ni, Al–Mg and pure Al replicated foams [16–18] and closed-cell foams [7,19–21]. Whilst most available micromechanics models for the creep of foams focus on uniaxial behavior, several phenomenological creep models have been developed for multiaxial creep [2,22,23]. It is noted that both theoretical and experimental results reveal that the secondary creep rate is sensitive to the foam relative density and a modified bending model with precise relative density expression is necessary. Open-cell foams usually have a complex microstructure consisting of a network of non-uniform ligaments. Those ligaments constitute the edges of randomly packed cells that evolve during the foaming process. Various finite element models from simple cubic model [5] and regular Kelvin model [24,25] to more realistic micro-computed tomography models [26,27] have been developed to mimic the microstructure of open-cell foams. In general, the more realistic the models are, the more complex and computational time consuming they are. For simplicity, 3D Voronoi models [11,28–30] with 128 cells (instead of 27 cells by Huang and Gibson [11]) are used in this paper to simulate the multiaxial secondary creep of open-cell foams. The effects of relative density, temperature, cell irregularity, and multiaxial stress state on foam creep properties are explored. By accounting for the presence of mass at strut nodes, the creep expression (2) by Gibson and Ashby [5] is adapted to model the simulated uniaxial compressive creep behavior of Voronoi foams. Subsequently, the biaxial and axisymmetric creep responses of Voronoi foams are calculated using the FE Voronoi model. Finally, the phenomenological elastic-plastic

Fig. 1. Schematic of typical three-dimensional open-cell Voronoi foam model.

constitutive model developed by Chen and Lu [31] is extended for the FE simulated multiaxial creep behavior of Voronoi foams. 2. Finite element Voronoi foam model A series of 3D random Voronoi foam models with 128 cells are constructed using the software compiled by Gan et al. [30]. The commercially available finite element analysis (FEA) code ABAQUS is adopted to simulate the time-dependent creep behavior of the Voronoi foams. The constituent struts in the models are assumed to have a constant circular cross section and are meshed with the Timoshenko beam elements (i.e., element type B32 in ABAQUS). According to Chen et al. [32], periodical boundary conditions are employed to ensure that the models yield representative global behavior. For illustration, one of the 3D random Voronoi models is shown in Fig. 1. Engineering strains and engineering stresses are used to characterize the deformations of the Voronoi foams. Power law Eq. (1) without primary creep is adopted for the solid material of the foams, with A = 1.95 × 103 MPa−n s−1 , n = 4.0 and Q = 173 KJ mol−1 [12]. Other relevant properties of the solid material used are: ideal gas constant R = 8.314 J mol−1 K−1 , Young’s modulus Es = 69 GPa, and Poisson’s ratio s = 0.35. For simplicity, all the struts in the Voronoi foam models are assumed to have the same and constant circular cross section with area S, although, in real open-cell foams, the area of the crosssection may vary along the strut length. The relative density  is of primary importance in describing the mechanical properties of cellular foams. In various studies [6,10,11,13,28,29], the relative density of a highly porous open-cell foam is determined by the

li as: cross-section area S and the total length of struts ¯ =

S

V0

li

=

r 2



L3

li

(5)

where r is the radius of circular cross-section of struts and L is the edge length of the Voronoi foam model. Gan et al. [30] and Wang and Cuitino [33] showed that the volume of intersection points of struts was calculated repeatedly in Eq. (5). The simplification in Eq. (5) is suitable for foams with low

17

7

15

6

13

εGA-b /εGA-b-2

(ρ−ρ)/ρ (%)

Z.G. Fan et al. / Materials Science and Engineering A 540 (2012) 83–88

11 Fitted curve model1 model2 model3 model4

9 7 5 3

4

5

6 ρ (%)

7

8

9

10

Fig. 2. Relative difference of two definitions of relative density for Voronoi foams (i.e., Eqs. (5) and (6)).

relative densities, whereas the error increases as the foam relative density is increased. Therefore, a modified expression for the relative density of Voronoi foams was suggested by Gan et al. [30] as: √ √

27r 2 ( li − 2 2rN) + 46 6Nr 3 (6) = 27L3 where N is the number of vertices in the Voronoi foam. In the following simulations, the relative density of a Voronoi foam model is changed by varying the radius r of its struts. To demonstrate the difference between Eqs. (5) and (6), four random Voronoi models are constructed. Their relative densities are calculated in accordance with Eqs. (5) and (6), respectively, with the relative difference between them shown in Fig. 2. It is seen from Fig. 2 that the discrepancy between Eqs. (5) and (6) increases with the foam relative density increasing. Specifically, the value of relative difference can be up to about 10% when the foam relative density equals 5%. Moreover, since the secondary creep rate of the low density open-cell foam follows a power law against the foam relative density (as seen in Eq. (2)), a power index greater that 1 will increase the influence of the simplification of the foam relative density on its secondary creep rate. Therefore, for consistence, Eq. (6) is used in all subsequent simulations. In fact, a relationship between the relative densities with and without taking into account strut node mass can be obtained by fitting the data plotted in Fig. 2, as: ¯ = (1 + 0.590.57 )

(7)

which is shown as solid line in Fig. 2. With the new definition of the relative density ¯ in Eq. (6) for Voronoi foams, Eq. (2) can then be modified to be

4 3

 C (2n + 1)  n  5 n

−(3n+1)/2

(1 + 0.590.57 )

0s

ε U (s-1)

a

(8)

-6

10

-7

3

4

5

6

ρ (%)

7

8

9

10

3. Uniaxial creep It has been shown by Zhu and Windle [29] that cell regularity affects significantly the elastic properties of open-cell foams. Firstly, we will also examine the effect of cell irregularity on the creep rate of Voronoi foams. The parameter ˛ used to quantify the degree of cell irregularity of Voronoi foam models is [29]: ⎧ √  1/3 ⎪ ⎨ d0 = 6 √V0 2 2m (9) ı ⎪ ⎩˛ = d0 -6

10

Model1 Model2 Model3 Model4

Model5 Model6 average

10

2

in which the effect of mass at strut joints has been taken into account. Eq. (8) is also a modified version of Eq. (2) and will be labeled in the present study as the GA-b-2 model. To show the remarkably high sensitivity of the secondary foam creep rate to foam relative density and the difference between GAb and GA-b-2, the curves of the ratios of the secondary foam creep rates predicted by Eq. (2) (i.e., GA-b) to that calculated by Eq. (8) (i.e., GA-b-2) against foam relative densities are plotted in Fig. 3. The stress exponents of the three typical cell materials marked in Fig. 3 are referred to Refs. [2,6,14]. With the increase of the foam relative density and stress exponent, the ratios (i.e., the y-axis in Fig. 3) increase quickly. The secondary foam creep rate calculated by GA-b-2 model is lower than that predicted by GA-b model, especially for foams having higher relative densities and greater stress exponents. Comparison with GA-b-2 model, e.g., GA-b model overpredicted the value of the secondary creep rates of the open-cell foam having stress exponent of 7.9 and relative density of 9% by a factor of about 5.6.

b Model1 Model2 Model3 Model4

1

Fig. 3. Curves of the ratios of the secondary foam creep rates predicted by Eq. (2) (i.e., GA-b) to that calculated by Eq. (8) (i.e., GA-b-2) against foam relative densities. The stress exponents of the three typical cell materials marked in Fig. 3 are referred to Refs. [2,6,14].

ε U (s-1)

ε˙ U C4 = n+2 ε˙ 0s

5

1 2

n=4.0, Al-6101 n=5.8, Ni-8Al n=7.9, 316 Stainless Steel

2

3 1

85

Model5 Model6 average

-7

10

α =0.4

α =0.8

-8

-8

10

10 0

0.01

0.02

0.03

εc (-)

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

εc (-)

Fig. 4. Calculated uniaxial compressive creep curves of Voronoi foams with different degrees of cell regularity: (a) ˛ = 0.4; (b) ˛ = 0.8 (stress:  = 0.05 MPa; temperature: T = 350 ◦ C; relative density:  = 5%).

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Z.G. Fan et al. / Materials Science and Engineering A 540 (2012) 83–88 -5

-2

10

10

-3

FEA GA-b GA-b-1 GA-b-2

-4

ε U (s-1)

10

-5

10

-6

10

-7

σ =0.42MPa

-8

T =250 C

10

o

10

-9

10

2

3

4

-6

10

-7

10

ε U (s-1)

10

FEA GA-b GA-b-1 GA-b-2

-8

10

-9

10 10

-10

10

-11

T =250oC ρ =9%

0.1

5 6 7 8 910

ρ (%)

1

σ (MPa)

Fig. 5. Relative density dependence of the secondary foam creep rates under uniaxial compressive stress calculated by FE method (hollow symbols) and using Eqs. (2), (4) and (8) (lines) (stress magnitude:  = 0.42 MPa; temperature: T = 250 ◦ C).

Fig. 6. Stress dependence of the secondary foam creep rates under uniaxial compressive loading calculated by FE method (hollow symbols) and using Eqs. (2), (4) and (8) (lines) (relative density:  = 9%; temperature: T = 250 ◦ C).

Here, d0 is the minimum distance between any two adjacent nuclei in a regular lattice with m identical tetrakaidecahedral cells in volume V0 , and ı is the minimum distance between any two adjacent nuclei in a random Voronoi model. By doing so, ˛ represents the degree of cell regularity of a Voronoi model, with ˛ = 1 corresponding to regular tetrakaidecahedral foams. Fig. 4 presents the simulated uniaxial compressive creep curves in terms of strain rate versus strain for Voronoi foams having various degrees of cell irregularity; the foam relative density is fixed at 5%, the magnitude of the uniaxial compression is 0.05 MPa, and the environmental temperature is 350 ◦ C. Note that all the Voronoi models simulated have random structures. As numerical scatters in predictions obtained with different models are inevitable, it is necessary to construct a number of Voronoi models having the same relative density as well as cell irregularity for a given loading condition. In Fig. 4, different symbols denote FE predictions with different Voronoi models (labeled by Model 1–6) whilst the solid lines refer to their averaged values. It can be seen from Fig. 4 that predicted foam creep behaviors from different Voronoi models are similar but numerically slightly different, consistent with the averaged values. Moreover, cell irregularity is found to have a moderate effect on the FEA predictions, with the degree of dispersion decreasing slowly as the cell regularity is increased. Therefore, hereinafter, Voronoi models with ˛ = 0.8 and 128 cells are used to calculate foam creep responses. To diminish the influence of randomness of Voronoi structures, results averaged from six nominally identical random samples are reported in what follows. As shown in Fig. 4, the creep rate is strain dependent, varying slowly in the strain range between 0.5% and 5%. In fact, it is found that the corresponding creep rate defined at 1% is very close to the averaged value of the considered strain range and is employed as the strain rate definition and averaged for all models [12]. The predicted dependence of the so-defined secondary foam creep rate upon relative density, stress and temperature is shown separately in Figs. 5–7, respectively. Fig. 5 gives the secondary foam creep rate (denoted by symbols) as a function of the relative density under uniaxial compression with stress magnitude of 0.42 MPa and temperature of 250 ◦ C. As the relative density is increased, the creep rate decreases rapidly. Fig. 6 plots the average secondary foam creep rate as a function of the uniaxial compressive stress for foams having a relative density 9%, with temperature fixed at 250 ◦ C. As the magnitude of the uniaxial stress increases, the secondary foam creep rate increases rapidly. The effect of temperature on the creep rate is shown in Fig. 7 for foams having a relative density 9% and subjected to

uniaxial compressive stress of 0.42 MPa. As the reciprocal of absolute temperature increases, the secondary foam creep rate decreases. The logarithmic secondary foam creep rates are linearly proportional to the reciprocal of absolute temperature. In addition, from Figs. 5 and 6, the logarithmic secondary foam creep rate is found to depend almost linearly on the logarithmic relative density and the logarithmic applied stress. Also included in Figs. 5–7 are predictions obtained using the GA-b, Ga-b-1 and GA-b-2 models, i.e., expressions (2), (4) and (8). The modified creep rate expression (8) (i.e., GA-b-2) is seen to agree better with the FEA results, compared to GA-b and GA-b-1. Since the volume of intersection points of struts was calculated repeatedly, as previously discussed, the bending model of Gibson and Ashby [5] moderately over-predicted the secondary creep rates of low density open-cell foams. Boonyongmaneerat and Dunand [8] also over-predicted the secondary creep rates of low density open-cell foams, especially when the relative density is very low. The experimental results included in Fig. 7 are taken from Andrews et al. [6]. The secondary foam creep rates predicted by the finite element Voronoi models and the modified creep expression (8) have overall good agreement with the experimental measurements, although slightly lower than the averaged experimental results. The discrepancy is believed to be attributed to the presence of various processing-induced defects in real metal foams which can greatly increase the secondary foam creep rate [11,13]. It is thus expected that a theoretical model without accounting for the

ε U (s-1)

-1

10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 -10 10 -11 10 1.2

σ =0.42MPa ρ =9% FEA GA-b GA-b-1 GA-b-2 Experimental data 1.4

1.6

1.8

2

2.2

-3 -1

1/T (10 K ) Fig. 7. Temperature dependence of the secondary foam creep rates under uniaxial compressive loading calculated by FE method (hollow symbols) and using Eqs. (2), (4) and (8) (lines) (stress:  = 0.42 MPa; relative density:  = 9%). Experimental data for Al6061-T6 foams taken from Ref. [6] are also plotted as full symbols.

Z.G. Fan et al. / Materials Science and Engineering A 540 (2012) 83–88

a

7

15

b

6

12

5

FEA Phenomenological model

4 3

σ 33=0.5Mpa

2

Biaxial loading

ε33 / ε33U

ε33 / ε33U

87

FEA Phenomenological model

9

σ 33=0.5Mpa

6

Axisymmetric loading 3

1 0

0 -1

-0.5

0

0.5

-1

1

-0.5

k2=σ22/σ33

0

0.5

1

k1=k2=σ22/σ33

Fig. 8. Comparison of the predictions from the phenomenological creep model and the finite element Voronoi foam models on the normalized multiaxial secondary foam / 0); (b) axisymmetric loading (k1 = k2 ). creep rate with T = 275 ◦ C and  = 9%: (a) biaxial loading (k1 = 0 and k2 =

influence of defects would underestimate the creep rates of foams, as do the present Voronoi foam models and the modified model GA-b-2.

ˆ can be derived from the uniaxial creep where the function f () result (8) as: f () ˆ = ε˙ 0s

4. Multiaxial creep



C4 C5 (2n + 1) n+2 n

n 

−(3n+1)/2



(1 + 0.590.57 )

ˆ 0s

n

(15) In addition to the uniaxial behaviors, the multiaxial creep of 3D Voronoi foams is also theoretically investigated. To this end, built upon the analysis of Chen and Lu [31] on constitutive modeling for cellular foams, a characteristic stress ˆ is introduced as: ˆ 2 ≡



1 1 + (/3)

2

e 2 + 2 m 2



With Eqs. (14) and (15), the multiaxial secondary foam creep rate can then be expressed as:

(10)

ε˙ ij ε˙ U -k-k

where e and m are the Mises effective stress and mean stress, respectively, and

⎧

⎪ ⎨  = 1 ( −  )2 + ( −  )2 + ( −  )2 e 1 2 2 3 3 1 2 ⎪ 1 ⎩ m =

3

=

ˆ ∂ˆ f () = f (kk ) ∂ij --



ˆ kk --

n

∂ˆ , (i = 1, 2, 3; k = 1, 2, 3) ∂ij

(16)

where the Einstein summation convention is not invoked for k- . For a proportional loading satisfying:

 (11)

(1 + 2 + 3 )

11 22 , k2 = 33 33 |11 | ≤ |22 | ≤ |33 | k1 =

(17)

the multiaxial secondary foam creep rate (16) reduces to: ε˙ 33 ε˙ U 33

 =

2





9(1 − 2) 2(1 + )

(12)

Gan et al. [30] have shown that the elastic Poisson ratio of the foam, , is given by:  = s + (0.5 − s )

 1−  1 + 14

(13)

where s is the Poisson ratio of the solid material of the foam. It should be pointed out that, under uniaxial loading, the characteristic stress defined in (10) reduces to the uniaxial stress. Following the flow rule, the secondary foam creep rate under multiaxial stressing may be described as [22,23]: ∂ˆ ε˙ ij = f () ˆ ∂ij

2

2

1 + (/3)

where i (i = 1, 2, 3) are the principal stresses;  is a material constant defined by 2 =

2

1 − (k1 + k2 )/2 + (/3) (1 + k1 + k2 ) · k1 + k2 − k1 k2 − k1 − k2 + 1 + (/3) (k1 + k2 + 1)

(14)



2 (n+1)/2



2 (n−1)/2

(18)

Based on the developed 3D Voronoi foam models, FE simulations of multiaxial foam creep behaviors are conducted and compared with those predicted by the phenomenological model of (18). Fig. 8 shows the normalized secondary foam creep rates under biaxial / 0) and axisymmetric (k1 = k2 ) loadings, where the (k1 = 0 and k2 = solid lines refer to the phenomenological model (18) and the solid symbols denote the Voronoi model predictions. In Fig. 8, the relative density of 9% and cell irregularity of ˛ = 0.8 are adopted for the Voronoi models, with temperature T = 275 ◦ C. Under uniaxial loading (i.e., k2 = 0 in Fig. 8), as expected, the normalized creep rate equals 1 for both theoretical and numerical predictions. For biaxial and axisymmetric loadings, it is evident from Fig. 8 that the overall agreement between the predictions obtained with the multiaxial phenomenological creep model (18) and the FE Voronoi foam model is very well. 5. Conclusions Three-dimensional finite element Voronoi foam models have been developed to simulate the uniaxial and multiaxial secondary creep behaviors of open-cell foams. The effects of relative density, temperature, cell irregularity, and multiaxial stress state on the

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Z.G. Fan et al. / Materials Science and Engineering A 540 (2012) 83–88

creep properties have been explored. Obtained results reveal the dependence of the uniaxial secondary foam creep rate on the relative density, temperature, and stress. The predicted dependence is found to be well captured by a modified version of the uniaxial creep rate model by Gibson and Ashby [5]. Cell irregularity in the Voronoi foam models only has a moderate effect on the creep rate. For multiaxial creep, the phenomenological elastoplastic constitutive model of Chen and Lu [31] is extended to include the rate dependent effect. The model predictions for uniaxial, biaxial and multiaxial secondary creep rates agree well with those obtained using the finite element Voronoi foam models.

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Acknowledgements

[19]

This work is supported by the National Natural Sciences Foundation of China (Nos. 11072127 and 10832002), the National Basic Research Program of China (Nos. 2011CB610305 and 2010CB832700), and the PhD Program of Ministry of Education of China (20110002110069).

[20]

References [1] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Prog. Mater. Sci. 43 (1998) 171–221. [2] M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley, Metal Foams: A Design Guide, Butterworth, Heinemann, 2000. [3] J. Banhart, Prog. Mater. Sci. 46 (2001) 559–632. [4] T.J. Lu, M. Kepets, A.P. Dowling, Sci. China Ser. E 51 (2008) 1803–1811. [5] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, 2nd ed., Cambridge University Press, Cambridge, UK, 1997.

[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

E.W. Andrews, L.J. Gibson, M.F. Ashby, Acta Mater. 47 (1999) 2853–2863. E.W. Andrews, J.S. Huang, L.J. Gibson, Acta Mater. 47 (1999) 2927–2935. Y. Boonyongmaneerat, D.C. Dunand, Acta Mater. 57 (2009) 1373–1384. A.M. Hodge, D.C. Dunand, Metall. Mater. Trans. A 34 (2003) 2353–2363. E.W. Andrews, L.J. Gibson, Mater. Sci. Eng. A 303 (2001) 120–126. J.S. Huang, L.J. Gibson, Mater. Sci. Eng. A 339 (2003) 220–226. S.M. Oppenheimer, D.C. Dunand, Acta Mater. 55 (2007) 3825–3834. A. Ajdari, P. Canavan, H. Nayeb-Hashemi, G. Warner, Mater. Sci. Eng. A 499 (2009) 434–439. H. Choe, D.C. Dunand, Acta Mater. 52 (2004) 1283–1295. H. Choe, D.C. Dunand, Mater. Sci. Eng. A 384 (2004) 184–193. F. Diologent, Y. Conde, R. Goodall, A. Mortensen, Philos. Mag. 89 (2009) 1121–1139. F. Diologent, R. Goodall, A. Mortensen, Acta Mater. 57 (2009) 830–837. S. Soubielle, F. Diologent, L. Salvo, A. Mortensen, Acta Mater. 59 (2011) 440–450. P. Zhang, M. Haag, O. Kraft, A. Wanner, E. Arzt, Philos. Mag. A 82 (2002) 2895–2907. M. Haag, A. Wanner, H. Clemens, P. Zhang, O. Kraft, E. Arzt, Metall. Mater. Trans. A 34 (2003) 2809–2817. O. Couteau, D.C. Dunand, Mater. Sci. Eng. A 488 (2008) 573–579. C. Chen, N.A. Fleck, M.F. Ashby, Adv. Eng. Mater. 4 (2002) 777–780. O. Kesler, L.K. Crews, L.J. Gibson, Mater. Sci. Eng. A 341 (2003) 264–272. H.X. Zhu, N.J. Mills, J. Mech. Phys. Solids 47 (1999) 1437–1457. W.Y. Jang, S. Kyriakides, Int. J. Solids Struct. 46 (2009) 635–650. A.D. Brydon, S.G. Bardenhagen, E.A. Miller, G.T. Seidler, J. Mech. Phys. Solids 53 (2005) 2638–2660. C. Veyhl, I.V. Belova, G.E. Murch, T. Fiedler, Mater. Sci. Eng. A 528 (2011) 4550–4555. V. Shulmeister, M. Van der Burg, E. Van der Giessen, R. Marissen, Mech. Mater. 30 (1998) 125–140. H.X. Zhu, A.H. Windle, Acta Mater. 50 (2002) 1041–1052. Y.X. Gan, C. Chen, Y.P. Shen, Int. J. Solids Struct. 42 (2005) 6628–6642. C. Chen, T.J. Lu, Int. J. Solids Struct. 37 (2000) 7769–7786. C. Chen, T.J. Lu, N.A. Fleck, J. Mech. Phys. Solids 47 (1999) 2235–2272. Y. Wang, A.M. Cuitino, J. Mech. Phys. Solids 48 (2000) 961–988.