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Sample size effect on the mechanical behavior of aluminum foam
International Journal of Mechanical Sciences 151 (2019) 622–638
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Sample size effect on the mechanical behavior of aluminum foam Yue Zhang a,b, Tao Jin a,c, Shiqiang Li a,c, Dong Ruan b, Zhihua Wang a,c,∗, Guoxing Lu b,∗∗ a
Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, VIC 3122, Australia c Shanxi Key Laboratory of Material Strength and Structural Impact, Taiyuan 030024, China b
a r t i c l e
i n f o
Keywords: Metal foam Sample size effect 3D Voronoi model Mechanical properties Compression, shear, and bending
a b s t r a c t The aim of this work is to explore the effect of sample size in three different directions (length, height, and thickness of a sample block), with respect to its cell size, on the mechanical properties of closed-cell aluminum foams. 3D Voronoi models were used to represent a real foam block, and finite element (FE) analysis was performed to simulate the mechanical properties of samples under three different loading conditions (uniaxial compression, shear, and bending). The numerical results match the experimental results. The stiffness and strength of samples with different sizes were normalized by those properties of samples with the maximum size. The normalized stiffness and strength were expressed as functions of thickness of weak cell layers (less constraint cells) and that of the strong boundary layers (strong constraint cells). Furthermore, a parametric study was performed to investigate the effects of the thicknesses of weak cell layers and strong boundary layers on the mechanical properties of aluminum foam. It is concluded that both the stiffness and strength increase with sample size under compression and bending, while they decrease with the increase of sample height under shear. The minimum size of samples characterizing bulk materials under different loading conditions was determined.
1. Introduction Metal foam has been applied in many engineering fields as lightweight and protective structures because of its high specific strength and great ability of energy absorption. The wide application in industries inspires researchers and experts to explore the mechanical properties of this material [1–4]. By using a method based on classical continuum theory, the properties of metal foam are considered as being related to relative density only [5]. However, in some investigations, cell geometries such as cell size and cell morphology were found to affect the mechanical behavior of cellular solids [6,7]. Moreover, the cell size of cellular materials for industrial applications is generally between 1 to 10 mm. The relatively small side length of industrial components may only contain several cells [8] and the mechanical properties vary with the number of cells or component size. Therefore, the effect of sample size should be considered. The effect of sample size or cell size on the mechanical properties of cellular solids has been reported. Jin et al. [9] investigated the effect of cell number on the in-plane and out-of-plane stiffnesses and plateau stresses of aluminum hexagonal honeycombs. They developed a relationship between cell number and mechanical behavior of honeycombs in three different directions. Rakow and Waas [10] utilized digital image correlation (DIC) to study the effects of sample size, rel∗ ∗∗
ative density and strain rate on the shear response of closed-cell aluminum foam. Their experimental results showed that the effect of sample size on the shear strength of foam was negligible. Chen and Fleck [11] found that the compressive properties of open-cell aluminum foam were independent of cell size. Jing et al. [12] investigated the effect of sample thickness on the properties of aluminum foam under dynamic compressive loading. They also found the compressive properties of open-cell aluminum foam increase with the increasing cell sizes of samples [13]. Moreover, the dominant factor for cell size effect has also been discussed. Onck et al. [8,14–16] investigated the effect of sample size on the mechanical behavior of aluminum foam in four different loading cases by using theoretical analysis, experiments and FE simulation of 2D Voronoi models. In the case of compression, the shape of sample was square prisms whose height was twice the width. In shear tests, the thickness of sample was 50 mm, while the length and height of sample varied with a constant ratio of length/height = 12. It was concluded that sample size effect stemmed from either weak cell layers for compression, indentation and bending or strong boundary layers for shear. Jeon and Asahina [17] explored the compressive properties of closedcell foams without or with evident structural defects such as missing cells and collapsed cells. They found that the yield strength of samples without defects was independent of sample size while Young’s mod-
Corresponding author at: Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China. Corresponding authors. E-mail addresses: [email protected], [email protected] (Z. Wang), [email protected] (G. Lu).
https://doi.org/10.1016/j.ijmecsci.2018.12.019 Received 27 August 2018; Received in revised form 2 December 2018; Accepted 11 December 2018 Available online 11 December 2018 0020-7403/© 2018 Published by Elsevier Ltd.
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 1. Schematic diagrams of three different loading cases: (a) uniaxial compression; (b) shear; (c) bending.
Table 1 Dimensions of samples for different loading cases. (all in mm).
Compression Shear Bending
X-direction (L)
Y-direction (T)
Z-direction (H)
6, 10, 14, 16, 18, 20 L/H = 3 L/H = 3
Same as X-direction 4, 8, 12, 16, 20 4, 8, 12, 16, 20
Same as X-direction 4, 8, 12, 16, 20 4, 8, 12, 16, 20
2. Numerical simulation 2.1. Voronoi model Some researchers established FE model of foams by using X-Ray computed tomography [20,21] while others used Voronoi model [8,18]. In the current work, a 3D Voronoi model is used to represent real foams. Similar to foaming process, nucleation points are distributed randomly in a certain volume V, and the number of nucleation points is N. To avoid the existence of tiny cells, a minimum distance between two nucleation points in the Voronoi model is defined, as follows, )1∕3 √ ( 6 𝑉 𝑑min = (1 − 𝑘)𝑑0 = (1 − 𝑘) (1) √ 2 2𝑁
ulus increased as sample size increased. On the other hand, both the Young’s modulus and yield stress of samples with defects increased with sample size. Defects played an important role in size effect. Li et al. [18] proposed a new statistical method to establish the relationship between the microstructure parameters and the macro-properties of foams by using FE simulation of a 3D Voronoi model. It was found that the percentage of ‘weak cell’ led to the cell size effect. Forest et al. [19] employed Weibull statistical analysis to interpret the variations of yield strengths and corresponding scatter of sample with different sizes. Considering the results in the literature mentioned above, the effect of sample size in thickness direction on properties of sample was rarely reported. In the current work, the sample size effects on quasi-static compressive, shear and bending properties are further investigated. The numerical simulations on the mechanical properties of aluminum foam with different sample sizes subjected to three different loadings are carried out by using 3D Voronoi models. Subsequently, the mechanism of sample size effect is analyzed and the relationship between sample size and mechanical behavior of aluminum foam is further developed. A parametric study has been performed to investigate the effects of thicknesses of weak cell layers and strong boundary layers on mechanical properties of aluminum foam.
where d0 is the distance between the two adjacent nucleation points in a regular tetrakaidecahedron model and k is the irregularity of cell shapes. The value of irregularity, k, ranges from 1 (completely random) to 0 (regular). In the current work, the value of k was chosen to be 0.2 [22,23]. Matlab software was utilized to place N nucleation points in the volume V and generate the coordinates of the points, lines and surfaces, which could be used to construct the geometric Voronoi model. Fig. 1 shows the Voronoi models for different loading conditions. The mean radius of cells is 1.241 mm, which is calculated from Eq. (2). 𝑁=
𝑉 (4𝜋∕3)𝑟3
(2)
where N and V are as defined in Eq. (1), and r is the mean equivalent radius of cells, i.e., the mean radius of sphere whose volume is equal to cell. The sample for uniaxial compression was cube to prevent buckling during deformation [17]. Its maximum side length was 623
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 2. Illustration of two criteria of the accuracy of quasi-static FE simulation: (a) artificial strain energy to total internal energy ratio history curve; (b) stress equilibrium history curve.
Table 2 Material properties of the parent material (aluminum). Density 2700 kg/m
3
Young’s modulus
Poisson’s ratio
Yield strength
Tangent modulus
70 GPa
0.3
80 MPa
700 MPa
L (X-direction) × T (Y-direction) × H (Z-direction) = 20 × 20 × 20 mm (Fig. 1a). For shear (Fig. 1b) and bending (Fig. 1c), the maximum sample size was L × T × H = 60 × 20 × 20 mm. The dimensions of samples for different loading cases are listed in Table 1. It should be noted that when the size effect in one direction was investigated, the dimensions in the other two directions were fixed at their maximum values.
as e and its history is plotted in Fig. 2. Meanwhile, the kinetic energy should not be more than 5% of total internal energy to avoid introducing inertia effect, and it can be estimated by the difference between the stresses at the fixed end and loading end [24]:
2.2. FE model and calculation
𝜎𝑙𝑜𝑎𝑑𝑖𝑛𝑔 − 𝜎𝑓 𝑖𝑥𝑒𝑑 𝑄= ( ) 𝜎𝑙𝑜𝑎𝑑𝑖𝑛𝑔 + 𝜎𝑓 𝑖𝑥𝑒𝑑 ∕2
The geometric Voronoi models aforementioned were meshed by using Hypermesh, and a commercial software Ls-Dyna was used to perform the finite element calculation. To reduce computational time, mass scaling was employed in simulation, in which the minimum time step size was set as 10−7 s [24]. Moreover, considering the accuracy of quasistatic process in explicit simulation, the artificial strain energy introduced by mass scaling should be less than 10% of total internal energy. The ratio of artificial strain energy to total internal energy is defined
where 𝜎 loading and 𝜎 fixed are stresses at loading end and fixed end, and the relationship between Q and time can be seen in Fig. 2. Shell elements with in-plane single point integration were employed in Voronoi FE models. The sensitivity of characteristic length of elements was calculated and the characteristic length was set as 0.3 mm. A Single Surface Contact was employed to simulate the contact among cell walls, and the coefficient of static friction was 0.02 [7]. The thickness of shell elements, t, can be calculated by the following Eq. (4) [7]:
(3)
Fig. 3. (a) A photograph of uniaxial compression test setup; (b) experimental and simulated stress-strain curves of cubic samples with a side length of 20 mm. 624
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 4. Stress-strain curves of samples with different size under various loading conditions: (a) compression; (b)(c) shear; (d)(e) bending (stress-rotation curves).
𝑉𝑓 𝑜𝑎𝑚 ( ∗ ) 𝜌 ∕ 𝜌𝑠 𝑡= ∑ 𝑆𝑖
A bilinear strain-hardening material model was employed for the parent material of Voronoi models. The values of the parameters for the material are listed in Table 2. In the simulation of uniaxial compression, a sample was placed between two rigid platens without any transverse constraint. The top platen moved at a constant velocity along the vertical direction to compress the sample (Fig. 1a), while the bottom platen was fully fixed. The velocity of the top platen varied with the sample size to maintain a constant strain rate of 10−3 /s.
(4)
where Vfoam is the volume of a sample; ΣSi is the total area of the surfaces in a foam sample and 𝜌∗ /𝜌s is the relative density of a sample. In the current work, the relative density of each sample employed in the simulation was constant, 20%. The element thickness was uniform but with various values for different samples to ensure the same relative density for all samples. 625
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 5. Normalized Young’s modulus (a) and yield strength (b) of the samples under compression vs. side length of the samples normalized by mean cell size, L/d.
Fig. 6. Normalized shear modulus (a) and shear strength (b) of the samples vs. sample height normalized by mean cell size, H/d.
Fig. 7. Normalized shear modulus (a), and shear strength (b) of the samples vs. sample thickness normalized by mean cell size, T/d.
For the shearing cases, a sample was also placed between two rigid platens. The top platen moved at a constant velocity along the Xdirection, while the bottom platen was fully fixed (Fig. 1b). The nodes on the top and bottom surfaces of the sample were tied to both the rigid platens.
For the bending cases, two rigid platens perpendicular to the Xdirection were adhered to all the nodes on the left and right sides of the sample. One of the rigid platens was fully constrained while the other rotated about the neutral axis of the fixed platen at a constant angular velocity.
626
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 8. Normalized yield strength under bending loading of the samples vs. (a) sample height normalized by mean cell size, H/d; (b) sample thickness normalized by mean cell size, T/d.
2.3. Experiments and validation of FE models Uniaxial compression tests of closed-cell aluminum foam (fabricated by Luoyang Ship Material Research Institution, Luoyang, China) were performed at a constant strain rate of 10−3 /s by using a universal test machine (Instron 5969). The experimental setup (Fig. 3a) was the same as that in the numerical simulation. Cubic foam samples had a relative density of 20% and side lengths of samples were the same as those used in numerical simulation for uniaxial compression, as listed in Table 1. Fig. 3(b) shows the stress-strain curves of samples obtained from both the simulation and tests of the cubic samples with a side length of 20 mm. The simulated result (except Young’s modulus, which will be discussed in Section 4.1) matches reasonably well with the experimental results of three repeated tests, which indicates that the FE model is valid. 3. Results Fig. 9. Stress-strain curves of samples with maximum size under compression while different values of the friction coefficients were employed.
In the current FE work, five 3D Voronoi models have been established for each sample size and each loading case. Average values are presented in Figs. 4–8 with error bars. Fig. 4 shows the stress-strain curves of samples with different sizes for three different loading cases (for bending, the stress-rotation curves is shown in Fig. 4, and the rotation, 𝜃, is the angle that the loading platen rotated by). The value of
Fig. 10. Normalized Young’s modulus (a) and yield strength (b) of the samples under compression vs. side length of the samples normalized by mean cell size, L/d, while various values of the friction coefficients were employed. 627
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 11. Compressive stress (in Pa) contours in the samples with three different sizes: (a) L/d ≈ 4; (b) L/d ≈ 6; (c) L/d ≈ 8. (d remains 2.482 mm.) Sections are Section 1 shown in Fig. 1(b).
Fig. 12. Effective plastic strain contours in the samples with three different sizes: (a) L/d ≈ 4; (b) L/d ≈ 6; (c) L/d ≈ 8. (d remains 2.482 mm.) Sections are Section 2 shown in Fig. 1(b).
modulus of each sample was calculated from the slope of the stress-strain curve in the elastic regime and the yield strength was the intersection point of the tangents of elastic regime and plastic regime of stress-strain curves. Modulus and yield strength of various size samples were normalized with the bulk properties of the same material, which were obtained from the sufficiently large samples (with the maximum size) for each loading condition.
good agreements (less than 10% discrepancy) between simulated and experimental results indicate the reliability of the FE models as well. The normalized Young’s modulus, E/Ebulk , and normalized yield strength, 𝜎 c /𝜎 bulk , of the samples increase with the length of sample sides, L. However, the slope of the normalized properties versus the ratio of length to mean cell size, L/d (d = 2r), changes when this ratio is larger than a certain value. Note that the value of L/d approximately indicates the number of cells contained in a sample in that direction. The normalized Young’s modulus plateaus when L/d ≥ 7, while the normalized yield strength converges when L/d ≥ 6.
3.1. Uniaxial compression Fig. 5 illustrates the comparison between experimental and simulated results under uniaxial compression. The results from the previous studies are also included. It is evident that sample size has an effect on the uniaxial compressive property of the Voronoi models. In addition,
3.2. Shear In this section, the effects of sample size on shear behavior of foam in both the Y-direction and Z-direction were investigated. Fig. 6 shows the 628
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 13. Shear stress (in Pa) contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases). Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
effect of sample height, H, sample size in the Z-direction on the normalized shear stiffness, G/Gbulk , and normalized shear strength, 𝜏 y /𝜏 bulk , of the Voronoi models. Both the normalized shear stiffness and normalized shear strength exhibit a decreasing trend when the value of H increases, which is a converse observation compared with compression cases. As seen in Fig. 6, sample height has an effect on the shear behavior of the Voronoi models when the ratio of sample height to the mean cell size H/d is less than 6. Experimental results in literature [16] manifested that the normalized shear strength also declined with the increase of sample height, which is similar to the present FE results for smaller values of H/d. Both the shear stiffness and strength of the current Voronoi model converge at a large value of H/d. Fig. 7 demonstrates the relationship between the sample thickness, T, sample size in the Y-direction and shear properties of the Voronoi models. In contrast to the effect of sample height, sample thickness has a positive effect on the normalized shear stiffness, G/Gbulk , and normalized shear strength, 𝜏 y /𝜏 bulk . The effect of sample thickness vanishes when the ratio of sample thickness to mean cell size is large, i.e., T/d ≥ 6.
bending yield strength of the Voronoi model is: 𝜎𝑦−𝑏𝑒𝑛𝑑𝑖𝑛𝑔 = 6𝑀𝑏 ∕𝑇 𝐻 2
(5)
where Mb is the bending moment corresponding to the beginning of nonlinearity on a moment-rotation angle curve, T and H are as defined in Section 3.2. It can be seen in Fig. 8 that the bending yield strength increases with both the sample height and thickness. Moreover, when the ratio of sample height to mean cell size H/d ≥ 6 or the ratio of sample thickness to mean cell size T/d ≥ 5, the bending yield strength is hardly affected by sample height or sample thickness.
4. Discussions 4.1. Overestimate in Young’s modulus
3.3. Bending
The value of Young’s modulus of sample obtained from simulation is higher than the experimental results, which is shown in Fig. 3. The dominant factors of this might be:
The effects of sample size in both the Y-direction and Z-direction on bending behavior of the Voronoi model are explored in this section. The
(1) The melt-foaming process was used to manufacture the foam investigated in this work. This process induces small pores in cell walls 629
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 14. Effective plastic strain contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases). Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
E/Ebulk and normalized yield strengths, 𝜎 c /𝜎 bulk of samples with different sizes when various friction coefficients were employed. It can be seen that friction coefficient between the samples and rigid plates has an effect on the normalized properties of the samples, especially when sample size is small. Normalized Young’s moduli and yield strengths plateau out with the increase of L/d.
of foam material, which results in a lower stiffness of cell walls than that of FE model; (2) There are some structural defects in actual aluminum foams such as missing cells and collapsed cells. Furthermore, there exist some curved cell walls, which leads to a decrease of stiffness of samples compared with that in the FE model. (3) In the simulations, at the shared edges of cell walls, the thicknesses are overestimated due to the overlap of different elements of different cell walls. This problem locally increases the stiffness compared with the actual material [25].
4.3. Compression Fig. 11 shows the stress contours at the onset of plastic stage for three selected sample sizes under uniaxial compression. Cells around the edge experience much lower stress than those in the central region. The reason is that the constraint on the inner cells is greater than that on the edge cells. Those edge cells with less constraint may be considered as ‘weak cell’. Furthermore, these ‘weak cells’ are even stress-free or nearly stress-free. Fig. 12 shows the effective plastic strain contours at the onset of plastic stage for three selected samples of different sizes under uniaxial compression. It can be seen that deformation is not uniform and it localizes in the cells. The most severely localized deformation occurs at the top and bottom boundary of the sample where the cells are incomplete. The stiffness of those cells is lower than that of the cells in the central region.
4.2. Effect of friction The effect of friction between the foam and the rigid plates is discussed in this section. The friction coefficient between samples and rigid plates 𝜇 was assumed as 0.02 [7], 0.1, 0.2, and 0.4, respectively. Fig. 9 demonstrates the stress-strain curves of the maximum sized sample with different values of the friction coefficient. It is evident that with an increase of the friction coefficient, both the apparent modulus and yield strength are increasing. Modulus and yield strength of various sized samples were normalized by respect to the bulk properties of the same material, which were obtained from the sufficiently large samples (of the maximum size). Fig. 10 shows the normalized Young’s moduli, 630
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 15. Shear stress (in Pa) contours in samples with three different thicknesses and in different directions: (a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
For stiffness, the weak cell layers include the cells with reduced stiffness (located near the stress-free boundary) and cells with zero stiffness (located at the stress-free boundary). The volume fraction of weak cell layers decreases with the increase of sample size, which leads to the sample size effect on the stiffness of samples. Onck et al. divided weak cell layers into three categories: (i) the layers between the core of a sample and zero stiffness boundary, which could represent the reduced stiffness. The thickness of these layers was nd, and the Young’s modulus was mEbulk ; (ii) the corners of layers with reduced stiffness and lower Young’s modulus m2 Ebulk ; (iii) zero stiffness layers, whose thickness was pd. Then, the normalized stiffness of material could be expressed as [15]
For compressive strength, Jeon and Asahina [17] reported that the strength of the samples with structural defects was more sensitive to sample size than that of the samples without defects (shown in Fig. 5b). Although there were no evident structure defects such as missing cells or collapsed cells in the current 3D Voronoi models, there were ellipsoidal cells with T-shaped cell-wall intersection. These cells deformed severely in localization band [26], which could be named as ‘easily deformed cells’. In smaller samples, the volume fraction of stress-free or nearly stress-free boundary (considered as weak cell layers) is greater. Therefore, the constraint surrounding easily deformed cells is less, and first localization deformation band forms more easily in a small sample than that in large one, which leads to a lower yield strength of sample. In other words, the existence of weak cell layers results in sample size effect on the strength of samples. The thickness of the weak cell layer is assumed to be wd on average (d is the mean diameter of cell). Then, the normalized compressive yield strength can be expressed as Eq. (7) and plotted in Fig. 5(b) when 𝑤 = 14 [8].
( ) ( )( ) ( )2 𝐸 𝑑 𝑑 2 𝑑 𝑑 𝑑 𝑑 1 − 2𝑛 − 2𝑝 + 4𝑛2 𝑚2 = 1 − 2𝑛 − 2𝑝 + 4𝑛𝑚 𝐸𝑏𝑢𝑙𝑘 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 (6) where m is the reduced stiffness factor for the boundary layer of thickness, and the range of value of m is 0 < m < 1; nd, and pd is the thickness of the stress-free boundary layer with no stiffness. Eq. (6) is plotted in Fig. 5(a) with m = 0.85, n = 0.5, p = 0.25 [15].
𝜎
𝜎𝑏𝑢𝑙𝑘 631
=
(𝐿∕𝑑 − 2𝑤∕𝑑 )2 (𝐿 ∕ 𝑑 )2
(7)
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 16. Effective plastic strain contours in samples with three different thicknesses and in different directions: (a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
4.4. Shear
crease of sample height, which leads to lower stiffness and shear strength of samples with larger height. As for the effect of sample thickness (dimension in the Y-direction), the dominate factor is those cell layers with less constraint (i.e., weak cell layers) because the volume percentage of strong boundary layers does not change. The thickness of weak cell layers, wd, does not change with sample thickness (Figs. 15 and 16). Compared with the effect of height, thickness of a sample has a positive effect on its shear behavior, because of the declining volume fraction of weak cell layers in sample with increasing thickness. Consequently, the normalized shear strength can be calculated by introducing the volume fraction of strong boundary layers and weak cell layers:
Figs. 13 and 14 illustrate the shear stress and effective plastic strain contours at the onset of plastic stage for three samples with different heights, respectively. The locations of sections in Figs. 13 and 14 are illustrated in Fig. 1(b). In Fig. 13, the shear stresses in the top and bottom zones of a sample are higher than that in the center, especially in shorter samples. This is because the nodes at top and bottom surfaces of samples are tied to the rigid platens, which are subjected to more constraints. Therefore, compared with the core of a sample, the higher stress zones (top and bottom) can be regarded as ‘strong boundary layer’. The thickness of strong boundary layer is sd, which varies slightly in samples with different sizes. Consequently, the volume fraction of strong boundary layer can be calculated as: 𝑠𝑑 × 𝐿 × 𝑇 × 2 2𝑠𝑑 2𝑠 𝑉𝑠𝑡𝑟 % = = = 𝐻 ×𝐿×𝑇 𝐻 𝐻∕𝑑
(𝑇 ∕𝑑 − 2𝑤)(𝐿∕𝑑 − 2𝑤) 𝜏 2𝑠 = + 𝜏𝑏𝑢𝑙𝑘 𝐻∕𝑑 (𝑇 ∕𝑑 )(𝐿∕𝑑 ) Eq. (9) is plotted in Figs. 6(b) and 7(b) when 𝑠 = spectively.
(8)
(9) 1 4
and 𝑤 = 14 , re-
4.5. Bending
where the meanings of L, H, T and d have been mentioned previously. The volume fraction of strong boundary layers decreases with the in-
Figs. 17 and 18 demonstrate the von Mises stress and effective plastic strain contours for samples with different heights, respectively, 632
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International Journal of Mechanical Sciences 151 (2019) 622–638
Fig. 17. Von Mises stress (in Pa) contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
expressed as Eq. (10) and plotted in Fig. 8(a) and (b) when 𝑤 = 14 .
while Figs. 19 and 20 are for samples with different thicknesses, respectively. It can be seen that each sample is subjected to compression below the neutral surface and tension above the neutral surface. The four surfaces parallel to the X-direction of a sample are free boundaries. The cells at top and bottom surfaces can deform more easily than those at the center of a sample because of less constraint. Therefore, cells located on the four surfaces parallel to the X-direction can be regarded as ‘weak cell layers’ though their stress is neither zero nor nearly zero. In addition, larger stress is produced in the top and bottom layers than at the center of a sample due to the nature of bending. It should be noted that the dominant factor of producing high stress zone in bending cases is different from that in shear cases. In bending, high stress at the top and bottom surfaces is due to the large strain, while in shear, top and bottom surfaces of a sample are tied to the loading platens and the strong constraint results in high stress. As a result of greater volume fraction of weak cell layers, strength of a sample with small size in height or thickness is lower than that of a sample with large size. Similar to that in compression, the thickness of weak cell layers is assumed to be wd, and 𝜎 y − bending can be calculated by using Eq. (5). Therefore, the normalized bending strength can be
𝜎𝑦−𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝜎𝑏𝑢𝑙𝑘−𝑏𝑒𝑛𝑑𝑖𝑛𝑔
=
6𝑀𝑏 ∕𝑇 𝐻 2 6𝑀𝑏 ∕(𝑇 − 2𝑤𝑑 )(𝐻 − 2𝑤𝑑 )
2
=
(𝑇 ∕𝑑 − 2𝑤)(𝐻∕𝑑 − 2𝑤)2 (𝑇 ∕𝑑 )(𝐻∕𝑑 )2 (10)
4.6. Effects of the thickness of two types of layers When the thicknesses of both the weak cell layers and strong boundary layers were assumed to be 14 𝑑 (where d is the mean diameter of cells), Eqs. (6–7) and (9–10) either underestimated or overestimated the foam properties, as shown in Figs. 4–7. This might be due to the approximation of the thickness of either weak cell layers or strong boundary layers. Therefore, different values of the thicknesses of weak cell layers and strong boundary layers were employed in Eqs. (6–7) and (9–10) to explore the effect of these parameters. Fig. 21 shows the effect of the thickness of weak cell layers on normalized Young’s modulus and normalized yield strength of samples under compression. With the increase in thickness of zero-stiffness layer, pd, the predicted normalized Young’s modulus decreases. Similarly, normalized strength increases when thicknesses of weak cell layers, wd, de633
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Fig. 18. Effective plastic strain contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
clines. It should be noted that the agreements of test results, simulation results and prediction are better when p = 0.1 and w = 0.1. This is because the bulk properties used for normalization in the experiments and simulation are different from that in theoretical analysis. In experiments or simulation, the properties of a sample with the maximum size studied were used as bulk properties. However, the properties of the material without any weak cell layers or strong boundary layers were regarded as the bulk properties in the theoretical analysis. Therefore, smaller p or w leads to the bulk properties used in experiments and simulation approaching theoretical bulk properties. Fig. 22 illustrates the effects of thicknesses of weak cell layers and strong boundary layers on the predicted normalized strength of samples under shear loading. The larger thickness of strong boundary layers results in larger shear strength, while shear strength of samples decreases with the increasing thickness of weak cell layers. It should be noted that when the effect of strong boundary layers thicknesses was investigated, the thicknesses of weak cell layers were fixed as 14 𝑑, and vice versa. Therefore, the effect of the thickness of weak cell layers is not significant when sample height varies. The agreements of experimental, simulated and predicted results are better when s = 0.1 and w = 0.25 (Fig. 22a). The reason is the same as that mentioned above when discussing the effect of the thickness of weak cell layers on compressive properties. However, when the thickness of samples changes, the prediction matches best the current simulated results when s = 0.25 and w = 0.25.
Fig. 23 demonstrates that the predicted normalized strength increases with the decrease in the thicknesses of weak cell layers. The reason is similar to that for compression and shear cases, which is the difference in the bulk material properties used for simulation and theoretical analysis. The prediction better matches the current simulation when w = 0.1.
4.7. Statistical analysis In this section, Weibull distributions [27] were employed to describe the results of the uniaxial compression simulations. Using Weibull distribution, the cumulative probability for a sample with volume V to yield at a compressive stress, 𝜎, can be given as [19,28]: [ ( )𝑚 ] 𝑉 𝜎 𝑃𝑅 = 1 − exp − (11) 𝑉 0 𝜎0 where V0 is the reference volume of a sample and m is the Weibull modulus. The reference stress, 𝜎 0 , is the value of stress giving a cumulative probability of 1 − e − 1 = 0.63 for a sample with volume V0 . Because the samples used in the simulations are cubic, Eq. (1) can be rewritten as: [ ( ) ( ) ] 3 𝑚 𝐿 𝜎 𝑃𝑅 = 1 − exp − (12) 𝐿0 𝜎0 where L and L0 are the length and reference length of a sample. 634
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Fig. 19. Von Mises stress (in Pa) contours in samples with three different thicknesses and in different directions:(a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
The cumulative probability of yield for a given applied stress corresponding to the rank i can be estimated as: 𝑖 𝑃𝑅𝑖 = (13) 𝑁 +1 where N is the total number of samples simulated. Due to the heterogeneity of foam material, the value of reference length L0 cannot be too small. Therefore, the value of L0 was assumed to be 16 mm, and the properties of samples with this side length are considered as bulk properties. The least squares method was used to fit the values of 𝜎 0 and m. Fig. 24 shows the values obtained by fitting 𝜎 0 and m from the simulation results of different size samples. It can be seen that both the values of 𝜎 0 and m increase with the increasing value of L/d, and the values are relatively stable (discrepancies are less than 10%) when L/d ≥ 6.
and experimental results show that stiffness and strength of aluminum foams vary with sample size and plateau when the sample size is sufficiently large. •
•
5. Conclusion Numerical simulations using 3D Voronoi models have been carried out to investigate the effects of sample size on the mechanical properties of foams subjected to compression, shear and bending, respectively. Uniaxial compressive tests have been conducted on aluminum foam with a relatively density of 20% and various sample sizes, whose results have been employed to validate the numerical models. Both the numerical
•
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For compression cases, Young’s modulus and yield strength of samples increase with side length of samples because the volume fraction of weak cell layers decreases with side length of samples. The normalized Young’s modulus plateaus when L/d ≥ 7, while the normalized yield strength converges when L/d ≥ 6. For shear cases, decreasing sample height, H, leads to an increase in both stiffness and strength of samples because the volume fraction of higher stress zone with strong constrain (strong boundary layers) is larger in smaller samples. Conversely, the thickness of sample, T, has a positive effect on properties of sample due to the decrease of weak cell layers (the volume fraction of strong boundary layers remains constant for thickness changing case). When H/d ≥ 6 and T/d ≥ 6, the difference between the stiffness or strength of samples and bulk properties is less than 5%. For bending cases, the strength of samples increases with the sample size. The four surfaces parallel to X-direction can be considered as weak cell layers and affect the mechanical response of samples. The yield strength of sample can be used to represent the bulk strength when H/d ≥ 6 and T/d ≥ 5 of sample.
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Fig. 20. Effective plastic strain contours in samples with three different thicknesses and in different directions: (a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
Fig. 21. The predictions of normalized Young’s modulus (a), and yield strength (b) of sample vs. side length of samples normalized by mean cell size, L/d.
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Fig. 22. The predictions of normalized shear yield strength of sample vs. (a) sample height normalized by mean cell size, H/d; (b) sample thickness normalized by mean cell size, T/d.
Fig. 23. The predictions of normalized bending yield strength of samples vs. (a) sample height normalized by mean cell size, H/d; (b) sample thickness normalized by mean cell size, T/d.
Furthermore, the effects of thicknesses of weak cell layers and strong boundary layers have been discussed. Acknowledgements This work is sponsored by the National Natural Science Foundation of China (Grant Nos. 11572214, 11772215), the “1331 Project” fund, Key Innovation Teams of Shanxi Province, the Top Young Academic Leaders of Shanxi, Opening foundation for State Key Laboratory for Strength and Vibration of Mechanical Structures, China Scholarship Council, and Postgraduate Innovation Project of Shanxi Province (No. 2017BY040). Their financial supports are gratefully acknowledged. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge: Cambridge University Press; 1997. doi:10.1017/CBO9781139878326. [2] Miller RE. A continuum plasticity model for the constitutive and indentation behaviour of foamed metals. Int J Mech Sci 2000;42:729–54. doi:10.1016/S0020-7403(99)00021-1. [3] Deshpande VS, Fleck NA. Isotropic constitutive models for metallic.pdf. J Mech Phys Solids 2000;48:1253–83. doi:10.1016/S0022-5096(99)00082-4. [4] Jing L, Su X, Chen De, Yang F, Zhao L. Experimental and numerical study of sandwich beams with layered-gradient foam cores under low-velocity impact. Thin-Walled Struct 2019;135:227–44. doi:10.1016/j.tws.2018.11.011. [5] Nguyen VD, Noels L. Computational homogenization of cellular materials. Int J Solids Struct 2014;51:2183–203. doi:10.1016/j.ijsolstr.2014.02.029. [6] Ramamurty U, Paul A. Variability in mechanical properties of a metal foam. Acta Mater 2004;52:869–76. doi:10.1016/j.actamat.2003.10.021.
Fig. 24. Values of 𝜎 0 and m of simulation results under compression vs. side length of the samples normalized by mean cell size, L/d.
•
The relationship between properties of samples and the thickness of weak cell layers and strong boundary layers has been developed. 637
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[7] Zheng Z, Yu J, Li J. Dynamic crushing of 2D cellular structures: a finite element study. Int J Impact Eng 2006;32:650–64. doi:10.1016/j.ijimpeng.2005.05.007. [8] Tekoglu C, Gibson LJ, Pardoen T, Onck PR. Size effects in foams: experiments and modeling. Prog Mater Sci 2011;56:109–38. doi:10.1016/j.pmatsci.2010.06.001. [9] Jin T, Zhou Z, Wang Z, Wu G, Shu X. Experimental study on the effects of specimen in-plane size on the mechanical behavior of aluminum hexagonal honeycombs. Mater Sci Eng A 2015;635:23–35. doi:10.1016/j.msea.2015.03.053. [10] Rakow JF, Waas AM. Size effects and the shear response of aluminum foam. Mech Mater 2005;37:69–82. doi:10.1016/j.mechmat.2003.12.002. [11] Chen C, Fleck NAA. Size effects in the constrained deformation of metallic foams. J Mech Phys Solids 2002;50:955–77. doi:10.1016/S0022-5096(01)00128-4. [12] Jing L, Su X, Yang F, Ma H, Zhao L. Compressive strain rate dependence and constitutive modeling of closed-cell aluminum foams with various relative densities. J Mater Sci 2018;53:14739–57. doi:10.1007/s10853-018-2663-z. [13] Jing L, Wang Z, Zhao L. The dynamic response of sandwich panels with cellular metal cores to localized impulsive loading. Compos Part B Eng 2016;94:52–63. doi:10.1016/j.compositesb.2016.03.035. [14] Onck PR. Scale effects in cellular metals. MRS Bull 2003;28:279–83. doi:10.1557/mrs2003.81. [15] Onck PR, Andrews E, Gibson L. Size effects in ductile cellular solids. Part I: modeling. Int J Mech Sci 2001;43:681–99. doi:10.1016/S0020-7403(00)00042-4. [16] Andrews E, Gioux G, Onck P, Gibson L. Size effects in ductile cellular solids. Part II: experimental results. Int J Mech Sci 2001;43:701–13. doi:10.1016/S0020-7403(00)00043-6. [17] Jeon I, Asahina T. The effect of structural defects on the compressive behavior of closed-cell Al foam. Acta Mater 2005;53:3415–23. doi:10.1016/j.actamat.2005.04.010. [18] Li L, Xue P, Chen Y, Butt HSU. Insight into cell size effects on quasi-static and dynamic compressive properties of 3D foams. Mater Sci Eng A 2015;636:60–9. doi:10.1016/j.msea.2015.03.052.
[19] Blazy JS, Marie-Louise A, Forest S, Chastel Y, Pineau A, Awade A, et al. Deformation and fracture of aluminium foams under proportional and non proportional multiaxial loading: statistical analysis and size effect. Int J Mech Sci 2004;46:217–44. doi:10.1016/j.ijmecsci.2004.03.005. [20] Maire E. X-ray tomography applied to the characterization of highly porous materials. Annu Rev Mater Res 2012;42:163–78. doi:10.1146/annurev– matsci-070511-155106. [21] Petit C, Meille S, Maire E. Cellular solids studied by x-ray tomography and finite element modeling - a review. J Mater Res 2013;28:2191–201. doi:10.1557/jmr.2013.97. [22] Zhu HX, Hobdell JR, Windle AH. Effects of cell irregularity on the elastic properties of open cell foams. Acta Mater 2000;48:4893–900. [23] Zhu HX, Windle AH. Effects of cell irregularity on the high strain compression of open-cell foams. Acta Mater 2002;50:1041–52. doi:10.1016/S1359-6454(01)00402-5. [24] Zhang X, Wu Y, Tang L, Liu Z, Jiang Z, Liu Y, et al. Modeling and computing parameters of three-dimensional Voronoi models in nonlinear finite element simulation of closed-cell metallic foams. Mech Adv Mater Struct 2016;0:0. doi:10.1080/15376494.2016.1190426. [25] Caty O, Maire E, Youssef S, Bouchet R. Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements. Acta Mater 2008;56:5524–34. doi:10.1016/j.actamat.2008.07.023. [26] Forest S, Blazy JS, Chastel Y, Moussy F. Continuum modeling of strain localization phenomena in metallic foams. J Mater Sci 2005;40:5903–10. doi:10.1007/s10853-005-5041-6. [27] Weibull W. A statistical distribution function of wide applicability. J Appl Mech 1951;103:293–7. [28] McCullough KYG, Fleck NA, Ashby MF. Uniaxial stress-strain behaviour of aluminum alloy foams. Acta Mater 1999;47:2323–30. doi:10.1016/S1359-6454(99)00128-7.
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Sample size effect on the mechanical behavior of aluminum foam Yue Zhang a,b, Tao Jin a,c, Shiqiang Li a,c, Dong Ruan b, Zhihua Wang a,c,∗, Guoxing Lu b,∗∗ a
Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, VIC 3122, Australia c Shanxi Key Laboratory of Material Strength and Structural Impact, Taiyuan 030024, China b
a r t i c l e
i n f o
Keywords: Metal foam Sample size effect 3D Voronoi model Mechanical properties Compression, shear, and bending
a b s t r a c t The aim of this work is to explore the effect of sample size in three different directions (length, height, and thickness of a sample block), with respect to its cell size, on the mechanical properties of closed-cell aluminum foams. 3D Voronoi models were used to represent a real foam block, and finite element (FE) analysis was performed to simulate the mechanical properties of samples under three different loading conditions (uniaxial compression, shear, and bending). The numerical results match the experimental results. The stiffness and strength of samples with different sizes were normalized by those properties of samples with the maximum size. The normalized stiffness and strength were expressed as functions of thickness of weak cell layers (less constraint cells) and that of the strong boundary layers (strong constraint cells). Furthermore, a parametric study was performed to investigate the effects of the thicknesses of weak cell layers and strong boundary layers on the mechanical properties of aluminum foam. It is concluded that both the stiffness and strength increase with sample size under compression and bending, while they decrease with the increase of sample height under shear. The minimum size of samples characterizing bulk materials under different loading conditions was determined.
1. Introduction Metal foam has been applied in many engineering fields as lightweight and protective structures because of its high specific strength and great ability of energy absorption. The wide application in industries inspires researchers and experts to explore the mechanical properties of this material [1–4]. By using a method based on classical continuum theory, the properties of metal foam are considered as being related to relative density only [5]. However, in some investigations, cell geometries such as cell size and cell morphology were found to affect the mechanical behavior of cellular solids [6,7]. Moreover, the cell size of cellular materials for industrial applications is generally between 1 to 10 mm. The relatively small side length of industrial components may only contain several cells [8] and the mechanical properties vary with the number of cells or component size. Therefore, the effect of sample size should be considered. The effect of sample size or cell size on the mechanical properties of cellular solids has been reported. Jin et al. [9] investigated the effect of cell number on the in-plane and out-of-plane stiffnesses and plateau stresses of aluminum hexagonal honeycombs. They developed a relationship between cell number and mechanical behavior of honeycombs in three different directions. Rakow and Waas [10] utilized digital image correlation (DIC) to study the effects of sample size, rel∗ ∗∗
ative density and strain rate on the shear response of closed-cell aluminum foam. Their experimental results showed that the effect of sample size on the shear strength of foam was negligible. Chen and Fleck [11] found that the compressive properties of open-cell aluminum foam were independent of cell size. Jing et al. [12] investigated the effect of sample thickness on the properties of aluminum foam under dynamic compressive loading. They also found the compressive properties of open-cell aluminum foam increase with the increasing cell sizes of samples [13]. Moreover, the dominant factor for cell size effect has also been discussed. Onck et al. [8,14–16] investigated the effect of sample size on the mechanical behavior of aluminum foam in four different loading cases by using theoretical analysis, experiments and FE simulation of 2D Voronoi models. In the case of compression, the shape of sample was square prisms whose height was twice the width. In shear tests, the thickness of sample was 50 mm, while the length and height of sample varied with a constant ratio of length/height = 12. It was concluded that sample size effect stemmed from either weak cell layers for compression, indentation and bending or strong boundary layers for shear. Jeon and Asahina [17] explored the compressive properties of closedcell foams without or with evident structural defects such as missing cells and collapsed cells. They found that the yield strength of samples without defects was independent of sample size while Young’s mod-
Corresponding author at: Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China. Corresponding authors. E-mail addresses: [email protected], [email protected] (Z. Wang), [email protected] (G. Lu).
https://doi.org/10.1016/j.ijmecsci.2018.12.019 Received 27 August 2018; Received in revised form 2 December 2018; Accepted 11 December 2018 Available online 11 December 2018 0020-7403/© 2018 Published by Elsevier Ltd.
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Fig. 1. Schematic diagrams of three different loading cases: (a) uniaxial compression; (b) shear; (c) bending.
Table 1 Dimensions of samples for different loading cases. (all in mm).
Compression Shear Bending
X-direction (L)
Y-direction (T)
Z-direction (H)
6, 10, 14, 16, 18, 20 L/H = 3 L/H = 3
Same as X-direction 4, 8, 12, 16, 20 4, 8, 12, 16, 20
Same as X-direction 4, 8, 12, 16, 20 4, 8, 12, 16, 20
2. Numerical simulation 2.1. Voronoi model Some researchers established FE model of foams by using X-Ray computed tomography [20,21] while others used Voronoi model [8,18]. In the current work, a 3D Voronoi model is used to represent real foams. Similar to foaming process, nucleation points are distributed randomly in a certain volume V, and the number of nucleation points is N. To avoid the existence of tiny cells, a minimum distance between two nucleation points in the Voronoi model is defined, as follows, )1∕3 √ ( 6 𝑉 𝑑min = (1 − 𝑘)𝑑0 = (1 − 𝑘) (1) √ 2 2𝑁
ulus increased as sample size increased. On the other hand, both the Young’s modulus and yield stress of samples with defects increased with sample size. Defects played an important role in size effect. Li et al. [18] proposed a new statistical method to establish the relationship between the microstructure parameters and the macro-properties of foams by using FE simulation of a 3D Voronoi model. It was found that the percentage of ‘weak cell’ led to the cell size effect. Forest et al. [19] employed Weibull statistical analysis to interpret the variations of yield strengths and corresponding scatter of sample with different sizes. Considering the results in the literature mentioned above, the effect of sample size in thickness direction on properties of sample was rarely reported. In the current work, the sample size effects on quasi-static compressive, shear and bending properties are further investigated. The numerical simulations on the mechanical properties of aluminum foam with different sample sizes subjected to three different loadings are carried out by using 3D Voronoi models. Subsequently, the mechanism of sample size effect is analyzed and the relationship between sample size and mechanical behavior of aluminum foam is further developed. A parametric study has been performed to investigate the effects of thicknesses of weak cell layers and strong boundary layers on mechanical properties of aluminum foam.
where d0 is the distance between the two adjacent nucleation points in a regular tetrakaidecahedron model and k is the irregularity of cell shapes. The value of irregularity, k, ranges from 1 (completely random) to 0 (regular). In the current work, the value of k was chosen to be 0.2 [22,23]. Matlab software was utilized to place N nucleation points in the volume V and generate the coordinates of the points, lines and surfaces, which could be used to construct the geometric Voronoi model. Fig. 1 shows the Voronoi models for different loading conditions. The mean radius of cells is 1.241 mm, which is calculated from Eq. (2). 𝑁=
𝑉 (4𝜋∕3)𝑟3
(2)
where N and V are as defined in Eq. (1), and r is the mean equivalent radius of cells, i.e., the mean radius of sphere whose volume is equal to cell. The sample for uniaxial compression was cube to prevent buckling during deformation [17]. Its maximum side length was 623
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Fig. 2. Illustration of two criteria of the accuracy of quasi-static FE simulation: (a) artificial strain energy to total internal energy ratio history curve; (b) stress equilibrium history curve.
Table 2 Material properties of the parent material (aluminum). Density 2700 kg/m
3
Young’s modulus
Poisson’s ratio
Yield strength
Tangent modulus
70 GPa
0.3
80 MPa
700 MPa
L (X-direction) × T (Y-direction) × H (Z-direction) = 20 × 20 × 20 mm (Fig. 1a). For shear (Fig. 1b) and bending (Fig. 1c), the maximum sample size was L × T × H = 60 × 20 × 20 mm. The dimensions of samples for different loading cases are listed in Table 1. It should be noted that when the size effect in one direction was investigated, the dimensions in the other two directions were fixed at their maximum values.
as e and its history is plotted in Fig. 2. Meanwhile, the kinetic energy should not be more than 5% of total internal energy to avoid introducing inertia effect, and it can be estimated by the difference between the stresses at the fixed end and loading end [24]:
2.2. FE model and calculation
𝜎𝑙𝑜𝑎𝑑𝑖𝑛𝑔 − 𝜎𝑓 𝑖𝑥𝑒𝑑 𝑄= ( ) 𝜎𝑙𝑜𝑎𝑑𝑖𝑛𝑔 + 𝜎𝑓 𝑖𝑥𝑒𝑑 ∕2
The geometric Voronoi models aforementioned were meshed by using Hypermesh, and a commercial software Ls-Dyna was used to perform the finite element calculation. To reduce computational time, mass scaling was employed in simulation, in which the minimum time step size was set as 10−7 s [24]. Moreover, considering the accuracy of quasistatic process in explicit simulation, the artificial strain energy introduced by mass scaling should be less than 10% of total internal energy. The ratio of artificial strain energy to total internal energy is defined
where 𝜎 loading and 𝜎 fixed are stresses at loading end and fixed end, and the relationship between Q and time can be seen in Fig. 2. Shell elements with in-plane single point integration were employed in Voronoi FE models. The sensitivity of characteristic length of elements was calculated and the characteristic length was set as 0.3 mm. A Single Surface Contact was employed to simulate the contact among cell walls, and the coefficient of static friction was 0.02 [7]. The thickness of shell elements, t, can be calculated by the following Eq. (4) [7]:
(3)
Fig. 3. (a) A photograph of uniaxial compression test setup; (b) experimental and simulated stress-strain curves of cubic samples with a side length of 20 mm. 624
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Fig. 4. Stress-strain curves of samples with different size under various loading conditions: (a) compression; (b)(c) shear; (d)(e) bending (stress-rotation curves).
𝑉𝑓 𝑜𝑎𝑚 ( ∗ ) 𝜌 ∕ 𝜌𝑠 𝑡= ∑ 𝑆𝑖
A bilinear strain-hardening material model was employed for the parent material of Voronoi models. The values of the parameters for the material are listed in Table 2. In the simulation of uniaxial compression, a sample was placed between two rigid platens without any transverse constraint. The top platen moved at a constant velocity along the vertical direction to compress the sample (Fig. 1a), while the bottom platen was fully fixed. The velocity of the top platen varied with the sample size to maintain a constant strain rate of 10−3 /s.
(4)
where Vfoam is the volume of a sample; ΣSi is the total area of the surfaces in a foam sample and 𝜌∗ /𝜌s is the relative density of a sample. In the current work, the relative density of each sample employed in the simulation was constant, 20%. The element thickness was uniform but with various values for different samples to ensure the same relative density for all samples. 625
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Fig. 5. Normalized Young’s modulus (a) and yield strength (b) of the samples under compression vs. side length of the samples normalized by mean cell size, L/d.
Fig. 6. Normalized shear modulus (a) and shear strength (b) of the samples vs. sample height normalized by mean cell size, H/d.
Fig. 7. Normalized shear modulus (a), and shear strength (b) of the samples vs. sample thickness normalized by mean cell size, T/d.
For the shearing cases, a sample was also placed between two rigid platens. The top platen moved at a constant velocity along the Xdirection, while the bottom platen was fully fixed (Fig. 1b). The nodes on the top and bottom surfaces of the sample were tied to both the rigid platens.
For the bending cases, two rigid platens perpendicular to the Xdirection were adhered to all the nodes on the left and right sides of the sample. One of the rigid platens was fully constrained while the other rotated about the neutral axis of the fixed platen at a constant angular velocity.
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Fig. 8. Normalized yield strength under bending loading of the samples vs. (a) sample height normalized by mean cell size, H/d; (b) sample thickness normalized by mean cell size, T/d.
2.3. Experiments and validation of FE models Uniaxial compression tests of closed-cell aluminum foam (fabricated by Luoyang Ship Material Research Institution, Luoyang, China) were performed at a constant strain rate of 10−3 /s by using a universal test machine (Instron 5969). The experimental setup (Fig. 3a) was the same as that in the numerical simulation. Cubic foam samples had a relative density of 20% and side lengths of samples were the same as those used in numerical simulation for uniaxial compression, as listed in Table 1. Fig. 3(b) shows the stress-strain curves of samples obtained from both the simulation and tests of the cubic samples with a side length of 20 mm. The simulated result (except Young’s modulus, which will be discussed in Section 4.1) matches reasonably well with the experimental results of three repeated tests, which indicates that the FE model is valid. 3. Results Fig. 9. Stress-strain curves of samples with maximum size under compression while different values of the friction coefficients were employed.
In the current FE work, five 3D Voronoi models have been established for each sample size and each loading case. Average values are presented in Figs. 4–8 with error bars. Fig. 4 shows the stress-strain curves of samples with different sizes for three different loading cases (for bending, the stress-rotation curves is shown in Fig. 4, and the rotation, 𝜃, is the angle that the loading platen rotated by). The value of
Fig. 10. Normalized Young’s modulus (a) and yield strength (b) of the samples under compression vs. side length of the samples normalized by mean cell size, L/d, while various values of the friction coefficients were employed. 627
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Fig. 11. Compressive stress (in Pa) contours in the samples with three different sizes: (a) L/d ≈ 4; (b) L/d ≈ 6; (c) L/d ≈ 8. (d remains 2.482 mm.) Sections are Section 1 shown in Fig. 1(b).
Fig. 12. Effective plastic strain contours in the samples with three different sizes: (a) L/d ≈ 4; (b) L/d ≈ 6; (c) L/d ≈ 8. (d remains 2.482 mm.) Sections are Section 2 shown in Fig. 1(b).
modulus of each sample was calculated from the slope of the stress-strain curve in the elastic regime and the yield strength was the intersection point of the tangents of elastic regime and plastic regime of stress-strain curves. Modulus and yield strength of various size samples were normalized with the bulk properties of the same material, which were obtained from the sufficiently large samples (with the maximum size) for each loading condition.
good agreements (less than 10% discrepancy) between simulated and experimental results indicate the reliability of the FE models as well. The normalized Young’s modulus, E/Ebulk , and normalized yield strength, 𝜎 c /𝜎 bulk , of the samples increase with the length of sample sides, L. However, the slope of the normalized properties versus the ratio of length to mean cell size, L/d (d = 2r), changes when this ratio is larger than a certain value. Note that the value of L/d approximately indicates the number of cells contained in a sample in that direction. The normalized Young’s modulus plateaus when L/d ≥ 7, while the normalized yield strength converges when L/d ≥ 6.
3.1. Uniaxial compression Fig. 5 illustrates the comparison between experimental and simulated results under uniaxial compression. The results from the previous studies are also included. It is evident that sample size has an effect on the uniaxial compressive property of the Voronoi models. In addition,
3.2. Shear In this section, the effects of sample size on shear behavior of foam in both the Y-direction and Z-direction were investigated. Fig. 6 shows the 628
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Fig. 13. Shear stress (in Pa) contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases). Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
effect of sample height, H, sample size in the Z-direction on the normalized shear stiffness, G/Gbulk , and normalized shear strength, 𝜏 y /𝜏 bulk , of the Voronoi models. Both the normalized shear stiffness and normalized shear strength exhibit a decreasing trend when the value of H increases, which is a converse observation compared with compression cases. As seen in Fig. 6, sample height has an effect on the shear behavior of the Voronoi models when the ratio of sample height to the mean cell size H/d is less than 6. Experimental results in literature [16] manifested that the normalized shear strength also declined with the increase of sample height, which is similar to the present FE results for smaller values of H/d. Both the shear stiffness and strength of the current Voronoi model converge at a large value of H/d. Fig. 7 demonstrates the relationship between the sample thickness, T, sample size in the Y-direction and shear properties of the Voronoi models. In contrast to the effect of sample height, sample thickness has a positive effect on the normalized shear stiffness, G/Gbulk , and normalized shear strength, 𝜏 y /𝜏 bulk . The effect of sample thickness vanishes when the ratio of sample thickness to mean cell size is large, i.e., T/d ≥ 6.
bending yield strength of the Voronoi model is: 𝜎𝑦−𝑏𝑒𝑛𝑑𝑖𝑛𝑔 = 6𝑀𝑏 ∕𝑇 𝐻 2
(5)
where Mb is the bending moment corresponding to the beginning of nonlinearity on a moment-rotation angle curve, T and H are as defined in Section 3.2. It can be seen in Fig. 8 that the bending yield strength increases with both the sample height and thickness. Moreover, when the ratio of sample height to mean cell size H/d ≥ 6 or the ratio of sample thickness to mean cell size T/d ≥ 5, the bending yield strength is hardly affected by sample height or sample thickness.
4. Discussions 4.1. Overestimate in Young’s modulus
3.3. Bending
The value of Young’s modulus of sample obtained from simulation is higher than the experimental results, which is shown in Fig. 3. The dominant factors of this might be:
The effects of sample size in both the Y-direction and Z-direction on bending behavior of the Voronoi model are explored in this section. The
(1) The melt-foaming process was used to manufacture the foam investigated in this work. This process induces small pores in cell walls 629
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Fig. 14. Effective plastic strain contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases). Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
E/Ebulk and normalized yield strengths, 𝜎 c /𝜎 bulk of samples with different sizes when various friction coefficients were employed. It can be seen that friction coefficient between the samples and rigid plates has an effect on the normalized properties of the samples, especially when sample size is small. Normalized Young’s moduli and yield strengths plateau out with the increase of L/d.
of foam material, which results in a lower stiffness of cell walls than that of FE model; (2) There are some structural defects in actual aluminum foams such as missing cells and collapsed cells. Furthermore, there exist some curved cell walls, which leads to a decrease of stiffness of samples compared with that in the FE model. (3) In the simulations, at the shared edges of cell walls, the thicknesses are overestimated due to the overlap of different elements of different cell walls. This problem locally increases the stiffness compared with the actual material [25].
4.3. Compression Fig. 11 shows the stress contours at the onset of plastic stage for three selected sample sizes under uniaxial compression. Cells around the edge experience much lower stress than those in the central region. The reason is that the constraint on the inner cells is greater than that on the edge cells. Those edge cells with less constraint may be considered as ‘weak cell’. Furthermore, these ‘weak cells’ are even stress-free or nearly stress-free. Fig. 12 shows the effective plastic strain contours at the onset of plastic stage for three selected samples of different sizes under uniaxial compression. It can be seen that deformation is not uniform and it localizes in the cells. The most severely localized deformation occurs at the top and bottom boundary of the sample where the cells are incomplete. The stiffness of those cells is lower than that of the cells in the central region.
4.2. Effect of friction The effect of friction between the foam and the rigid plates is discussed in this section. The friction coefficient between samples and rigid plates 𝜇 was assumed as 0.02 [7], 0.1, 0.2, and 0.4, respectively. Fig. 9 demonstrates the stress-strain curves of the maximum sized sample with different values of the friction coefficient. It is evident that with an increase of the friction coefficient, both the apparent modulus and yield strength are increasing. Modulus and yield strength of various sized samples were normalized by respect to the bulk properties of the same material, which were obtained from the sufficiently large samples (of the maximum size). Fig. 10 shows the normalized Young’s moduli, 630
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Fig. 15. Shear stress (in Pa) contours in samples with three different thicknesses and in different directions: (a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
For stiffness, the weak cell layers include the cells with reduced stiffness (located near the stress-free boundary) and cells with zero stiffness (located at the stress-free boundary). The volume fraction of weak cell layers decreases with the increase of sample size, which leads to the sample size effect on the stiffness of samples. Onck et al. divided weak cell layers into three categories: (i) the layers between the core of a sample and zero stiffness boundary, which could represent the reduced stiffness. The thickness of these layers was nd, and the Young’s modulus was mEbulk ; (ii) the corners of layers with reduced stiffness and lower Young’s modulus m2 Ebulk ; (iii) zero stiffness layers, whose thickness was pd. Then, the normalized stiffness of material could be expressed as [15]
For compressive strength, Jeon and Asahina [17] reported that the strength of the samples with structural defects was more sensitive to sample size than that of the samples without defects (shown in Fig. 5b). Although there were no evident structure defects such as missing cells or collapsed cells in the current 3D Voronoi models, there were ellipsoidal cells with T-shaped cell-wall intersection. These cells deformed severely in localization band [26], which could be named as ‘easily deformed cells’. In smaller samples, the volume fraction of stress-free or nearly stress-free boundary (considered as weak cell layers) is greater. Therefore, the constraint surrounding easily deformed cells is less, and first localization deformation band forms more easily in a small sample than that in large one, which leads to a lower yield strength of sample. In other words, the existence of weak cell layers results in sample size effect on the strength of samples. The thickness of the weak cell layer is assumed to be wd on average (d is the mean diameter of cell). Then, the normalized compressive yield strength can be expressed as Eq. (7) and plotted in Fig. 5(b) when 𝑤 = 14 [8].
( ) ( )( ) ( )2 𝐸 𝑑 𝑑 2 𝑑 𝑑 𝑑 𝑑 1 − 2𝑛 − 2𝑝 + 4𝑛2 𝑚2 = 1 − 2𝑛 − 2𝑝 + 4𝑛𝑚 𝐸𝑏𝑢𝑙𝑘 𝐿 𝐿 𝐿 𝐿 𝐿 𝐿 (6) where m is the reduced stiffness factor for the boundary layer of thickness, and the range of value of m is 0 < m < 1; nd, and pd is the thickness of the stress-free boundary layer with no stiffness. Eq. (6) is plotted in Fig. 5(a) with m = 0.85, n = 0.5, p = 0.25 [15].
𝜎
𝜎𝑏𝑢𝑙𝑘 631
=
(𝐿∕𝑑 − 2𝑤∕𝑑 )2 (𝐿 ∕ 𝑑 )2
(7)
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Fig. 16. Effective plastic strain contours in samples with three different thicknesses and in different directions: (a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(b); sections of (b)(d)(f) are Section 2 shown in Fig. 1(b).
4.4. Shear
crease of sample height, which leads to lower stiffness and shear strength of samples with larger height. As for the effect of sample thickness (dimension in the Y-direction), the dominate factor is those cell layers with less constraint (i.e., weak cell layers) because the volume percentage of strong boundary layers does not change. The thickness of weak cell layers, wd, does not change with sample thickness (Figs. 15 and 16). Compared with the effect of height, thickness of a sample has a positive effect on its shear behavior, because of the declining volume fraction of weak cell layers in sample with increasing thickness. Consequently, the normalized shear strength can be calculated by introducing the volume fraction of strong boundary layers and weak cell layers:
Figs. 13 and 14 illustrate the shear stress and effective plastic strain contours at the onset of plastic stage for three samples with different heights, respectively. The locations of sections in Figs. 13 and 14 are illustrated in Fig. 1(b). In Fig. 13, the shear stresses in the top and bottom zones of a sample are higher than that in the center, especially in shorter samples. This is because the nodes at top and bottom surfaces of samples are tied to the rigid platens, which are subjected to more constraints. Therefore, compared with the core of a sample, the higher stress zones (top and bottom) can be regarded as ‘strong boundary layer’. The thickness of strong boundary layer is sd, which varies slightly in samples with different sizes. Consequently, the volume fraction of strong boundary layer can be calculated as: 𝑠𝑑 × 𝐿 × 𝑇 × 2 2𝑠𝑑 2𝑠 𝑉𝑠𝑡𝑟 % = = = 𝐻 ×𝐿×𝑇 𝐻 𝐻∕𝑑
(𝑇 ∕𝑑 − 2𝑤)(𝐿∕𝑑 − 2𝑤) 𝜏 2𝑠 = + 𝜏𝑏𝑢𝑙𝑘 𝐻∕𝑑 (𝑇 ∕𝑑 )(𝐿∕𝑑 ) Eq. (9) is plotted in Figs. 6(b) and 7(b) when 𝑠 = spectively.
(8)
(9) 1 4
and 𝑤 = 14 , re-
4.5. Bending
where the meanings of L, H, T and d have been mentioned previously. The volume fraction of strong boundary layers decreases with the in-
Figs. 17 and 18 demonstrate the von Mises stress and effective plastic strain contours for samples with different heights, respectively, 632
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Fig. 17. Von Mises stress (in Pa) contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
expressed as Eq. (10) and plotted in Fig. 8(a) and (b) when 𝑤 = 14 .
while Figs. 19 and 20 are for samples with different thicknesses, respectively. It can be seen that each sample is subjected to compression below the neutral surface and tension above the neutral surface. The four surfaces parallel to the X-direction of a sample are free boundaries. The cells at top and bottom surfaces can deform more easily than those at the center of a sample because of less constraint. Therefore, cells located on the four surfaces parallel to the X-direction can be regarded as ‘weak cell layers’ though their stress is neither zero nor nearly zero. In addition, larger stress is produced in the top and bottom layers than at the center of a sample due to the nature of bending. It should be noted that the dominant factor of producing high stress zone in bending cases is different from that in shear cases. In bending, high stress at the top and bottom surfaces is due to the large strain, while in shear, top and bottom surfaces of a sample are tied to the loading platens and the strong constraint results in high stress. As a result of greater volume fraction of weak cell layers, strength of a sample with small size in height or thickness is lower than that of a sample with large size. Similar to that in compression, the thickness of weak cell layers is assumed to be wd, and 𝜎 y − bending can be calculated by using Eq. (5). Therefore, the normalized bending strength can be
𝜎𝑦−𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝜎𝑏𝑢𝑙𝑘−𝑏𝑒𝑛𝑑𝑖𝑛𝑔
=
6𝑀𝑏 ∕𝑇 𝐻 2 6𝑀𝑏 ∕(𝑇 − 2𝑤𝑑 )(𝐻 − 2𝑤𝑑 )
2
=
(𝑇 ∕𝑑 − 2𝑤)(𝐻∕𝑑 − 2𝑤)2 (𝑇 ∕𝑑 )(𝐻∕𝑑 )2 (10)
4.6. Effects of the thickness of two types of layers When the thicknesses of both the weak cell layers and strong boundary layers were assumed to be 14 𝑑 (where d is the mean diameter of cells), Eqs. (6–7) and (9–10) either underestimated or overestimated the foam properties, as shown in Figs. 4–7. This might be due to the approximation of the thickness of either weak cell layers or strong boundary layers. Therefore, different values of the thicknesses of weak cell layers and strong boundary layers were employed in Eqs. (6–7) and (9–10) to explore the effect of these parameters. Fig. 21 shows the effect of the thickness of weak cell layers on normalized Young’s modulus and normalized yield strength of samples under compression. With the increase in thickness of zero-stiffness layer, pd, the predicted normalized Young’s modulus decreases. Similarly, normalized strength increases when thicknesses of weak cell layers, wd, de633
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Fig. 18. Effective plastic strain contours in samples with three different heights and in different directions: (a)(b) H/d ≈ 2; (c)(d) H/d ≈ 5; (e)(f) H/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
clines. It should be noted that the agreements of test results, simulation results and prediction are better when p = 0.1 and w = 0.1. This is because the bulk properties used for normalization in the experiments and simulation are different from that in theoretical analysis. In experiments or simulation, the properties of a sample with the maximum size studied were used as bulk properties. However, the properties of the material without any weak cell layers or strong boundary layers were regarded as the bulk properties in the theoretical analysis. Therefore, smaller p or w leads to the bulk properties used in experiments and simulation approaching theoretical bulk properties. Fig. 22 illustrates the effects of thicknesses of weak cell layers and strong boundary layers on the predicted normalized strength of samples under shear loading. The larger thickness of strong boundary layers results in larger shear strength, while shear strength of samples decreases with the increasing thickness of weak cell layers. It should be noted that when the effect of strong boundary layers thicknesses was investigated, the thicknesses of weak cell layers were fixed as 14 𝑑, and vice versa. Therefore, the effect of the thickness of weak cell layers is not significant when sample height varies. The agreements of experimental, simulated and predicted results are better when s = 0.1 and w = 0.25 (Fig. 22a). The reason is the same as that mentioned above when discussing the effect of the thickness of weak cell layers on compressive properties. However, when the thickness of samples changes, the prediction matches best the current simulated results when s = 0.25 and w = 0.25.
Fig. 23 demonstrates that the predicted normalized strength increases with the decrease in the thicknesses of weak cell layers. The reason is similar to that for compression and shear cases, which is the difference in the bulk material properties used for simulation and theoretical analysis. The prediction better matches the current simulation when w = 0.1.
4.7. Statistical analysis In this section, Weibull distributions [27] were employed to describe the results of the uniaxial compression simulations. Using Weibull distribution, the cumulative probability for a sample with volume V to yield at a compressive stress, 𝜎, can be given as [19,28]: [ ( )𝑚 ] 𝑉 𝜎 𝑃𝑅 = 1 − exp − (11) 𝑉 0 𝜎0 where V0 is the reference volume of a sample and m is the Weibull modulus. The reference stress, 𝜎 0 , is the value of stress giving a cumulative probability of 1 − e − 1 = 0.63 for a sample with volume V0 . Because the samples used in the simulations are cubic, Eq. (1) can be rewritten as: [ ( ) ( ) ] 3 𝑚 𝐿 𝜎 𝑃𝑅 = 1 − exp − (12) 𝐿0 𝜎0 where L and L0 are the length and reference length of a sample. 634
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Fig. 19. Von Mises stress (in Pa) contours in samples with three different thicknesses and in different directions:(a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
The cumulative probability of yield for a given applied stress corresponding to the rank i can be estimated as: 𝑖 𝑃𝑅𝑖 = (13) 𝑁 +1 where N is the total number of samples simulated. Due to the heterogeneity of foam material, the value of reference length L0 cannot be too small. Therefore, the value of L0 was assumed to be 16 mm, and the properties of samples with this side length are considered as bulk properties. The least squares method was used to fit the values of 𝜎 0 and m. Fig. 24 shows the values obtained by fitting 𝜎 0 and m from the simulation results of different size samples. It can be seen that both the values of 𝜎 0 and m increase with the increasing value of L/d, and the values are relatively stable (discrepancies are less than 10%) when L/d ≥ 6.
and experimental results show that stiffness and strength of aluminum foams vary with sample size and plateau when the sample size is sufficiently large. •
•
5. Conclusion Numerical simulations using 3D Voronoi models have been carried out to investigate the effects of sample size on the mechanical properties of foams subjected to compression, shear and bending, respectively. Uniaxial compressive tests have been conducted on aluminum foam with a relatively density of 20% and various sample sizes, whose results have been employed to validate the numerical models. Both the numerical
•
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For compression cases, Young’s modulus and yield strength of samples increase with side length of samples because the volume fraction of weak cell layers decreases with side length of samples. The normalized Young’s modulus plateaus when L/d ≥ 7, while the normalized yield strength converges when L/d ≥ 6. For shear cases, decreasing sample height, H, leads to an increase in both stiffness and strength of samples because the volume fraction of higher stress zone with strong constrain (strong boundary layers) is larger in smaller samples. Conversely, the thickness of sample, T, has a positive effect on properties of sample due to the decrease of weak cell layers (the volume fraction of strong boundary layers remains constant for thickness changing case). When H/d ≥ 6 and T/d ≥ 6, the difference between the stiffness or strength of samples and bulk properties is less than 5%. For bending cases, the strength of samples increases with the sample size. The four surfaces parallel to X-direction can be considered as weak cell layers and affect the mechanical response of samples. The yield strength of sample can be used to represent the bulk strength when H/d ≥ 6 and T/d ≥ 5 of sample.
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Fig. 20. Effective plastic strain contours in samples with three different thicknesses and in different directions: (a)(b) T/d ≈ 2; (c)(d) T/d ≈ 5; (e)(f) T/d ≈ 8. (d is 2.482 mm for all the cases.) Sections of (a)(c)(e) are Section 1 shown in Fig. 1(c); sections of (b)(d)(f) are Section 2 shown in Fig. 1(c).
Fig. 21. The predictions of normalized Young’s modulus (a), and yield strength (b) of sample vs. side length of samples normalized by mean cell size, L/d.
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Fig. 22. The predictions of normalized shear yield strength of sample vs. (a) sample height normalized by mean cell size, H/d; (b) sample thickness normalized by mean cell size, T/d.
Fig. 23. The predictions of normalized bending yield strength of samples vs. (a) sample height normalized by mean cell size, H/d; (b) sample thickness normalized by mean cell size, T/d.
Furthermore, the effects of thicknesses of weak cell layers and strong boundary layers have been discussed. Acknowledgements This work is sponsored by the National Natural Science Foundation of China (Grant Nos. 11572214, 11772215), the “1331 Project” fund, Key Innovation Teams of Shanxi Province, the Top Young Academic Leaders of Shanxi, Opening foundation for State Key Laboratory for Strength and Vibration of Mechanical Structures, China Scholarship Council, and Postgraduate Innovation Project of Shanxi Province (No. 2017BY040). Their financial supports are gratefully acknowledged. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge: Cambridge University Press; 1997. doi:10.1017/CBO9781139878326. [2] Miller RE. A continuum plasticity model for the constitutive and indentation behaviour of foamed metals. Int J Mech Sci 2000;42:729–54. doi:10.1016/S0020-7403(99)00021-1. [3] Deshpande VS, Fleck NA. Isotropic constitutive models for metallic.pdf. J Mech Phys Solids 2000;48:1253–83. doi:10.1016/S0022-5096(99)00082-4. [4] Jing L, Su X, Chen De, Yang F, Zhao L. Experimental and numerical study of sandwich beams with layered-gradient foam cores under low-velocity impact. Thin-Walled Struct 2019;135:227–44. doi:10.1016/j.tws.2018.11.011. [5] Nguyen VD, Noels L. Computational homogenization of cellular materials. Int J Solids Struct 2014;51:2183–203. doi:10.1016/j.ijsolstr.2014.02.029. [6] Ramamurty U, Paul A. Variability in mechanical properties of a metal foam. Acta Mater 2004;52:869–76. doi:10.1016/j.actamat.2003.10.021.
Fig. 24. Values of 𝜎 0 and m of simulation results under compression vs. side length of the samples normalized by mean cell size, L/d.
•
The relationship between properties of samples and the thickness of weak cell layers and strong boundary layers has been developed. 637
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