- Email: [email protected]
Insight into cell size effects on quasi-static and dynamic compressive properties of 3D foams
Materials Science & Engineering A 636 (2015) 60–69
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Insight into cell size effects on quasi-static and dynamic compressive properties of 3D foams L. Li, P. Xue n, Y. Chen, H.S.U. Butt School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 10 January 2015 Received in revised form 13 March 2015 Accepted 14 March 2015 Available online 31 March 2015
The aim of this paper is to investigate the cell size effects on the quasi-static and dynamic compressive properties of foams. Firstly, 3D Voronoi model is constructed to represent real foams, and then verified by the experiments conducted in our laboratory. By comparing the load vs. displacement curves, it is shown that the established model works well in studying the compressive properties of 3D foams. Furthermore, a new statistical method is proposed to establish the relationship between the microstructure parameters and the macro-properties of foams. This method can be used to clearly explain the size effect on the quasi-static and dynamic compressive properties of foams. Finally, the static and dynamic responses of the foams under compression are numerically implemented, and the size effects are investigated by considering the models with fixed density but different cell sizes. The results show that with increasing the cell size, Young's modulus of foams increases, whilst the plateau stress decreases under quasi-static compression. Under impact loading, the dynamic enhanced stress is insensitive to the cell sizes when the base material is rate independent, but is sensitive when the base material is rate dependent. The phenomena are also clearly explained by the proposed statistical method. & 2015 Elsevier B.V. All rights reserved.
Keywords: Foam Cell size effects Statistical method 3D Voronoi model
1. Introduction Foams have been used in various engineering fields, such as lightweight structures and energy-absorbing devices. Their behaviors, especially dynamic behavior, play important roles in these applications, and attract more and more attention in these years [1,2]. Generally, mechanical properties of foams can be predicted by the homogenization method [3,4], showing they are only determined by relative density of foams. The cell geometries (i.e., cell size, cell morphology, and so on) do not appear in the established equations through applying scaling laws. This method is based on the classical continuum theory, which works under an assumption that the microstructural length, or cell size, is infinitely small compared to the size of macrostructure. If not satisfying this assumption, the homogenization method should not be used. Then, a more generalized Cosserat homogenization method using a representative volume element (RVE) was proposed [5,6]. However, it should be noticed that the performance of the foam is directly related to the size of the RVE. When the RVE
n Correspondence to: School of Aeronautics, Northwestern Polytechnical University, 127 Youyixilu, 118 mailbox, Xi'an 710072, PR China. Tel.: þ 86 29 88493386. E-mail address: [email protected] (P. Xue).
http://dx.doi.org/10.1016/j.msea.2015.03.052 0921-5093/& 2015 Elsevier B.V. All rights reserved.
size and the cell size were of the same order, size effect appeared. In addition, for some engineering applications, the cell size of the foams used is between 1 and 10 mm, it is not uncommon to have components with dimensions of only a few cell sizes; thus, cell size effects have to be taken into account [7]. There are some papers in the literatures discussing cell size effects of foams in these years. Nieh et al. [8] studied the compressive properties of open-cell aluminum foams with different density and morphology, and found that it is cell morphology that affects the strength of foams rather than cell size. Chen [9] studied the effect of cellular microstructure on the mechanical properties of open-cell aluminum foams and found that cell size has a negligible effect on the compressive properties (elastic modulus and bending strength). However, Xia et al. [10] investigated the effects of cell size on quasi-static compressive properties of Mg alloy foams and concluded that foams with smaller cell size show higher strength. From the results mentioned above, it is seen that the conclusions on the cell size effects are conflicting or confusing. The confusion may related to several factors including misunderstanding the underlying mechanisms responsible for cell size effects. Therefore, some further studies have been conducted. Onck and his co-workers [11–13] established a 2D Voronoi model, and explained that the cell size effect comes from the weak
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
boundary layers located at the stress-free edges, in which the cells are more compliant or carrying less load. Andrews et al. [14,15] conducted their experimental investigation on metal foams and found that the structural defects, such as partially coupled cells, missing cells, and collapsed cells, lead to the cell size effects. However, the findings from Onck's group mentioned above came from a 2D Voronoi model and may not be enough to explain the cell size effects. On the other hand, experimental study is hard to strictly distinguish the contribution from individual microstructural defect due to their sample difference in microstructure. Furthermore, most of the findings are based on the quasi-static studies; there are limited reports about the cell size effects on dynamic properties of foams. Foams usually exhibit a dynamic stress enhancement under impact condition. Qiu [16] studied three kinds of lattices under dynamic loading and concludes that configuration has little effect on dynamic stress enhancement. Whether the cell size has effect on dynamic properties of foams is seldom studied. In this paper, the cell size effect on quasi-static and dynamic compressive properties of foams are investigated. Firstly, 3D Voronoi models are constructed to represent real foams, and then the Voronoi model is verified by experiments. Then, the static and dynamic responses of the foams under compression are numerically implemented based on the Voronoi models, and the cell size effect is analyzed. Afterwards, a new method is proposed to establish the relationship between the statistically microstructure parameters and the macro-properties of the foam. Finally, the method is applied to explain the cell size effect on the static and dynamic properties of the foams.
61
Table 1 Material parameters of the base material. Density (kg/m3)
Young's modulus (GPa)
Tangent modulus (GPa)
Yield stress (MPa)
Poisson's ratio
2700
69
0.47
76
0.35
thickness of cell walls, as follows: Pn A Ut ρn =ρs ¼ i ¼ 1 i V
ð2Þ
in which ρn is the density of foam, ρs is the density of base material. Ai and V are the area and volume of the cell wall, respectively. n is the number of cell walls and t is the cell wall thickness. The foam model is set between two rigid plates, and the top plate moving toward the supporting plate at a velocity, as shown in Fig. 1a. Both static and dynamic friction coefficient is set to 0.1, which has been proved to cause little influence on the simulation results [18]. The compression process is implemented using commercialized software LS-DYNA. Fig. 1b gives the nominal stress–strain curve of the voronoi model with density of 300 kg/m3 under quasi-static loading, where the nominal stress is calculated from the contact force divided by the contact area, while the nominal strain is calculated by dividing the moving displacement by the original height of the Voronoi model. 2.3. Validation of FE models
2. Numerical modeling 2.1. 3D Voronoi technique 3D Voronoi tessellations can well represent foams as its technique is akin to the foaming process [17]. It has been proven that the topological structure of real foams shows coincidence with Voronoi model. In the modeling process, a 3D Voronoi model can be constructed by randomly placing N nucleation points in a 3D block, named block A, with the distance between any two nuclei kept lager than a minimum allowable distance, r. For each nucleus, there is a region around it where any location is closer to this nucleus than any other nucleus. The block then can be divided into N cells, and the boundaries of all cells constitute the so called δ-Voronoi diagram (simply called Voronoi diagram in this paper) with a degree of irregularity of δ ¼ r=d, where d is defined as the distance between the two nuclei in a regular tetrakaidecahedral foam model with N cells in the volume V. d is given by pffiffiffi 13 6 V pffiffiffi ð1Þ d¼ 2 2N
2.2. Numerical modeling A 3D Voronoi model with uniform cell wall thickness is constructed by Voronoi tessellations to represent aluminum alloy closed-cell foam. Due to random allocation of foam cells, three models are constructed for a given cell size, and the simulation results obtained in the paper are mean values from the three models. The three numerical models are discretized by shell elements with same mesh size. The bilinear strain-hardening relationship is used to represent the true stress–strain relationship of the base material. The material properties are listed in Table 1. The different relative densities are obtained by changing the
In order to validate the FE models, quasi-static tests were conducted using a universal test machine at a loading speed of 5 mm/min, as shown in Fig. 2b. ALPORAS aluminum foam with density of 300 kg/m3 and base material of aluminum alloy (Al þ1.5%Caþ1.5%Ti) is shown in Fig. 2a. The nominal stress–strain curve was obtained from the displacement and the force data. The results are compared with the simulation results, as shown in Fig. 3. Good agreement indicates the reliability and accuracy of the developed FE models.
3. Key parameters describing crushable properties of foam 3.1. Densification strain and plateau stress The plateau stress and densification strain are two key parameters to characterize the energy absorption of cellular materials. The existing empirical approximate methods in determining the two parameters are given in [19]. The densification strain (εd ) is determined either by truncation of the plateau region and a rapid rise in stress with further strain or determined by [20]
εd ¼ 1 1:4
ρn ρs
ð3Þ
However, they lack consistency in extracting the densification strain from a nominal stress–strain curve. The method given below can avoid such a problem. In order to precisely determine the parameters, the energybased method [21] was proposed to calculate the densification strain. Energy absorption efficiency E is defined as the energy absorbed up to a given nominal strain εa , normalized by the corresponding stress value σ ðεa Þ: R εa σ ðεÞdε ð4Þ Eðεa Þ ¼ εcr σ ðεa Þ
62
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
Nominal stress (MPa)
10
8
6
4
2
0 0.0
0.2
0.4
0.6
0.8
Nominal strain Fig. 1. (a) 3D Voronoi foam model and (b) stress vs. strain curve of the Voronoi foam under quasi-static compression.
Fig. 2. (a) ALPORAS aluminum foam sample and (b) experimental setup.
10
0.0
0.2
0.4
0.6
0.8
10
6
4
4
2
2
Energy absorption effeciency
6
70 60
0.3
50 40
0.2 30 20 0.1 10
0
0 0.0
0.2
0.4
0.6
0.8
Nominal strain Fig. 3. Comparison of stress–strain response through experiment (—) and numerical simulations (-.-.-).
Nominal stress (MPa)
Nominal stress (MPa)
Quasi-static compression 0.4
8
8
densification strain 0.0
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Nominal strain Fig. 4. Stress vs. strain curve and the energy absorption efficiency vs. strain curve.
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
The plateau stress under quasi-static compression is then determined by R εd σ ðεÞdε σ pl ¼ εcr ð6Þ εd εcr 3.2. Locking strain Under high speed impact, localized deformation can be clearly observed on metal foams. It is interesting to notice that the velocity of the material point on the foam suffers rapid and finite change as stress wave propagation, and the deformation in the foam forms a narrow band, called a ‘shock front’, as shown in Fig. 5, where Vi is the prescribed impact velocity. Ahead of the shock front, the material is assumed to hold still. And behind the shock front, the material is fully compacted. When the shock wave reaches to the stationary side, the shock wave will be reflected and the material is compressed again. In order to describe the behavior of localized deformation caused by inertial effect, Ried and Peng [22] proposed a shock wave theoretical model. Tan and his coworkers [23] introduced the model to explain the dynamic response of metal foams. The model has been further developed, such as RLHPL, EPPL and so on [24,25]. In applying these models, the predicted stresses are all sensitive to the locking strain. Besides, their densification strains are obtained from the quasi-static stress vs. strain curve, leading to an over estimation of the dynamic stress. Actually, the deformation of the foam under impact is totally different from that under quasi-static compression, resulting in the locking strain not same for the two cases. In this section we will propose a new method to determine the locking strain for dynamic loading condition. During compression of the foam, the forces at the impact side and stationary side vary with the displacement, and the two curves are shown in Fig. 6. When the foam is fully compressed, the intersecting point of the stress of the impact side and the stationary side can be defined as the locking strain εnlocking , as shown in Fig. 6.
100
stress at impact side stress at stationary side
80
Nominal stress (MPa)
where εcr is the strain at the initial yield point corresponding to the first peak of the crushing stress–strain curve. Densification strain is the strain when the energy absorption efficiency is the maximum, as shown in Fig. 4: dEðεÞ ¼0 ð5Þ dε ε ¼ εd
63
60
40
20
locking strain 0 0.0
0.2
0.4
0.6
0.8
1.0
Nominal strain Fig. 6. Intersecting point of the stresses at impact side and stationary side.
different cell number, as shown in Fig. 7, are established. The size of model is about the same order of the cell size. The numerical calculations are conducted to study the cell size effects under quasi-static compression and impact condition. 4.1. Quasi-static compression Two boundary conditions are considered. One is free boundary condition, which is defined as that the foam is compressed without transverse constrain, and the other is constrained boundary condition, which refers to the case when the foam is compressed, the transverse deformation on the surface is not allowed. Fig. 8 shows Young's modulus obtained from the load vs. displacement curve for the model with the cell numbers 256, 512 and 1000. Under both boundary conditions, it is found that Young's modulus decreases with the increase of the cell number for a given relative density. Fig. 9 shows the plateau stress increase with the increase of the cell number. From the above results, it can be concluded that Young's modulus and the plateau stress of the foam are cell size dependent. It can be also seen from Fig. 8 that cell size effects are more obvious at a large density. When the density of foam is small, i.e. 135 kg/m3, Young's modulus as well as plateau stresses change little with the cell size changes. Comparing (a) and (b) in Figs. 8 and 9, it can be seen that the constrained boundary strengthens Young's modulus and the plateau stress of the foam.
4. Numerical investigation on cell size effects In order to investigate the size effect of the foam, the 3D Voronoi foam models with dimension 10 10 10 mm3 but
Fig. 5. Schematic drawing showing the shock front.
4.2. Impact It has been reported that the stress enhancement under impact is one of the outstanding features of dynamic properties of foams. The stress enhancement generally comes from four aspects, i.e. the strain rate sensitivity of the base material, the inner pressure of the air entrapped in the cell, the micro-inertia effect and the shock wave effect (or inertia effect) [26]. In order to identify the cell size effects on dynamic properties of foams, two cases are considered in this section, i.e., base material being strain rate dependent and strain rate independent. In FE simulation, the rate dependent base material is described by the well known Cowper–Symonds model, as given in Eq. (7). 1=p σ dys ¼ σ sys 1 þ ε=c ð7Þ where σ dys denotes the dynamic yield stress; σ sys denotes the quasistatic yield stress and ε is the strain rate. For aluminum alloy, referring to Reference [27], c¼ 6500 and p ¼4.
64
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
0.18
0.18
0.16
0.16
0.14
0.14
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
*
Normalized Young's modulus (E /E S )
Fig. 7. 3D Voronoi models with the same size of 10 10 10 mm3 but different cell numbers: (a) 1000 cells, (b) 512 cells and (c) 256 cells.
0.00
0.00 200
400
600
800
1000
200
Number of cells 135 kg/m 3
270 kg/m 3
400
600
800
1000
Number of cells 405 kg/m 3
540 kg/m 3
675 kg/m 3
Fig. 8. Normalized Young's modulus vs. number of cells for: (a) free boundary condition and (b) constrained boundary condition.
Applying a constant velocity of 100 m/s in the vertical direction on the Voronoi foam models with 512 cells under free boundary condition, Fig. 10 shows the comparison of load (P) vs. displacement curves at the impact side of the three foam cases. The stress at the impact side can be obtained from the load divided by the loading area. It is found that the plateau stress is different for the three cases. This implies that the plateau stress enhances due to the inertia effect and strain rate dependence (micro-inertia effect is not considered). From Fig. 10, it is possible to identify the contribution from the inertia effect and strain rate effect of the base material to the dynamic behavior of the foam. Similarly, the impact processes are simulated on the Voronoi foam models with 256 and 1000 cells. By deducting the quasi-
static load vs. displacement curves from the dynamic curve at the impact side, two groups of results are obtained to show the cell size effect under impact condition, as shown in Fig. 11. The upper three curves are for the foam whose base material is strain rate dependent, while the lower ones are strain rate independent. It is shown that the cell size has little effect on dynamic enhanced load when the base material is strain rate independent. However, when the base material of the foam is rate dependent, foams with larger cell number, or small size, show higher dynamic enhanced load. Considering what affects the stress enhancement, therefore, it can be concluded that the inertia effect is not sensitive to cell size, whilst, rate dependent effect of the base material is sensitive to cell size.
Plateau stress (MPa)
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
65
0 200
400
600
800
1000
200
Number of cells 135 kg/m
3
400
600
800
1000
Number of cells
270 kg/m
3
405 kg/m
3
540 kg/m
3
675 kg/m
3
Fig. 9. Plateau stress of foam models with: (a) free boundary condition and (b) constrained boundary condition, plotted against number of cells.
7500
Quasi-static Rate independent base material Rate dependent base material
6000
0.06
0.05
Percentage (%)
P (KN)
4500
3000
1500
0.04
0.03
0.02
0.01
0 0
2
4
6
8
10 0.00
Displacement (mm)
0.5
Fig. 10. Comparison of load vs. displacement curves for Voronoi foams with 512 cells, and the relative density of 0.15.
1.5
2.0
2.5
3.0
0.20
5000
, , ,
4000
256 cells 512 cells 1000cells 0.15
Percentage (%)
Enhanced load (KN)
1.0
Length of cell edges (mm)
3000
rate independent base material 2000
rate dependent base material
0.10
0.05
1000
0.00
0 0
2
4
6
Displacement (mm) Fig. 11. Comparison of enhanced load vs. displacement curves for three cell size foam models under the impact velocity of 100 m/s.
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2
Area of cell walls (mm ) Fig. 12. Statistical histogram of (a) length of cell edges and (b) area of cell walls for Voronoi models with 512 cells.
66
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
Fig. 13. Comparison of undeformed cell and deformed cell.
0.05
5. New explanation on cell size effects through a statistical analysis
LogNormal fit 0.04
In this section, a new method based on a statistical analysis of cell geometrical parameter is proposed. A Voronoi model is established with a given cells, e.g. 512. Assuming le represents the length of cell edge, and Ai the area of a cell wall. Through statistical analysis, the length of cell edge and the area of a cell wall are obtained. The distributions of the two parameters are given in Fig. 12. Then, a displacement is applied on the surface of the foam model. The deformed cells during deformation are shown in Fig. 13. The cell edge bending and cell wall stretching can be identified. By equating the external work done to the elastic internal energy, the normalized Young's modulus can be obtained [20]: En t4 t ¼ α 4 þβ l Es f l
Percentage (%)
5.1. Statistical study on cell geometrical parameter of foams
weak cells
0.03
t/le=xa 0.02
rigid cells 0.01
0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ð8Þ
0.05
where En and Es are Young's modulus of foam material and the base material of foam, respectively; α and β are constants, and t is the thickness of cell walls. lf ispthe ffiffiffi size of cell walls obtained by the square root of cell wall area A. It should be noted that the thickness of edges, t 0 , is larger than that of the cell wall due to the cell walls overlap at the edges, or t 0 ¼ λtwhereλ is large than 1. In Eq. (8), λis included by the parameter α. With increasing the displacement, the cells deform plastically. Plastic hinges and cell walls stretching happen during the plastic deformation. The angle of rotation at the plastic hinges is proportional to displacement. By equating the external work done to the plastic internal energy the normalized plateau stress can be obtained as
0.04
ð9Þ
0.9
1.0
LogNormal fit
e
σ pl t3 0t ¼ α0 3 þ β lf σ ys le
0.8
t/le
Percentage (%)
weak cells 0.03
t/l f=x b
0.02
rigid cells 0.01
0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
t/l f Fig. 14. Statistic histogram of (a) t/le and (b) t/lf of 3DVoronoi foam with 256 cells (each data point is the mean value obtained from the three models).
0
where α0 and β are also constants. From Eqs. (8) and (9), it can be seen that Young's modulus and plateau stresses vary with the ratio of thickness to length. For a given Voronoi model, statistical distribution of the ratio of thickness to length can be obtained, as shown in Fig. 14.
From Fig. 14, it is found that the Lognormal function can well fit the envelope of the distribution, t/lf or t/le, and can be described as
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
0.05
0.05
67
0.05 1000 cells 512 cells
256 cells
0.04
0.04
0.02
0.01
0.04
Percentage (%)
Percentage (%)
Percentage (%)
LogNormal fitting curve
0.03
LogNormal fitting curve
LogNormal fitting curve
0.03
0.02
0.01
0.00 0.2
0.4
0.6
0.8
1.0
0.02
0.01
0.00 0.0
0.03
0.00 0.0
0.2
0.4
t/l e
0.6
0.8
0.0
1.0
0.2
0.4
0.05
0.05
0.6
1000 cells
LogNormal fitting curve
0.04
0.02
Percentage (%)
0.03
0.03
0.02
0.01
0.01
0.00 0.2
0.4
0.6
0.8
1.0
0.03
0.02
0.01
0.00 0.0
LogNormal fitting curve
0.04
LogNormal fitting curve
Percentage (%)
Percentage (%)
0.04
1.0
0.05 512 cells
256 cells
0.8
t/l e
t/le
0.00 0.0
0.2
t/l f
0.4
0.6
0.8
1.0
0.0
0.2
t/l f
0.4
0.6
0.8
1.0
t/l f
Fig. 15. Histograms and fitting curves of: (a) t/le and (b) t/lf for cell numbers 256, 512 and 1000.
5.2. New explanation of cell size effects
Table 2 Parameter xc for different cell size foams. Cell number
256
512
1000
xc (for t/le) xc (for t/lf)
0.1372 0.1496
0.1217 0.1415
0.1052 0.1332
follows: y¼
x xc 2 A pffiffiffiffiffiffiffiffiffie 2ð w Þ w π =2
ð10Þ
wherexc describes the mean value of t/le or t/lf, and w is the standard width of t/l (referring to t/le or t/lf). In elastic deformation, the stiffness of foam relies on the mean value of t/l; thus we can use the parameter xc to evaluate Young's modulus of the foam. During the deformation process, larger cell edges and cell walls will deform ahead of small ones. Or the smaller cells behave relatively rigidly, named ‘rigid cells’ in the paper, while the deformed cells are named ‘week cells’. The cells of a foam model can be divided into ‘rigid cells’ and ‘week cells’, separating by xa and xb as show in Fig. 14. These ‘rigid cells’ contribute little to the plastic deformation; therefore, cells to sustain the load reduce, resulting in a decrease in the plateau stress.
Cell size effects have been reported, but the explanations for the phenomena are mainly from the boundary or defect difference among the foam models with different cell size. In this section, a new explanation about cell size effects will be provided based on the statistical analysis as mentioned in Section 5.1.
5.2.1. Explanation of cell size effects on Young's modulus Fig. 15 shows the distributions of t/le and t/lf for cell numbers 256, 512 and 1000. The envelope of the distribution is described by Eq. (10). By curve fitting, the parameter xc is determined, as given in Table 2. It is found that xc increases as the cell number increases. According to the statistical analysis in Section 5.1, Young's modulus of the foam is proportional to parameter xc. Thus, it is found that the effect of cell size on Young's modulus of foam comes from the different distributions of t/l e and t/lf.
5.2.2. Explanation of cell size effects on plateau stress Fig. 16 gives the comparison of Log-normal fitting curves of t/le and t/lf for the cell numbers 256, 512 and 1000. It is shown that the percentage of ‘rigid cells’ is the lowest for the foam models with the smallest cell size. As mentioned above that these ‘rigid cells’ contribute little to the plastic deformation, therefore, cells to sustain the load reduce, resulting a decrease in the plateau stress.
68
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
5
4
Percentage (%)
where σ 0 is the stress at the stationary side, and ðρ0 v20 =εnlocking Þ is the dynamic stress enhancement or said as inertia effect enhancement. Generally, plateau stress under quasi-static loading (σ pl ) is used to represent σ 0 in Eq. (11) when the base material of foam is rate independent, thus Eq. (11) can be rewritten as
256 ce lls 512 ce lls 1000 cells
3
σ d ¼ σ pl þ
2
0 0.2
0.4
0.6
0.8
1.0
t/le 4.0
σ dpl t3 0t ¼ α0 3 þ β d lf σ ys le
2 56 ce lls 5 12 ce lls 1 00 0 cells
3.5
ð13Þ
Substituting Eq. (7) into Eq. (13), σ dpl can be obtained as ! 1=p 3 0t d s 0t σ pl ¼ σ ys 1 þ ε=c α 3 þβ lf l
3.0
Percentage (%)
ð12Þ
However, when the base material is rate dependent, defining a dynamic plateau stress, σ pl d , to represent the stress at the stationary side, which is not the same as the plateau stress under quasi-static loading (σ pl ). Then, Eq. (9) can be written as
1
0.0
ρ0 v20 εnlocking
2.5
ð14Þ
e
2.0
It can be rewritten as 1=p σ dpl ¼ σ pl 1 þ ε=c
1.5 1.0
ð15Þ
0.5
Thus, when the base material is rate dependent, the dynamic stress at the impact side can be written as
0.0 0.0
0.2
0.4
0.6
0.8
1.0
t/lf
σ d ¼ σ pl þ σ pl ε=c
Fig. 16. Comparison of Lognormal fitting curves: (a) t/le and (b) t/lf, for cell numbers 256, 512 and 1000.
FE results from S.Pattofatto's paper predicted results numerical results based on Voronoi model
Relative stress enhancement
6 5 4 3 2 1 0 40
60
80
100
120
140
160
180
200
220
Impact velocity Fig. 17. Comparison of relative stress enhancement among the predicted results, numerical results and results found in [28].
Therefore, with the cell size decreases the plateau stress increases, which is shown in Fig. 9. 5.2.3. Explanation of cell size effects on dynamic enhanced stress Reid and Peng [22] proposed a shock wave theory to explain the stress enhancement at the impact side. The shock wave theory relied on an idealized rigid-perfectly plastic-locking (RPPL) model, and the base material of foams was assumed as rate independent. The dynamic stress at the impact side can be determined as
σd ¼ σ0 þ
ρ0 v20 εnlocking
ð11Þ
1=p
þ
ρ0 v20 εnlocking
ð16Þ
Eqs. (14) and (16) give the dynamic stress at the impact side when the base material is rate independent and rate dependent, respectively. The dynamic enhanced stress at the impact side can be written as follows: 9 8 ρ0 v20 > > > Rate independent > = < εnlocking ð17Þ σ d σ pl ¼ 2 ρ v 1=p > > > þ εn 0 0 Rate dependent > ; : σ pl ε =c locking In order to validate Eq. (17), the predicted dynamic enhanced stress for foams with rate dependent base material are compared with the following numerical results based on Voronoi models, meanwhile compared with the results obtained in Pattofatto's paper [28]. The density of foam is set to 245 kg/m3, and plateau stress of the foam is 1.7 MPa, in accordance with Pattofatto's paper. Four impact velocities (56, 100, 150 and 200 m/s) are considered. The comparison of relative stress enhancement ððσ d σ pl Þ=σ pl Þ is shown in Fig. 17. It can be found that the predicted results show good agreement with the numerical results and the results found in [28], especially when the impact velocity is high. With the decrease of impact velocity, the numerical results become larger than the other two results, due to micro-inertia effect in Voronoi models. At high impact velocity, foam cells do not have enough time to expand in transversal direction, resulting in the micro-inertia effect becoming week. From Eq. (17), it can be found that the difference of dynamic 1=p enhanced stress lies in the termσ pl ε =c . As aforementioned, the plateau stress under quasi-static loading varies with the cell sizes. Under impact condition, the size effect on the dynamic enhanced stress is enlarged by multiplying the coefficient ε =c 1=p . Meanwhile, it is seen that the reason of the cell size effects on dynamic plateau stress enhancement obtained in Fig. 11 is also the difference in the percentage of the ‘rigid cells’.
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
69
6. Conclusions
Reference
The influence of cell size on Young's module and plateau stress under quasi-static loading and dynamic enhanced stress under impact loading is investigated. By statistically studying the cell geometrical parameter of foams, a new explanation on cell size effect is proposed. The conclusion can be draw as follows:
[1] G.X. Lu, T.X. Yu, Energy Absorption of Structures and Materials, Cambridge University Press, Cambridge, 2003. [2] G. Sun, G. Li, S. Hou, S. Zhou, W. Li, Q. Li, Mater. Sci. Eng. A 527 (2010) 1911–1919. [3] P.G. Kevrekidis, I.G. Kevrekidis, A.R. Bishop, E.S. Titi., Phys. Rev. E 65 (2002) 1–13. [4] V.D. Nguyen, L. Noels., Int. J. Solids Struct. 51 (2014) 2183–2203. [5] V. Aleksey, A. Miroshnichenko, E. Andrey, S.V. Dmitriev, Eur. J. Mech. A 46 (2014) 96–105. [6] Rajesh S. Kumar, David L. McDowell, Int. J. Solids Struct. 41 (2004) 7399–7422pp 41 (2004) 7399–7422. [7] C. Telcoglu, L.J. Gibson, T. Pardoen, P.R. Onck, Prog. Mater. Sci. 56 (2011) 109–138. [8] T.G. Nieh, K. Higashi, J. Wadsworth, Mater. Sci. Eng. A 283 (2000) 105–110pp 283 (2000) 105–110. [9] C. Chen, N.A. Fleck., J. Mech. Phys. Solids 50 (2002) 955–977. [10] X.C. Xia, W.M. Zhao, X.Z. Feng, H. Feng, X. Zhang, Mater. Des. 49 (2013) 19–24. [11] C. Tekoglu, P.R. Onck, J. Mech. Phys. Solids 56 (2008) 3541–3564. [12] P.R. Onck, E.W. Andrews, L.J. Gibson, Int. J. Mech. Sci. 43 (2001) 681–699. [13] P.R. Onck., MRS Bull. 28 (2003) 279–283. [14] C. Tekoglu, P.R. Onck, J. Mater. Sci. 40 (2005) 5911–5917. [15] E.W. Andrews, G. Gioux, L.J. Gibson, Int. J. Mech. Sci. 43 (2001) 701–713. [16] X.M. Qiu, J. Zhang, T.X. Yu., Int. J. Impact Eng. 36 (2009) 1231–1241. [17] K.W. Mahin, k. Hanson, J.W. Morris, Acta Mater. 28 (1980) 443–453. [18] Y.Z. Song, Z.H. Wang, L.M. Zhao, J. Luo, Mater. Des. 31 (2010) 4281–4289pp 31 (2010) 4281–4289. [19] A. Paul, U. Ramamurty, Mater. Sci. Eng. A 281 (2000) 1–7pp 281 (2000) 1–7. [20] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, New York, 1997. [21] P.J. Tan, S.R. Reid, J.J. Harrigan, Z. Zou, S. Li, J. Mech. Phys. Solids 53 (2005) 2174–2205. [22] S.R. Reid, C. Peng, Int. J. Impact Eng. 19 (1997) 531–570. [23] P.J. Tan, S.R. Reid, J.J. Harrigan, Z. Zou, S. Li, J. Mech. Phys. Solids 53 (2005) 2206–2230. [24] Z.J. Zheng, Y.D. Liu, J.L. Yu, S.R. Reid, Int. J. Impact Eng. 42 (2012) 66–79. [25] S.L. Lopatnikov, B.A. Gama, M.J. Haque, C. Krauthauser, J.W. Gillespie Jr., M. Guden, I.W. Hall, Compos. Struct., 61, 200361–71. [26] H. Zhao, I. Elnasri, H.J. Li, Adv. Eng. Mater. 8 (2006) 877–883. [27] X.Y. Su, T.X. Yu, S.R. Reid, Int. J. Impact Eng. 16 (1995) 673–689. [28] S. Pattofatto, I. Elnasri, H. Zhao, H. Tsitsiris, F. Hild, Y. Girard, J. Mech. Phys. Solids 55 (2007) 2672–2686.
Densification strain, plateau stress and locking strain are three
key parameters describing foam crushable properties. The energy-efficiency method is used to determine the densification strain and plateau stress. A shock wave based method is proposed to determine the locking strain. With the cell sizes decrease, Young's modulus of foams at a given density decrease, while the quasi-static plateau stresses increase. The cell size has little effect on dynamic enhanced stress when the base material of the foam is strain rate independent. However, when the base material is strain rate dependent, foams with larger cell number, or small size, show higher dynamic enhanced stress. A new statistical method is proposed to establish the relationship between the microstructure parameters and the macroproperties of foams. And the method is applied to explain the cell size effects. The effect of cell size on Young's modulus of foam is due to the different distributions of the ratio (thickness to length of cell walls); ‘rigid cells’ behaving rigidly during the deformation lead to cell size effects on plateau stress. Similarly, ‘rigid cells’ are also the dominate reason that causes cell size effects on dynamic enhanced stress.
Acknowledgment The authors highly acknowledge the financial support from National Nature Science Foundation of China under Grant nos. 11472226 and 11072202.
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Insight into cell size effects on quasi-static and dynamic compressive properties of 3D foams L. Li, P. Xue n, Y. Chen, H.S.U. Butt School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 10 January 2015 Received in revised form 13 March 2015 Accepted 14 March 2015 Available online 31 March 2015
The aim of this paper is to investigate the cell size effects on the quasi-static and dynamic compressive properties of foams. Firstly, 3D Voronoi model is constructed to represent real foams, and then verified by the experiments conducted in our laboratory. By comparing the load vs. displacement curves, it is shown that the established model works well in studying the compressive properties of 3D foams. Furthermore, a new statistical method is proposed to establish the relationship between the microstructure parameters and the macro-properties of foams. This method can be used to clearly explain the size effect on the quasi-static and dynamic compressive properties of foams. Finally, the static and dynamic responses of the foams under compression are numerically implemented, and the size effects are investigated by considering the models with fixed density but different cell sizes. The results show that with increasing the cell size, Young's modulus of foams increases, whilst the plateau stress decreases under quasi-static compression. Under impact loading, the dynamic enhanced stress is insensitive to the cell sizes when the base material is rate independent, but is sensitive when the base material is rate dependent. The phenomena are also clearly explained by the proposed statistical method. & 2015 Elsevier B.V. All rights reserved.
Keywords: Foam Cell size effects Statistical method 3D Voronoi model
1. Introduction Foams have been used in various engineering fields, such as lightweight structures and energy-absorbing devices. Their behaviors, especially dynamic behavior, play important roles in these applications, and attract more and more attention in these years [1,2]. Generally, mechanical properties of foams can be predicted by the homogenization method [3,4], showing they are only determined by relative density of foams. The cell geometries (i.e., cell size, cell morphology, and so on) do not appear in the established equations through applying scaling laws. This method is based on the classical continuum theory, which works under an assumption that the microstructural length, or cell size, is infinitely small compared to the size of macrostructure. If not satisfying this assumption, the homogenization method should not be used. Then, a more generalized Cosserat homogenization method using a representative volume element (RVE) was proposed [5,6]. However, it should be noticed that the performance of the foam is directly related to the size of the RVE. When the RVE
n Correspondence to: School of Aeronautics, Northwestern Polytechnical University, 127 Youyixilu, 118 mailbox, Xi'an 710072, PR China. Tel.: þ 86 29 88493386. E-mail address: [email protected] (P. Xue).
http://dx.doi.org/10.1016/j.msea.2015.03.052 0921-5093/& 2015 Elsevier B.V. All rights reserved.
size and the cell size were of the same order, size effect appeared. In addition, for some engineering applications, the cell size of the foams used is between 1 and 10 mm, it is not uncommon to have components with dimensions of only a few cell sizes; thus, cell size effects have to be taken into account [7]. There are some papers in the literatures discussing cell size effects of foams in these years. Nieh et al. [8] studied the compressive properties of open-cell aluminum foams with different density and morphology, and found that it is cell morphology that affects the strength of foams rather than cell size. Chen [9] studied the effect of cellular microstructure on the mechanical properties of open-cell aluminum foams and found that cell size has a negligible effect on the compressive properties (elastic modulus and bending strength). However, Xia et al. [10] investigated the effects of cell size on quasi-static compressive properties of Mg alloy foams and concluded that foams with smaller cell size show higher strength. From the results mentioned above, it is seen that the conclusions on the cell size effects are conflicting or confusing. The confusion may related to several factors including misunderstanding the underlying mechanisms responsible for cell size effects. Therefore, some further studies have been conducted. Onck and his co-workers [11–13] established a 2D Voronoi model, and explained that the cell size effect comes from the weak
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
boundary layers located at the stress-free edges, in which the cells are more compliant or carrying less load. Andrews et al. [14,15] conducted their experimental investigation on metal foams and found that the structural defects, such as partially coupled cells, missing cells, and collapsed cells, lead to the cell size effects. However, the findings from Onck's group mentioned above came from a 2D Voronoi model and may not be enough to explain the cell size effects. On the other hand, experimental study is hard to strictly distinguish the contribution from individual microstructural defect due to their sample difference in microstructure. Furthermore, most of the findings are based on the quasi-static studies; there are limited reports about the cell size effects on dynamic properties of foams. Foams usually exhibit a dynamic stress enhancement under impact condition. Qiu [16] studied three kinds of lattices under dynamic loading and concludes that configuration has little effect on dynamic stress enhancement. Whether the cell size has effect on dynamic properties of foams is seldom studied. In this paper, the cell size effect on quasi-static and dynamic compressive properties of foams are investigated. Firstly, 3D Voronoi models are constructed to represent real foams, and then the Voronoi model is verified by experiments. Then, the static and dynamic responses of the foams under compression are numerically implemented based on the Voronoi models, and the cell size effect is analyzed. Afterwards, a new method is proposed to establish the relationship between the statistically microstructure parameters and the macro-properties of the foam. Finally, the method is applied to explain the cell size effect on the static and dynamic properties of the foams.
61
Table 1 Material parameters of the base material. Density (kg/m3)
Young's modulus (GPa)
Tangent modulus (GPa)
Yield stress (MPa)
Poisson's ratio
2700
69
0.47
76
0.35
thickness of cell walls, as follows: Pn A Ut ρn =ρs ¼ i ¼ 1 i V
ð2Þ
in which ρn is the density of foam, ρs is the density of base material. Ai and V are the area and volume of the cell wall, respectively. n is the number of cell walls and t is the cell wall thickness. The foam model is set between two rigid plates, and the top plate moving toward the supporting plate at a velocity, as shown in Fig. 1a. Both static and dynamic friction coefficient is set to 0.1, which has been proved to cause little influence on the simulation results [18]. The compression process is implemented using commercialized software LS-DYNA. Fig. 1b gives the nominal stress–strain curve of the voronoi model with density of 300 kg/m3 under quasi-static loading, where the nominal stress is calculated from the contact force divided by the contact area, while the nominal strain is calculated by dividing the moving displacement by the original height of the Voronoi model. 2.3. Validation of FE models
2. Numerical modeling 2.1. 3D Voronoi technique 3D Voronoi tessellations can well represent foams as its technique is akin to the foaming process [17]. It has been proven that the topological structure of real foams shows coincidence with Voronoi model. In the modeling process, a 3D Voronoi model can be constructed by randomly placing N nucleation points in a 3D block, named block A, with the distance between any two nuclei kept lager than a minimum allowable distance, r. For each nucleus, there is a region around it where any location is closer to this nucleus than any other nucleus. The block then can be divided into N cells, and the boundaries of all cells constitute the so called δ-Voronoi diagram (simply called Voronoi diagram in this paper) with a degree of irregularity of δ ¼ r=d, where d is defined as the distance between the two nuclei in a regular tetrakaidecahedral foam model with N cells in the volume V. d is given by pffiffiffi 13 6 V pffiffiffi ð1Þ d¼ 2 2N
2.2. Numerical modeling A 3D Voronoi model with uniform cell wall thickness is constructed by Voronoi tessellations to represent aluminum alloy closed-cell foam. Due to random allocation of foam cells, three models are constructed for a given cell size, and the simulation results obtained in the paper are mean values from the three models. The three numerical models are discretized by shell elements with same mesh size. The bilinear strain-hardening relationship is used to represent the true stress–strain relationship of the base material. The material properties are listed in Table 1. The different relative densities are obtained by changing the
In order to validate the FE models, quasi-static tests were conducted using a universal test machine at a loading speed of 5 mm/min, as shown in Fig. 2b. ALPORAS aluminum foam with density of 300 kg/m3 and base material of aluminum alloy (Al þ1.5%Caþ1.5%Ti) is shown in Fig. 2a. The nominal stress–strain curve was obtained from the displacement and the force data. The results are compared with the simulation results, as shown in Fig. 3. Good agreement indicates the reliability and accuracy of the developed FE models.
3. Key parameters describing crushable properties of foam 3.1. Densification strain and plateau stress The plateau stress and densification strain are two key parameters to characterize the energy absorption of cellular materials. The existing empirical approximate methods in determining the two parameters are given in [19]. The densification strain (εd ) is determined either by truncation of the plateau region and a rapid rise in stress with further strain or determined by [20]
εd ¼ 1 1:4
ρn ρs
ð3Þ
However, they lack consistency in extracting the densification strain from a nominal stress–strain curve. The method given below can avoid such a problem. In order to precisely determine the parameters, the energybased method [21] was proposed to calculate the densification strain. Energy absorption efficiency E is defined as the energy absorbed up to a given nominal strain εa , normalized by the corresponding stress value σ ðεa Þ: R εa σ ðεÞdε ð4Þ Eðεa Þ ¼ εcr σ ðεa Þ
62
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
Nominal stress (MPa)
10
8
6
4
2
0 0.0
0.2
0.4
0.6
0.8
Nominal strain Fig. 1. (a) 3D Voronoi foam model and (b) stress vs. strain curve of the Voronoi foam under quasi-static compression.
Fig. 2. (a) ALPORAS aluminum foam sample and (b) experimental setup.
10
0.0
0.2
0.4
0.6
0.8
10
6
4
4
2
2
Energy absorption effeciency
6
70 60
0.3
50 40
0.2 30 20 0.1 10
0
0 0.0
0.2
0.4
0.6
0.8
Nominal strain Fig. 3. Comparison of stress–strain response through experiment (—) and numerical simulations (-.-.-).
Nominal stress (MPa)
Nominal stress (MPa)
Quasi-static compression 0.4
8
8
densification strain 0.0
0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Nominal strain Fig. 4. Stress vs. strain curve and the energy absorption efficiency vs. strain curve.
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
The plateau stress under quasi-static compression is then determined by R εd σ ðεÞdε σ pl ¼ εcr ð6Þ εd εcr 3.2. Locking strain Under high speed impact, localized deformation can be clearly observed on metal foams. It is interesting to notice that the velocity of the material point on the foam suffers rapid and finite change as stress wave propagation, and the deformation in the foam forms a narrow band, called a ‘shock front’, as shown in Fig. 5, where Vi is the prescribed impact velocity. Ahead of the shock front, the material is assumed to hold still. And behind the shock front, the material is fully compacted. When the shock wave reaches to the stationary side, the shock wave will be reflected and the material is compressed again. In order to describe the behavior of localized deformation caused by inertial effect, Ried and Peng [22] proposed a shock wave theoretical model. Tan and his coworkers [23] introduced the model to explain the dynamic response of metal foams. The model has been further developed, such as RLHPL, EPPL and so on [24,25]. In applying these models, the predicted stresses are all sensitive to the locking strain. Besides, their densification strains are obtained from the quasi-static stress vs. strain curve, leading to an over estimation of the dynamic stress. Actually, the deformation of the foam under impact is totally different from that under quasi-static compression, resulting in the locking strain not same for the two cases. In this section we will propose a new method to determine the locking strain for dynamic loading condition. During compression of the foam, the forces at the impact side and stationary side vary with the displacement, and the two curves are shown in Fig. 6. When the foam is fully compressed, the intersecting point of the stress of the impact side and the stationary side can be defined as the locking strain εnlocking , as shown in Fig. 6.
100
stress at impact side stress at stationary side
80
Nominal stress (MPa)
where εcr is the strain at the initial yield point corresponding to the first peak of the crushing stress–strain curve. Densification strain is the strain when the energy absorption efficiency is the maximum, as shown in Fig. 4: dEðεÞ ¼0 ð5Þ dε ε ¼ εd
63
60
40
20
locking strain 0 0.0
0.2
0.4
0.6
0.8
1.0
Nominal strain Fig. 6. Intersecting point of the stresses at impact side and stationary side.
different cell number, as shown in Fig. 7, are established. The size of model is about the same order of the cell size. The numerical calculations are conducted to study the cell size effects under quasi-static compression and impact condition. 4.1. Quasi-static compression Two boundary conditions are considered. One is free boundary condition, which is defined as that the foam is compressed without transverse constrain, and the other is constrained boundary condition, which refers to the case when the foam is compressed, the transverse deformation on the surface is not allowed. Fig. 8 shows Young's modulus obtained from the load vs. displacement curve for the model with the cell numbers 256, 512 and 1000. Under both boundary conditions, it is found that Young's modulus decreases with the increase of the cell number for a given relative density. Fig. 9 shows the plateau stress increase with the increase of the cell number. From the above results, it can be concluded that Young's modulus and the plateau stress of the foam are cell size dependent. It can be also seen from Fig. 8 that cell size effects are more obvious at a large density. When the density of foam is small, i.e. 135 kg/m3, Young's modulus as well as plateau stresses change little with the cell size changes. Comparing (a) and (b) in Figs. 8 and 9, it can be seen that the constrained boundary strengthens Young's modulus and the plateau stress of the foam.
4. Numerical investigation on cell size effects In order to investigate the size effect of the foam, the 3D Voronoi foam models with dimension 10 10 10 mm3 but
Fig. 5. Schematic drawing showing the shock front.
4.2. Impact It has been reported that the stress enhancement under impact is one of the outstanding features of dynamic properties of foams. The stress enhancement generally comes from four aspects, i.e. the strain rate sensitivity of the base material, the inner pressure of the air entrapped in the cell, the micro-inertia effect and the shock wave effect (or inertia effect) [26]. In order to identify the cell size effects on dynamic properties of foams, two cases are considered in this section, i.e., base material being strain rate dependent and strain rate independent. In FE simulation, the rate dependent base material is described by the well known Cowper–Symonds model, as given in Eq. (7). 1=p σ dys ¼ σ sys 1 þ ε=c ð7Þ where σ dys denotes the dynamic yield stress; σ sys denotes the quasistatic yield stress and ε is the strain rate. For aluminum alloy, referring to Reference [27], c¼ 6500 and p ¼4.
64
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
0.18
0.18
0.16
0.16
0.14
0.14
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
*
Normalized Young's modulus (E /E S )
Fig. 7. 3D Voronoi models with the same size of 10 10 10 mm3 but different cell numbers: (a) 1000 cells, (b) 512 cells and (c) 256 cells.
0.00
0.00 200
400
600
800
1000
200
Number of cells 135 kg/m 3
270 kg/m 3
400
600
800
1000
Number of cells 405 kg/m 3
540 kg/m 3
675 kg/m 3
Fig. 8. Normalized Young's modulus vs. number of cells for: (a) free boundary condition and (b) constrained boundary condition.
Applying a constant velocity of 100 m/s in the vertical direction on the Voronoi foam models with 512 cells under free boundary condition, Fig. 10 shows the comparison of load (P) vs. displacement curves at the impact side of the three foam cases. The stress at the impact side can be obtained from the load divided by the loading area. It is found that the plateau stress is different for the three cases. This implies that the plateau stress enhances due to the inertia effect and strain rate dependence (micro-inertia effect is not considered). From Fig. 10, it is possible to identify the contribution from the inertia effect and strain rate effect of the base material to the dynamic behavior of the foam. Similarly, the impact processes are simulated on the Voronoi foam models with 256 and 1000 cells. By deducting the quasi-
static load vs. displacement curves from the dynamic curve at the impact side, two groups of results are obtained to show the cell size effect under impact condition, as shown in Fig. 11. The upper three curves are for the foam whose base material is strain rate dependent, while the lower ones are strain rate independent. It is shown that the cell size has little effect on dynamic enhanced load when the base material is strain rate independent. However, when the base material of the foam is rate dependent, foams with larger cell number, or small size, show higher dynamic enhanced load. Considering what affects the stress enhancement, therefore, it can be concluded that the inertia effect is not sensitive to cell size, whilst, rate dependent effect of the base material is sensitive to cell size.
Plateau stress (MPa)
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
65
0 200
400
600
800
1000
200
Number of cells 135 kg/m
3
400
600
800
1000
Number of cells
270 kg/m
3
405 kg/m
3
540 kg/m
3
675 kg/m
3
Fig. 9. Plateau stress of foam models with: (a) free boundary condition and (b) constrained boundary condition, plotted against number of cells.
7500
Quasi-static Rate independent base material Rate dependent base material
6000
0.06
0.05
Percentage (%)
P (KN)
4500
3000
1500
0.04
0.03
0.02
0.01
0 0
2
4
6
8
10 0.00
Displacement (mm)
0.5
Fig. 10. Comparison of load vs. displacement curves for Voronoi foams with 512 cells, and the relative density of 0.15.
1.5
2.0
2.5
3.0
0.20
5000
, , ,
4000
256 cells 512 cells 1000cells 0.15
Percentage (%)
Enhanced load (KN)
1.0
Length of cell edges (mm)
3000
rate independent base material 2000
rate dependent base material
0.10
0.05
1000
0.00
0 0
2
4
6
Displacement (mm) Fig. 11. Comparison of enhanced load vs. displacement curves for three cell size foam models under the impact velocity of 100 m/s.
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2
Area of cell walls (mm ) Fig. 12. Statistical histogram of (a) length of cell edges and (b) area of cell walls for Voronoi models with 512 cells.
66
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
Fig. 13. Comparison of undeformed cell and deformed cell.
0.05
5. New explanation on cell size effects through a statistical analysis
LogNormal fit 0.04
In this section, a new method based on a statistical analysis of cell geometrical parameter is proposed. A Voronoi model is established with a given cells, e.g. 512. Assuming le represents the length of cell edge, and Ai the area of a cell wall. Through statistical analysis, the length of cell edge and the area of a cell wall are obtained. The distributions of the two parameters are given in Fig. 12. Then, a displacement is applied on the surface of the foam model. The deformed cells during deformation are shown in Fig. 13. The cell edge bending and cell wall stretching can be identified. By equating the external work done to the elastic internal energy, the normalized Young's modulus can be obtained [20]: En t4 t ¼ α 4 þβ l Es f l
Percentage (%)
5.1. Statistical study on cell geometrical parameter of foams
weak cells
0.03
t/le=xa 0.02
rigid cells 0.01
0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ð8Þ
0.05
where En and Es are Young's modulus of foam material and the base material of foam, respectively; α and β are constants, and t is the thickness of cell walls. lf ispthe ffiffiffi size of cell walls obtained by the square root of cell wall area A. It should be noted that the thickness of edges, t 0 , is larger than that of the cell wall due to the cell walls overlap at the edges, or t 0 ¼ λtwhereλ is large than 1. In Eq. (8), λis included by the parameter α. With increasing the displacement, the cells deform plastically. Plastic hinges and cell walls stretching happen during the plastic deformation. The angle of rotation at the plastic hinges is proportional to displacement. By equating the external work done to the plastic internal energy the normalized plateau stress can be obtained as
0.04
ð9Þ
0.9
1.0
LogNormal fit
e
σ pl t3 0t ¼ α0 3 þ β lf σ ys le
0.8
t/le
Percentage (%)
weak cells 0.03
t/l f=x b
0.02
rigid cells 0.01
0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
t/l f Fig. 14. Statistic histogram of (a) t/le and (b) t/lf of 3DVoronoi foam with 256 cells (each data point is the mean value obtained from the three models).
0
where α0 and β are also constants. From Eqs. (8) and (9), it can be seen that Young's modulus and plateau stresses vary with the ratio of thickness to length. For a given Voronoi model, statistical distribution of the ratio of thickness to length can be obtained, as shown in Fig. 14.
From Fig. 14, it is found that the Lognormal function can well fit the envelope of the distribution, t/lf or t/le, and can be described as
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
0.05
0.05
67
0.05 1000 cells 512 cells
256 cells
0.04
0.04
0.02
0.01
0.04
Percentage (%)
Percentage (%)
Percentage (%)
LogNormal fitting curve
0.03
LogNormal fitting curve
LogNormal fitting curve
0.03
0.02
0.01
0.00 0.2
0.4
0.6
0.8
1.0
0.02
0.01
0.00 0.0
0.03
0.00 0.0
0.2
0.4
t/l e
0.6
0.8
0.0
1.0
0.2
0.4
0.05
0.05
0.6
1000 cells
LogNormal fitting curve
0.04
0.02
Percentage (%)
0.03
0.03
0.02
0.01
0.01
0.00 0.2
0.4
0.6
0.8
1.0
0.03
0.02
0.01
0.00 0.0
LogNormal fitting curve
0.04
LogNormal fitting curve
Percentage (%)
Percentage (%)
0.04
1.0
0.05 512 cells
256 cells
0.8
t/l e
t/le
0.00 0.0
0.2
t/l f
0.4
0.6
0.8
1.0
0.0
0.2
t/l f
0.4
0.6
0.8
1.0
t/l f
Fig. 15. Histograms and fitting curves of: (a) t/le and (b) t/lf for cell numbers 256, 512 and 1000.
5.2. New explanation of cell size effects
Table 2 Parameter xc for different cell size foams. Cell number
256
512
1000
xc (for t/le) xc (for t/lf)
0.1372 0.1496
0.1217 0.1415
0.1052 0.1332
follows: y¼
x xc 2 A pffiffiffiffiffiffiffiffiffie 2ð w Þ w π =2
ð10Þ
wherexc describes the mean value of t/le or t/lf, and w is the standard width of t/l (referring to t/le or t/lf). In elastic deformation, the stiffness of foam relies on the mean value of t/l; thus we can use the parameter xc to evaluate Young's modulus of the foam. During the deformation process, larger cell edges and cell walls will deform ahead of small ones. Or the smaller cells behave relatively rigidly, named ‘rigid cells’ in the paper, while the deformed cells are named ‘week cells’. The cells of a foam model can be divided into ‘rigid cells’ and ‘week cells’, separating by xa and xb as show in Fig. 14. These ‘rigid cells’ contribute little to the plastic deformation; therefore, cells to sustain the load reduce, resulting in a decrease in the plateau stress.
Cell size effects have been reported, but the explanations for the phenomena are mainly from the boundary or defect difference among the foam models with different cell size. In this section, a new explanation about cell size effects will be provided based on the statistical analysis as mentioned in Section 5.1.
5.2.1. Explanation of cell size effects on Young's modulus Fig. 15 shows the distributions of t/le and t/lf for cell numbers 256, 512 and 1000. The envelope of the distribution is described by Eq. (10). By curve fitting, the parameter xc is determined, as given in Table 2. It is found that xc increases as the cell number increases. According to the statistical analysis in Section 5.1, Young's modulus of the foam is proportional to parameter xc. Thus, it is found that the effect of cell size on Young's modulus of foam comes from the different distributions of t/l e and t/lf.
5.2.2. Explanation of cell size effects on plateau stress Fig. 16 gives the comparison of Log-normal fitting curves of t/le and t/lf for the cell numbers 256, 512 and 1000. It is shown that the percentage of ‘rigid cells’ is the lowest for the foam models with the smallest cell size. As mentioned above that these ‘rigid cells’ contribute little to the plastic deformation, therefore, cells to sustain the load reduce, resulting a decrease in the plateau stress.
68
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
5
4
Percentage (%)
where σ 0 is the stress at the stationary side, and ðρ0 v20 =εnlocking Þ is the dynamic stress enhancement or said as inertia effect enhancement. Generally, plateau stress under quasi-static loading (σ pl ) is used to represent σ 0 in Eq. (11) when the base material of foam is rate independent, thus Eq. (11) can be rewritten as
256 ce lls 512 ce lls 1000 cells
3
σ d ¼ σ pl þ
2
0 0.2
0.4
0.6
0.8
1.0
t/le 4.0
σ dpl t3 0t ¼ α0 3 þ β d lf σ ys le
2 56 ce lls 5 12 ce lls 1 00 0 cells
3.5
ð13Þ
Substituting Eq. (7) into Eq. (13), σ dpl can be obtained as ! 1=p 3 0t d s 0t σ pl ¼ σ ys 1 þ ε=c α 3 þβ lf l
3.0
Percentage (%)
ð12Þ
However, when the base material is rate dependent, defining a dynamic plateau stress, σ pl d , to represent the stress at the stationary side, which is not the same as the plateau stress under quasi-static loading (σ pl ). Then, Eq. (9) can be written as
1
0.0
ρ0 v20 εnlocking
2.5
ð14Þ
e
2.0
It can be rewritten as 1=p σ dpl ¼ σ pl 1 þ ε=c
1.5 1.0
ð15Þ
0.5
Thus, when the base material is rate dependent, the dynamic stress at the impact side can be written as
0.0 0.0
0.2
0.4
0.6
0.8
1.0
t/lf
σ d ¼ σ pl þ σ pl ε=c
Fig. 16. Comparison of Lognormal fitting curves: (a) t/le and (b) t/lf, for cell numbers 256, 512 and 1000.
FE results from S.Pattofatto's paper predicted results numerical results based on Voronoi model
Relative stress enhancement
6 5 4 3 2 1 0 40
60
80
100
120
140
160
180
200
220
Impact velocity Fig. 17. Comparison of relative stress enhancement among the predicted results, numerical results and results found in [28].
Therefore, with the cell size decreases the plateau stress increases, which is shown in Fig. 9. 5.2.3. Explanation of cell size effects on dynamic enhanced stress Reid and Peng [22] proposed a shock wave theory to explain the stress enhancement at the impact side. The shock wave theory relied on an idealized rigid-perfectly plastic-locking (RPPL) model, and the base material of foams was assumed as rate independent. The dynamic stress at the impact side can be determined as
σd ¼ σ0 þ
ρ0 v20 εnlocking
ð11Þ
1=p
þ
ρ0 v20 εnlocking
ð16Þ
Eqs. (14) and (16) give the dynamic stress at the impact side when the base material is rate independent and rate dependent, respectively. The dynamic enhanced stress at the impact side can be written as follows: 9 8 ρ0 v20 > > > Rate independent > = < εnlocking ð17Þ σ d σ pl ¼ 2 ρ v 1=p > > > þ εn 0 0 Rate dependent > ; : σ pl ε =c locking In order to validate Eq. (17), the predicted dynamic enhanced stress for foams with rate dependent base material are compared with the following numerical results based on Voronoi models, meanwhile compared with the results obtained in Pattofatto's paper [28]. The density of foam is set to 245 kg/m3, and plateau stress of the foam is 1.7 MPa, in accordance with Pattofatto's paper. Four impact velocities (56, 100, 150 and 200 m/s) are considered. The comparison of relative stress enhancement ððσ d σ pl Þ=σ pl Þ is shown in Fig. 17. It can be found that the predicted results show good agreement with the numerical results and the results found in [28], especially when the impact velocity is high. With the decrease of impact velocity, the numerical results become larger than the other two results, due to micro-inertia effect in Voronoi models. At high impact velocity, foam cells do not have enough time to expand in transversal direction, resulting in the micro-inertia effect becoming week. From Eq. (17), it can be found that the difference of dynamic 1=p enhanced stress lies in the termσ pl ε =c . As aforementioned, the plateau stress under quasi-static loading varies with the cell sizes. Under impact condition, the size effect on the dynamic enhanced stress is enlarged by multiplying the coefficient ε =c 1=p . Meanwhile, it is seen that the reason of the cell size effects on dynamic plateau stress enhancement obtained in Fig. 11 is also the difference in the percentage of the ‘rigid cells’.
L. Li et al. / Materials Science & Engineering A 636 (2015) 60–69
69
6. Conclusions
Reference
The influence of cell size on Young's module and plateau stress under quasi-static loading and dynamic enhanced stress under impact loading is investigated. By statistically studying the cell geometrical parameter of foams, a new explanation on cell size effect is proposed. The conclusion can be draw as follows:
[1] G.X. Lu, T.X. Yu, Energy Absorption of Structures and Materials, Cambridge University Press, Cambridge, 2003. [2] G. Sun, G. Li, S. Hou, S. Zhou, W. Li, Q. Li, Mater. Sci. Eng. A 527 (2010) 1911–1919. [3] P.G. Kevrekidis, I.G. Kevrekidis, A.R. Bishop, E.S. Titi., Phys. Rev. E 65 (2002) 1–13. [4] V.D. Nguyen, L. Noels., Int. J. Solids Struct. 51 (2014) 2183–2203. [5] V. Aleksey, A. Miroshnichenko, E. Andrey, S.V. Dmitriev, Eur. J. Mech. A 46 (2014) 96–105. [6] Rajesh S. Kumar, David L. McDowell, Int. J. Solids Struct. 41 (2004) 7399–7422pp 41 (2004) 7399–7422. [7] C. Telcoglu, L.J. Gibson, T. Pardoen, P.R. Onck, Prog. Mater. Sci. 56 (2011) 109–138. [8] T.G. Nieh, K. Higashi, J. Wadsworth, Mater. Sci. Eng. A 283 (2000) 105–110pp 283 (2000) 105–110. [9] C. Chen, N.A. Fleck., J. Mech. Phys. Solids 50 (2002) 955–977. [10] X.C. Xia, W.M. Zhao, X.Z. Feng, H. Feng, X. Zhang, Mater. Des. 49 (2013) 19–24. [11] C. Tekoglu, P.R. Onck, J. Mech. Phys. Solids 56 (2008) 3541–3564. [12] P.R. Onck, E.W. Andrews, L.J. Gibson, Int. J. Mech. Sci. 43 (2001) 681–699. [13] P.R. Onck., MRS Bull. 28 (2003) 279–283. [14] C. Tekoglu, P.R. Onck, J. Mater. Sci. 40 (2005) 5911–5917. [15] E.W. Andrews, G. Gioux, L.J. Gibson, Int. J. Mech. Sci. 43 (2001) 701–713. [16] X.M. Qiu, J. Zhang, T.X. Yu., Int. J. Impact Eng. 36 (2009) 1231–1241. [17] K.W. Mahin, k. Hanson, J.W. Morris, Acta Mater. 28 (1980) 443–453. [18] Y.Z. Song, Z.H. Wang, L.M. Zhao, J. Luo, Mater. Des. 31 (2010) 4281–4289pp 31 (2010) 4281–4289. [19] A. Paul, U. Ramamurty, Mater. Sci. Eng. A 281 (2000) 1–7pp 281 (2000) 1–7. [20] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, New York, 1997. [21] P.J. Tan, S.R. Reid, J.J. Harrigan, Z. Zou, S. Li, J. Mech. Phys. Solids 53 (2005) 2174–2205. [22] S.R. Reid, C. Peng, Int. J. Impact Eng. 19 (1997) 531–570. [23] P.J. Tan, S.R. Reid, J.J. Harrigan, Z. Zou, S. Li, J. Mech. Phys. Solids 53 (2005) 2206–2230. [24] Z.J. Zheng, Y.D. Liu, J.L. Yu, S.R. Reid, Int. J. Impact Eng. 42 (2012) 66–79. [25] S.L. Lopatnikov, B.A. Gama, M.J. Haque, C. Krauthauser, J.W. Gillespie Jr., M. Guden, I.W. Hall, Compos. Struct., 61, 200361–71. [26] H. Zhao, I. Elnasri, H.J. Li, Adv. Eng. Mater. 8 (2006) 877–883. [27] X.Y. Su, T.X. Yu, S.R. Reid, Int. J. Impact Eng. 16 (1995) 673–689. [28] S. Pattofatto, I. Elnasri, H. Zhao, H. Tsitsiris, F. Hild, Y. Girard, J. Mech. Phys. Solids 55 (2007) 2672–2686.
Densification strain, plateau stress and locking strain are three
key parameters describing foam crushable properties. The energy-efficiency method is used to determine the densification strain and plateau stress. A shock wave based method is proposed to determine the locking strain. With the cell sizes decrease, Young's modulus of foams at a given density decrease, while the quasi-static plateau stresses increase. The cell size has little effect on dynamic enhanced stress when the base material of the foam is strain rate independent. However, when the base material is strain rate dependent, foams with larger cell number, or small size, show higher dynamic enhanced stress. A new statistical method is proposed to establish the relationship between the microstructure parameters and the macroproperties of foams. And the method is applied to explain the cell size effects. The effect of cell size on Young's modulus of foam is due to the different distributions of the ratio (thickness to length of cell walls); ‘rigid cells’ behaving rigidly during the deformation lead to cell size effects on plateau stress. Similarly, ‘rigid cells’ are also the dominate reason that causes cell size effects on dynamic enhanced stress.
Acknowledgment The authors highly acknowledge the financial support from National Nature Science Foundation of China under Grant nos. 11472226 and 11072202.