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Effects of cell irregularity on the high strain compression of open-cell foams
Acta Materialia 50 (2002) 1041–1052 www.actamat-journals.com
Effects of cell irregularity on the high strain compression of open-cell foams H.X. Zhu 1, A.H. Windle * Department of Materials Science and Metallurgy, Cambridge University, Pembroke Street, Cambridge CB2 3QZ, UK Received 15 June 2001; received in revised form 30 October 2001; accepted 30 October 2001
Abstract The high strain compression of low-density open-cell polymer foams has been modelled by finite element analysis. We used a Voronoi method to generate periodic structures with different degrees of randomness of the cell size and shape, then to investigate the influence of this randomness on the response of Voronoi open-cell foams to high strain compression. It is found that, although the reduced compressive stress–strain relationship and the Poisson’s ratio vary in different directions for individual samples, the models are, on average, isotropic. A highly irregular foam has a larger tangential modulus at very low strains and a lower effective stress at high compressive strains than a more regular foam. The geometrical properties were investigated and used to predict the compressive stress–strain relationships for random open-cell foams with different degrees of cell regularity. For irregular low density foams, strut bending and twisting (the “springs-in-parallel” model) dominate the mechanical response at low strains and strut buckling (the “springs-in-series” model) becomes the main deformation mechanism at large compressive strains. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Foams; High strain compression; Computer simulation; Microstructure
1. Introduction Low-density open-cell foams are widely used in engineering applications such as lightweight structural sandwich panels or components designed for absorbing impact energy. Research works on the mechanical properties and deformation mechanisms of foams have received wide attention [1]. * Corresponding author. E-mail address: [email protected] (A.H. Windle). 1 Present address: Polymer and Colloids Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK.
Mechanical models are usually based on models of cell structure. Unit cell models have proved to be useful in understanding some of the key aspects of the mechanical behaviour of foams [1–10] and, as long as the mechanical properties of the solid are known, this method can give the full response of the foam or the honeycomb subjected to a stress or a strain. Based on the regular Kelvin cell structure, Zhu et al. [4] derived all the three independent elastic constants as functions of the Young’s modulus of the solid material, the relative density of the foam and the shape of the cell edge cross-section, and found that such structure is nearly isotropic. They [4,5] also found that, if a Kelvin foam is compressed in the
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[111] direction, strut twisting plays almost the same important role as strut bending does in the foam deformation. However, most cellular materials usually have random cells. Thus, despite their proven utility, unit cell models do not accurately represent the microstructure of most real foams. To better represent the microstructure, random Voronoi technique and finite element analysis have been used in modelling the mechanical properties of foams [11–15]. Shulmeister et al. [12] have modelled foam structure by random Voronoi cells using finite element analysis. They started from regular bcc and fcc lattice nuclei distributions, subsequently gave the nuclei positions an increasing random offset and constructed the random structure using the Voronoi procedure. However, all the struts at the boundary are normal to the boundary face giving a much stiffer structure. Their simulated compressive stress–strain relationship [12] for random open-cell foams is about 6 times larger than the theoretical results [5] for a regular Kelvin foam. Also, as their model is not periodic, they could not have been able to apply periodic boundary conditions in the FEA analysis. As is well known, boundary conditions can greatly influence the mechanical response of a structure, and even mixed boundary conditions tend to underestimate the mechanical properties of foams [13–15]. In our previous work [14], we found that the periodic Voronoi foams are averagely isotropic, and the cell regularity has a strong effect on the elastic properties of Voronoi foams. The aim of this work is to determine the influence of disorder in foam cell size and shape on the high strain compression of open-cell foams. We have constructed 3D periodic random structures with different degrees of irregularity, and applied finite element analysis to derive the compressive stress–strain relationships. The geometrical properties of random foams have also been investigated and used to predict the reduced compressive stress–strain relationships for foams with varying degrees of regularities. 2. The definition of the regularity 2.1. The construction of periodic random Voronoi foams Cellular solids are usually formed by nucleation and growth of cells. If all cells nucleate randomly
in space at the same time and grow at the same linear rate, the resulting structure is a Voronoi foam. The central field V0, which is surrounded by 26 equivalent boxes, is taken to be a cube in the present study. An orthogonal coordinate system is chosen, with the origin at one corner of the cube, and nucleation points are created in the cube by generating the x, y and z coordinates independently from the pseudo-random numbers between 0 and 1. After the first point is specified, each subsequent random point is accepted only if it is greater than a minimum allowable distance d from any existing point (i.e., it does not fall within an alreadydeposited ball with diameter d), until n nuclei are seeded in the cube. The periodic Voronoi foam sample used in the FEA model is the “unit” cell of an infinite periodic foam (Fig. 1(a)). A dedicated code has been developed to construct the periodic random Voronoi structures. 2.2. The definition of the regularity The fully ordered limit of the 3D Voronoi tessellation is effectively a cubic lattice of tetrakaidecahedral cells. Such cells have a surface area very close to the minimum for a given cell volume. To construct a regular lattice with n identical tetrakaidecahedral cells in the volume V0, the minimum distance d0 between any two adjacent nuclei is given by d0 ⫽
冑6 2
冉冑 冊 V0
2n
1/3
(1)
To construct a random Voronoi tessellation with n cells in the volume V0, the maximum d (the minimum distance between any two nuclei) should be less than d0; otherwise, it is impossible to obtain n random cells. The regularity of a 3D Voronoi tessellation can be measured by [16]: a⫽
d d0
(2)
For a regular lattice with tetrakaidecahedral cells, d equals d0 and a is 1. For a completely random Voronoi tessellation, d equals 0 and thus a is 0. Due to the fact that it is very difficult to construct n cells in a large sample with a regu-
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tinuously as the vertices are approached. For simplicity of the model, all the struts are assumed to have the same and constant Plateau Border crosssection with area A. Consequently, the foam relative density r is determined by the cross-section area A and specified by
冘 N
A r⫽
li
i⫽1
V0
(3)
where li are the cell strut lengths and N is the total number of cell struts. The ABAQUS standard package was used to perform the FEA analysis. Each strut was modelled, depending on its length, with 1 to 5 Timoshenko beam elements (ABAQUS element type B32). The solid polymer material was treated as linear elastic and the Young’s modulus of the struts, Es, was set to 1.0 and the Poisson’s ratio to 0.5. To derive the uniaxial compressive stress– strain relationship, displacement boundary conditions were imposed in either the x or the y directions. 3.2. Boundary conditions
Fig. 1. (a) An undeformed random Voronoi foam with 27 complete cells. (b) The deformed structure with periodic boundary conditions.
larity parameter a larger than 0.7, we only present simulated results for foams with a regularity a up to 0.7.
3. Computational aspects and results 3.1. General methodology All struts in the foam are represented mechanically by beams that are rigidly connected in vertices. In real foams, the cross-section of the struts is a Plateau Border and the area of the cross-section is variable along the strut length, thickening con-
Before performing any finite element analysis, one should choose the most suitable boundary conditions. The best boundary conditions should lead to the average global behaviour of the 3D foams, and avoid any large localised deformation near the boundaries of the mesh. Silva and Gibson have analysed the uniaxial high strain compression of non-periodic 2D foams [11]. Displacement boundary conditions were imposed in the loading direction in their model. The nodes along the “fixed” boundary and the “displaced” boundary were constrained from translating in the loading direction and from in-plane rotation, but were free to translate in the non-loading direction. In our analysis, each sample is a periodic one, it is a “unit” cell taken from an infinite sample (see Fig. 1(a)). It has been shown that the periodic boundary conditions are the most suitable [13–15] for periodic samples. Thus, only periodic boundary conditions are used in this paper. The periodic boundary conditions assume that the cor-
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responding nodes on the opposite struts of the mesh have the same expansions in the normal directions, the same displacements in the other directions, and the same rotations in all the directions (see Fig. 1(b)). 3.3. Mesh sensitivity In order to determine the size of a “unit” cell that is suitable for compression to a large strain and can also supply an accurate solution, a mesh sensitivity study was performed by changing the total number n of the cells for samples having a regularity parameter a=0.1. It is found that the larger the number n, the greater the possibility of instability during compression. A number of random samples were compressed for each different number of cells, n=27, 64 and 125. Each Voronoi foam was generated using a different list of random numbers and had the same relative density of 0.01. The reduced compressive stress–strain relationships (see Eq. (4)) are shown in Fig. 2(a), (b) and (c). The larger the number of cells n, the smaller the variations in the compressive stress–strain relationships of different samples. However, the mean results of the samples having different number of cells are almost the same, as shown in Fig. 2(d). Considering the compressibility, the stability and the accuracy, we fix the number of cells at n=27 in the following finite element analysis. 3.4. Isotropic properties 20–80 random samples with regularity parameter a=0.7 and relative density r=0.01 were compressed in the x and y directions. Due to the fact that some samples became unstable at small strains, only the results of about 10 samples have been taken into account. Figure 3(a) and (b) show the mean reduced compressive stresses and the standad deviations against the compressive strains in the y and x directions. Similarly, Fig. 4(a) and (b) show the mean Poisson’s ratios n21 and n23 and their standard deviations against the compressive strain in the y direction (the 2 direction) for foams having regularity parameters a=0.3 and a=0.5. Although the reduced compressive stresses and the Poisson’s ratios varied over a considerable range
in different directions for individual samples, their mean results in different directions are almost the same, and Figs 3 and 4 suggest that the models, on average, are isotropic. 3.5. Effects of cell regularity on the high strain compression of open-cell foams 20 or more samples have been simulated for foams with different degrees of regularity. All the samples have a fixed relative density of 0.01. To simplify the results, the effective stresses of the foams were reduced by dividing them by r2, the square of the foam relative density, and by the Young’s modulus Es of the solid from which the foam is produced, and given by s¯ ⫽
s Esr2
(4)
Due to the fact that some samples became unstable at small strains, only the results of about 10–20 samples have been taken into account for each degree of regularity. Figure 5(a) shows the simulated mean reduced compressive stress–strain relationships of foams having different degrees of regularity, the finite element simulated results for the Kelvin foam (a=1.0) in the [111] direction [3] are also presented for comparison. Figure 5(b) displays the predicted stress–strain relationships by Eq. (17) (see the discussion section) for foams having different degrees of regularity, the theoretical results for the Kelvin foam (a=1.0) in the [111] direction [5] are also included for comparison. Figures 5(a) and (b) indicate that a highly irregular foam has a larger tangential modulus at small strains and a lower effective stress at high compressive strains than a more regular foam. Although an irregular foam has a smaller Poisson’s ratio at large compressive strains than a perfect regular foam, Figs 4(a) and (b) suggest that cell irregularity has very little influence on the mean Poisson’s ratios of random foams. 3.6. Effects of foam relative density For foams having a regularity a=0.7 and relative densities r=0.01, r=0.02, r=0.04 and r=0.08, Figs
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Fig. 2. Effects of unit-cell size on the reduced compressive stress–strain relationships of random Voronoi foams having a regularity parameter a=0.1. (a) N=125, (b) N=64, (c) N=27, (d) the mean results of the samples having numbers of cells 125, 64 and 27.
6 and 7 present the reduced stresses and the Poisson’s ratios against the compressive strains. The theoretical results for a perfect regular foam are also included for comparison. For a given compressive strain, the reduced stress of an open-cell foam decreases with increasing relative density. However, although the Poisson’s ratio of an opencell foam decreases with increasing relative density at low compressive strains, it increases with increasing relative density at large strains (see Fig. 7).
4. Discussion For a perfect regular low density open-cell foam (i.e. a Kelvin foam and a=1.0) with a constant Plateau Border strut cross-section, Zhu et al. [5] have derived the reduced compressive stress–strain relationship in the [111] direction:
s0 ⫽ f(e)
(5)
or e ⫽ F(s0)
(6)
Both equations (5) and (6) describe the same relationship, as shown in Fig. 8. However, an irregular foam has a larger reduced Young’s modulus than a perfect regular foam, and the higher the cell irregularity, the larger will be the reduced Young’s modulus E¯ (E¯ ⫽ E / Esr2), as shown in Fig. 9 [14]. Consequently, an irregular foam is stiffer than a more regular foam at low compressive strains (see Fig. 5(a)). As discussed in [14], for low density foams, strut bending and twisting are the main deformation mechanisms at small strains (e tends to 0.0). Although many different geometrical properties could affect the mechanical properties of random foams, the effects of cell irregularity on the elastic properties (Young’s modulus) can be explained by the “springs-in-parallel” model [14].
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Fig. 3. The reduced stresses against the compressive strains of foams having a regularity parameter a=0.7, compared with the predicted results. (a) compressed in the y direction, (b) compressed in the x direction.
Fig. 4. The Poisson’s ratio n21 against the compressive strains, compared with n23 and the theoretical results of a perfect regular foam in the [111] direction [5]. (a) For foams with regularity parameter a=0.3, (b) for foams with regularity parameter a=0.5.
If the “springs-in-parallel” model was the only deformation mechanism in the high strain compression of irregular foams, the reduced stress– strain relationship of an irregular low density open cell foam could be approximately expressed by
is why an irregular foam has a lower reduced stress than a more regular foam at high compressive strains (see Fig. 5(a)). If “strut buckling” was the only deformation mechanism, the cells in the foam could be treated as “springs-in-series” (that means every cell in the sample has the same compressive stress, but a different compressive strain). For a single uniform rod, the buckling force is given by
sp ⫽ j(e) ⫽ E¯ f(e)
(7)
For a foam with regularity parameter a=0.3, the relationship (Eq. (7)) is shown in Fig. 8, where e is the compressive strain of the foam. The above relationship (Eq. (7)) depends on the value of E¯ , i.e. on the cell regularity parameter a. However, at high compressive strains, the “strut buckling” did play a very important role, and that
P⫽
KA2Es l2
(8)
where A is the cross-sectional area of the rod, l is the rod length, Es is the Young’s modulus, and K
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Fig. 6. Effects of relative density on the reduced compressive stress–strain relationships of foams having regularity parameter a=0.7, compared with the theoretical results of a perfect regular foam (a=1.0) in the [111] direction [5] and the predicted results of a random foam with a=0.7.
Fig. 5. Effects of cell regularity on the mean reduced stress– strain relationships, compared with the predicted results (by Eq. (17)) for random Voronoi foams. The finite element simulated result for the Kelvin foam (a=1.0) in the [111] direction [3] and the theoretical results for a perfect regular foam in the [111] direction [5] are also presented for comparison. (a) Simulated mean results; (b) predicted results.
is a constant depending on the shape of the rod cross-section and the restriction conditions at the two ends of the rod. Thus, for a Kelvin cell (or foam) with a relative density r=0.01, the reduced buckling stress can be expressed as s0 ⫽
s0 CA20 ⫽ 2 Esr2 l20V2/3 0 r
(9)
where A0 is the strut cross-sectional area, l0 is the strut length, r is the overall relative density of the foam, V0 is the cell volume and C is a function
Fig. 7. Effects of relative density on the Poisson’s ratio of foams having regularity parameter a=0.7, compared with the theoretical results of a perfect regular foam (a=1.0 and r=0.01) in the [111] direction [5].
depending on the shape of the rod cross-section, the restriction conditions of the two ends of the rod, and the foam compressive strain. Similarly, the reduced buckling stress for an irregular cell may be expressed as si ⫽
CA2i 2 l V2/3 i r 2 i
(10)
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cell which equals the mean volume of all the cells in the random sample; Ai is assumed to be the same for all the cell struts in the random sample, which depends on the regularity parameter a and is different from A0 if a⫽1.0. However, s0 is a function of the compressive strain of the foam [5] (see Eq. (5) and Fig. 8). Substituting equation (5) to (11) gives si ⫽ or Fig. 8. The reduced compressive stress–strain relationships in the [111] direction: f(e) for open-cell Kelvin foam with a constant Plateau Border strut cross-section [5]; j(e) and f(e) for foams with regularity parameter a=0.3.
冉冊
A2i l20 V0 A20l2i Vi
2/3
f(ei)
(12)
冉 冊 A20 bs A2i i i
ei ⫽ F
(13)
In Eq. (13), the function F is the same as in Eq. (6), but the variable is (A20 / A2i )bisi, ei is the compressive strain of cell i, and bi ⫽
冉冊
l2i Vi l20 V0
2/3
(14)
According to the “springs-in-series” model, the reduced compressive stress si of cell i equals the overall reduced compressive stress ss (s means “inseries”) of the foam. Thus, the compressive strain of a random foam can be expressed by
冘 n
e⫽
i⫽1
Fig. 9. Effects of cell irregularity on the reduced Young’s modulus of random Voronoi foams having a constant relative density r=0.01 [14].
where si is the reduced compressive stress of cell i; Ai, li and Vi are respectively the strut cross-sectional area, the mean strut length and the volume of cell i; r is the overall relative density of the foam and equals 0.01; and the function C is assumed to be the same as in Eq. (9). Substituting Eq. (9) to Eq. (10) gives si ⫽
冉冊
A2i l20 V0 A20l2i Vi
2/3
s0
(11)
In the above equation, V0 is the volume of a regular
冘冉 n
eiPV(bi) ⫽
F
i⫽1
冊
A20 b s P (b ) A2i i s V i
(15)
where PV(bi) is the volume probability of the cells having a value of the parameter bi in a random foam with regularity a. Based on the statistical analysis of 105 cells for random Voronoi foams with different degrees of cell regularity, the volume probability functions PV(bi) have been obtained, as shown in Fig. 10, where bi was grouped in equal intervals of 0.1. In Eq. (15), when F[(A20 / A2i )biss] equals 0.95 (i.e. the compressive strain ei of cell i equals 0.95), we assume that the struts in the cell become contacted and the cell cannot be compressed any more. So, we set the maximum value of F[(A20 / A2i )biss] as 0.95. Eq. (15) can also be expressed as ss ⫽ f(e)
(16)
Both equations (15) and (16) describe the same
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Fig. 10. The volume probability distribution of cells with values of function b (Eq. (14)) in random Voronoi foams having different values of the regularity parameters: (a) a=0.0, (b) a=0.3, (c) a=0.5, (d) a=0.7. The data b were grouped in equal intervals of width 0.1.
relationship, which depends on the cell regularity parameter a. For foams with a=0.3, the relationship (16) is shown in Fig. 8. The simulated (computational) results (Fig. 5(a)) indicate that at small compressive strains, strut stretching, bending and twisting (the “springs-inparallel” model) are the dominating deformation mechanisms, and an irregular foam has a larger tangential modulus than a perfect regular foam, those are in agreement with our previous findings [14]. However, at larger compressive strains, “strut buckling” (the “springs-in-series” model) plays a very important role, that is why a highly irregular foam has a lower reduced compressive stress than a perfect regular foam. Based on equations (7) and (16), the reduced compressive stress–strain relationship of an irregular foam can be predicted by s∗ ⫽ G(e) ⫽ [j(e)]1⫺g[f(e)]g
(17)
where g is a weight function of the role which the
“strut buckling” mechanism (the “springs-in-series” model) plays in the foam deformation, and given by g ⫽ 0.15 ⫹ 2.425e⫺0.6a2e
(18)
The predicted results by Eq. (17) are shown in Fig. 5(b), the theoretical results [5] for a perfect regular foam in the [111] direction are also presented for comparison. Both the simulations and the predictions are limited to an effective foam compressive strain up to 0.6. When the compressive strain e tends to zero, the parameter g should also tend to zero, and the predicted reduced Young’s modulus (the initial tangential modulus) should tend to E (Fig. 9). Hence, there is an error of about 18 percent between the simulated reduced Young’s modulus E (Fig. 9) and the result predicted by Eq. (17). However, Fig. 5(a) and (b) indicate that the predicted reduced stress–strain relationships are in good agreement with the simulated results for foams with varying degrees of regularity. As
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reported in [3], to compress a Kelvin foam (a=1.0) in the [111] direction, both the finite element simulated result [3] and the theoretically predicted result [5] are in very good agreement (see Fig. 5(a) and (b)). Figures 11 and 12 present the deformed structures for Voronoi foams with regularities a=0.0 and a=0.7, suggesting that, “strut buckling” (“springs-in-series”) gradually becomes the dominating deformation mechanism with increasing compressive strain; and the more irregular the foam (i.e., the smaller the regularity parameter a), the more important a role will the “strut buckling” mechanism play at a fixed compressive strain. That is exactly what Eq. (18) describes. However, for a fully random Voronoi foam (a=0.0), g will be larger than 1.0 when the foam compressive strain e is larger than 0.35, (see Eq. (18)). That means even the “spring-in-series” model is stiffer than the irregular foams at larger compressive strains. Figures 11 and 12 also show that the cells in the low density random open-cell foams could have a compressive strain which is larger than 1.0 (the ABAQUS standard 3D beam elements cannot deal
with the strut contact problem). For a perfect regular foam, the reduced Young’s modulus E ⫽ 1.0 [4,14], both the functions j(e) (Eq. (7)) and f(e) (Eq. (16)) will reduce to f(e) (Eq. (5)), hence, Eq. (17) also gives the right results. Shulmeister et al. [12] have modelled the high strain compression of open-cell foams by the Voronoi technique and finite element analysis. They assumed that all the struts have the same constant circular cross-section. They did not consider the effects of degrees of cell irregularity on the compressive stress–strain relationships. However, the struts are normal to the boundary face in their models, this makes their models (samples) much stiffer than ours, and their simulated compressive stress–strain relationships are about 6 times larger than the theoretical results f(e) for a perfect regular foam in the [111] direction [5]. Figure 4(a) and (b) show that the initial mean Poisson’s ratios of random foams with different degrees of regularity are very close to the theoretical result 0.5 [4], this is in agreement with our previous work [14]. However, a random foam has a much smaller Poisson’s ratio than a perfect regular
Fig. 11. The deformed structures of a random Voronoi sample with regularity parameter a=0.0: (a) e=0.16; (b) e=0.262; (c) e=0.414; (d) e=0.602.
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Fig. 12. The undeformed and the deformed structures of a random Voronoi sample with regularity parameter a=0.7: (a) e=0.0; (b) e=0.211; (c) e=0.348; (d) e=0.586.
foam at large compressive strains, that is because all the cells in a perfect regular foam have the same deformation pattern, and the individual cells in a random foam deform in different patterns. Although an irregular foam has a much smaller Poisson’s ratio than a perfect foam at large compressive strains, the regularity parameter a has very little influence on the Poisson’s ratio of random foams. The big jump of the Poisson’s ratios from the value of a perfect regular foam (a=1.0) to the smaller value of an irregular foam (a is smaller than 1.0) is because any irregularity could cause the loss of the “same uniform deformation pattern” in the cells of a perfect foam. Figure 6 indicates that the reduced stress decreases with increasing foam relative density r, this is in agreement with [14]. However, due to the fact that the beam theory has been used in the finite element analysis, the cell struts should be thin and long, and the foam relative density should be less than 0.03 [14]. According to the theoretical analysis [5], assuming that the solid is a polymer which becomes significantly non-linear at a strain of
0.075, to compress a foam to a strain of 0.5 and to keep the maximum strain in the solid polymer less than 0.075, the foam relative density r should be less than 0.00134. Figure 7 shows that the initial Poisson’s ratio decreases with increasing foam relative density r, this is also in agreement with [14]. However, at large compressive strains, a foam with a higher relative density has a Poisson’s ratio larger than that of a foam with a lower relative density. This is reasonable, if the relative density tends to 1.0, the Poisson’s ratio of the foam will tend to the value of the solid material.
5. Conclusions The analysis on the perfect regular foam [4,5] has been extended to random Voronoi foams; the linear elastic analysis on random open cell foams [14] has been extended to geometrical nonlinearity. Periodic samples with varying degrees of regularity have been generated by the Voronoi technique [16]. The solid material was assumed to be
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linear elastic, all the struts were treated as uniform straight beams, and the ABAQUS standard program was used in the finite element simulation. Consequently, the analyses were limited to models having very low relative density. The results indicate that a highly irregular foam has a larger tangential modulus at low strains and a lower reduced stress at high compressive strains than a more regular foam. For irregular low density random foams, strut bending and twisting (the “springs-in-parallel” model) dominate the mechanical response at low strains and strut buckling (the “springs-inseries”) becomes the main deformation mechanism at high compressive strains. The Poisson’s ratio of a low density irregular foam is initially very close to 0.5 (the theoretical result for a regular foam [4]) and becomes much smaller than that of a regular foam [5] at high compressive strains. The regularity parameter a has very little influence on the Poisson’s ratio for irregular foams. Although the reduced compressive stress–strain relationship and the Poisson’s ratio vary in different directions for individual samples, the models are, on average, isotropic. For foams with varying degrees of regularity parameter a, the reduced compressive stress decreases with the increment of foam relative density in all the range of compressive strains, however, with increasing relative density, the Poisson’s ratio reduces at low compressive strains and increases at large compressive strains (Fig. 7). The compressive stress–strain relationship for a regular Kelvin foam [5] and the geometrical properties of 3D random Voronoi tessellations [16] have been used to predict the high strain compression behaviour for random foams with varying degrees of regularity, and the predicted results are in good agreement with those of the simulations.
Acknowledgements The authors wish to thank Drs Anthony Cunningham and John R. Hobdell of the Huntsman Corporation for their continued support and intellectual contributions to this work. They also wish to acknowledge the University of Cambridge for the use of their SGI High Performance Computing Facility Origin 2000 computer and associated staff time.
References [1] Gibson LJ, Ashby MF. Cellular solids: Structure and properties. Cambridge: Cambridge University Press, 1997. [2] Warren WE, Kraynik AM. J Appl Mech 1988;55:341. [3] Kraynik AM. Foams and Emulsions. In: Sadoc JF, River N, editors. Prepared for NATO/ASI meeting, May 1997. Kluwer Academic Publishers; 1999:259–86. [4] Zhu HX, Knott JF, Mills NJ. J Mech Phys Solids 1997;45:319. [5] Zhu HX, Mills NJ, Knott JF. J Mech Phys Solids 1997;45:1875. [6] Zhu HX, Mills NJ. J Mech Phys Solids 1999;47:1437. [7] Mills NJ, Zhu HX. J Mech Phys Solids 1999;47:669. [8] Zhu HX, Mills NJ. Int J Solids Struct 2000;37:1931. [9] Papka SD, Kyriakides S. J Mech Phys Solids 1994;42:1499. [10] Mills NJ, Gilchrist A. J Eng Mater and Tech 2000;122:67. [11] Silva MJ, Gibson LJ. Int J Mech Sci 1997;39:549. [12] Shulmeister V, Van der Burg MWD, Van der Giessen E, Marissen V. Mech Mater 1998;30:125. [13] Chen C, Lu TJ, Fleck NA. J Mech Phys Solids 1999;47:2235. [14] Zhu HX, Hobdell JR, Windle AH. Acta Mater 2000;48:4893. [15] Zhu HX, Hobdell JR, Windle AH. J Mech Phys Solids 2001;49:857. [16] Zhu HX, Thorpe SM and Windle AH, 2001, submitted for publication.
Effects of cell irregularity on the high strain compression of open-cell foams H.X. Zhu 1, A.H. Windle * Department of Materials Science and Metallurgy, Cambridge University, Pembroke Street, Cambridge CB2 3QZ, UK Received 15 June 2001; received in revised form 30 October 2001; accepted 30 October 2001
Abstract The high strain compression of low-density open-cell polymer foams has been modelled by finite element analysis. We used a Voronoi method to generate periodic structures with different degrees of randomness of the cell size and shape, then to investigate the influence of this randomness on the response of Voronoi open-cell foams to high strain compression. It is found that, although the reduced compressive stress–strain relationship and the Poisson’s ratio vary in different directions for individual samples, the models are, on average, isotropic. A highly irregular foam has a larger tangential modulus at very low strains and a lower effective stress at high compressive strains than a more regular foam. The geometrical properties were investigated and used to predict the compressive stress–strain relationships for random open-cell foams with different degrees of cell regularity. For irregular low density foams, strut bending and twisting (the “springs-in-parallel” model) dominate the mechanical response at low strains and strut buckling (the “springs-in-series” model) becomes the main deformation mechanism at large compressive strains. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Foams; High strain compression; Computer simulation; Microstructure
1. Introduction Low-density open-cell foams are widely used in engineering applications such as lightweight structural sandwich panels or components designed for absorbing impact energy. Research works on the mechanical properties and deformation mechanisms of foams have received wide attention [1]. * Corresponding author. E-mail address: [email protected] (A.H. Windle). 1 Present address: Polymer and Colloids Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK.
Mechanical models are usually based on models of cell structure. Unit cell models have proved to be useful in understanding some of the key aspects of the mechanical behaviour of foams [1–10] and, as long as the mechanical properties of the solid are known, this method can give the full response of the foam or the honeycomb subjected to a stress or a strain. Based on the regular Kelvin cell structure, Zhu et al. [4] derived all the three independent elastic constants as functions of the Young’s modulus of the solid material, the relative density of the foam and the shape of the cell edge cross-section, and found that such structure is nearly isotropic. They [4,5] also found that, if a Kelvin foam is compressed in the
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[111] direction, strut twisting plays almost the same important role as strut bending does in the foam deformation. However, most cellular materials usually have random cells. Thus, despite their proven utility, unit cell models do not accurately represent the microstructure of most real foams. To better represent the microstructure, random Voronoi technique and finite element analysis have been used in modelling the mechanical properties of foams [11–15]. Shulmeister et al. [12] have modelled foam structure by random Voronoi cells using finite element analysis. They started from regular bcc and fcc lattice nuclei distributions, subsequently gave the nuclei positions an increasing random offset and constructed the random structure using the Voronoi procedure. However, all the struts at the boundary are normal to the boundary face giving a much stiffer structure. Their simulated compressive stress–strain relationship [12] for random open-cell foams is about 6 times larger than the theoretical results [5] for a regular Kelvin foam. Also, as their model is not periodic, they could not have been able to apply periodic boundary conditions in the FEA analysis. As is well known, boundary conditions can greatly influence the mechanical response of a structure, and even mixed boundary conditions tend to underestimate the mechanical properties of foams [13–15]. In our previous work [14], we found that the periodic Voronoi foams are averagely isotropic, and the cell regularity has a strong effect on the elastic properties of Voronoi foams. The aim of this work is to determine the influence of disorder in foam cell size and shape on the high strain compression of open-cell foams. We have constructed 3D periodic random structures with different degrees of irregularity, and applied finite element analysis to derive the compressive stress–strain relationships. The geometrical properties of random foams have also been investigated and used to predict the reduced compressive stress–strain relationships for foams with varying degrees of regularities. 2. The definition of the regularity 2.1. The construction of periodic random Voronoi foams Cellular solids are usually formed by nucleation and growth of cells. If all cells nucleate randomly
in space at the same time and grow at the same linear rate, the resulting structure is a Voronoi foam. The central field V0, which is surrounded by 26 equivalent boxes, is taken to be a cube in the present study. An orthogonal coordinate system is chosen, with the origin at one corner of the cube, and nucleation points are created in the cube by generating the x, y and z coordinates independently from the pseudo-random numbers between 0 and 1. After the first point is specified, each subsequent random point is accepted only if it is greater than a minimum allowable distance d from any existing point (i.e., it does not fall within an alreadydeposited ball with diameter d), until n nuclei are seeded in the cube. The periodic Voronoi foam sample used in the FEA model is the “unit” cell of an infinite periodic foam (Fig. 1(a)). A dedicated code has been developed to construct the periodic random Voronoi structures. 2.2. The definition of the regularity The fully ordered limit of the 3D Voronoi tessellation is effectively a cubic lattice of tetrakaidecahedral cells. Such cells have a surface area very close to the minimum for a given cell volume. To construct a regular lattice with n identical tetrakaidecahedral cells in the volume V0, the minimum distance d0 between any two adjacent nuclei is given by d0 ⫽
冑6 2
冉冑 冊 V0
2n
1/3
(1)
To construct a random Voronoi tessellation with n cells in the volume V0, the maximum d (the minimum distance between any two nuclei) should be less than d0; otherwise, it is impossible to obtain n random cells. The regularity of a 3D Voronoi tessellation can be measured by [16]: a⫽
d d0
(2)
For a regular lattice with tetrakaidecahedral cells, d equals d0 and a is 1. For a completely random Voronoi tessellation, d equals 0 and thus a is 0. Due to the fact that it is very difficult to construct n cells in a large sample with a regu-
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tinuously as the vertices are approached. For simplicity of the model, all the struts are assumed to have the same and constant Plateau Border crosssection with area A. Consequently, the foam relative density r is determined by the cross-section area A and specified by
冘 N
A r⫽
li
i⫽1
V0
(3)
where li are the cell strut lengths and N is the total number of cell struts. The ABAQUS standard package was used to perform the FEA analysis. Each strut was modelled, depending on its length, with 1 to 5 Timoshenko beam elements (ABAQUS element type B32). The solid polymer material was treated as linear elastic and the Young’s modulus of the struts, Es, was set to 1.0 and the Poisson’s ratio to 0.5. To derive the uniaxial compressive stress– strain relationship, displacement boundary conditions were imposed in either the x or the y directions. 3.2. Boundary conditions
Fig. 1. (a) An undeformed random Voronoi foam with 27 complete cells. (b) The deformed structure with periodic boundary conditions.
larity parameter a larger than 0.7, we only present simulated results for foams with a regularity a up to 0.7.
3. Computational aspects and results 3.1. General methodology All struts in the foam are represented mechanically by beams that are rigidly connected in vertices. In real foams, the cross-section of the struts is a Plateau Border and the area of the cross-section is variable along the strut length, thickening con-
Before performing any finite element analysis, one should choose the most suitable boundary conditions. The best boundary conditions should lead to the average global behaviour of the 3D foams, and avoid any large localised deformation near the boundaries of the mesh. Silva and Gibson have analysed the uniaxial high strain compression of non-periodic 2D foams [11]. Displacement boundary conditions were imposed in the loading direction in their model. The nodes along the “fixed” boundary and the “displaced” boundary were constrained from translating in the loading direction and from in-plane rotation, but were free to translate in the non-loading direction. In our analysis, each sample is a periodic one, it is a “unit” cell taken from an infinite sample (see Fig. 1(a)). It has been shown that the periodic boundary conditions are the most suitable [13–15] for periodic samples. Thus, only periodic boundary conditions are used in this paper. The periodic boundary conditions assume that the cor-
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responding nodes on the opposite struts of the mesh have the same expansions in the normal directions, the same displacements in the other directions, and the same rotations in all the directions (see Fig. 1(b)). 3.3. Mesh sensitivity In order to determine the size of a “unit” cell that is suitable for compression to a large strain and can also supply an accurate solution, a mesh sensitivity study was performed by changing the total number n of the cells for samples having a regularity parameter a=0.1. It is found that the larger the number n, the greater the possibility of instability during compression. A number of random samples were compressed for each different number of cells, n=27, 64 and 125. Each Voronoi foam was generated using a different list of random numbers and had the same relative density of 0.01. The reduced compressive stress–strain relationships (see Eq. (4)) are shown in Fig. 2(a), (b) and (c). The larger the number of cells n, the smaller the variations in the compressive stress–strain relationships of different samples. However, the mean results of the samples having different number of cells are almost the same, as shown in Fig. 2(d). Considering the compressibility, the stability and the accuracy, we fix the number of cells at n=27 in the following finite element analysis. 3.4. Isotropic properties 20–80 random samples with regularity parameter a=0.7 and relative density r=0.01 were compressed in the x and y directions. Due to the fact that some samples became unstable at small strains, only the results of about 10 samples have been taken into account. Figure 3(a) and (b) show the mean reduced compressive stresses and the standad deviations against the compressive strains in the y and x directions. Similarly, Fig. 4(a) and (b) show the mean Poisson’s ratios n21 and n23 and their standard deviations against the compressive strain in the y direction (the 2 direction) for foams having regularity parameters a=0.3 and a=0.5. Although the reduced compressive stresses and the Poisson’s ratios varied over a considerable range
in different directions for individual samples, their mean results in different directions are almost the same, and Figs 3 and 4 suggest that the models, on average, are isotropic. 3.5. Effects of cell regularity on the high strain compression of open-cell foams 20 or more samples have been simulated for foams with different degrees of regularity. All the samples have a fixed relative density of 0.01. To simplify the results, the effective stresses of the foams were reduced by dividing them by r2, the square of the foam relative density, and by the Young’s modulus Es of the solid from which the foam is produced, and given by s¯ ⫽
s Esr2
(4)
Due to the fact that some samples became unstable at small strains, only the results of about 10–20 samples have been taken into account for each degree of regularity. Figure 5(a) shows the simulated mean reduced compressive stress–strain relationships of foams having different degrees of regularity, the finite element simulated results for the Kelvin foam (a=1.0) in the [111] direction [3] are also presented for comparison. Figure 5(b) displays the predicted stress–strain relationships by Eq. (17) (see the discussion section) for foams having different degrees of regularity, the theoretical results for the Kelvin foam (a=1.0) in the [111] direction [5] are also included for comparison. Figures 5(a) and (b) indicate that a highly irregular foam has a larger tangential modulus at small strains and a lower effective stress at high compressive strains than a more regular foam. Although an irregular foam has a smaller Poisson’s ratio at large compressive strains than a perfect regular foam, Figs 4(a) and (b) suggest that cell irregularity has very little influence on the mean Poisson’s ratios of random foams. 3.6. Effects of foam relative density For foams having a regularity a=0.7 and relative densities r=0.01, r=0.02, r=0.04 and r=0.08, Figs
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Fig. 2. Effects of unit-cell size on the reduced compressive stress–strain relationships of random Voronoi foams having a regularity parameter a=0.1. (a) N=125, (b) N=64, (c) N=27, (d) the mean results of the samples having numbers of cells 125, 64 and 27.
6 and 7 present the reduced stresses and the Poisson’s ratios against the compressive strains. The theoretical results for a perfect regular foam are also included for comparison. For a given compressive strain, the reduced stress of an open-cell foam decreases with increasing relative density. However, although the Poisson’s ratio of an opencell foam decreases with increasing relative density at low compressive strains, it increases with increasing relative density at large strains (see Fig. 7).
4. Discussion For a perfect regular low density open-cell foam (i.e. a Kelvin foam and a=1.0) with a constant Plateau Border strut cross-section, Zhu et al. [5] have derived the reduced compressive stress–strain relationship in the [111] direction:
s0 ⫽ f(e)
(5)
or e ⫽ F(s0)
(6)
Both equations (5) and (6) describe the same relationship, as shown in Fig. 8. However, an irregular foam has a larger reduced Young’s modulus than a perfect regular foam, and the higher the cell irregularity, the larger will be the reduced Young’s modulus E¯ (E¯ ⫽ E / Esr2), as shown in Fig. 9 [14]. Consequently, an irregular foam is stiffer than a more regular foam at low compressive strains (see Fig. 5(a)). As discussed in [14], for low density foams, strut bending and twisting are the main deformation mechanisms at small strains (e tends to 0.0). Although many different geometrical properties could affect the mechanical properties of random foams, the effects of cell irregularity on the elastic properties (Young’s modulus) can be explained by the “springs-in-parallel” model [14].
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Fig. 3. The reduced stresses against the compressive strains of foams having a regularity parameter a=0.7, compared with the predicted results. (a) compressed in the y direction, (b) compressed in the x direction.
Fig. 4. The Poisson’s ratio n21 against the compressive strains, compared with n23 and the theoretical results of a perfect regular foam in the [111] direction [5]. (a) For foams with regularity parameter a=0.3, (b) for foams with regularity parameter a=0.5.
If the “springs-in-parallel” model was the only deformation mechanism in the high strain compression of irregular foams, the reduced stress– strain relationship of an irregular low density open cell foam could be approximately expressed by
is why an irregular foam has a lower reduced stress than a more regular foam at high compressive strains (see Fig. 5(a)). If “strut buckling” was the only deformation mechanism, the cells in the foam could be treated as “springs-in-series” (that means every cell in the sample has the same compressive stress, but a different compressive strain). For a single uniform rod, the buckling force is given by
sp ⫽ j(e) ⫽ E¯ f(e)
(7)
For a foam with regularity parameter a=0.3, the relationship (Eq. (7)) is shown in Fig. 8, where e is the compressive strain of the foam. The above relationship (Eq. (7)) depends on the value of E¯ , i.e. on the cell regularity parameter a. However, at high compressive strains, the “strut buckling” did play a very important role, and that
P⫽
KA2Es l2
(8)
where A is the cross-sectional area of the rod, l is the rod length, Es is the Young’s modulus, and K
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Fig. 6. Effects of relative density on the reduced compressive stress–strain relationships of foams having regularity parameter a=0.7, compared with the theoretical results of a perfect regular foam (a=1.0) in the [111] direction [5] and the predicted results of a random foam with a=0.7.
Fig. 5. Effects of cell regularity on the mean reduced stress– strain relationships, compared with the predicted results (by Eq. (17)) for random Voronoi foams. The finite element simulated result for the Kelvin foam (a=1.0) in the [111] direction [3] and the theoretical results for a perfect regular foam in the [111] direction [5] are also presented for comparison. (a) Simulated mean results; (b) predicted results.
is a constant depending on the shape of the rod cross-section and the restriction conditions at the two ends of the rod. Thus, for a Kelvin cell (or foam) with a relative density r=0.01, the reduced buckling stress can be expressed as s0 ⫽
s0 CA20 ⫽ 2 Esr2 l20V2/3 0 r
(9)
where A0 is the strut cross-sectional area, l0 is the strut length, r is the overall relative density of the foam, V0 is the cell volume and C is a function
Fig. 7. Effects of relative density on the Poisson’s ratio of foams having regularity parameter a=0.7, compared with the theoretical results of a perfect regular foam (a=1.0 and r=0.01) in the [111] direction [5].
depending on the shape of the rod cross-section, the restriction conditions of the two ends of the rod, and the foam compressive strain. Similarly, the reduced buckling stress for an irregular cell may be expressed as si ⫽
CA2i 2 l V2/3 i r 2 i
(10)
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cell which equals the mean volume of all the cells in the random sample; Ai is assumed to be the same for all the cell struts in the random sample, which depends on the regularity parameter a and is different from A0 if a⫽1.0. However, s0 is a function of the compressive strain of the foam [5] (see Eq. (5) and Fig. 8). Substituting equation (5) to (11) gives si ⫽ or Fig. 8. The reduced compressive stress–strain relationships in the [111] direction: f(e) for open-cell Kelvin foam with a constant Plateau Border strut cross-section [5]; j(e) and f(e) for foams with regularity parameter a=0.3.
冉冊
A2i l20 V0 A20l2i Vi
2/3
f(ei)
(12)
冉 冊 A20 bs A2i i i
ei ⫽ F
(13)
In Eq. (13), the function F is the same as in Eq. (6), but the variable is (A20 / A2i )bisi, ei is the compressive strain of cell i, and bi ⫽
冉冊
l2i Vi l20 V0
2/3
(14)
According to the “springs-in-series” model, the reduced compressive stress si of cell i equals the overall reduced compressive stress ss (s means “inseries”) of the foam. Thus, the compressive strain of a random foam can be expressed by
冘 n
e⫽
i⫽1
Fig. 9. Effects of cell irregularity on the reduced Young’s modulus of random Voronoi foams having a constant relative density r=0.01 [14].
where si is the reduced compressive stress of cell i; Ai, li and Vi are respectively the strut cross-sectional area, the mean strut length and the volume of cell i; r is the overall relative density of the foam and equals 0.01; and the function C is assumed to be the same as in Eq. (9). Substituting Eq. (9) to Eq. (10) gives si ⫽
冉冊
A2i l20 V0 A20l2i Vi
2/3
s0
(11)
In the above equation, V0 is the volume of a regular
冘冉 n
eiPV(bi) ⫽
F
i⫽1
冊
A20 b s P (b ) A2i i s V i
(15)
where PV(bi) is the volume probability of the cells having a value of the parameter bi in a random foam with regularity a. Based on the statistical analysis of 105 cells for random Voronoi foams with different degrees of cell regularity, the volume probability functions PV(bi) have been obtained, as shown in Fig. 10, where bi was grouped in equal intervals of 0.1. In Eq. (15), when F[(A20 / A2i )biss] equals 0.95 (i.e. the compressive strain ei of cell i equals 0.95), we assume that the struts in the cell become contacted and the cell cannot be compressed any more. So, we set the maximum value of F[(A20 / A2i )biss] as 0.95. Eq. (15) can also be expressed as ss ⫽ f(e)
(16)
Both equations (15) and (16) describe the same
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Fig. 10. The volume probability distribution of cells with values of function b (Eq. (14)) in random Voronoi foams having different values of the regularity parameters: (a) a=0.0, (b) a=0.3, (c) a=0.5, (d) a=0.7. The data b were grouped in equal intervals of width 0.1.
relationship, which depends on the cell regularity parameter a. For foams with a=0.3, the relationship (16) is shown in Fig. 8. The simulated (computational) results (Fig. 5(a)) indicate that at small compressive strains, strut stretching, bending and twisting (the “springs-inparallel” model) are the dominating deformation mechanisms, and an irregular foam has a larger tangential modulus than a perfect regular foam, those are in agreement with our previous findings [14]. However, at larger compressive strains, “strut buckling” (the “springs-in-series” model) plays a very important role, that is why a highly irregular foam has a lower reduced compressive stress than a perfect regular foam. Based on equations (7) and (16), the reduced compressive stress–strain relationship of an irregular foam can be predicted by s∗ ⫽ G(e) ⫽ [j(e)]1⫺g[f(e)]g
(17)
where g is a weight function of the role which the
“strut buckling” mechanism (the “springs-in-series” model) plays in the foam deformation, and given by g ⫽ 0.15 ⫹ 2.425e⫺0.6a2e
(18)
The predicted results by Eq. (17) are shown in Fig. 5(b), the theoretical results [5] for a perfect regular foam in the [111] direction are also presented for comparison. Both the simulations and the predictions are limited to an effective foam compressive strain up to 0.6. When the compressive strain e tends to zero, the parameter g should also tend to zero, and the predicted reduced Young’s modulus (the initial tangential modulus) should tend to E (Fig. 9). Hence, there is an error of about 18 percent between the simulated reduced Young’s modulus E (Fig. 9) and the result predicted by Eq. (17). However, Fig. 5(a) and (b) indicate that the predicted reduced stress–strain relationships are in good agreement with the simulated results for foams with varying degrees of regularity. As
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reported in [3], to compress a Kelvin foam (a=1.0) in the [111] direction, both the finite element simulated result [3] and the theoretically predicted result [5] are in very good agreement (see Fig. 5(a) and (b)). Figures 11 and 12 present the deformed structures for Voronoi foams with regularities a=0.0 and a=0.7, suggesting that, “strut buckling” (“springs-in-series”) gradually becomes the dominating deformation mechanism with increasing compressive strain; and the more irregular the foam (i.e., the smaller the regularity parameter a), the more important a role will the “strut buckling” mechanism play at a fixed compressive strain. That is exactly what Eq. (18) describes. However, for a fully random Voronoi foam (a=0.0), g will be larger than 1.0 when the foam compressive strain e is larger than 0.35, (see Eq. (18)). That means even the “spring-in-series” model is stiffer than the irregular foams at larger compressive strains. Figures 11 and 12 also show that the cells in the low density random open-cell foams could have a compressive strain which is larger than 1.0 (the ABAQUS standard 3D beam elements cannot deal
with the strut contact problem). For a perfect regular foam, the reduced Young’s modulus E ⫽ 1.0 [4,14], both the functions j(e) (Eq. (7)) and f(e) (Eq. (16)) will reduce to f(e) (Eq. (5)), hence, Eq. (17) also gives the right results. Shulmeister et al. [12] have modelled the high strain compression of open-cell foams by the Voronoi technique and finite element analysis. They assumed that all the struts have the same constant circular cross-section. They did not consider the effects of degrees of cell irregularity on the compressive stress–strain relationships. However, the struts are normal to the boundary face in their models, this makes their models (samples) much stiffer than ours, and their simulated compressive stress–strain relationships are about 6 times larger than the theoretical results f(e) for a perfect regular foam in the [111] direction [5]. Figure 4(a) and (b) show that the initial mean Poisson’s ratios of random foams with different degrees of regularity are very close to the theoretical result 0.5 [4], this is in agreement with our previous work [14]. However, a random foam has a much smaller Poisson’s ratio than a perfect regular
Fig. 11. The deformed structures of a random Voronoi sample with regularity parameter a=0.0: (a) e=0.16; (b) e=0.262; (c) e=0.414; (d) e=0.602.
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Fig. 12. The undeformed and the deformed structures of a random Voronoi sample with regularity parameter a=0.7: (a) e=0.0; (b) e=0.211; (c) e=0.348; (d) e=0.586.
foam at large compressive strains, that is because all the cells in a perfect regular foam have the same deformation pattern, and the individual cells in a random foam deform in different patterns. Although an irregular foam has a much smaller Poisson’s ratio than a perfect foam at large compressive strains, the regularity parameter a has very little influence on the Poisson’s ratio of random foams. The big jump of the Poisson’s ratios from the value of a perfect regular foam (a=1.0) to the smaller value of an irregular foam (a is smaller than 1.0) is because any irregularity could cause the loss of the “same uniform deformation pattern” in the cells of a perfect foam. Figure 6 indicates that the reduced stress decreases with increasing foam relative density r, this is in agreement with [14]. However, due to the fact that the beam theory has been used in the finite element analysis, the cell struts should be thin and long, and the foam relative density should be less than 0.03 [14]. According to the theoretical analysis [5], assuming that the solid is a polymer which becomes significantly non-linear at a strain of
0.075, to compress a foam to a strain of 0.5 and to keep the maximum strain in the solid polymer less than 0.075, the foam relative density r should be less than 0.00134. Figure 7 shows that the initial Poisson’s ratio decreases with increasing foam relative density r, this is also in agreement with [14]. However, at large compressive strains, a foam with a higher relative density has a Poisson’s ratio larger than that of a foam with a lower relative density. This is reasonable, if the relative density tends to 1.0, the Poisson’s ratio of the foam will tend to the value of the solid material.
5. Conclusions The analysis on the perfect regular foam [4,5] has been extended to random Voronoi foams; the linear elastic analysis on random open cell foams [14] has been extended to geometrical nonlinearity. Periodic samples with varying degrees of regularity have been generated by the Voronoi technique [16]. The solid material was assumed to be
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linear elastic, all the struts were treated as uniform straight beams, and the ABAQUS standard program was used in the finite element simulation. Consequently, the analyses were limited to models having very low relative density. The results indicate that a highly irregular foam has a larger tangential modulus at low strains and a lower reduced stress at high compressive strains than a more regular foam. For irregular low density random foams, strut bending and twisting (the “springs-in-parallel” model) dominate the mechanical response at low strains and strut buckling (the “springs-inseries”) becomes the main deformation mechanism at high compressive strains. The Poisson’s ratio of a low density irregular foam is initially very close to 0.5 (the theoretical result for a regular foam [4]) and becomes much smaller than that of a regular foam [5] at high compressive strains. The regularity parameter a has very little influence on the Poisson’s ratio for irregular foams. Although the reduced compressive stress–strain relationship and the Poisson’s ratio vary in different directions for individual samples, the models are, on average, isotropic. For foams with varying degrees of regularity parameter a, the reduced compressive stress decreases with the increment of foam relative density in all the range of compressive strains, however, with increasing relative density, the Poisson’s ratio reduces at low compressive strains and increases at large compressive strains (Fig. 7). The compressive stress–strain relationship for a regular Kelvin foam [5] and the geometrical properties of 3D random Voronoi tessellations [16] have been used to predict the high strain compression behaviour for random foams with varying degrees of regularity, and the predicted results are in good agreement with those of the simulations.
Acknowledgements The authors wish to thank Drs Anthony Cunningham and John R. Hobdell of the Huntsman Corporation for their continued support and intellectual contributions to this work. They also wish to acknowledge the University of Cambridge for the use of their SGI High Performance Computing Facility Origin 2000 computer and associated staff time.
References [1] Gibson LJ, Ashby MF. Cellular solids: Structure and properties. Cambridge: Cambridge University Press, 1997. [2] Warren WE, Kraynik AM. J Appl Mech 1988;55:341. [3] Kraynik AM. Foams and Emulsions. In: Sadoc JF, River N, editors. Prepared for NATO/ASI meeting, May 1997. Kluwer Academic Publishers; 1999:259–86. [4] Zhu HX, Knott JF, Mills NJ. J Mech Phys Solids 1997;45:319. [5] Zhu HX, Mills NJ, Knott JF. J Mech Phys Solids 1997;45:1875. [6] Zhu HX, Mills NJ. J Mech Phys Solids 1999;47:1437. [7] Mills NJ, Zhu HX. J Mech Phys Solids 1999;47:669. [8] Zhu HX, Mills NJ. Int J Solids Struct 2000;37:1931. [9] Papka SD, Kyriakides S. J Mech Phys Solids 1994;42:1499. [10] Mills NJ, Gilchrist A. J Eng Mater and Tech 2000;122:67. [11] Silva MJ, Gibson LJ. Int J Mech Sci 1997;39:549. [12] Shulmeister V, Van der Burg MWD, Van der Giessen E, Marissen V. Mech Mater 1998;30:125. [13] Chen C, Lu TJ, Fleck NA. J Mech Phys Solids 1999;47:2235. [14] Zhu HX, Hobdell JR, Windle AH. Acta Mater 2000;48:4893. [15] Zhu HX, Hobdell JR, Windle AH. J Mech Phys Solids 2001;49:857. [16] Zhu HX, Thorpe SM and Windle AH, 2001, submitted for publication.