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Elastic properties of equilibrium foams
Acta Materialia 113 (2016) 11e18
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Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Full length article
Elastic properties of equilibrium foams € ll, S. Hallstro € m* J. Ko KTH Royal Institute of Technology, Department of Aeronautical and Vehicle Engineering, SE-100 44, Stockholm, Sweden
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 May 2015 Received in revised form 8 January 2016 Accepted 9 January 2016
Stochastic equilibrium finite element (FE) foam models are used to study the influence of relative density and distribution of solid material between cell walls and edges on the elastic properties of foam materials. It is first established that the models contain a sufficient number of cells to ascertain isotropy and numerically and statistically robust results. It is then found that the elastic moduli are very weakly coupled to cell size variation in the models, when the latter is varied systematically. The influence from relative density and distribution of solid on the elastic parameters is considerably stronger. Analytical estimates from the literature, based on idealized cell models and dimension analysis, are matched by fitting coefficients to the FE results, providing good qualitative but relatively poor quantitative correlation. An expansion of the analytical coupling functions is then suggested in order to reduce their level of idealization. The expanded formulation shows virtually perfect agreement with the numerical results for almost the whole range of relative densities and distributions of solid in the FE parameter study. The presented analytical expression is believed to be general and provide accurate estimates of the elastic properties of a wide range of foam materials, provided that their bulk material properties and micro structure can be established. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Stochastic Cellular materials Relative density Distribution of solid
1. Introduction A relatively simple but yet powerful model for relating the constitutive properties of solid foams to their cellular structure is the cubic model suggested by Gibson and Ashby [1]. Several scaling laws were derived from this model, relating different mechanical properties of foams to their relative density and the distribution of solid material between their cell walls and edges. The estimates of elastic properties are based on the assumptions that the bending stiffness of cell walls can be neglected and that the cell edges only carry bending loads. Form factor coefficients are used to compensate for geometrical differences between the model and true foams. It is however generally difficult to match the models to specific real materials since there is no direct analytical way to determine the form factor coefficients for a given real foam. Stochastic methods have been used in the past to generate realistic foam model geometries, e.g. Refs. [2e8], and to compute homogenized elastic properties taking the foam geometry, relative density and bulk material properties into account [2,4e6,8]. Stochastic models are necessary to capture the random and amorphous nature of real foam materials for which cell sizes and shapes
* Corresponding author. http://dx.doi.org/10.1016/j.actamat.2016.01.025 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
could vary significantly. A model example from the present study is presented as two different representative volume elements (RVE) in Fig. 1, one that better illustrates the foam structure and one showing the model as implemented into the finite element (FE) software. Such modeling approaches are however challenging since they involve a relatively high level of model complexity and require substantial and thorough computing to obtain robust and reliable results. The modeling techniques used in the work mentioned above vary some in terms of model constitutions, methods of model generation as well as in the level of mechanical sophistication when managing e.g. varying foam parameters, boundary conditions and size effects. Stochastic foam models with different relative densities have been presented earlier but no systematic study has been found that shows how the constitutive properties scale with the relative density and the distribution of solid material in these models. The convergence of such properties with respect to the number of cells in the models has not received much attention in the past either, although a common conception seems to be that many cells are needed. In many previous studies the number of cells used in the models appear to be governed by what could be afforded computationally while the convergence as such is often not discussed. In the present work a modeling technique specifically
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Fig. 1. Illustration of a stochastic foam model with 100 cells. a) whole-cell RVE visualizing the foam morphology and b) cubic RVE with mesh constituting the corresponding FE model.
developed to enable stochastic variation of foam parameters is used to determine constitutive properties of foam materials. Then the relations between such parameters and the homogenized mechanical properties of the models are studied. The work specifically examines the convergence of the Young's and shear moduli, and the Poisson's ratio. Then the effects from cell volume distribution, relative density and distribution of solid on these properties are investigated and presented. The results from the parameter study are used to fit coefficients to the scaling laws suggested by Gibson and Ashby [1]. A generalization of the scaling laws is finally suggested for better agreement with the modeling results over the studied ranges of relative density and distribution of solid between the cell walls and edges. Throughout the work the models are relaxed through numerical minimization of cell surface area in combination with topology transitions, similar to the relaxation that would take place naturally in real liquid foams. An essential feature of the foam models used in the study is thus that they are at equilibrium from a surface energy point of view. 2. Method Stochastic foam models are generated as described in a previous study by the authors [7] where the Surface Evolver software [9] is used to bring the models to surface energy equilibrium. The models are periodic in three dimensions (3D) and combinations of prescribed displacements and periodic boundary conditions are applied at all model boundaries in order to fully comply with a RVE approach. The initial geometries come from Voronoi partitions [10] around seed points defined by the centra of equi-sized spheres at various levels of packing. The sphere populations are generated using Jodry & Tory's packing algorithm [11]. Some benefits of the algorithm are that it is computationally efficient and controlled by one single parameter only, making the procedure well defined and easily repeatable although the resulting sphere populations are random. Surface Evolver uses a gradient method to iteratively evolve the generated Voronoi partitions into dry foams with a minimum total surface area. When the minimum is reached, topology transitions are triggered in areas where cell edges have become short and tend to vanish. Thereafter the surface area minimization is continued. These two steps are altered repeatedly until global topological convergence is reached and an equilibrium dry foam structure is obtained.
The minimization of surface area is performed without changing the volume of individual cells. The cell volume variation coming from the previous Voronoi partitioning is thus conserved. Constitutive properties are then computed through FE analysis in Abaqus employing 6-noded Kirchoff type (STRI65) shell elements with constant thickness. The effect of material concentrations along cell edges is investigated by relocating material from the cell wall shell elements to 3-noded shear-flexible (B32) beam elements placed along the cell edges. Such struts are introduced along all cell wall edges assuming constant three-cuspid cross sections [12], illustrated in Fig. 2, where the cross section area is given by
A¼
pffiffiffi p 3 r2 : 2
(1)
The moment of inertia of a strut is then axisymmetric and given by
I¼
1 pffiffiffi 20 3 11p r 4 : 24
(2)
The thickness of the cell walls is reduced and values of A and I corresponding to the relocation of material are assigned to the beam elements along the cell edges. The cell wall material is assumed to have a Poisson's ratio ns ¼ 0.3 in the FE analysis. The results coming out from the models are eventually used to fit co0 0 efficients C1, C1 , C2 and C2 to the scaling laws given by Gibson and Ashby [1], suggesting that
2 0 E r r ¼ C1 f2 þ C1 ð1 fÞ Es rs rs
Fig. 2. Illustration of the cross-section used for the struts.
(3)
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
2 0 G r r ¼ C2 f2 þ C2 ð1 fÞ Es rs rs
13
(4)
where E* and r* are the Young's modulus and the density of the foam, respectively. Es and rs are the corresponding properties for the (solid) raw material, and f is the distribution of solid, i.e. the fraction of solid material accumulated along the cell wall edges. 3. Results and discussion 3.1. Model convergence To determine the minimum number of cells needed in the models for the constitutive properties to converge a total of 2000 models consisting of 8e250 cells are generated and analyzed. For each model type the mean value and standard deviation from 10 models are derived. All models in the convergence study are based on random close packed (RCP) spheres generating foams with a coefficient of variation (CV) of cell volume in the range 0.04e0.05 and a relative density r =rs ¼ 0:10, using shell elements only (f ¼ 0) with a uniform cell wall thickness in each model. The results presented in Figs. 3e5 indicate that a good level of convergence is achieved, both for mean values and standard deviations, using 85 cells or more in the models. The number of cells necessary for convergence obviously depends on how the models are built and how loads and model boundaries are handled in the analysis. The relatively low number of cells necessary for convergence in this study can likely be attributed to the relaxation and/or that the periodic loading and boundary conditions are applied to the models with great concern. These issues are described more thoroughly in the following paragraph. Roberts and Garboczi [2] who also used a Voronoi modeling approach, but without relaxation, stated that at least 100 cells were needed for convergence of macroscopic properties of their closed-cell foam models but did not present any variational data. Redenbach et al. [6] and Chen et al. [8] presented results indicating a much slower convergence rate, both for monodisperse and strongly polydisperse models based on Laguerre tessellations. It is not clear to the authors what causes the difference in convergence between the current study and those of Redenbach et al. [6] and Chen et al. [8] but there are some plausible
Fig. 3. Relative Young's modulus versus the number of cells in the model, at 0.10 relative density.
Fig. 4. Relative shear modulus versus the number of cells in the model, at 0.10 relative density.
Fig. 5. Poisson's ratio versus the number of cells in the model, at 0.10e relative density.
explanations. Firstly the Laguerre models are not brought to equilibrium in either of the previous studies and it is known that relaxation in Surface Evolver changes the model topology significantly for Voronoi models, by e.g. eliminating unnaturally small faces and short edges from the cell structure [3,13]. (Voronoi tessellations constitute special cases of the more general Laguerre tessellations but to the authors knowledge no study has yet presented results from closed-cell equilibrium foam models based on Laguerre partitioning.) Such unnatural topological features will otherwise remain or have to be eliminated by other means of manipulation. Typically they are then removed with mesh-tidying operations that alter the connectivity without resolving the morphological frustration. Secondly the periodic displacements in the current study are applied on node-pairs across the RVE boundaries so that the strain between each node-pair is identical without imposing uniform displacement on the whole RVE boundaries [14]. The periodic conditions are thus considerably smoother than in many other studies. Thirdly 3-noded shell elements are used in Ref. [6] while 6-noded elements are used in the present work. (Details of the FE models are unfortunately absent in
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Ref. [8].) The CV of the three Young's moduli E1, E2 and E3 of each model is calculated to check the (an)isotropy. The results from all 2000 simulations, presented in Fig. 6, show that the models are relatively isotropic when they contain 85 cells or more. The results also appear to be consistent with observations from a previous study [13], looking at the convergence of cell topology parameters, but the resolution was then lower since only models containing 50, 100 or 150 cells were studied. 3.2. Spread in cell volume The cell volume distribution has significant effect on some key morphology parameters such as the number of faces per cell and the spread in cell face area and cell edge length, see e.g. Refs. [3,13]. To study the influence from cell volume distribution on the constitutive properties, 50 models consisting of 100 cells each are generated with the CV of cell volume ranging from 0.04 to 0.25, again with a relative density r =rs ¼ 0:10, f ¼ 0, and uniform cell wall thickness. The results, presented in Figs. 7e9, indicate that the Young's and shear moduli, as well as the Poisson's ratio are virtually independent of the spread in cell volume in the studied range. Note that the results also indicate that the convergence for the more polydisperse models is good since they would otherwise be expected to show greater spread. The results deviate from what was reported earlier for polydisperse foam models built from Laguerre partitions [6,8], where a significant reduction of stiffness with increasing polydispersity was found. The divergence could be due to the relatively low polydispersity in the current work but it is believed that it rather is the relaxation of the models that makes the difference.
Fig. 7. Relative Young's modulus versus the coefficient of variation of the cell volume.
3.3. Relative density and distribution of solid The effect from relocating material to the cell edges, i.e. from the distribution of solid, is investigated by introducing beam elements along the cell edges of the 50 models used in the cell volume distribution study. The distribution of solid f is then defined as the share of material located in the beams. Various levels of distribution of solid are considered at four different relative densities meaning that either none, parts of, or all
Fig. 6. Anisotropy measured as the coefficient of variation CV of E1, E2, and E3; mean values () of 50 models for each number of cells N and corresponding standard deviations.
Fig. 8. Relative shear modulus versus the coefficient of variation of the cell volume.
Fig. 9. Poisson's ratio versus the coefficient of variation of the cell volume.
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
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of the material is relocated from the shell elements to the introduced beam elements; the latter case creating open cell foams without cell walls. The results presented in Figs. 10 and 11 show that the elastic properties vary strongly with both the relative density and the distribution of solid. The FE simulations are notably consistent with insignificant standard deviations between the results from the 50 models computed for each case. 0 0 The coefficients C1, C1 , C2 and C2 in eqs. (3) and (4) are fitted to the FE model results using the least square method, giving the following relations
2 E r r ¼ 0:418f2 þ 0:356ð1 fÞ Es rs rs
(5)
2 G r r ¼ 0:153f2 þ 0:135ð1 fÞ : Es rs rs
(6) Fig. 11. Relative shear modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
The ratios
C2 ¼ 0:366 C1
(7)
0
C2 0 ¼ 0:379 C1
(8)
are close to 3/8 and comply well with the constitutive ratio
G 1 ¼ E 2ð1 þ n Þ
(9)
when assuming n z0:33 [1]. There is however substantial mismatch between the FE results and eqs. (5) and (6) where the equations underestimate the stiffnesses for high distributions of solid and overestimate them for low distributions of solid. For the worst combination in the parameter study, i.e. a relative density of 0.025 and f ¼ 1, the difference is 58%. Apparently the form of eqs. (3) and (4) is too idealised to fully relate the load-carrying mechanisms of the foam to its relative density and distribution of solid. The equations are based on the assumption that the stiffness (or rather the compliance) of an open cell foam in absence of cell walls
(f ¼ 1) is governed by the bending stiffness of cell struts only, without any contribution from stretching of the struts. In addition it is assumed that bending of cell walls does not contribute significantly to the deformation of closed cell foams in comparison to stretching of the same walls. As it turns out, these assumptions are overly simplistic and it is therefore here suggested to expand eqs. (3) and (4) to a more general form, taking also bending of cell walls and stretching of cell struts into account. The equations are generalised to the form
2 E r r ¼ f1 ðfÞ þ g1 ðfÞ Es rs rs
(10)
2 G r r ¼ f2 ðfÞ þ g2 ðfÞ Es rs rs
(11)
and expanded by introducing general quadratic functions for fi(f) and general linear functions for gi(f), according to
fi ðfÞ ¼ ai f2 þ bi f þ ci
(12)
gi ðfÞ ¼ di f þ ei :
(13)
The ten coefficients in eqs. (12) and (13) can then be determined using least square fits to FE results again, resulting in the following expressions
Fig. 10. Relative Young's modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
f1 ðfÞ ¼ 0:172f2 0:0427f þ 0:233
(14)
g1 ðfÞ ¼ 0:282f þ 0:311
(15)
f2 ðfÞ ¼ 0:0632f2 0:0140f þ 0:0942
(16)
g2 ðfÞ ¼ 0:108f þ 0:117
(17)
to be inserted into eqs. (10) and (11). Figs. 12 and 13 show the results from eqs. (14)e(17) together with the corresponding terms from eqs. (5) and (6), and single values extracted from the FE model results. The latter are obtained from best (least square) fits of the FE data, for each specific f value, to couples of values corresponding to fi(f) and gi(f) in eqs. (10) and
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€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
Fig. 12. f1(f) and g1(f) (see eq. (10)) from eqs. (5), (14) and (15), and corresponding best fits to the FE model data.
Fig. 13. f2(f) and g2(f) (see eq. (11)) from eqs. (6), (16) and (17), and corresponding best fits to the FE model data.
(11). As seen, the match between the expanded formulation and the model data is very good. The fit between the model data and eqs. (5) and (6) is strongly limited by the fact that the quadratic dependence on the relative density in the equations vanishes when f/0, while the FE model data show no such tendency. The linear terms in eqs. (5) and (6) correlate better but are also somewhat limited by the fact that they vanish when f/1. The FE model data presented in Figs. 12 and 13 indicate that the quadratic terms of the relative density in eqs. (10) and (11) are important for the global properties, even when f ¼ 0, implying that there is a deformation mechanism in the models that is not accounted for in eqs. (3) and (4). Correspondingly the linear terms have a certain effect on the global properties even when f ¼ 1. In Figs. 14 and 15 eqs. (10) and (11) are presented together with results from the simulations. The agreement is very good for all relative densities except 0.025 for which the results still differ significantly, indicating that the higher order polynomial functions provide a much better but still not perfect match with the results
Fig. 14. Relative Young's modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
Fig. 15. Relative shear modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
from the simulations. The distribution of solid is here treated as a pure model parameter representing the fraction of solid material in the beam elements at the cell edges. That is however not identical with the fraction of material contained in the struts of a real foam for several reasons. Firstly the distinction between struts and cell walls is generally ambiguous. Calibrating the distribution of solid with respect to geometry data from real foams is therefore cumbersome, and more so the higher the relative density. Secondly, the modeling approach, where cell faces are represented by shell elements and cell struts by beam elements, inherently creates material overlap along the cell edges. Even for f ¼ 0, with only cell faces in the model, there is some inaccuracy related to the use of 2D finite elements in the analysis. Since typically three cell walls meet at virtually dihedral angles the nodes along each cell edge are shared and the walls thereby jointly contribute with significant bending stiffness to the edge. Local
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
material overlaps also occur, as in any 2D FE model where elements meet at an angle. Using 3D solid elements in the kind of models used in this study is however not feasible with the computational capacity currently available. In conclusion the distribution of solid is a quite simplistic parameter for expressing the true material distribution in a foam material but it is still conceptually relevant and practical when comparing different foam structures. 3.4. Poisson's ratio The Poisson's ratio is also extracted from the models and presented in Fig. 16. It increases with the distribution of solid and decreases, virtually asymptotically towards about 0.3, when the relative density increases. For open cell foams (f ¼ 1), n* seems to approach 0.5 when the relative density approaches zero. The effective Poisson's ratio of a foam can be described by
n ¼ f ððr =rs Þ; ns Þ *
(18)
[15]. That n seems to approach 0.5 when for an open cell foam is in agreement with earlier reported results for Kelvin foams [16e18]. Its deviation from 0.3 can partly be explained by diminishing influence from ns when the relative density approaches zero [15]. For equilibrium foams n z0:33 does not seem to be a valid approximation for high distributions of solid, especially not in combination with low relative densities, thus compromising the compliance with the constitutive ratio in eq. (9). These cases are also the ones showing the largest deviation when comparing eqs. (5) and (6) with the model data. As mentioned above ns ¼ 0.3 is assumed in all models. It was chosen somewhat arbitrary in this study and is not believed to have a big influence on the results. For the beam elements there is obviously no effect at all but for the shell elements there is potentially some influence on the constitutive foam properties due to the in-plane coupling effects. It was not systematically examined but some spot checks of the models gave at hand that E* increases and decreases by less than 4% when ns is changed to 0.2 and 0.4, respectively. Since Es is kept constant a reduction of ns implies that Gs increases and vice versa, so the inverse effect from ns on E* seems reasonable. The effect is r =rs /0
Fig. 16. Poisson's ratio as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models. The dashed lines in the figure are added for visual aid only.
17
however very weak, which is in agreement with previously reported analytically results [19] suggesting that the influence from ns on the Young's modulus of isotropic foams containing spheroid cells should be insignificant. 4. Conclusions Stochastic equilibrium finite element models containing 85 cells are shown to provide converged results of elastic properties and models containing 100 cells are used throughout the presented work. It is likely that the relatively low number of cells required can be attributed to the relaxation of the models in combination with careful management of the periodicity of the model geometry, boundary conditions and load introduction. In the investigated range, spread in cell volume has insignificant impact on the elastic properties. The elastic constants are however strongly affected by the relative density and the distribution of solid. Calibrating the distribution of solid is challenging since it is a model parameter that origins from the use of beams and shells rather than an easily measurable property in real foam materials. No single set of coefficients could be found for Gibson and Ashby's scaling laws [1] to satisfactory match the model data for varying distribution of solid. When the scaling laws are modified to also include lower order terms, a set of polynomials could be found that bring the scaling laws to much better agreement with the model data for the investigated range of relative density at all levels of distribution of solid. Although one may argue that the morphology of real structural foams differs from that of liquid foams, due to e.g. viscosity and gradual solidification during the foaming process, incorporating principles of minimisation of energy in the topology makes the models more representative of real foams than if such considerations are totally absent. Acknowledgements The results presented in this paper are partly based on research funded by the Swedish Research Council, grant No. 50576001. References [1] L.J. Gibson, M.F. Ashby, Cellular Solids, 2nd edition, Cambridge University Press, 1997. [2] A.P. Roberts, E.J. Garboczi, Elastic moduli of model random three-dimensional closed-cell cellular solids, Acta Mater. 49 (2001) 189e197. [3] A.M. Kraynik, D.A. Reinelt, F. Van Swol, Structure of random monodisperse foam, Phys. Rev. E 67 (2003) 031403. €dt, Numerical simulation of me[4] F. Fischer, G.T. Lim, U.A. Handge, V. Altsta chanical properties of cellular materials using computed tomography analysis, J. Cell. Plast. 8 (2006) 441e460. €uble, Prediction of material behaviour of [5] R. Schlimper, M. Rinker, R. Scha closed cell rigid foams via mesoscopic modelling, in: Proceedings of ICCM-17, Edinburgh, UK, 2009. €, Laguerre tessellations for elastic stiffness [6] C. Redenbach, I. Shklyar, H. Andra simulations of closed foams with strongly varying cell sizes, Int. J. Eng. Sci. 50 (2012) 70e78. €ll, S. Hallstro €m, Generation of periodic stochastic foam models for nu[7] J. Ko merical analysis, J. Cell. Plast. 50 (2014) 37e54. [8] Y. Chen, R. Das, M. Battley, Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams, Int. J. Solids Struct. 52 (2015) 150e164. [9] K. Brakke, The surface evolver, Exp. Math. 1 (2) (1992) 141e165. tres continus a la the orie des [10] G. Voronoi, Nouvelles applications des parame me me moire. recherches sur les paralle lloe dres formes quadratiques. deuxie primitifs, J. Reine Angew. Math. 134 (1908) 198e287. [11] W.S. Jodrey, E.M. Tory, Computer simulation of close random packing of equal spheres, Phys. Rev. A 32 (4) (1985) 2347e2351. [12] W.-Y. Jang, A.M. Kraynik, S. Kyriakides, On the microstructure of open-cell foams and its effects on elastic properties, Int. J. Solids Struct. 45 (2008) 1845e1875. €ll, S. Hallstro € m, Influence of sphere packing fraction on polydispercity and [13] J. Ko morphology of voronoi-partitioned foam models, J. Cell. Plast. (2016), http://
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dx.doi.org/10.1177/0021955X16644892 in press. €ll, Modelling of Rigid Foams (PhD thesis), KTH Royal Institute of Tech[14] J. Ko nology, June 2015. [15] A.P. Roberts, A.J. Garboczi, Computation of the linear elastic properties of random porous materials with a wide variety of microstructure, Proc. Math. Phys. Eng. Sci. 458 (2002) 1033e1054. [16] H.X. Zhu, J.F. Knott, N.J. Mills, Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells, J. Mech. Phys. Solids 45 (3) (1997)
319e343. [17] W.E. Warren, A.M. Kraynik, Linear elastic behavior of a low-density Kelvin foam with open cells, J. Appl. Mech. 64 (1997) 787e794. [18] L. Gong, S. Kyriakides, W.-Y. Yang, Compressive response of open-cell foams. part i: morphology and elastic properties, Int. J. Solids Struct. 42 (2005) 1355e1379. [19] R.M. Christensen, Effective properties of composite materials containing voids, Proc. Math. Phys. Sci. 440 (1993) 461e473.
Contents lists available at ScienceDirect
Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Full length article
Elastic properties of equilibrium foams € ll, S. Hallstro € m* J. Ko KTH Royal Institute of Technology, Department of Aeronautical and Vehicle Engineering, SE-100 44, Stockholm, Sweden
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 May 2015 Received in revised form 8 January 2016 Accepted 9 January 2016
Stochastic equilibrium finite element (FE) foam models are used to study the influence of relative density and distribution of solid material between cell walls and edges on the elastic properties of foam materials. It is first established that the models contain a sufficient number of cells to ascertain isotropy and numerically and statistically robust results. It is then found that the elastic moduli are very weakly coupled to cell size variation in the models, when the latter is varied systematically. The influence from relative density and distribution of solid on the elastic parameters is considerably stronger. Analytical estimates from the literature, based on idealized cell models and dimension analysis, are matched by fitting coefficients to the FE results, providing good qualitative but relatively poor quantitative correlation. An expansion of the analytical coupling functions is then suggested in order to reduce their level of idealization. The expanded formulation shows virtually perfect agreement with the numerical results for almost the whole range of relative densities and distributions of solid in the FE parameter study. The presented analytical expression is believed to be general and provide accurate estimates of the elastic properties of a wide range of foam materials, provided that their bulk material properties and micro structure can be established. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Stochastic Cellular materials Relative density Distribution of solid
1. Introduction A relatively simple but yet powerful model for relating the constitutive properties of solid foams to their cellular structure is the cubic model suggested by Gibson and Ashby [1]. Several scaling laws were derived from this model, relating different mechanical properties of foams to their relative density and the distribution of solid material between their cell walls and edges. The estimates of elastic properties are based on the assumptions that the bending stiffness of cell walls can be neglected and that the cell edges only carry bending loads. Form factor coefficients are used to compensate for geometrical differences between the model and true foams. It is however generally difficult to match the models to specific real materials since there is no direct analytical way to determine the form factor coefficients for a given real foam. Stochastic methods have been used in the past to generate realistic foam model geometries, e.g. Refs. [2e8], and to compute homogenized elastic properties taking the foam geometry, relative density and bulk material properties into account [2,4e6,8]. Stochastic models are necessary to capture the random and amorphous nature of real foam materials for which cell sizes and shapes
* Corresponding author. http://dx.doi.org/10.1016/j.actamat.2016.01.025 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
could vary significantly. A model example from the present study is presented as two different representative volume elements (RVE) in Fig. 1, one that better illustrates the foam structure and one showing the model as implemented into the finite element (FE) software. Such modeling approaches are however challenging since they involve a relatively high level of model complexity and require substantial and thorough computing to obtain robust and reliable results. The modeling techniques used in the work mentioned above vary some in terms of model constitutions, methods of model generation as well as in the level of mechanical sophistication when managing e.g. varying foam parameters, boundary conditions and size effects. Stochastic foam models with different relative densities have been presented earlier but no systematic study has been found that shows how the constitutive properties scale with the relative density and the distribution of solid material in these models. The convergence of such properties with respect to the number of cells in the models has not received much attention in the past either, although a common conception seems to be that many cells are needed. In many previous studies the number of cells used in the models appear to be governed by what could be afforded computationally while the convergence as such is often not discussed. In the present work a modeling technique specifically
12
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
Fig. 1. Illustration of a stochastic foam model with 100 cells. a) whole-cell RVE visualizing the foam morphology and b) cubic RVE with mesh constituting the corresponding FE model.
developed to enable stochastic variation of foam parameters is used to determine constitutive properties of foam materials. Then the relations between such parameters and the homogenized mechanical properties of the models are studied. The work specifically examines the convergence of the Young's and shear moduli, and the Poisson's ratio. Then the effects from cell volume distribution, relative density and distribution of solid on these properties are investigated and presented. The results from the parameter study are used to fit coefficients to the scaling laws suggested by Gibson and Ashby [1]. A generalization of the scaling laws is finally suggested for better agreement with the modeling results over the studied ranges of relative density and distribution of solid between the cell walls and edges. Throughout the work the models are relaxed through numerical minimization of cell surface area in combination with topology transitions, similar to the relaxation that would take place naturally in real liquid foams. An essential feature of the foam models used in the study is thus that they are at equilibrium from a surface energy point of view. 2. Method Stochastic foam models are generated as described in a previous study by the authors [7] where the Surface Evolver software [9] is used to bring the models to surface energy equilibrium. The models are periodic in three dimensions (3D) and combinations of prescribed displacements and periodic boundary conditions are applied at all model boundaries in order to fully comply with a RVE approach. The initial geometries come from Voronoi partitions [10] around seed points defined by the centra of equi-sized spheres at various levels of packing. The sphere populations are generated using Jodry & Tory's packing algorithm [11]. Some benefits of the algorithm are that it is computationally efficient and controlled by one single parameter only, making the procedure well defined and easily repeatable although the resulting sphere populations are random. Surface Evolver uses a gradient method to iteratively evolve the generated Voronoi partitions into dry foams with a minimum total surface area. When the minimum is reached, topology transitions are triggered in areas where cell edges have become short and tend to vanish. Thereafter the surface area minimization is continued. These two steps are altered repeatedly until global topological convergence is reached and an equilibrium dry foam structure is obtained.
The minimization of surface area is performed without changing the volume of individual cells. The cell volume variation coming from the previous Voronoi partitioning is thus conserved. Constitutive properties are then computed through FE analysis in Abaqus employing 6-noded Kirchoff type (STRI65) shell elements with constant thickness. The effect of material concentrations along cell edges is investigated by relocating material from the cell wall shell elements to 3-noded shear-flexible (B32) beam elements placed along the cell edges. Such struts are introduced along all cell wall edges assuming constant three-cuspid cross sections [12], illustrated in Fig. 2, where the cross section area is given by
A¼
pffiffiffi p 3 r2 : 2
(1)
The moment of inertia of a strut is then axisymmetric and given by
I¼
1 pffiffiffi 20 3 11p r 4 : 24
(2)
The thickness of the cell walls is reduced and values of A and I corresponding to the relocation of material are assigned to the beam elements along the cell edges. The cell wall material is assumed to have a Poisson's ratio ns ¼ 0.3 in the FE analysis. The results coming out from the models are eventually used to fit co0 0 efficients C1, C1 , C2 and C2 to the scaling laws given by Gibson and Ashby [1], suggesting that
2 0 E r r ¼ C1 f2 þ C1 ð1 fÞ Es rs rs
Fig. 2. Illustration of the cross-section used for the struts.
(3)
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
2 0 G r r ¼ C2 f2 þ C2 ð1 fÞ Es rs rs
13
(4)
where E* and r* are the Young's modulus and the density of the foam, respectively. Es and rs are the corresponding properties for the (solid) raw material, and f is the distribution of solid, i.e. the fraction of solid material accumulated along the cell wall edges. 3. Results and discussion 3.1. Model convergence To determine the minimum number of cells needed in the models for the constitutive properties to converge a total of 2000 models consisting of 8e250 cells are generated and analyzed. For each model type the mean value and standard deviation from 10 models are derived. All models in the convergence study are based on random close packed (RCP) spheres generating foams with a coefficient of variation (CV) of cell volume in the range 0.04e0.05 and a relative density r =rs ¼ 0:10, using shell elements only (f ¼ 0) with a uniform cell wall thickness in each model. The results presented in Figs. 3e5 indicate that a good level of convergence is achieved, both for mean values and standard deviations, using 85 cells or more in the models. The number of cells necessary for convergence obviously depends on how the models are built and how loads and model boundaries are handled in the analysis. The relatively low number of cells necessary for convergence in this study can likely be attributed to the relaxation and/or that the periodic loading and boundary conditions are applied to the models with great concern. These issues are described more thoroughly in the following paragraph. Roberts and Garboczi [2] who also used a Voronoi modeling approach, but without relaxation, stated that at least 100 cells were needed for convergence of macroscopic properties of their closed-cell foam models but did not present any variational data. Redenbach et al. [6] and Chen et al. [8] presented results indicating a much slower convergence rate, both for monodisperse and strongly polydisperse models based on Laguerre tessellations. It is not clear to the authors what causes the difference in convergence between the current study and those of Redenbach et al. [6] and Chen et al. [8] but there are some plausible
Fig. 3. Relative Young's modulus versus the number of cells in the model, at 0.10 relative density.
Fig. 4. Relative shear modulus versus the number of cells in the model, at 0.10 relative density.
Fig. 5. Poisson's ratio versus the number of cells in the model, at 0.10e relative density.
explanations. Firstly the Laguerre models are not brought to equilibrium in either of the previous studies and it is known that relaxation in Surface Evolver changes the model topology significantly for Voronoi models, by e.g. eliminating unnaturally small faces and short edges from the cell structure [3,13]. (Voronoi tessellations constitute special cases of the more general Laguerre tessellations but to the authors knowledge no study has yet presented results from closed-cell equilibrium foam models based on Laguerre partitioning.) Such unnatural topological features will otherwise remain or have to be eliminated by other means of manipulation. Typically they are then removed with mesh-tidying operations that alter the connectivity without resolving the morphological frustration. Secondly the periodic displacements in the current study are applied on node-pairs across the RVE boundaries so that the strain between each node-pair is identical without imposing uniform displacement on the whole RVE boundaries [14]. The periodic conditions are thus considerably smoother than in many other studies. Thirdly 3-noded shell elements are used in Ref. [6] while 6-noded elements are used in the present work. (Details of the FE models are unfortunately absent in
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€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
Ref. [8].) The CV of the three Young's moduli E1, E2 and E3 of each model is calculated to check the (an)isotropy. The results from all 2000 simulations, presented in Fig. 6, show that the models are relatively isotropic when they contain 85 cells or more. The results also appear to be consistent with observations from a previous study [13], looking at the convergence of cell topology parameters, but the resolution was then lower since only models containing 50, 100 or 150 cells were studied. 3.2. Spread in cell volume The cell volume distribution has significant effect on some key morphology parameters such as the number of faces per cell and the spread in cell face area and cell edge length, see e.g. Refs. [3,13]. To study the influence from cell volume distribution on the constitutive properties, 50 models consisting of 100 cells each are generated with the CV of cell volume ranging from 0.04 to 0.25, again with a relative density r =rs ¼ 0:10, f ¼ 0, and uniform cell wall thickness. The results, presented in Figs. 7e9, indicate that the Young's and shear moduli, as well as the Poisson's ratio are virtually independent of the spread in cell volume in the studied range. Note that the results also indicate that the convergence for the more polydisperse models is good since they would otherwise be expected to show greater spread. The results deviate from what was reported earlier for polydisperse foam models built from Laguerre partitions [6,8], where a significant reduction of stiffness with increasing polydispersity was found. The divergence could be due to the relatively low polydispersity in the current work but it is believed that it rather is the relaxation of the models that makes the difference.
Fig. 7. Relative Young's modulus versus the coefficient of variation of the cell volume.
3.3. Relative density and distribution of solid The effect from relocating material to the cell edges, i.e. from the distribution of solid, is investigated by introducing beam elements along the cell edges of the 50 models used in the cell volume distribution study. The distribution of solid f is then defined as the share of material located in the beams. Various levels of distribution of solid are considered at four different relative densities meaning that either none, parts of, or all
Fig. 6. Anisotropy measured as the coefficient of variation CV of E1, E2, and E3; mean values () of 50 models for each number of cells N and corresponding standard deviations.
Fig. 8. Relative shear modulus versus the coefficient of variation of the cell volume.
Fig. 9. Poisson's ratio versus the coefficient of variation of the cell volume.
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
15
of the material is relocated from the shell elements to the introduced beam elements; the latter case creating open cell foams without cell walls. The results presented in Figs. 10 and 11 show that the elastic properties vary strongly with both the relative density and the distribution of solid. The FE simulations are notably consistent with insignificant standard deviations between the results from the 50 models computed for each case. 0 0 The coefficients C1, C1 , C2 and C2 in eqs. (3) and (4) are fitted to the FE model results using the least square method, giving the following relations
2 E r r ¼ 0:418f2 þ 0:356ð1 fÞ Es rs rs
(5)
2 G r r ¼ 0:153f2 þ 0:135ð1 fÞ : Es rs rs
(6) Fig. 11. Relative shear modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
The ratios
C2 ¼ 0:366 C1
(7)
0
C2 0 ¼ 0:379 C1
(8)
are close to 3/8 and comply well with the constitutive ratio
G 1 ¼ E 2ð1 þ n Þ
(9)
when assuming n z0:33 [1]. There is however substantial mismatch between the FE results and eqs. (5) and (6) where the equations underestimate the stiffnesses for high distributions of solid and overestimate them for low distributions of solid. For the worst combination in the parameter study, i.e. a relative density of 0.025 and f ¼ 1, the difference is 58%. Apparently the form of eqs. (3) and (4) is too idealised to fully relate the load-carrying mechanisms of the foam to its relative density and distribution of solid. The equations are based on the assumption that the stiffness (or rather the compliance) of an open cell foam in absence of cell walls
(f ¼ 1) is governed by the bending stiffness of cell struts only, without any contribution from stretching of the struts. In addition it is assumed that bending of cell walls does not contribute significantly to the deformation of closed cell foams in comparison to stretching of the same walls. As it turns out, these assumptions are overly simplistic and it is therefore here suggested to expand eqs. (3) and (4) to a more general form, taking also bending of cell walls and stretching of cell struts into account. The equations are generalised to the form
2 E r r ¼ f1 ðfÞ þ g1 ðfÞ Es rs rs
(10)
2 G r r ¼ f2 ðfÞ þ g2 ðfÞ Es rs rs
(11)
and expanded by introducing general quadratic functions for fi(f) and general linear functions for gi(f), according to
fi ðfÞ ¼ ai f2 þ bi f þ ci
(12)
gi ðfÞ ¼ di f þ ei :
(13)
The ten coefficients in eqs. (12) and (13) can then be determined using least square fits to FE results again, resulting in the following expressions
Fig. 10. Relative Young's modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
f1 ðfÞ ¼ 0:172f2 0:0427f þ 0:233
(14)
g1 ðfÞ ¼ 0:282f þ 0:311
(15)
f2 ðfÞ ¼ 0:0632f2 0:0140f þ 0:0942
(16)
g2 ðfÞ ¼ 0:108f þ 0:117
(17)
to be inserted into eqs. (10) and (11). Figs. 12 and 13 show the results from eqs. (14)e(17) together with the corresponding terms from eqs. (5) and (6), and single values extracted from the FE model results. The latter are obtained from best (least square) fits of the FE data, for each specific f value, to couples of values corresponding to fi(f) and gi(f) in eqs. (10) and
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€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
Fig. 12. f1(f) and g1(f) (see eq. (10)) from eqs. (5), (14) and (15), and corresponding best fits to the FE model data.
Fig. 13. f2(f) and g2(f) (see eq. (11)) from eqs. (6), (16) and (17), and corresponding best fits to the FE model data.
(11). As seen, the match between the expanded formulation and the model data is very good. The fit between the model data and eqs. (5) and (6) is strongly limited by the fact that the quadratic dependence on the relative density in the equations vanishes when f/0, while the FE model data show no such tendency. The linear terms in eqs. (5) and (6) correlate better but are also somewhat limited by the fact that they vanish when f/1. The FE model data presented in Figs. 12 and 13 indicate that the quadratic terms of the relative density in eqs. (10) and (11) are important for the global properties, even when f ¼ 0, implying that there is a deformation mechanism in the models that is not accounted for in eqs. (3) and (4). Correspondingly the linear terms have a certain effect on the global properties even when f ¼ 1. In Figs. 14 and 15 eqs. (10) and (11) are presented together with results from the simulations. The agreement is very good for all relative densities except 0.025 for which the results still differ significantly, indicating that the higher order polynomial functions provide a much better but still not perfect match with the results
Fig. 14. Relative Young's modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
Fig. 15. Relative shear modulus as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models.
from the simulations. The distribution of solid is here treated as a pure model parameter representing the fraction of solid material in the beam elements at the cell edges. That is however not identical with the fraction of material contained in the struts of a real foam for several reasons. Firstly the distinction between struts and cell walls is generally ambiguous. Calibrating the distribution of solid with respect to geometry data from real foams is therefore cumbersome, and more so the higher the relative density. Secondly, the modeling approach, where cell faces are represented by shell elements and cell struts by beam elements, inherently creates material overlap along the cell edges. Even for f ¼ 0, with only cell faces in the model, there is some inaccuracy related to the use of 2D finite elements in the analysis. Since typically three cell walls meet at virtually dihedral angles the nodes along each cell edge are shared and the walls thereby jointly contribute with significant bending stiffness to the edge. Local
€ll, S. Hallstro €m / Acta Materialia 113 (2016) 11e18 J. Ko
material overlaps also occur, as in any 2D FE model where elements meet at an angle. Using 3D solid elements in the kind of models used in this study is however not feasible with the computational capacity currently available. In conclusion the distribution of solid is a quite simplistic parameter for expressing the true material distribution in a foam material but it is still conceptually relevant and practical when comparing different foam structures. 3.4. Poisson's ratio The Poisson's ratio is also extracted from the models and presented in Fig. 16. It increases with the distribution of solid and decreases, virtually asymptotically towards about 0.3, when the relative density increases. For open cell foams (f ¼ 1), n* seems to approach 0.5 when the relative density approaches zero. The effective Poisson's ratio of a foam can be described by
n ¼ f ððr =rs Þ; ns Þ *
(18)
[15]. That n seems to approach 0.5 when for an open cell foam is in agreement with earlier reported results for Kelvin foams [16e18]. Its deviation from 0.3 can partly be explained by diminishing influence from ns when the relative density approaches zero [15]. For equilibrium foams n z0:33 does not seem to be a valid approximation for high distributions of solid, especially not in combination with low relative densities, thus compromising the compliance with the constitutive ratio in eq. (9). These cases are also the ones showing the largest deviation when comparing eqs. (5) and (6) with the model data. As mentioned above ns ¼ 0.3 is assumed in all models. It was chosen somewhat arbitrary in this study and is not believed to have a big influence on the results. For the beam elements there is obviously no effect at all but for the shell elements there is potentially some influence on the constitutive foam properties due to the in-plane coupling effects. It was not systematically examined but some spot checks of the models gave at hand that E* increases and decreases by less than 4% when ns is changed to 0.2 and 0.4, respectively. Since Es is kept constant a reduction of ns implies that Gs increases and vice versa, so the inverse effect from ns on E* seems reasonable. The effect is r =rs /0
Fig. 16. Poisson's ratio as function of relative density for different distributions of solid. Each marker shows average values and standard deviations for 50 models. The dashed lines in the figure are added for visual aid only.
17
however very weak, which is in agreement with previously reported analytically results [19] suggesting that the influence from ns on the Young's modulus of isotropic foams containing spheroid cells should be insignificant. 4. Conclusions Stochastic equilibrium finite element models containing 85 cells are shown to provide converged results of elastic properties and models containing 100 cells are used throughout the presented work. It is likely that the relatively low number of cells required can be attributed to the relaxation of the models in combination with careful management of the periodicity of the model geometry, boundary conditions and load introduction. In the investigated range, spread in cell volume has insignificant impact on the elastic properties. The elastic constants are however strongly affected by the relative density and the distribution of solid. Calibrating the distribution of solid is challenging since it is a model parameter that origins from the use of beams and shells rather than an easily measurable property in real foam materials. No single set of coefficients could be found for Gibson and Ashby's scaling laws [1] to satisfactory match the model data for varying distribution of solid. When the scaling laws are modified to also include lower order terms, a set of polynomials could be found that bring the scaling laws to much better agreement with the model data for the investigated range of relative density at all levels of distribution of solid. Although one may argue that the morphology of real structural foams differs from that of liquid foams, due to e.g. viscosity and gradual solidification during the foaming process, incorporating principles of minimisation of energy in the topology makes the models more representative of real foams than if such considerations are totally absent. Acknowledgements The results presented in this paper are partly based on research funded by the Swedish Research Council, grant No. 50576001. References [1] L.J. Gibson, M.F. Ashby, Cellular Solids, 2nd edition, Cambridge University Press, 1997. [2] A.P. Roberts, E.J. Garboczi, Elastic moduli of model random three-dimensional closed-cell cellular solids, Acta Mater. 49 (2001) 189e197. [3] A.M. Kraynik, D.A. Reinelt, F. Van Swol, Structure of random monodisperse foam, Phys. Rev. E 67 (2003) 031403. €dt, Numerical simulation of me[4] F. Fischer, G.T. Lim, U.A. Handge, V. Altsta chanical properties of cellular materials using computed tomography analysis, J. Cell. Plast. 8 (2006) 441e460. €uble, Prediction of material behaviour of [5] R. Schlimper, M. Rinker, R. Scha closed cell rigid foams via mesoscopic modelling, in: Proceedings of ICCM-17, Edinburgh, UK, 2009. €, Laguerre tessellations for elastic stiffness [6] C. Redenbach, I. Shklyar, H. Andra simulations of closed foams with strongly varying cell sizes, Int. J. Eng. Sci. 50 (2012) 70e78. €ll, S. Hallstro €m, Generation of periodic stochastic foam models for nu[7] J. Ko merical analysis, J. Cell. Plast. 50 (2014) 37e54. [8] Y. Chen, R. Das, M. Battley, Effects of cell size and cell wall thickness variations on the stiffness of closed-cell foams, Int. J. Solids Struct. 52 (2015) 150e164. [9] K. Brakke, The surface evolver, Exp. Math. 1 (2) (1992) 141e165. tres continus a la the orie des [10] G. Voronoi, Nouvelles applications des parame me me moire. recherches sur les paralle lloe dres formes quadratiques. deuxie primitifs, J. Reine Angew. Math. 134 (1908) 198e287. [11] W.S. Jodrey, E.M. Tory, Computer simulation of close random packing of equal spheres, Phys. Rev. A 32 (4) (1985) 2347e2351. [12] W.-Y. Jang, A.M. Kraynik, S. Kyriakides, On the microstructure of open-cell foams and its effects on elastic properties, Int. J. Solids Struct. 45 (2008) 1845e1875. €ll, S. Hallstro € m, Influence of sphere packing fraction on polydispercity and [13] J. Ko morphology of voronoi-partitioned foam models, J. Cell. Plast. (2016), http://
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dx.doi.org/10.1177/0021955X16644892 in press. €ll, Modelling of Rigid Foams (PhD thesis), KTH Royal Institute of Tech[14] J. Ko nology, June 2015. [15] A.P. Roberts, A.J. Garboczi, Computation of the linear elastic properties of random porous materials with a wide variety of microstructure, Proc. Math. Phys. Eng. Sci. 458 (2002) 1033e1054. [16] H.X. Zhu, J.F. Knott, N.J. Mills, Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells, J. Mech. Phys. Solids 45 (3) (1997)
319e343. [17] W.E. Warren, A.M. Kraynik, Linear elastic behavior of a low-density Kelvin foam with open cells, J. Appl. Mech. 64 (1997) 787e794. [18] L. Gong, S. Kyriakides, W.-Y. Yang, Compressive response of open-cell foams. part i: morphology and elastic properties, Int. J. Solids Struct. 42 (2005) 1355e1379. [19] R.M. Christensen, Effective properties of composite materials containing voids, Proc. Math. Phys. Sci. 440 (1993) 461e473.