Polymeric foams, a class of cellular materials, are attractive engineering materials, as a direct result of their outstanding energy absorbing, acoustic, insulating, and specific mechanical properties. The macroscopic mechanical behavior of polymeric foams is the result of a subtle interplay between the intrinsic material behavior of the polymer base material and the complex microstructure resulting from their process history. Understanding and predicting the structure–property relation of these materials is of importance for optimization of products as well as developing constitutive models.1, 2
For an elastomeric foam, the stress–strain response (Fig. 1) is initially governed by elastic bending of the cell walls, followed by a long plateau dominated by elastic-buckling of these cell walls. Finally, a stiff response is present due to the full collapse of the foam's structure.3
The mechanical behavior of these cellular materials has been analyzed with different methods. At first, analytical expressions and design criteria have been developed relating the mechanical properties of cellular materials to the mechanics of periodic unit cells,3, 4 periodic honeycomb structures,5 and periodic tetrakaidecahedral cells (referred to as Kelvin cells).6–8 These analytical expressions relate bulk mechanical properties of the constituent material and the relative density (or other morphological properties) of the cellular materials to the macroscopic mechanical properties. However, they neglect the influence of the more realistic structure with complex morphological details of polymeric foams, like cell size distribution (irregular structures), nonuniform cell wall thickness (over the length of a cell wall), cell shape anisotropy, cell wall misalignment, and so forth.
To incorporate the effects of the complex structure of foams, finite element (FE) analyses have been used in many studies on foams and composites to determine the structure–property relationship.5, 9–20 In these studies, the artificially generated FE models were based on regular structures,5, 9–14 for example, perfect honeycomb structures and regular Kelvin cells, where others used irregular structures to more realistically represent the microstructure, mostly based on Voronoi tessellation.10, 11, 14–19 These studies conclude that small details of the foam morphology, like cell irregularity, cell wall thickness variations, and cell shape anisotropy strongly influence the mechanical responses of these structures. Consequently, it can be concluded that accurate structure–property relationships can only be achieved by using realistic microstructures.
Detailed information became available with the development of X-ray computed tomography (CT), which enables a direct and nondestructive 3D characterization of the foam's microstructure.21, 22 The geometry results from a reconstruction of the X-ray projections taken from different angles and consists of voxel data containing gray values corresponding either to the solid material of the microstructure or to the contained gas. After filtering and segmentation, by choosing a proper threshold value, the final geometry is obtained. This can then be used for morphology studies or as a starting point for FE analyses.
An example of the first is the study of Jang et al.23 where this technique was employed to determine the morphological properties of a polymeric and an aluminum open-celled foam. Subsequently, they used this information as input for an FE model based on periodic regular Kelvin cells or irregular models based on the Surface Evolver software24 and predicted the elastic properties. Drawback of such an approach is that the real microstructure with its imperfections is not captured in sufficient detail.
A convenient solution is to base the FE model directly on the segmented microstructure obtained by X-ray CT.25–30 In the first studies, this has been successfully done to determine the linear elastic response of cellular materials, like bone,25 bread,26 metal, and polymeric foams.29, 30 In recent studies, elastoplastic material models were incorporated in FE analyses and the response of a larger strain regime (εln < 0.1) was successfully compared with the experimental data.27, 28 However, in these studies, the influence of FE model size within the large strain regime is not addressed. The required size of the FE model to be representative was obtained for the artificial regular models, where at least 82 cells in the FE model should be used, and for artificial irregular structures, at least 192 cells in the FE model should be used.5, 11
In the current study, the goal is to determine, for realistic microsctructures as obtained with X-ray CT-based modeling, the influence of (i) element type, (ii) element size, and (iii) the model size on the mechanical response. For this purpose, 2D slices are taken from a 3D partially close-celled foam. Since a 3D close-celled foam is chosen, the cells will be closed in 2D slices. The advantage of using 2D models from a 3D model is the ability to include large deformations and incorporate contact between elements while still considering a large number of cells. These models are then subjected to a compression load to determine the mechanical response. Although in most studies the analyses are linear elastic29, 30 or nonlinear up to a strain of εln < 0.1,27, 28 in this study, a larger strain regime (up to εln = 0.6) is examined. To do so, nonlinear FE analyses are employed by incorporating a large strain framework as well as contact between elements. Results of in situ compression experiments performed during X-ray CT are used to qualitatively validate the FE results.
To determine the influences of various aspects of a hybrid numerical–experimental approach, 2D structures are used to reduce the computational cost. To do so, a structure is needed that is fully connected in 2D, that is, in which each cell wall is connected to a vertex in 2D corresponding to a 3D configuration, where all pores are isolated and not interconnected. Therefore a close-celled, PVC/NBR foam (ENSOLITE®), is used. Since the mechanical properties of the polymer material used in this study are not known, a hyperelastic behavior is assumed according to
with σ the Cauchy stress tensor, J the volume change factor, I the unity tensor, the deviatoric part of the isochoric left Cauchy-Green deformation tensor, K the bulk modulus of 3750 MPa, and G the shear modulus of 334 MPa.31 Although these properties do not correspond with the constituent material of the actual foam, they are representative for this class of polymers.
Scanning electron microscope (SEM) micrographs (Fig. 2) revealed that the smallest dimension of the foam, the cell wall thickness, is of the order of 4μm. To describe every detail with at least four voxels, a resolution of at least 1μm would be needed, resulting in an X-ray CT reconstruction as shown in Figure 3, containing around 150,000 voxels.
As a consequence, the maximum theoretical model, which is determined by the size of the X-ray detector (2236 × 2236 pixels), is then 2.24 × 2.24 mm2. The SEM images also showed that the largest cells can span over 600μm, leading to the limitation of ∼42 – 62 cells in the model, which might be not enough to be representative to determine bulk properties. Also, the number of voxels strongly increases when, with the same resolution, larger structures would be made. To circumvent these limitations, a lower resolution of 3.75 μm is taken, to be able to increase the number of cells in the model and to limit the computational size of the models. At this resolution, the cell walls are still visible. As a result, the density of the foam model will not be compared with that of the actual foam, but rather the microstructure is used as a realistic representation.
The geometry of the close-celled foam is characterized by using an X-ray (absorption) CT system (the Nanotom system of GE Sensing and Inspection Technology), where the resolution was set at 3.75 μm and a volume of 1200 × 336 × 1200 voxels is reconstructed (partly shown in Fig. 4). Reconstructions are taken from the middle of the scanned volume to prevent boundary effects (the level of noise increases toward the boundary of the scanned volume). A slice is then taken at a randomly chosen position (in y-direction shown in Fig. 4). From this slice, six different model sizes are taken at different randomly chosen positions, where the models on an average contain 172, 142, 122, 82, 52, and 32 cells (depicted by the boxes in Fig. 4). The models are referred to as CF1 to CF6, where the former represents the model with on an average of 32 cells and the latter 172 cells. In total, eight different slices are taken, resulting in eight models for each model size. A threshold value is then chosen such that the largest model (172 cells) has a volume fraction of 36%. Above this threshold, the fine scale features did not change significantly. This threshold is then taken as a representative and is applied to all other model sizes. As a result of using a low resolution, the density is at least three times higher than the foam's density.
Using continuum elements to describe the polymer material, for this volume fraction, is justified by Caty et al.28 The resulting models consist of hexahedron elements, tetrahedron elements, or a combination of them, where the faces of the elements for each model are converted into 2D elements (triangular and quadrilateral elements), resulting in 2D plane strain models. In this study, the conversion of the segmented volume into FE models is based on direct voxel conversion for models based on hexahedral elements.25 For the other element types, the volumetric marching cubes algorithm32 was used, which is based on the marching cube algorithm.33 For the analyses, MSC.MARC is used and, since contact between elements is incorporated, linear elements are used, and the assumed strain formulation is invoked because of the bending behavior of this type of elements. Linear elements are used since the contact formulation is more optimized for these elements. Furthermore, a Hermann-element formulation is used since a hyperelastic material model (with a relatively high bulk modulus) is used, which can lead to locking effects. In Figure 5, the discretization of a detail of a structure using quadrilateral or triangular elements or a combination of both is given.
From Figure 5(a), it is clear that models based on only quadrilateral elements25 cannot accurately describe contact effects, since the elements at the boundary of cell walls result in a jagged boundary description. For the other types of discretization [Fig. 5(b,c)], a smoother description of the boundaries is obtained due to the volume marching cube algorithm.
After reconstruction of the CT projections, the largest structures, with a voxel size of 3.75 μm, contain over 600,000 voxels. To investigate the influence of discretization and to limit the number of elements, models with different element sizes (7.5, 11.75, and 15 μm) are used. This results in 324,083 triangular elements for the largest model and three to four elements over the cell wall thickness for the smallest element size. At first, the 3D volume data are rescaled and filtered with a Gaussian filter. Then, for each element size, a threshold value for the largest model is determined so that a volume fraction of 36% remains. This threshold is then applied to the smaller models with a corresponding element size. The result of increasing voxel size is graphically represented in Figure 6. Clearly visible in Figure 6 is the loss of detail when the element size is increased and the removal of noise for the larger models due to smoothing. This occurs when small features and noise have dimensions smaller than the element size and therefore are lost during discretization. By increasing the element size, the total structure is thickened due to volume fraction equivalency. The loss of detail and noise results in the addition of extra material to the remaining cell walls. Furthermore, due to loss of detail, locally, edges are not connected any more, as shown in Figure 6 by the boxes.
It was shown by Chen et al.10 that the use of periodic boundary conditions is preferential to determine the mechanical behavior of these structures. Unfortunately, models based on the approach of X-ray CT modeling of foams cannot be combined with periodic boundary conditions. Therefore, the models are loaded quasi-statically in compression with a rigid plate, where frictionless contact between the plate and the polymer material is accounted for. The left and bottom boundaries are subjected to symmetry boundary conditions. Nodes that initially or during deformation coincide with these boundaries cannot penetrate these boundaries. The right boundary is left free, as will be the case in a compression experiment. Since large deformations and a complex microstructure are concerned, a geometrically nonlinear formulation is used with an updated Lagrange procedure. The pressure build up due to compression of the gas phase in cells is neglected.3
The variation of the local volume fraction of the models with model size is represented in Figure 7. The volume fraction is given as a function of the mean number of cells present in the models. For all element sizes, the largest spread in volume fraction is found for the smallest models, which is due to local variation in the microstructure. The standard deviation for the smallest model with the coarsest mesh is the largest because of the loss of detail and cell wall connections. A strong decrease of the standard deviation with increasing model size is found, for models larger or equal to CF3 (8.312 cells).
Large Strain Response and Element Type
A typical stress–strain behavior found with FE analyses, of a model with size CF3 and a resolution of 15 μm, is shown in Figure 8, for models based on triangular elements and a combination of triangular and quadrilateral elements. From Figure 8 it is clear that there is almost no difference between these models. The characteristic mechanical behavior of foams is found in these analyses, where a linear elastic response at small strains (εln ≤ 0.02) is followed by a less stiff regime (0.09 ≤ εln ≤ 0.39) due to elastic buckling of cell walls. Finally, a stiff response is found due to collapse of most cells and contact between cell walls.
The deformation of a model with size CF3 is given in Figure 9 and the corresponding stress–strain response was given in Figure 8. The onset of the plateau regime occurs at a true strain of εln = 0.1 and corresponds to the first cell walls starting to buckle, as is visible in Figure 9, indicated by the boxes. Thereafter, at higher strains, the cell corresponding to this cell wall is fully collapsed and other cells start to collapse. Finally, the response stiffens due to the collapse of most of the cells. The onset of densification is relatively low, which is due to the relatively high volume fraction of these structures. For all the simulations, the densification strain varied between εln = 0.3 and 0.6.
To qualitatively validate the results shown in Figure 9, the deformation mechanisms in an open-celled polyurethane (PUR) foam were determined from in situ compression experiments during X-ray CT. This type of foam is chosen since the structure of the 2D FE models corresponds to that of a 3D open-celled foam.
For the in situ compression experiments, a poly(methyl methacrylate) (PMMA) loading apparatus was used, which is X-ray accessible over 360°. The uniaxial compression was displacement controlled with a microspindle (resolution of 50 μm). Samples of 5 mm in diameter and 7 mm in height were milled with a hollow saw from a foam slab that was cooled with nitrogen.
The in situ deformation of the open-celled foam in 3D is given in Figure 10. Initially, the cell walls are straight, where after a deformation of εln = 10.5%, the first cell walls start to deform by bending. At a strain of εln = 16.3%, some cell walls buckle as shown in Figure 10. The same deformation mechanisms were found with the FE simulations, as shown in Figure 9, at nearly the same deformation loads.
Element and Model Size
Next, the influence of discretization and model size on the mechanical behavior are discussed. Therefore, the effective modulus E* is calculated between 0 ≤ εln ≤ 0.02 and normalized with the Young's modulus E of the linearized behavior of the polymer base material. The effective normalized modulus is then scaled with the corresponding volume fraction vf to reduce the effect of the variation of local volume fraction (see Fig. 7). The average effective normalized modulus E*/(Evf) as a function of the mean number of cells in the model is shown in Figure 11. The dashed horizontal line is the average effective normalized modulus of CF6 (7.5 μm), which is taken here as a reference. From these results, it is clear that the smallest model size leads to an overestimation of the modulus and shows the largest spread. Because of the small number of cells in the model, either relatively stiff or compliant behavior is found. By increasing the model size, and consequently including more cells in the model, a converging stiffness is found and at least 112 – 122 cells should be in the model, for accurately estimating the effective Young's modulus. This is in agreement with the conclusions of Onck et al.5 for regular structures and those of Li et al.11 for Voronoi structures.
From Figure 11, it is also clear that for almost all model sizes, the models with a resolution of 11.25 μm show the stiffest response. This scale being stiffer than the scale of 7.5 μm is attributed to the addition of material in the remaining cell walls, as a result of the loss of detail, noise removal, and the volume fraction equivalency. Note that this effect partly originates from the choice of a high volume fraction. One would expect that for the same reason the scale with the lowest resolution (15 μm) shows an even stiffer response. However, due to loss of connectivity in these models, some cell walls do not contribute to the stiffness (see Fig. 6). Similarly, fractured cell walls have also been found to lead to a reduced effective modulus.10, 19
To determine the effects of model size and element size on the large strain behavior, a mean stress–strain response and corresponding standard deviation are determined. Figure 12(a) shows the mechanical response for models with size CF4 and a resolution of 7.5 μm taken from eight different slices. The gray band represents the standard deviation, where the solid line indicates the mean response. Since for some models the simulations did not converge at a certain strain, some discontinuities in the solid lines and gray bands are present. As expected, this gray band decreased in width by increasing the model size due to the smaller spread in volume fraction and the decreasing influence of the boundaries by increasing the volume (not shown here).
Properties such as plateau modulus and densification strain are hard to determine uniquely, since in some cases there is no clear transition between different regimes. Still, to compare the regime after linear elasticity, the macroscopic stress–strain curves are given in Figure 12(b). The responses of models with size CF4, discretized with different element sizes, are shown. The data show that the effective mechanical behavior of the model with the middle resolution and that of the coarsest model are approximately equal and larger than that of the finest model. This can be accounted to the buckling of the cell walls. Since for a coarse model, detail is lost, the remaining cell walls are thickened and need more force to buckle. Within almost all responses it is found that models with 15 and 11.75 μm element size respond the same and more stiff than the finest element size in the elastic buckling regime. Therefore, there is a balance between cell wall thickening and the loss of connectivity for both scales, resulting in almost the same level in the elastic buckling regime.
The influence of the model size on the mean macroscopic response is shown in Figure 13 for the smallest element size of 7.5 μm, since the models containing larger elements respond too stiff, as shown in Figure 12(b). For clarity, no standard deviation bandwidth is shown. From this figure, it is clear that at least 122 – 142 cells in the model should be taken to describe a representative stress–strain behavior within the large strain regime. Studies on artificial structures report values between 82 for regular structures5 to 192 cells for irregular structures11 for 2D models.
X-ray CT is successfully used to characterize the microstructure of a close-celled polymeric foam. 2D FE models are derived from the segmented reconstructions and used together with a hyperelastic material model to determine the influence for this approach of (i) element type, (ii) element size, and (iii) model size on the mechanical response determined with FE analyses. The results show no significant difference between the use of triangular elements or a combination of triangular and quadrilateral elements. On the other hand, it is shown that the element size significantly influences the mechanical response. The models are based on volume equivalency for all element sizes, so smaller elements are able to describe more detail than larger elements. Because of the loss of detail, the remaining structure is thickened resulting in a stiffer response. However, too coarse elements cause loss of connectivity in cell walls, reducing the stiffness as is also seen in studies of fractures cell walls.10, 19 It was found that at least three to four elements over the thickness should be taken for this type of structures. The study on the effect of model size revealed that at least 112 – 122 cells should be taken in the model to predict elastic properties, whereas for the large strain regime, 122 – 142 cells should be taken. Furthermore, a qualitative agreement was found with deformations observed during in situ compression experiments on an open-celled foam during X-ray CT. Deformation mechanisms such as bending in the linear elastic regime and buckling in the plateau regime were present in both experiments and FE results.
In this study, the volume fraction of the foam was artificially increased to be able to still perform a study on model size. It is believed that the same conclusions hold for the model size when using this approach for lower volume fractions. It is known that the initial shape of the cell size and their distribution through the structure influence the mechanical behavior of the structures.3, 10, 11, 16, 17 However, in this study, a realistic structure is taken where the cells are randomly distributed, which is a result of their process history. Also, it should be mentioned that previous studies showed that the intrinsic material behavior strongly influences the effective mechanical behavior.9, 27, 28, 34 Plasticity causes localization in the structure, and therefore, one can anticipate that, by incorporating plasticity, larger models may be needed.10, 19, 34 Therefore, future studies should incorporate the viscoplastic behavior of the polymer base material.
This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs (under grant number 07345).