As was stated in the Introduction, the primary purpose of this paper was to study random porous models made up of elongated objects, to see the difference pore morphology made in elastic properties, and to make comparisons between 2-D and 3-D. Figures 1, 2, 3, 4 displayed the differences between the two 2-D models. The elastic differences seemed mostly to be controlled by the amount of solid overlap at a given porosity. Figure 5 showed a small example of the capability of the models, of how the overlap area could be considered to be a different solid phase and so change the overall properties as the properties of this phase changed. In real materials, the properties of this phase could change as the constituents of the solid backbone had different bonding criteria, for example.
Figures 6, 7, 8, 9, 10 presented the elastic moduli results for the 3-D models, and a comparison of the extrapolated O-HV 3-D results with various experimental measurements. It seemed clear that extrapolation to the infinite resolution limit was necessary in order to get reasonable comparison to experiment. The agreement with experiment was reasonable, indicating that the kind of simple model studied can reproduce the important points of the real microstructure. The different 3-D models gave quite similar elastic results, unlike the case in 2-D. This implies that trying to choose between various 2-D models by comparing their elastic properties to 3-D experimental results may be difficult, as the percolation thresholds and microstructure, and therefore elastic differences, between the models may be exaggerated in 2-D as compared to 3-D.
In the whole area of relating microstructure and elastic properties, an important question for many people is the relation between 2-D and 3-D. A typical problem is predicting the elastic moduli of a real material, given only a 2-D image of the microstructure from some kind of micrograph. If the various phases are assigned elastic moduli, then the 2-D elastic moduli can be predicted, using the programs described in this paper or another, web-based package [27], which is similar. The question then becomes: of what value are these computed moduli? What relation do they have to the real, 3-D elastic moduli?
The models studied in this paper can perhaps shed some light on this question. We can first compute the elastic properties of a 3-D model to use as our "experiment." Then we can take 2-D slices of that model, taking these as our 2-D "micrographs," and compute their elastic moduli. These two sets of data can then be compared, to see if operating on the micrographs can tell us anything quantitative about the 3-D elastic properties.
But before carrying out this procedure, some qualitative insight can be brought to this question by using the 2-D model microstructures as intermediates between 2-D slices of 3-D microstructures and the 3-D microstructures themselves. Figure 11 shows a visual comparison between slices from the 3-D models (Figs. 11a,c,e, corresponding to Figs. 6a,b,c) and images of the direct 2-D models (Figs. 11b,d,f) (note that the 2-D O-R model was not previously discussed, but is assembled analogously to the 3-D version). It is important to recall that in the 2-D models, all bars are oriented in the plane, while in the 3-D models, many bars are not oriented in the plane. A planar slice in 3-D, then, will cut through the mid-section of many bars, resulting in small pieces in the slice that tend not to be well-connected in 2-D. This will result in a loss of stiffness at the same porosity, as compared to the direct 2-D models. So at equal porosities, the direct 2-D models should be stiffer than the 2-D slices of the equivalent 2-D microstructures.
Figure 11: Comparison of 2-D slices of the 3-D models (Fig. 6) and the corresponding direct 2-D model (Fig. 1): (a) O-HV (3-D), (b) O-HV (2-D), (c) HCSS-HV (3-D), (d) HCSS-HV (2-D), (e) O-R (3-D), and (f) O-R (2-D). Dark gray = pores, medium gray = unoverlapped solids, white = overlapped solid regions.
One can compare the 3-D and 2-D models by comparing their connectivities. The O-HV 3-D model reveals a percolation threshold of about 80 % porosity. The 2-D O-HV model had a percolation threshold of about 60 %. Therefore, at the same porosity, the 3-D O-HV model will be better connected, and therefore stiffer, than the 2-D O-HV model. This applies as well for the HCSS models. So if the 2-D models are stiffer than the 2-D slices, and the 3-D models are stiffer than the 2-D models, this implies that the 2-D slices will be less stiff than the 3-D microstructures that they come from. This is assuming, of course, that the elastic moduli of the backbones, in 2-D and in 3-D, are similar.
In order to carry out our program of quantitatively comparing 2-D computations to 3-D results, a question that must first be decided is: (1) What solid moduli should be used in the 2-D slices? A related question is: (2) How does one relate 2-D computed moduli to 3-D measurements?
One strategy for solving problem (1) would be to take the 3-D solid elastic moduli, make plane stress or plane strain moduli from them, and use them as input into the 2-D finite element computation of the material. Note that since either the programs used here or the OOF software [27] are digital image-based, they can easily operate on a real image that has been suitably processed to fit input requirements [27, 28]. A strategy for solving problem (2) might be to take the computed 2-D moduli, treat them as plane stress or plane strain moduli, and work backward to the 3-D equivalents, in order to compare to the 3-D results. This procedure, however, is not valid, as has been detailed in a recent paper [29]. The best one can do, then, is either to make plane stress or plane strain moduli from the measured 3-D solid backbone moduli and use them as the solid backbone moduli in the 2-D slices. One can then directly compare the 3-D measured moduli to the 2-D results.
Figure 12 shows the results of this procedure, where the values computed for K and G, the effective bulk and shear moduli, are compared, vs. porosity, for the full 3-D models and for 2-D slices of the 3-D models. Both choices of solid backbone moduli, plane strain and plane stress, were made for the 2-D slices. It can be seen in Fig. 12a and 12b that both the plane stress and plane strain solid moduli in the 2-D slices result in lower stiffnesses than for the full 3-D models (the "true experimental" result in this case), in agreement with the qualitative arguments presented above. So the plane stress or plane strain procedure outlined above does not work in this case, in the sense that it does not well reproduce the true 3-D results. We might guess from the results that it will not work in any case, at least not quantitatively, because of the differences in connectivity between 2-D and 3-D. However, for zero porosity samples, this procedure will work perfectly, so we expect that the procedure will work better and better as the porosity decreases.
Figure 12: Relative (a) bulk and (b) shear modulus vs. porosity for the full 3-D O-HV model and for 2-D slices. For the 3-D results, the solid moduli were the spherical average of typsum crystal properties, and for 2-D the solid moduli were plane stress and plane stain versions of the 3-D solid elastic moduli.