3-D Model Results

In Fig. 6 is shown 3-D views of the three 3-D models studied: O-HV (Fig. 6a), HCSS-HV (Fig. 6b), and O-R (Fig. 6c). The finite size and statistical fluctuation errors in these three models were similar, but how the digital resolution error scaled with resolution was different. This is because in the HV models, the shape of the bar was always properly represented, while in the O model, the pixel shape of the bars depended on the orientation of the bar.

Figure 6: Pictures of the 3-D models studies, same size bars: a) O-HV, b) HCSS-HV, and c) O-R. Dark gray = pore, medium gray = non-overlapped solids, white = overlapped solid regions.

To show the effect of randomness and the effect of orientation of the bars along the x, y, and z axes, the full elastic moduli tensor computed on one of the HCSS-HV runs in 3-D will be shown. The full tensor (Voigt notation) is given by (units in GPa):

We have shown above that the small cross terms in the upper right and lower left of the tensor do not contribute in the angular averaging process. They are included now just to show that these coefficients do exist, though their magnitude is only about 1 % or less of the major coefficients. They are not due to round-off error, because of their magnitude (much too large), and the fact that they appear symmetrically in eq. ( 16). Since each column in eq. (16) represents an independent run, and round-off errors are random, we would expect that if these elements were only a product of non-symmetric round-off errors, they would then appear less symmetric.

We now present the results for E for the three 3-D models in Fig. 7 at a system size of 100³. Recall that the HV models were rotationally averaged, while the random direction model was not. Surprisingly, all the data points for the three models seem to fall on roughly the same curve vs. porosity. There is little difference among the stiffnesses of the three models in 3-D, as opposed to 2-D, where there were significant differences (see Fig. 4a). This could be a sign that the percolation thresholds of the three models were similar. We did not pursue higher resolution studies of the percolation thresholds in 3-D to see if this was true. The companion computations for for all three models are also similar (not shown). So if K and G, which are a combination of E and , were plotted vs. porosity, there would still be little difference seen among the three models. Because of these results, we can focus on the O-HV model, as it is the simplest.

Figure 7: Young's modulus vs. porosity for the different 3-D models, 100³ systems (see Fig. 6). E_s = 45.7 GPa comes from the spherical average of the gypsum crystal elastic tensor.

Figure 8 shows the results for the Poisson's ratio for the O-HV model, with three different values of _s, at a size of 100³. Recall that _s = 0.33 is the isotropically averaged value for solid gypsum given in eq. (20), while the other two values are arbitrary. As porosity increases, three lines of points are formed, which tend to flow toward6the same fixed point of about 0.2 to 0.25. Note the three different line shapes (decreasing, flat, increasing) for the three different starting values for _s . This is non-intuitive behavior, and means that in the porosity limit where E goes to zero, whether at a non-zero or zero percolation threshold, the Poisson's ratio is determined by the structure of the material and not the value of _s [8]. However, as can be seen from Fig. 8, the value of _s does have a strong influence on the value of well away from this threshold. This flow diagram behavior seems to be universal, and has been seen in many model systems [2, 15, 21, 22]. In 3-D, unlike in 2-D, this behavior is probably not exact.

Figure 8: Effective value of vs. porosity for three different values of _s (0, 0.2 and 0.33) for the O-HV model, at a system size of 100³ pixels.

Since we wish to compare the 3-D model results to experiment, extrapolation to infinite resolution is of importance. Figure 9 shows the difference between the 100³ system and the infinite resolution extrapolation by displaying the results for E vs. porosity. The differences between the two sets of points is significant, showing the importance of extrapolation to get the correct results [15]. Also, Fig. 9 shows several sets of experimental results for gypsum plasters [23-26]. The best comparison is with the Acoustic method results, which measured the value of E with ultrasonic methods. The other experimental results used flexural tests to extract E as the slope of the stress-strain curve in the small strain limit, which usually give lower values than do ultrasonic techniques. Both sets of computations lie within the scatter between the different sets of experimental data, however. Note that the 100³ values tend to lie above all the experimental data, again showing the probable importance of extrapolation. The fact that at higher porosities, all the experimental data tends to fall above the model results indicates that the percolation behavior of how the real material approaches zero moduli is different from the simple overlapping object behavior of the models. It is important to be able to see how E varies with porosity in order to test a model. Single points of comparison are not enough to see if a model really faithfully captures how the properties vary with pore structure.

To set the O-HV 3-D model in context, Fig. 10 shows a comparison of the extrapolated Poisson's ratio data for the O-HV model with the results from other models taken from Ref. [15], all with _s = 0.2. The model results from Ref. [15] were also all extrapolated to the infinite N limit. The fact that the data spread out as the porosity increases is an indication that each model has its own flow diagram, and the fixed points towards which all values of tend to go as porosity increases are all somewhat different. These fixed points are definitely a function of microstructure. Also, there is increased statistical fluctuation and finite size uncertainty in the data as the percolation threshold is approached.

Figure 10: Comparison of the Poisson's ratio with increasing porosity for the O-HV model and other 3-D models computed by Roberts and Garboczi (14]. The value _s = 0.2 was used for all the models.