2-D Model Results

Figure 3 shows how the digital resolution error is dealt with for the O-HV model. The elastic moduli were computed at different resolutions and then plotted against the reciprocal of the number of pixels per side of the unit cell, 1/N. For example, if 600 21 x 3-size bars were used to construct a 200 x 200 pixel unit cell system, then to double the resolution, the same number of 42 x 6-size bars would be used for a 400 x 400 pixel unit cell system. The value of C₁₁ is shown in Fig. 3, as it was typical of all the other elastic moduli. The result was found previously that in the large N limit, where "large" is different for different models, the moduli scale like 1/N [15, 18]. By extrapolating to the N= limit (1/N=0), the true value for infinite resolution can be accurately approximated. This procedure has been checked for a single sphere in a matrix, where the true value is known [19], and was found to be accurate.

The three lines on Fig. 3, for the three different porosities, seem to be similar. However, if we examine the percentage difference between the 200² system values and the infinite limit extrapolated values, we find that this difference depends roughly linearly on the porosity (10 % at the lowest porosity, and about 40 % at the highest porosity). This is not hard to understand. If the porosity were zero, extrapolation would make no difference at all, as the solid elastic moduli are replicated exactly in each solid pixel. Even one pixel, representing the entire system, would give the correct elastic moduli. As porosity increases, the model becomes more and more random, with more intricate morphology, so that more and more resolution is needed to correctly represent complex shapes and compute the elastic properties. This was found to be true for the 3-D models as well.

Figure 3: Resolution influence on the value of C₁₁ for the 2-D O-HV model, at different porosity fractions. The three system sizes are (N² in pixels) 200², 400² and 1000² pixels.

All the 2-D models considered had similar asymptotic behavior. Since in 2-D, we were mainly interested in comparisons between the different microstructures, only non-extrapolated results were used, in order to save computational time. The extrapolated results are used later in the paper when comparing 3-D model results to experimental data, since extrapolation did make a significant difference in the computed quantities.

In all the models considered in this paper, errors due to the finite size effect and statistical fluctuation were much less than the error induced by digital resolution. When reasonably far away from the percolation threshold in any model, 2-D or 3-D, it was found adequate to use only one realization of each model, as these errors were only on the order of 1 % to 2 %. However, near the percolation threshold, the statistical fluctuation and finite size effect errors increased greatly. Since we are not very interested in behavior near the percolation threshold, at least not in this paper, no additional averaging was done to get less uncertain results in this regime. Consequently, the details of the elastic behavior near the percolation threshold have associated large error bars (not shown on the graphs).

Figure 4a shows the Young's modulus (isotropically averaged for the HV models) of the two different 2-D models, plotted vs. porosity. Note that the HCSS models tend to be somewhat less stiff than the overlapping models. Comparing with Fig. 2, one can see that at a given porosity, there is more overlap area for the overlapping models than for the HCSS models. Since this overlap area is what gives the models their stiffness, more overlap area translates to higher stiffness at the same porosity. Also, going along with this fact, the percolation thresholds for the models are different. The percolation threshold, which is the porosity at which the solid phase becomes geometrically disconnected, is somewhat higher for the O-HV model (about 0.6) than for the HCSS-HV model (0.45). That implies that at a given porosity, where both models are still percolated, the O-HV model will be better connected than the HCSS-HV model, causing it to be stiffer. This "better connectedness" is clearly also correlated closely with the overlap fraction (Fig. 2).

Figure 4b shows the Poisson's ratio for the 2-D models, plotted against porosity. Similar behavior is seen for both models. The solid Poisson's ratio, 0.45, may seem high, but note that eq. (5) allows values of in 2-D up to 1, when the allowable maximum in 3-D is only 1/2. Adding porosity seems to lower the effective Poisson's ratio. As is known in 2-D exactly, the Poisson's ratio tends to flow towards a fixed point, as the percolation threshold is approached. The data shown in Fig. 4b then implies that for both models, the value of this fixed point is between 0.2 and 0.3. The large uncertainties in the data near the percolation threshold do not allow any more specific statement to be made. If, however, the fixed point were 0.45 or higher, then adding porosity would make the Poisson's ratios increase with increasing porosity. We shall see an example of this kind of behavior in 3-D.

Figure 4a: Angularly averaged Young's modulus versus porosity of the 2-D models (200² pixels).

Figure 4b: Averaged Poisson's ratio versus porosity of the 2-D models (200 ² pixels).

Figure 5 shows an interesting example of what was briefly touched on in Section 2: the flexibility of digital models. For the O-HV and HCSS-HV models, the overlap area was separately labeled from the other solid parts, and allowed to have a Young's modulus different from the other parts of the solid backbone. This value was only allowed to be less than that of the bulk crystal backbone, as larger values were thought to be unphysical. The Poisson's ratio remained the same, however. Both systems were at a porosity of 27 %. The overlap fractions, defined as the percentage of solid material that belongs to two or more bars, is 35 % for the O-HV model and 31 % for the HCSS-HV model. In Fig. 5, the overall Young's modulus of each model, normalized by the Young's modulus when all solid parts had the same moduli (given for the two sets of data in Fig. 4a, 27 % porosity), is plotted vs. the fraction (less than one) of E_o/E_no for the solid phases (o = overlap, no = non-overlap). Because of the different normalization for each model, the two sets of data overlap at unity when the ratio E_o/E _no = 1. It is interesting to note that the two models both now fall on nearly the same curve, as opposed to Fig. 4a. Figure 5 shows a different kind of dependence on microstructure than does Fig. 4a, but the two graphs are not expected to be similar.

One should note that there is another method used for computing the effective elastic moduli of porous materials based on a finite difference scheme [20]. This works well on uniform solid phases, but is very difficult to implement when there is more than one solid phase with different elastic properties. For example, the computational results shown in Fig. 5 would have been very difficult to obtain using this method. Also, the finite element technique [4, 10] can handle any symmetry elastic moduli tensor, which is possible but difficult in the finite difference technique [ 20].