Discussion and Summary

One limitation of the D-EMT equations is our ability (or inability) to match the microstructure of composites [41]. In concrete, for example, several modeling and experimental studies have shown that in a typical concrete, the shell regions are themselves overlapping and percolating [37,42,3]. The form of D-EMT considered in this paper does not reflect this fact. The model microstructures used to test the D-EMT, therefore, were carefully constructed to have non-overlapping and therefore non-percolating shell regions. However, whether or not percolation of a phase significantly affects the overall properties depends on the contrast of its properties with those of the surrounding phases [5,43]. In concrete, the shell moduli are less than the matrix moduli by a factor of at most 2-3, which is not enough contrast for percolation to be important [5]. So this deficiency in D-EMT should not significantly affect the accuracy of D-EMT for the concrete problem [20].

The exact effect of this aspect of the D-EMT was explored by creating a microstructure like that shown in the lefthand side of Fig. 2. The same number and type of inclusions were used, but now the shells were allowed to be overlapping, although phase 3, the inclusions, were still not allowed to overlap. There was therefore the same area fraction for phase 3, but the matrix area fraction was 0.340, instead of 0.303, and the shell area fraction became 0.238, instead of 0.275. The shell overlaps caused a somewhat smaller shell area fraction, and thus a larger matrix area fraction.

Figure 9 shows the numerical results, along with two sets of D-EMT results. The first, the dashed lines, are just the D-EMT results shown in Fig. 5. These actually work quite well, with a maximum error of less than 10 % at the highest values of E₂/E₁. Most real concrete materials have 0.1 < E₂/E₁ < 2.0, and in this region the D-EMT compares well with the numerical results. The second set of D-EMT results is for the case of a matrix area fraction of 0.340, which matches the real microstructure. In the numerical solution of the D-EMT equations, the variable is the matrix area fraction, so it makes sense to use the actual known value for the overlapping shell microstructure. In this case, there is significantly better agreement with the numerical results, as good as that seen in Fig. 5.

Figure 9: Showing the D-EMT and numerical results for K and G for the system when the shells were allowed to overlap. The matrix area fraction is 0.340. The dashed lines are the D-EMT results from the non-overlapping (no) shell case. The solid lines are the D-EMT results for the overlapping (o) case, where the true matrix area fraction was used (0.340 instead of 0.303).

In the 2-D models, the shell thickness was such that the shell area fraction was comparable to the matrix area fraction. This is why there was a significant difference in area fractions between the non-overlapping and overlapping shell microstructures. This would be the case also in the 3-D models considered as well. However, in most concrete materials, the shell is very thin compared to the inclusion diameters. The difference between non-overlapping and overlapping shell volume fractions would be quite small. Figure 10 shows, for a typical concrete aggregate (inclusion) particle size distribution, the shell volume fraction if the shells are considered to be non-overlapping, and the overlapping shell volume fraction as given by the Lu-Torquato formulae [44], plotted vs. the volume fraction of the aggregate, which is the only phase volume fraction that is precisely known in concrete. The Lu-Torquato formulae have been shown to be very accurate for concrete problems [7,11]. The aggregate volume fraction in concrete is typically 60 % to 70 %. Since the three-phase model thickness of the ITZ in most concrete materials is 20 µm or less, it is seen that there is little difference between the shell volume fraction for the two cases, which implies that the fact of shell overlap in real concrete will not make much difference to the utility of the D-EMT equations in the elastic case.

Figure 10: Showing the shell volume fraction, c₂, vs. the inclusion volume fraction, c₃, for a 3-D case where the size distribution of the particles was that of the aggregates in a typical concrete mixture. The three sets of curves are labelled with the thickness of the shell phase in micrometers. The solid curves are the shell volume fractions assuming no overlaps, while the dashed curves are the Lu-Torquato estimate [44], which has been shown to be very accurate for overlapping shells.

Since EMT is an uncertain approximation (having relatively crude control of the underlying microstructure), the results of this paper are especially important in carefully showing the expected accuracy of the D-EMT equations. We have shown that the new form of D-EMT worked very well for the class of problems considered. Checking the D-EMT against models for a material like concrete, where the particle size distribution of the inclusions is quite a bit larger (about two orders of magnitude) than that studied here, has not been done. The main difficulty comes in finding a numerical representation of the inclusion structure using a digital image. The pixel width must be at most 5 µm, in order to barely resolve the 20 µm ITZ regions. To look at even a 1000 mm³ cubic sample, which is small for concrete, a 2000³ model must be considered, which would require 1600 Gbytes of memory, and require weeks or months of CPU time to run [25]. This run time would of course be diminished by using massively parallel computers, and the memory required could be reduced by using an adaptive meshing scheme, to optimize the finite element mesh and so use fewer elements. The errors incurred using the D-EMT for a material like concrete could be significantly larger. But the excellent agreement with the numerical data found here strongly suggests that successful extensions to concrete are possible.

Tables 4-7 in this paper should be useful for other researchers who wish to test various forms of EMT or other approximate formulas, by listing accurate data for the linear elastic properties of non-trivial random systems. Using the information contained in Tables 1-3, the microstructures can be recreated easily, in case new numerical methods need to be tested. Modern computers and computer methods can now be used for the quantitative testing of approximate micromechanics theories on non-trivial, non-analytic microstructures. This will allow a better sorting of various equations into areas of greatest usefulness, and should inspire the creation of better, more accurate choices among the various theories available now [41] and possible in the future.