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# D-EMT Results

The D-EMT equations were solved for the four (two 2-D, two 3-D) different microstructures for which finite element results were obtained. By varying both the microstructure and the shell stiffness, a range of data was obtained to provide a rigorous test of the D-EMT results.

Figures 5 and 6 show the comparison between the finite element results for the 2-D microstructures (symbols) and the numerically integrated D-EMT results (lines). Both graphs show excellent agreement, with the best agreement being at the greater matrix area fraction (Fig. 6). This is not surprising, as the largest matrix area fraction has the fewest inclusions, and so is closer to the dilute limit, where the D-EMT is virtually exact. However, even for the 0.3 matrix area fraction system, Fig. 5, the agreement is still very good. Tables 4 and 5 show the actual numbers in the graphs, for closer comparison and future reference. Note that as the matrix area fraction decreases, the moduli curves become steeper. This is because the shell area fraction (c2) is becoming larger, and so changing its moduli has a greater effect on the overall moduli.

 E2/E1 K(data) K(D-EMT) G(data) G(D-EMT) 0.1 0.320 0.319 0.186 0.187 0.3 0.645 0.643 0.364 0.378 0.5 0.855 0.854 0.479 0.495 0.7 1.005 1.006 0.564 0.580 0.9 1.120 1.123 0.631 0.644 1.5 1.351 1.360 0.772 0.775 2.5 1.568 1.580 0.910 0.897 3.5 1.702 1.714 0.995 0.971 4.5 1.797 1.809 1.055 1.023 5.5 1.872 1.883 1.101 1.063 6.5 1.935 1.944 1.138 1.096 7.5 1.988 1.997 1.168 1.123 8.5 2.036 2.044 1.194 1.147 9.5 2.079 2.086 1.216 1.168 10.5 2.119 2.125 1.236 1.186

 E2/E1 K(data) K(D-EMT) G(data) G(D-EMT) 0.1 0.401 0.397 0.226 0.227 0.3 0.665 0.663 0.370 0.378 0.5 0.812 0.811 0.449 0.460 0.7 0.910 0.910 0.502 0.513 0.9 0.979 0.981 0.540 0.551 1.5 1.109 1.114 0.614 0.621 2.5 1.218 1.226 0.677 0.680 3.5 1.280 1.289 0.713 0.713 4.5 1.321 1.332 0.737 0.735 5.5 1.353 1.365 0.755 0.752 6.5 1.378 1.391 0.768 0.765 7.5 1.399 1.413 0.778 0.776 8.5 1.418 1.432 0.787 0.785 9.5 1.434 1.449 0.795 0.793 10.5 1.449 1.464 0.801 0.800

Figure 7 shows the comparision between D-EMT and finite element results for the 3-D system with one size of inclusions. The agreement is excellent, as can also be seen in Table 6. Figure 8 shows the same kind of comparison but for the two-size sphere 3-D system. The agreement in this case between D-EMT and finite element data is almost as good as in Fig. 7. The data for this case is in Table 7. It should be recalled here that the finite element results for the two-size sphere 3-D system were not checked as thoroughly as were the other systems, so there could be a larger degree of error in these results. It was found in the one-size sphere 3-D system that increasing the system size for the larger values of E2 tended to decrease the overall moduli. If the numerical results in Fig. 8 were to drop by only a few percent, the already good agreement with the D-EMT results would be substantially improved.

 E2/E1 K(data) K(D-EMT) G(data) G(D-EMT) 0.1 0.455 0.436 0.233 0.238 0.3 0.613 0.602 0.300 0.311 0.5 0.720 0.713 0.346 0.357 0.7 0.799 0.795 0.381 0.392 0.9 0.861 0.858 0.410 0.420 1.0 0.887 0.886 0.422 0.432 2.0 1.058 1.063 0.510 0.517 3.0 1.157 1.160 0.565 0.569 4.0 1.214 1.224 0.603 0.606 5.0 1.260 1.271 0.633 0.635 6.0 1.296 1.307 0.657 0.657 7.0 1.326 1.336 0.678 0.676 8.0 1.350 1.360 0.695 0.692 9.0 1.371 1.381 0.710 0.706 10.0 1.390 1.399 0.723 0.718

 E2/E1 K(data) K(D-EMT) G(data) G(D-EMT) 0.1 0.346 0.327 0.183 0.189 0.3 0.537 0.526 0.269 0.283 0.5 0.676 0.668 0.331 0.346 0.7 0.785 0.779 0.381 0.396 0.9 0.874 0.869 0.423 0.437 1.2 0.983 0.980 0.476 0.487 1.5 1.071 1.075 0.521 0.533 2.0 1.189 1.187 0.584 0.589 4.0 1.479 1.475 0.755 0.746 6.0 1.650 1.639 0.868 0.846 8.0 1.772 1.753 0.953 0.919 10.0 1.866 1.837 1.022 0.975

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