D-EMT Results-Elastic moduli of a material containing composite inclusions: Effective medium theory and finite element computations

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.

**Figure 5:** Showing the D-EMT results for K and G compared to the numerical results for the 2-D 0.3 matrix area fraction model.

**Figure 6:** Showing the D-EMT results for K and G compared to the numerical results for the 2-D 0.5 matrix area fraction model.

**Table 4:** A list of numerical data and D-EMT results for the 2-D, 0.3 matrix area fraction system.
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

**Table 5:** A list of numerical data and D-EMT results for the 2-D, 0.5 matrix area fraction system.
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.

**Figure 7:** Showing the D-EMT results for K and G compared to the numerical results for the monosize sphere 3-D model. The matrix volume fraction is 0.668.

**Table 6:** A list of numerical data and D-EMT results for the 3-D, 0.668 matrix volume fraction, one sphere system (Fig. 7).
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

**Figure 8:** Showing the D-EMT results for K and G compared to the numerical results for the two-size sphere 3-D model. The matrix volume fraction is 0.517.

**Table 7:** A list of numerical data and D-EMT results for the 3-D, 0.517 matrix volume fraction, two sphere system (Fig. 8).
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