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Numerical Results

A series of five realizations of microstructures were created using the algorithm discussed in Sec. II, for a range of concentrations c₁ and c₂. The ratio of E₁/E₂ was held fixed at 10, and a range of Poisson ratios ₁ = ₂ was investigated. Fig. 2 shows the 3-D results for the composite Poisson's ratio, averaged over the 5 realizations, as a function of c1. The variation of the effective moduli between configurations was about 5%, as discussed above, with the Poisson's ratios sometimes varying up to 10% over the five realizations. The system size was 64³, and w = 5 in the Gaussian kernel of eq. (3). The solid lines are the EMT predictions of eqs. (4)-(7). The EMT results describe the numerical results well. The numerical results are in agreement with the EMT predictions for the behavior of the effective Poisson ratio, so that when ₁ = ₂ = 1/5, = 1/5 as well for all volume fractions. Also, when ₁ = ₂ > 1/5, 1/5 < < ₁ = ₂, and when ₁ = ₂ < 1/5, ₁ = ₂ < < 1/5. This result is also predicted by a "differential method" EMT [21]. The EMT also accurately fits the Young's modulus results, which are not shown.

Figure 2: Showing the 3-D effective Poisson ratios vs. phase fraction for a stiffness ratio of 10. The lines are the graphs of the EMT equations (4)-(7), and the symbols are numerical results for the Gaussian kernel-based microstructure of Fig. 1.

We have carried out the equivalent simulations in 2-D, for values of Poisson's ratio that are now accessible to our previous algorithm, and have confirmed the EMT results found before [1]. A 2-D version of the microstructure algorithm in eqs. (1)-(3) was used, with w = 5 and a 128 x 128 unit cell. Results are displayed in Fig. 3, with the agreement between the simulation data and the 2-D EMT [3] slightly worse than in the 3-D case shown in Fig. 2.

Figure 3: Showing the 2-D effective Poisson ratios vs. phase fraction for a stiffness ratio of 10. The lines are the graphs of the 2-D EMT equations [3], and the symbols are numerical results.