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Discussion and Conclusions

The result found is that when ₁ = ₂, there is a critical value of Poisson's ratio, ^*, that separates the vs. phase fraction graph into two distinct regions. Below ^*, the effective Poisson's ratio is such that ₁ = ₂ < < ^*, and above ^*, ₁ = ₂ > > ^* . This result was true for d-dimensional EMT's based on spherical inclusions, and for the quite general Gaussian kernel-based microstructure in 2-D and 3-D. The question may be raised: is this behavior independent of microstructure, at least for isotropic systems, or is it dependent on the spherical inclusion microstructure? The Gaussian kernel-based microstructure is not based on spherical inclusions, but in the small c₁ or c₂ limit, when either phase is discontinuous, the inclusions that result should be roughly spherical, since the kernel in eq. (3) is isotropic. We have checked this limit visually, by generating images, and by computing K and G in these limits. The asymptotic slopes for the effective values of K and G agree fairly well with the exact result for spherical particles [16]. Visually examining the microstructural images in this limit also reveals that there is indeed a rough sphericity present in the isolated particles of the dilute phase. This may be the reason why the sphere-based EMT seems to work so well.

One way to check for the behavior of the effective Poisson's ratio in other microstructures is by using a 3-D EMT for inclusions shaped like ellipsoids of revolution [17], and studying its predictions for highly non-spherical shapes. Judging by the good agreement between simulation and EMT in Fig. 2, we expect that a similarly-derived EMT for ellipsoids of revolution should be reasonably trustworthy, at least as long as there is not too much of an elastic contrast between phases, and so do not carry out the numerical computations. These computations would, however, be possible for our algorithm. The only problem would be the lack of resolution in 3-D, because of computer memory limitations, to adequately represent a sufficient number of ellipsoids to get good statistics for the random geometry.

Figure 4a shows the effective Poisson's ratio vs. phase fraction for prolate inclusions with an aspect ratio of 20, which is a very elongated ellipsoid, and a stiffness ratio of 10 (inclusion to matrix). The same picture is preserved as in Figs. 2 and 3. Figure 4b shows a vertically expanded view of the same results, showing the S-shaped behavior around ₁ = ₂ = 1/5. This behavior persists from about ₁ = ₂ = 0.18 to about ₁ = ₂ = 0.23 . Outside this region, no significant qualitative difference is seen from the previous EMT and numerical results. The EMT results for oblate ellipsoids are similar.

Figure 4: Showing the 3-D effective Poisson ratios vs. phase fraction for a stiffness ratio of 10, for prolate ellipsoidal inclusions with an aspect ratio of 20. The lines are the graphs of the full 3- D EMT equations (Berryman, 1980): a) full scale (top), b) vertical expanded scale showing region around ^* = 1/5 (bottom).

We therefore tentatively suggest that the behavior of the Poisson's ratio when the two phase Poisson's ratios of the composite are equal is generic. To a good approximation, this behavior does not depend on microstructure, with the critical value ₁ = ₂ = ^* = 1/(2d - 1) in d dimensions. We expect that generally, when ₁ = ₂ = > ^*, the value of the effective Poisson's ratio will decrease as the two phases are mixed, with a minimum value, dependent on the stiffness contrast between the two phases [1], that is bounded below by ₁ = ₂. When ₁ = ₂ < ^*, the value of the effective Poisson' ratio will increase as the two phases are mixed, with a maximum value that does not not exceed ₁ = ₂ and that also depends on the elastic stiffness contrast between the two phases.

Finally, the elastic algorithm presented in this paper has shown itself to be a straightforward and accurate way of computing the linear elastic properties of heterogeneous materials. The only limitation of the algorithm is if the maximum size digital image that can be handled by a given computer is large enough to adequately represent the microstructure of interest.