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Small contrast of properties

 

When the difference between the properties of a two phase material is small, a power series expansion that can be made for the effective properties in terms of this difference becomes useful. A conductivity result which is true for general two phase isotropic microstructures has been derived by Brown [16] and extended by Torquato [14]. The effective conductivity, to second order in the difference (σ2 − σ1), is given by:

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where d is the dimensionality and ci is the volume fraction of phase i. The coefficients for the O(σ2 − σ1)3, and higher order terms involve details of the microstructure, and are given in terms of various correlation functions over the phase geometry [14,16].

A 22 x 22 x 22 pixel cube (phase 2) centered in a 30 x 30 x 30 unit cell was used to test Brown's result for the case of small contrast between the two phase conductivities. For this case, we have c1 = 0.39437, and c2 = 0.60563. We take σ1 = 1 always, and vary σ2 between 1 and 1.3. Figure 5 shows the result, plotted against σ2, where the quantity σ1 + c2 (σ2 − σ1) has been subtracted from both the numerical and theoretical results. This has been done to show how the numerical results compare to the theoretical results in the quadratic term. The numerical data follows the theoretical line very closely. As the value of σ2 gets larger, there should also be contributions from the cubic term, which could account for the small differences between the numerical and theoretical results. For cubic symmetry, the effective conductivity tensor is isotropic, so Brown's results can be used to analyze this system.

  


Figure 5: Showing σ vs. σ2 when σ2 differs only slightly from σ1. The points are finite element data, and the straight line is Brown's exact expansion to second order in the contrast σ2 − σ1).

The same procedure can be carried out for the elastic moduli [17]. A simple way to derive this result, at least up to second order in the modulus contrasts, is to take Hashin's bounds [15], which bound the effective properties between an upper and lower limit, and expand them to second order in the modulus differences. These bounds are known to be exact to second order in the modulus differences, and in fact the upper and lower bounds agree exactly to this order. The 3-D results for the effective bulk modulus K and shear modulus G are:

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In 2-D, the results are

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Similar results as those for the small contrast in conductivity case are obtained when testing the accuracy of how well the finite element method computes the small contrast in elastic moduli case. The bounds themselves [15] can also be useful for checking results, since whatever effective elastic moduli are found for an n-phase composite must lie between the appropriate n-phase bounds. Just as in the electric case, the coefficients for the O( )3 and higher order terms involve details of the microstructure, and are given in terms of various correlation functions over the phase geometry [17].

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