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For isotropic two component mixtures, in 2-D or 3-D, there are exact results connecting field fluctuations and the effective conductivity [20]. For elastic problems, one can also exactly relate strain averages to the effective bulk and shear moduli [20], though this topic is not discussed in this manual.
For conductivity, one must first formulate the average over a phase:
The exact relation can then be stated simply:
where again σ is the effective conductivity. If σ is known analytically as a function of σj, then this differentiation can be readily carried out. If not, this derivative can always be evaluated numerically, by evaluating σ for σj ± δ, with δ being a small number.
Consider a system where σ is analytically known. The checkerboard microstructure, as shown in Fig. 6, can be evaluated by the Keller-Mendelson formula given above. Since the microstructure is clearly invariant under inversion of σ1 and σ2, the effective conductivity must then be σ = (σ1 σ2)1/2.
Figure 6: Showing the
checkerboard microstructure, with dark gray being phase 2 and light gray phase 1.
The exact formula then gives < E2 >2 / < E2 > = (σ1 / σ2)1/2. Fig. 7 shows the numerical results, compared to the exact theoretical results, for a 128 x 128 system for a variety of conductivity ratios. Excellent agreement is shown, with some systematic disagreement growing at larger values of the conductivity difference.
Figure 7: Showing the effective conductivity and average of the electric field magnitude
squared in phase 2 for the checkerboard, as a function of σ2. The points
are numerical finite element results, and the lines are the
exact results discussed in the
text. The system size was 128 x 128.
If we fix the conductivity ratio between the two phases to be 10, we can then examine how well the field averages and effective conductivity are computed as a function of system size L x L. Table 8 gives the data obtained.
| 16 | 3.4442 | 3.1623 | 8.9 | 0.4132 | 0.3162 | 30.7 |
| 32 | 3.3232 | 3.1623 | 5.1 | 0.3791 | 0.3162 | 19.9 |
| 64 | 3.2550 | 3.1623 | 2.9 | 0.3569 | 0.3162 | 12.9 |
| 128 | 3.2159 | 3.1623 | 1.7 | 0.3424 | 0.3162 | 8.3 |
| 256 | 3.1934 | 3.1623 | 1.0 | 0.3330 | 0.3162 | 5.3 |
| 512 | 3.1804 | 3.1623 | 0.6 | 0.3269 | 0.3162 | 3.4 |
| 1024 | 3.1728 | 3.1623 | 0.3 | 0.3230 | 0.3162 | 2.1 |
Table 8:
Size dependence (L x L system) of effective conductivity and field average for checkerboard,
as determined by finite element method. Each individual "check" of the checkerboard
is L/2 x L/2. The conductivity ratio
σ2 / σ1 = 10 .
Next: Equal shear modulus Up: Exact solutions for Previous: Keller and Mendelsen
