Conclusion

Throughout this paper, we have treated the finite element computation method as being perfectly accurate, so that comparisons of elastic results to experiment were solely a test of how well the reconstructed microstructure compared to the real microstructure. This is not exactly true, since there are numerical errors in the finite element method  [18,17]. These are small, however, and are generally of about the same size or less than the differences seen between model computations and experimental data for the elastic moduli. There are also statistical sampling errors associated with the finite size ($\approx$ 40 µm of the models we employ to estimate the elastic properties. Since this is much greater than the correlation length of the samples ($\approx$ 5 µm - see Fig. 7) we again assume these errors to be small. Therefore, the good agreement between model prediction and experimental data seen in this paper is good evidence that the model considered is indeed capturing the main aspects of the experimental microstructure.

We have compared various theoretical results to finite element computations of the effective Young's modulus E_e for non-particulate media: a W-Ag composite and two model media (overlapping spheres and a single-cut Gaussian random field). The generalized self-consistent method (derived for particulate composites) did not provide a good estimate of E_e for the bi-continuous materials considered here. The standard self-consistent method provided a good estimate for the single-cut GRF and W-Ag composite. Since the method predicts zero moduli for porosity above 50% but the solid phase of the single-cut GRF remains connected up to porosities of around 90% [37] such agreement cannot be general. Upper bounds, calculated using three-point statistical correlation functions, provided a good prediction at low contrast (E₁/ E₂ $E_2\approx$ 6) for each composite. When one of the phases was completely soft the bounds lost predictive value. Therefore, for general composites, it is important to employ numerical computations of the effective moduli. For accurate numerical prediction of composite properties it is important that a realistic model be used. Model-based statistical reconstruction, based on the Joshi-Quiblier-Adler approach, appears to be a viable route for microstructural simulation. However, it is important that the models underlying the procedure be capable of mimicking the composite microstructure. We have shown how several different models can be employed to find a useful reconstruction.

Acknowledgements

A.R. thanks the Australian-American Educational Foundation (Fulbright Commission) for financial support and the Department of Civil Engineering and Operations Research at Princeton University where this work was completed. We also thank the Partnership for High-Performance Concrete program of the National Institute of Standards and Technology for partial support of this work.