Analysis of analytical effective elastic moduli predictions

We now compare the different analytical predictions of effective moduli with the finite element data. We have chosen to study the two most common models and only consider the moduli appropriate for the W-Ag composite studied above. The results are shown in Fig. 12(a) for overlapping spheres and in Fig. 12(b) for the single level-cut GRF or excursion set of Quiblier [model N(c=0)]. The self-consistent method provides a very good estimate of E_e for model N(c=0), but not for overlapping spheres. This might be expected because the N(c=0) model is symmetric with respect to phase-interchange (like the SCM) while the overlapping sphere model is not. As stated above the application of the generalized SCM is difficult because it is not obvious which phase should be chosen as the 'inclusion' phase. For the overlapping sphere model the tungsten phase is comprised of spheres (at 80% volume fraction), so is the more likely choice for the inclusion phase. Nevertheless, we report both estimates (80% W inclusions and 20% Ag matrix or 20% Ag inclusions and 80% W matrix) for both models. For either choice, the GSCM fails to provide an accurate estimate. Indeed, above the melting point of silver the GSCM vanishes for the case 20% Ag matrix case since the matrix phase is now completely soft. For the overlapping sphere model the Beran, Molyneux, Milton and Phan-Thien bounds are calculated using the microstructure parameters $\zeta_1=0.52$ = 0.52, $\eta_1=0.42$ = 0.42 [44,9]. Below the melting point of silver (where the contrast between the phases is moderate) the upper bounds provide a very good estimate of the effective moduli.

A brief discussion of the effect of elastic contrast is necessary here. We have already noted that the analytical predictions of effective moduli do not explicitly depend on microstructure, but have a ``built-in" microstructure. The elastic contrast, the ratio between the phase moduli, will determine how sensitive the effective moduli actually are to microstructure. For example, in the case of a two-phase composite having equal shear moduli but different bulk moduli, there is a simple exact formula for the effective bulk modulus which is totally insensitive to microstructure [21]. In the case of small contrast, the effective moduli can be expressed exactly as a power series in the moduli differences  [42]. Up to second order in this difference, at any volume fraction, the coefficients of the power series are not dependent on anything but the volume fractions and the individual phase properties. Therefore at small contrast, analytical predictions of effective moduli that explicitly depend only on volume fractions and phase moduli should all work well.

Figure 12: Comparison of theory (predictions and bounds) with finite element (FEM) calculations for the Young's modulus of (a) the overlapping sphere model and (b) the single-cut GRF model [N(c=0), see Fig. 10]. The standard (SCM) and generalized (GSCM) self-consistent methods are shown, as are the Hashin and Shtrikman (HS) and Beran, Molyneux, Milton, and Phan-Thien (BMMP) bounds.

(a)

(b)