It has been mentioned above, qualitatively, how the cement particle phase might contribute to the overall elastic moduli at early hydration age. It is interesting to study quantitatively how each major phase contributes to the overall elastic moduli. One can develop a measure of this analytically. The overall elastic moduli are found by averaging each component of the stress tensor and strain tensor over the entire microstructure:
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| (2) |
and defining the effective elastic moduli via
 | (3) |
If one wants all the individual components of the elastic moduli tensor, then one can apply one component of the strain tensor at a time [17]. The average strain tensor is equal to the strain tensor applied to the periodic unit cell of the cement paste model [2-4]. One can then read off the components of the effective elastic moduli tensor by dividing the appropriate components of the average stress tensor by the value of the average strain. We can rewrite eq. (2) for the average stress tensor by:
 | (4) |
where the subscript m indicates the m'th distinct chemical phase out of N total phases, the integral is now taken only over the volume of the m'th phase, and cm is the volume fraction of phase m. We can compute the partial contributions of each phase to the overall effective elastic moduli, and express them as a fraction of the total elastic moduli. We can then write:
 | (5) |
where <K>m, for example, represents the average bulk modulus, as defined by eq. (4), in phase m. Dividing both sides of eq. (5) by the effective property, we then obtain
 | (6) |
where now the fraction of the bulk modulus supplied by phase m is equal to the product of cm, the volume fraction of phase m, and a coefficient km. A similar relation holds for the shear modulus.
To analyze how each phase fraction contributes to the overall elastic moduli, we have used the late age D cement results. Figure 12 shows the results, broken down into five parts. Graph (a) shows how the phase volume fractions vary with w/c ratio. The saturated pores and the residual cement vary the most, while the CH and C-S-H phases are fairly flat vs. w/c ratio. The cement phase includes only remnant clinker phases − C3S, C2S, C3A, and C4AF. All other phases, including any residual form of gypsum phase, are included in the "Other" phase. This joint phase must be fairly flat vs. w/c, since the rise in the saturated porosity with increasing w/c ratio approximately cancels out the corresponding decrease in residual cement, and the CH and C-S-H phase are, as said above, also fairly flat with w/c. Graph (b) shows how each of these five phases contribute to the overall bulk modulus, as a function of w/c ratio. At the lowest w/c ratios, the residual cement can contribute up to 40 % . At the highest values of w/c ratio, the C-S-H phase contributes the most, about 45 %, while the saturated porosity now contributes about 6 %. Remember that the water in the saturated porosity has a bulk modulus of 2.0 GPa, so it can contribute to the bulk modulus in dynamic modulus measurements. The CH phase contributes between 20 % and 30 % over all the range of w/c ratios studied, so it has to be classified as a major contributor to the bulk modulus. Figure 12c shows how each phase contributes to the overall shear modulus. The saturated porosity is of course missing from this graph, as the water cannot sustain a shear stress, so it cannot contribute to the shear stress average over the microstructure. At the lowest w/c ratio, w/c = 0.25, the residual cement contributes 40 % of the shear modulus, the C-S-H 28 %, and the CH and the "Other" phase each contribute about 16 %. The contributions of the CH and "Other" phase both increase slightly with w/c ratio, but the contribution of the C-S-H phase rises to 50 %, while the residual cement contribution falls off to near-zero since the volume fraction of residual cement also decreases sharply.
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Figure 12: (a) Volume fractions of phases, fractional phase contributions to (b) bulk and (c) shear moduli, and phase volume fraction coefficients for (d) bulk and (e) shear moduli, for the D cement at late ages of hydration over a range of w/c ratios.
There is an additional composite theory concept needed first before Figs. 12d and 12e can be discussed. The question to consider is − what is the value of the coefficients km (gm) in eq. (6)? In general, these coefficients can only be exactly known if the complete elastic solution for a random composite system were known, which is usually not the case. But in the dilute limit, these coefficients are exactly known, at least up to first order in volume fraction. For simplicity, consider the bulk modulus of a two phase composite, where both phases are elastically isotropic. When only a dilute amount of phase 2 is present, the composite bulk modulus Keff can be written exactly as a power series in c, the volume fraction of phase 2 [2, 3],
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(7) |
where Ki is the bulk modulus of phase i, and [K] is the intrinsic bulk modulus that depends on the shape of the phase 2 inclusion and the ratio K2 /K1 [2, 3, 41-43]. If one rewrites eq. (7) slightly, dividing both sides by Keff and showing the phase contributions more clearly, then up to first order in c one has exactly
 | (8) |
where c1 = 1 − c and c2 = c. Equation (8) looks exactly like eq. (6), except that now the coefficients k1 and k2 are known exactly: k1 = K1/Keff and k2 = (K1+[K])Keff. Each term consists of a factor, defined as km, times the volume fraction of that phase, as in eq. (6). Now consider the second line in eq. (8). It is known that if K1 = K2, then [K] = 0 [42]. In that case, the factor for phase 2, k2, which multiplies the phase 2 volume fraction, c2, is then equal to 1, and so is k1, since in this case Keff = K1. Though eq. (8) becomes a trivial identity, this exercise does show that the coefficients km become equal to one when the bulk modulus of a phase equals the overall effective bulk modulus. In the general, multi-phase case, we have the results of eq. (6).
When the factors km and gm are computed for a general composite, we might expect that the case when Km = Keff might be an important point for the value of km. In particular, the factors km could have the property that km = 1 when the moduli of the m'th phase equals the overall moduli, which is taken to be the effective "matrix phase." There is no reason to expect, however, that an exact result from the dilute limit will still hold in the multi-phase, non-dilute limit.
Figure 12d shows the values of km and the equivalent quantities for the shear modulus, gm, are plotted in Fig. 12e. All the values increase with w/c ratio, as the overall elastic moduli decrease. It is known that intrinsic moduli always increase as the ratio of the elastic moduli of the inclusion to the matrix phase increases. These quantities do the same. The phase elastic moduli are fixed, so as the "matrix" moduli decrease, the ratio increases. In Fig. 12d, only the C-S-H and the "Other" phase cross the value of 1. The C-S-H phase crosses the value of 1 at a w/c ratio of approximately 0.5. The bulk modulus of the cement paste at this point is K = 15 GPa, which is quite close to the value of the bulk modulus for C-S-H , K = 14.9 GPa (see Table 1). In Figure 12e, again only the C-S-H and "Other" phases cross the value of 1. The C-S-H phase coefficient crosses unity at w/c = 0.45. The cement paste shear modulus at w/c = 0.45 is G = 8.7, remarkably close to the C-S-H phase value of 8.96 GPa (see Table 1). The "Other" phase is a conglomerate of phases, some having CH-like elastic moduli and some having C-S-H-like elastic moduli, so it makes sense that the average stiffness of this phase is somewhat greater than that of C-S-H. Therefore the lines for this phase cross the value of 1 in Figs. 12d and 12e at values of w/c less than that for C-S-H.
Figures 12d and 12e also show that the values for CH and cement are always greater than 1. The largest cement paste moduli computed in these results was K = 24.8 GPa and G = 13.5 GPa, at w/c = 0.25, 56 d. The CH elastic moduli are K = 40 and G = 16, while those for cement are K = 105 and G = 45, all greater than the cement paste. Therefore the lines for these phases never cross 1.
If we think of the phase parameters above as something like intrinsic moduli, it is known that as the shape of an inclusion deviates from a sphere, the intrinsic moduli will increase [42]. Figure 13 shows the cross-sections of the L cement paste microstructures at late age hydration, where nearly full hydration has been achieved. The remnant cement particles in the lower w/c ratio pastes look less spherical than those shown for the 56 d D cement pastes in Fig. 4. It is possible that these phase parameters could be higher in the late-age L cement pastes because of this shape factor. This was checked, and it was found that the phase parameters for the L cement were only slightly larger than those for the D cements, in the expected direction, but not larger than the probable numerical uncertainties.
Figure 13: Microstructural slices for the simulated L cement pastes at nearly full hydration. From left: w/c = 0.25, w/c = 0.35, w/c = 0.50, and w/c = 0.60. Gray scale is similar to that in Fig. 4.
