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3-D analytical result for dilute sand mortars

If the concentration of sand is small enough so each sand grain is relatively unaffected by its neighbors, then the overall shrinkage can be calculated analytically for spherical or ellipsoidal sand grains. The calculated form for this dilute limit for any shape of sand grain is:

ε*/ εp = 1 + χcs       (4)

where cs is the volume fraction of sand. The factor χ, a dimensionless parameter in 3-D or 2- D, depends only on the moduli and unrestrained shrinkage ratios between the aggregate, ITZ, and bulk cement paste, and on the ratio of the interfacial zone volume (or area) to the sand volume (or area), for a single sand grain and interfacial zone. The derivation of χ for spherical sand grains is given in the Appendix, in both 3-D and 2-D. To our knowledge, this derivation, though simple, has not been previously published. Figure 6 shows the results of this calculation in 3-D for ε*/ εp, plotted vs. Ei/Ep for the same values of εi/ εp, shown in Fig. 4, using a volume fraction of c = 5.9% to match the experimental data discussed below. The qualitative shape of the curves is strikingly similar to the numerical curves shown in Figure 4, which tends to justify the overall results generated using the 2-D mortar model for 3-D. The equivalent calculation for 2-D in the Appendix also gives qualitatively similar results.

Figure 6: 3-dimensional analytical results for overall shrinkage at a sand area fraction of 5.9%.

The 3-D analytical result for the shrinkage of dilute sand systems can be used to analyze experiments, if data for such a low sand concentration is available. Such data was measured by Pickett [28] and summarized by Hansen [29]. This data can be used to quantitatively determine the average interfacial zone characteristics, at least for the mortars in Pickett's study [28].

The data set described by Pickett was for a type III portland cement with water/cement ratio=0.35, approximately 65% hydrated, using Ottawa sand. A 5.9% volume fraction of sand is low enough to be considered dilute. A rough size distribution for Ottawa sand was derived from ASTM C778 and is shown in Table 1. The modulus of the aggregate was determined by Hansen [29] to be 4.29 times the bulk paste for Pickett's samples. In this work, the Poisson's ratio of the aggregate is set at 0.2 and the ITZ and paste at 0.3, as with the preceding computations. The value of ε*/ εp was measured to be 0.93 [28,29]. To fit this value using the 3-D analytical result, the analytical result for χ was averaged over the sand size distribution shown in Table 1. A variety of combinations of Ei/Ep and εi/ εp were tried, with Ei/Ep < 1 and 0.2 < εi/ εp < 2. When 0.3 < Ei/Ep < 0.5, it was found that 0.8 < εi/ εp < 1.4, which seems physically reasonable given the discussion in the previous section. This result implies that intrinsic shrinkage of the interfacial zone cement paste is similar to the bulk cement paste, recalling that the experimental data probably contains inelastic shrinkage, which is not treated in the model at present.

The microstructural origin of this result is not obvious, as the volume fractions of the various phases, shrinking and restraining, are known to change rapidly throughout the interfacial zone. In the next section, results are presented from simulations of cement paste microstructure in the ITZ that help to explain this finding.

Next: Justification of unrestrained Up: Effects of interfacial Previous: Numerical results