Figure 8: Overall shrinkage in the cement paste microstructure model as a function of distance from the aggregate surface.
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After analyzing the composition of each of the slices, shrinkage analysis in 2-D was performed, in turn, on each slice. The moduli and intrinsic shrinkage values used are shown in Table 6. The Young's modulus and Poisson's ratio values for C-S-H were extracted from data acquired by Helmuth and Turk [23] for a 24-month-old pure C3S cement paste with a density of 2.044 g/cc which corresponds roughly to a w/c of 0.35. The C3S Young's modulus is that used by Hansen [29]. The moduli for CH was determined by Voigt and Reuss averaging [40] of the full elastic tensor of CH as determined by Brillouin scattering [41], and agrees well with experimental data obtained from compressed CH powders [42].
| Unnormalized | Normalized | |||||
| Phase | E(GPa) | Poisson's Ratio | Intrinsic Shrinkage | E | Poisson's Ratio | Intrinsic Shrinkage |
| C-S-H | 14 | 0.27 | i−1 | 1 | 0.27 | −1 |
| C3S | 75.9 | 0.3 | 0 | 5.42 | 0.3 | 0 |
| CH | 43 | 0.32 | 0 | 3.07 | 0.32 | 0 |
| pore | 0 | 0 | 0 | 0 | 0 | 0 |
Table 6: Values used in shrinkage analysis of cement paste microstructure model slices.
It is noted that computations of shrinkage using 2-D slices of the 3-D interfacial zone microstructure is not the equivalent of a full 3-D shrinkage computation on the entire microstructure, followed by analyzing each slice individually. However, the main value of this method is its feasibility, since we do not yet have a 3-D shrinkage algorithm, and it also serves to focus on each slice, eliminating interactions between different parts of the microstructure at different distances from the aggregate.
The unrestrained strains of the four phases determine the shrinking and non-shrinking phases in the composite. All of the shrinkage is assumed to take place in the C-S-H, while the C3S and CH are assumed to be non-shrinking. Porosity is also assumed to have no inherent shrinkage, though it may change shape or size as a result of deformation of C-S-H. Since porosity has zero static moduli, the value of its unrestrained shrinkage is irrelevant anyway.
Figure 8 is a graph of computed shrinkage as a function of distance from the aggregate. Comparison with Figure 7 shows a direct correlation between the amount of C-S-H and the amount of shrinkage, as expected. However, the most important value is derived from the average value of this region. Examination of Figure 8 shows that the value of bulk shrinkage, obtained far from the interface, coincides closely with the value of the average shrinkage of the interfacial region. This implies that if the interfacial transition zone is treated as a shell with uniform properties, as has been done in this paper, the average shrinkage in the ITZ is identical to the bulk paste, in accord with the data presented in Section IV above.
Figure 8: Overall shrinkage in the cement paste microstructure model as a function of distance from the aggregate surface.
Next: Summary Up: Justification of unrestrained Previous: Simulation of ITZ