Next: 3-D analytical result Up: Effects of interfacial Previous: Effects of interfacial

Numerical results

Computations of shrinkage were performed on model microstructures with a 55% sand area fraction, in which each sand grain had a diameter of 410 µm and an interfacial zone thickness of 20 µm. These values were chosen to simulate a typical mortar, with 410 µm being a compromise between the number and mass-weighted average diameters based on typical sand-size distributions [14]. The unrestrained shrinkage of the interfacial zone (ε_i) was allowed to vary from 25% to 500% of the paste shrinkage (_p), and the interfacial Young's modulus was varied so that 0.05 < E_i/E_p < 3. Again, values of E_i/E_p are included for the cases of lightweight and chemically treated aggregates.

Figure 4 shows the results for ε^*/ ε_p, as a function of E_i/E_p for different values of ε_i/ ε_p , where ε^* is the overall composite shrinkage. The shape of the curves is interesting. For a given intrinsic strain ratio, ε_i/ ε_p, there is a modulus ratio, E_i/E_p, which will minimize the shrinkage of the sample. This can be physically explained as follows.

Figure 4: Effect of interfacial/paste Young's modulus ratio on composite shrinkage with changing interfacial shrinkage for a sand particle diameter of 410 µm at a constant area fraction of 55%.

At very low interfacial moduli, the overall shrinkage increases sharply for any value of ε_i/ ε_p . In this limit, the very soft interfacial zone essentially decouples the restraining aggregate from the shrinking bulk cement paste phase. The bulk paste no longer "sees" the non-shrinking aggregate, and is able to shrink towards its intrinsic value; thus, ε^*/ ε_p tends toward 1. As the modulus of the interfacial zone increases, the bulk cement paste becomes more strongly coupled to the non-shrinking sand, decreasing the overall shrinkage. However, past a certain point, the increasing moduli of the interfacial zone cement paste causes the interface to act as a shrinking phase, which is well coupled to the bulk paste. The total paste phase can now better resist the restraining effect of the sand, thereby increasing the overall shrinkage and placing the aggregate in compression. The combination of these two mechanisms depends sensitively on the fact that, topologically, the bulk cement paste is physically separated from the sand by the interfacial zone phase. This topological fact is true in both 2-D and 3-D, and is the cause of the observed minima in the curves in Figure 4. For lower values of ε_i/ ε_p, the rise in ε^*/ ε_p with increasing E_i/E_p is very shallow but still numerically present.

The above analysis is confirmed in Table 4 by examining the average compressive stress per unit area of sand below, at, and above the minimum shown in Figure 4. As can be seen, the lowest compressive stress occurs when the paste is debonded from the sand particle at very low interfacial zone stiffness, indicating little influence between the paste and sand. As the minimum is approached, the compressive stress per sand pixel increases, indicating that the sand particle is under compression as the bulk paste begins to exert more influence on the sand particle. Above the minimum, the compressive stress in the sand continues to increase as the interfacial zone begins to exert its own intrinsic shrinkage upon the particle, simultaneously increasing both the compressive stress and the overall shrinkage.

This result implies that to minimize the shrinkage of mortar and concrete it may not necessarily be desirable in order to stiffen the ITZ significantly. The result is in direct contrast to strength considerations, where it is generally desirable to strengthen (and usually simultaneously stiffen) the ITZ as much as possible.

If knowledge of the shrinkage of the mortar specimen and its paste phase were known, the unrestrained shrinkage of the interfacial zone could be determined by comparing a 3-D version of Figure 4 to experimental data. Having determined the elastic modulus of the interfacial zone by one of the methods described above in Section IV, the normalized shrinkage ratio in Figure 4 could be compared to determine the unrestrained shrinkage that best fits the experimental data.

An additional consideration is that real mortars and concretes almost always contain air voids. When air voids are introduced, the amount of the shrinking phase (cement paste), is reduced, while no restraint is added since the moduli of the air pore are taken to be zero. In effect, the average elastic moduli of the cement paste are reduced because the porosity of the non-sand matrix is increased. To explore this effect, two series of seven simulated model mortars were created with sand area fractions ranging from 5% to 55%. In one series, 4% (by total area) air was added, replacing an equivalent amount of cement paste. The sand grains had a diameter of 510 µ m, and the air voids, a diameter of 110 µm. An interfacial zone of 20 µm was assigned to both [27]. The elastic properties and unrestrained shrinkage strains of the bulk paste, interfacial paste, and aggregate were: E_p=1.0, E_i=0.5, E_a=4.29; ν_p = ν_i = 0.3, ν_a = 0.2, ε_p = ε_i < 0, and ε_a = 0. The air voids were assumed to be empty, non-shrinking (ε_V = 0), and non-restraining (E_v=0) as noted above.

The results of these simulations are shown in Figure 5. Clearly, the presence of the air voids will decrease the magnitude of shrinkage with all sand contents (except zero), but the effect is increased at the higher inclusion areas. This is consistent with our hypothesis, since at the higher sand area fractions there is less paste initially, so that replacing a constant 4% volume fraction of bulk cement paste with air eliminates a greater relative amount of shrinking phase for these microstructures.

Figure 5: Effect of entrapped air on mortar shrinkage for a system of sand particles of diameter 510 µm at varying area coverages. One series contains a constant 4% area coverage of air voids of diameter 110 µm.

Next: 3-D analytical result Up: Effects of interfacial Previous: Effects of interfacial