*3.3. Shrinkage Modeling*

The shrinkage of concrete and mortar is affected by various mixing and curing conditions, and it must be measured for a long period of time. Therefore, various models have been proposed to predict shrinkage in advance. The reaction mechanisms of the AAMs, however, are different from those of the existing OPC-based mortar and concrete. They are also characterized by different binder reaction times and reaction rates. Therefore, a model that reflects the effects of materials must be selected to predict the shrinkage of the AAM mortar using a model. Hu et al. [37] applied various models to predict the autogenous and drying shrinkage of alkali-activated slag mortar and examined their suitability. They also reported that the exponential function model is suitable for predicting the autogenous and drying shrinkage of AAM mortar. In the exponential function model, the effects of materials and mixing can be reflected through the values of the constants. Thus, an exponential function model was applied to analyze the shrinkage of the AAM mortars mixed with the expansive additive, as follows:

$$
\varepsilon\_{(l)} = \varepsilon\_s \left[ 1 - a \cdot \exp(-bt) \right] \tag{2}
$$

In this exponential model, ε(*t*) is the shrinkage at *t* days of age and ε<sup>150</sup> is the shrinkage at 150 d of age. Both α and *b* are material constants, where α denotes the generated amount of shrinkage and *b* is the slope of generated shrinkage over time, and subscripts <sup>a</sup> and <sup>d</sup> refer to the autogenous shrinkage and drying shrinkage respectively. Table 6 summarizes the material constants and amount of shrinkage at 150 d of age for the AAM mortars mixed with the expansive additive using the exponential model. When the material constants for the autogenous shrinkage of the AAM mortars were analyzed, the value of α<sup>a</sup> for EA-0.0 was −0.499, while those for EA-2.5, EA-5.0, and EA-7.5, which used an expansive additive, ranged from −0.553 to −0.572. Thus, the values of α<sup>a</sup> for the AAM mortar specimens that used the expansive additive were lower. The values of *b*a, however, showed no significant difference despite a slight increase when the expansive additive was added. Moreover, when the material constants for the drying shrinkage of the AAM mortars were analyzed, it was found that the value of α<sup>d</sup> for EA-0.0 was −0.485, but those for the AAM mortar specimens that used the expansive additive ranged from −0.700 to −0.944. Thus, the use of the expansive additive significantly decreased the value of αd. The value of *b*d, however, exhibited no significant difference, similar to the case of autogenous shrinkage.

Figure 4 shows the relationships between the expansive additive content and the coefficients of the shrinkage model. While the material constants of autogenous shrinkage did not exhibit significant changes even when the expansive additive content increased, α<sup>d</sup> significantly decreased as the expansive additive content increased. This indicates that the CSA expansive additive has a larger impact on drying shrinkage than on autogenous shrinkage. An increase in the expansive additive content can compensate for and reduce shrinkage, but this mostly results from the expansion effect at early ages. The value of *b* was not affected by the expansive additive content. This is because the shrinkage reduction effect of the expansive additive was not significant in a long term.


**Table 6.** Material constants of the exponential function model for autogenous and drying shrinkage.

**Figure 4.** Relationships between the expansive additive content and material coefficients of the shrinkage model: (**a**) for coefficient "α", and (**b**) for coefficient "*b*".

#### *3.4. Shrinkage Stress*

To calculate the shrinkage stress of the AAM mortars, the prediction curves for the modulus of elasticity obtained by the ACI 209 model and the total shrinkage prediction curves of the AAM mortars obtained by the exponential function were used. The shrinkage stress of the AAM mortars (f(*sh*, *<sup>t</sup>*+Δ*t*)) was calculated by adding the stress caused by the shrinkage generated per unit time (Δ*fsh*) to the shrinkage stress acting on the AAM mortar (*f* (*sh*, *<sup>t</sup>*)).

$$f\_{\left(sh, t+\Delta t\right)} = f\_{\left(sh, t\right)} + \Delta f\_{\text{slv}} \tag{3}$$

$$
\Delta f\_{slt} = E \cdot \Delta \varepsilon\_{slt} \,. \tag{4}
$$

Figure 5 shows the stress generated by shrinkage at 30 min intervals (Δ*fsh*) and the shrinkage stress accumulated in the AAM mortars (*fsh*) over time. The shrinkage stress generated per unit time in Figure 5a shows that a large amount shrinkage occurred until 1 d of age, and thus, Δ*fsh* was also high even though the modulus of elasticity was low in that period. Δ*fsh* rapidly decreased after 1 d of age, but it showed a tendency to slowly increase as the age increased. This is because the modulus of elasticity increased with the age. For the AAM mortars, shrinkage stress continuously occurred until 150 d as total shrinkage increased. Δ*fsh* mostly ranged from 0.001–0.003 MPa, but it showed a higher range (0.004–0.006 MPa) within 60 d of age. Large shrinkage stress was observed in some sections because some differences were observed in Δε*sh* when the shrinkage obtained by the data logger was divided into 30 min intervals. The increase in the expansive additive content decreased the maximum value of Δ*fsh* at 1 d of age, but the expansive additive content could not significantly affect Δ*fsh* after 1 d of age.

**Figure 5.** Stress generated by shrinkage. (**a**) Stress generated per unit time (Δ*fsh*), and (**b**) shrinkage stress (*fsh*).

As shown in Figure 5b, the shrinkage stress accumulated in the AAM mortars (*fsh*) was hardly affected by the expansive additive content. At early ages, the shrinkage stress decreased as the expansive additive content increased. After 6 d of age, however, such tendency was not observed. For the AAM mortars, the shrinkage stress continuously increased until 150 d of age. Table 7 summarizes the cumulative shrinkage stress at 1, 28, and 150 d of age. The shrinkage stress at 1 d of age ranged from 0.54 to 0.26 MPa, which was approximately 12.1–6.1% of the shrinkage stress at 150 d of age. The shrinkage stress at 28 d of age ranged from 2.37 to 1.90 MPa, which was approximately 55.5–43.5% of the shrinkage stress at 150 d of age. These results show that relatively larger shrinkage stress occurred at early ages. After 28 d of age, however, continuous shrinkage stress was observed in the AAM mortars. This value increased until 150 d without reduction even when the expansive additive

was added. When the shrinkage stress values according to the expansive additive content at 1 and 150 d of age were compared, it was found that the shrinkage stress of the AAM mortars reduced by 29.6–51.9% at 1 d of age depending on the expansive additive content. However, the shrinkage stress reduction rate decreased to less than 5% at 150 d of age. This is because the expansive additive could not reduce the shrinkage stress in the long term as it reacted at early ages, resulting in an expansion, and could not generate further expansion thereafter.


