# Strength Correlation and Prediction of Engineered Cementitious Composites with Microwave Properties
## 4 Mathematical Modelling of Experimental Results
### Correlations between Compressive Strength and Conductance  
In this section, the correlations between compressive strength and electrical conductance are established via data mapping of (4) and (5). As the first step, effective conductance at 2 GHz is calculated using (4) and compressive strength is obtained using (5), both through the bi-exponential functions for the first 8 days. The compressive strength of all ECC specimens is developing while electrical conductance decays with time as shown in Figure 12a. All (three) strength-conductance curves (S-C curves for short) have similar trends in accordance with hydration process--reducing the free water whilst C-S-H gels are developed inside specimens. It is observed that the S-C curves of PE1 and PE2 have alike behaviors in the conductance range of 3.1 to 3.5, whereas the S-C curves of PE2 and PE3 have close behaviors in the conductance range of 3.6 to 5. Meanwhile, the S-C curves show that when conductance equals 3.54 both the PE1 and PE3 specimens yield a strength value of 24 MPa while PE2 exhibits a higher strength of about 28 MPa. This observation agrees with the strength values of ECC specimens on Day-3 as shown in Figures 6 and 9. As the second step, all ECC mixes with similar initial \(w/b\) ratios in the range of 0.20-0.30 are considered as a global set of S-C data from Day-1 to Day-8. A generic regression model is established by fitting all S-C data using a single function in bi-exponential form, as given by  
\[f_{c2}(g_{2})=3236\;exp(-1.53\cdot g_{2})+19.97\;exp(-0.14\cdot g_{2}) \tag{6}\]  
The bi-exponential regression function (6) fitted with data as shown in Figure 12b, where the goodness of fit is \(R^{2}\) = 0.8829 and RMSE = 3.138. Electrical conduction at high frequency such as 2 GHz occurs due to dielectric relaxation, dipolar loss, and ion transport through pore solution and is strappingly related to moisture content, porosity and ion concentration inside the CBMs. The reduction of effective conductance reflects the growth of C-S-H gels and thus the strength gains. Therefore, one can use this generic function to monitor the strength evolution of ECC specimens as long as the \(w/b\) ratio has fallen into the range of 0.20-0.30.  
As the final step for forming correlations between compressive strength and effective conductance, the effect of different \(w/b\) ratios on compressive strength is further considered. The authors developed a more accurate regression function (7), by taking the \(w/b\) ratio as a dependent variable, as given by  
\[f_{c2}(g_{2},\frac{w}{b})=\frac{K}{g_{2}^{(w/b)}-m} \tag{7}\]  
where \(K\) and \(m\) are the regression coefficients of (7).  
Figure 12: (a) Computed correlations between compressive strength and conductance directly using regression functions (4) and (5); (b) proposed regression function (6) for fitting of all S-C data.  
Figure 13 shows the performance of the proposed regression functions as a function of effective conductance at 2 GHz (g2). Function proposed for the PE1 (w/b = 0.20) mix is shown in Figure 13a whereas functions fitting individual data sets of PE2 (w/b = 0.255) and PE3 (w/b = 0.30) are shown in Figure 13b. Table 6 summarizes the fitting coefficients and goodness of fit for (7). The rational model of (7) gives more meaning yet a simpler form than the bi-exponential model of (6), both the coefficients K and m are varying linearly with increasing w/b, which can be used as indications for distinguishing water contents of ECC specimens.