# Strength Correlation and Prediction of Engineered Cementitious Composites with Microwave Properties
## 4 Mathematical Modelling of Experimental Results
### Compressive Strength Modelled by using Bi-Exponential Regressions  
Exponential regression methods are known to be well-suited for the fitting of growing/decaying data with time [13; 20; 21]. In particular, the mean compressive strength of cement concrete at times other than 28 days can be estimated, as long as the type of cement is known [33]. In this study, the measured results of compressive strength of ECCs, which grow with age for 28 days, were modelled by using the bi-exponential regression as given in (5).  
\[f_{c}(t)=S1\;exp\left(\frac{t}{d}\right)+S2\;exp\left(-\frac{t}{e}\right),\;0 \leq t\leq 28 \tag{5}\]  
where \(S1\) and \(S2\) are the initial values of the two exponential terms, respectively, and \(d\) and \(e\) are the regression coefficients. Table 5 summarizes all regression coefficients and the goodness of fit values, whereas the regression curves are plotted against age as shown in Figure 9.  
Figure 8: (**a**) Comparison of time constants for step responses; (**b**) Regression coefficients \(A1\) and \(c\) (time constant) versus \(w/b\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**ECC** & \(w/b\) & \(S1\) & \(S2\) & \(d\) & \(e\) & \(R^{2}\) \\ \hline PE1 (\(f_{c1}\)) & 0.200 & 41.9 & \(-41.82\) & 143.947 & 3.960 & 0.9998 \\ PE2 (\(f_{c2}\)) & 0.255 & 45.1 & \(-44.94\) & 534.759 & 3.347 & 0.9980 \\ PE3 (\(f_{c3}\)) & 0.300 & 30.68 & \(-30.63\) & 72.727 & 2.820 & 0.9996 \\ \hline \end{tabular}
\end{table}
Table 5: Regression coefficients and goodness of fit for Equation (5).  
Figure 9: Compressive strength modelled by using bi-exponential regression.  
For a better understanding the strength modelling using the bi-exponential regressions, the fitting curves (\(f_{cs21},f_{cs22},f_{cs23}\)) of (5) were resolved into two terms as shown in Figure 10a for comparison. We examined the fact that the first terms are the approximate ramp functions with small slopes, whereas the second terms are the exponential growth functions in which the strength curve has different growing rates due to different initial water contents. Through further analysis of the unit-step responses of the second terms of (5) as shown in Figure 10b by using the criterion of \(exp(-1)\), the values of time constants \(e_{1}\), \(e_{2}\), and \(e_{3}\) are verified as the same as that given in Table 5. However, the magnitudes of the time constants are in the reverse order of the initial water contents, i.e., \(e_{1}>e_{2}>e_{3}\). Figure 11a shows the variation of time constants versus the \(w/b\); this linear curve indicates the strength evolution is inversely proportion to the initial water content. Namely, the higher the \(w/b\) ratio of ECC specimen features, the faster the rate of strength evolution; however, it has a lower final strength. This statement can be further verified using (5) by putting \(t=28\) days to obtain the compressive strengths for PE1, PE2 and PE3 specimens, respectively, and compared with the measured data, as shown in Figure 11b. Moreover, the 28-day strength of bi-exponential functions exhibits a linear decay relation with \(w/b\), which corroborates the results for the normal cementitious materials, e.g., [20,21]. This decline relation is highly expected in this study of PVA-ECC specimens despite the unknown factor of water absorption from the PVA fibers.  
Figure 11: (**a**) Regression coefficient (time constant), \(e\) versus \(w/b\); (**b**) comparison of the 28-day strengths; measurement, regression values using (5).  
Figure 10: (**a**) Decomposition of bi-exponential regression for all specimens; (**b**) step response of second term of (5) that shows the differences in time constant.