## 4 Micromechanical Analysis  
Figure 5 shows a typical fiber bridging curve and conditions for pseudo strain-hardening behavior with multiple cracks, and controlled crack width [25]. [PERSON] and [PERSON] (1988) proposed the energy criterion for steady-state matrix-cracking of composites, which is expressed as Equation (4) [9]. The complementary energy (\(\dot{J_{b}}\): gray shaded area in Figure 5) in the fiber bridging curve must be higher than the matrix fracture energy (\(J_{tip}\): blue shaded area in Figure 5). It can be explained physically that the energy required to increase the crack width in the composite should be higher than that required for crack propagation in the matrix.  
\[J_{tip}\leq\sigma_{0}\delta_{0}-\int_{0}^{\delta_{0}}\sigma(\delta)d\delta \equiv\dot{J_{b}}, \tag{4}\]  
where \(\sigma_{0}\) is the maximum fiber bridging stress and \(\delta_{0}\) is the crack opening corresponding to \(\sigma_{0}\) on the fiber bridging curve.  
Figure 6 shows the fiber-bridging curves of each mixture, which were calculated theoretically by numerical analysis [26; 27] according to micromechanics based on micromechanical parameters (i.e., fiber, matrix, and fiber/matrix interfacial properties) obtained by si  
Figure 5: Typical fiber bridging curve for a strain hardening composite (adapted from [26]).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mixture** & \(K_{\text{m}}\) (**MPa-m\({}^{1/2}\)**) & \(J_{\text{tip}}\) (**J/m\({}^{2}\)**) \\ \hline HDCC0 & 1.065 \(\pm\) 0.06 & 8.2 \(\pm\) 0.15 \\ HDCC15 & 0.821 \(\pm\) 0.03 & 6.0 \(\pm\) 0.21\({}^{\prime}\) \\ HDCC30 & 0.679 \(\pm\) 0.04 & 4.1 \(\pm\) 0.17 \\ HDCC45 & 0.552 \(\pm\) 0.03 & 2.6 \(\pm\) 0.24 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Fracture toughness of HDCC mortar.  
fracture toughness tests. The complementary energies were also calculated from the theoretical fiber bridging curves listed in Table 6. Although peak fiber bridging stress decreased with an increase of the replacement ratio of cement with limestone powder, the complementary energy increased with an increase of the replacement ratio of cement with limestone powder.  
The energy performance index (\(R_{E}\)) proposed by Kanda and Li (2006) [PERSON] and Li (2006) Kanda and Li (2006) is defined as the ratio of the complementary energy to the fracture energy of a matrix. It is known that when \(R_{E}\) is over 3.0, multiple micro-cracking can be ensured. The \(R_{E}\) values of all the HDCC mixtures were higher than 3.0, properly satisfying the energy condition. The \(R_{E}\) value of the HDCC0 specimen was 4.8; however, this was the lowest value among all the specimens. On the other hand, the HDCC45 mixture exhibited the largest \(R_{E}\) (17.8), which is 3.7 times higher than that of the HDCC0 mixture. The \(R_{E}\) values increased sequentially as the replacement ratio of cement with limestone powder increased, which means that the potential of steady state cracking behavior increases with an increase of the replacement ratio. As described in Section 3.1, the stress performance index also increased with an increase in the replacement ratio. Overall, the replacement of cement with limestone powder increased the tensile strain capacity and the potential of saturated pseudo-strain hardening behavior, although it also induced some loss of strength. Micromechanical tests and analysis supported the tensile behavior and fundamental mechanisms of the HDCC mixtures investigated in this study.