# Mechanical and Fiber-Bridging Behavior of Slag-Based Composite with High Tensile Ductility
## 3 Results and Discussion
### Micromechanical Analysis  
In order to analyze the mechanism of the strain-hardening and highly ductile behavior of the composite, \(I_{SP}\) and \(I_{EP}\) were evaluated based on the micromechanical properties of the materials and by fiber-bridging analysis. A theoretical fiber-bridging curve of the composite was obtained based on fiber-bridging constitutive law, expressed in the form of Equation (5) [22]:  
\[\sigma_{B}(\delta)=\frac{4V_{f}}{\pi d_{f}}^{2}\int_{0}^{\frac{\pi}{2}}\int_{0} ^{\frac{L_{f}}{2}}P(\theta,L_{e},\delta)p(\theta)cos(\theta)dL_{e}d\theta \tag{5}\]  
Here, \(V_{f}\) is the fiber volume fraction and \(P(\theta,\ L_{e},\ \delta)) is the pull-out load of each fiber with respect to its inclined angle (\(\theta\)), embedment length (\(L_{e}\)), and crack opening displacement (\(\delta\)). \(p(\theta)\) is the probability density function with respect to the inclined angle of the fibers. Additional information for the analysis, such as spalling coefficient, fiber strength reduction factor, and snubbing coefficient, were adopted from the literature; those values are 500, 0.475, and 0.3, respectively [23; 24; 25]. Figure 8 shows the numerical procedure to obtain the fiber-bridging curve. First, crack opening displacement is assumed. Following that, for a given crack opening displacement, the stress induced on the fiber is determined with respect to the embedment length and inclined angle of the fibers. Next, based on the calculated level of induced stress, it is determined whether the fibers have been fractured or not. For a given crack opening displacement, the pullout loads for all fibers in the crack are numerically integrated. This procedure is repeated until the assumed fiber opening displacement reaches the final crack opening displacement. Figure 8: Numerical procedure for obtaining fiber-bridging curve. The theoretical fiber-bridging curve of the composite is shown in Figure 9. Given that PE fiber is hydrophobic, there was no fiber-bridging stress observed in the initial part of the curve. The maximum fiber-bridging stress was 8.7 MPa and corresponding crack opening was 44.1 \(\upmu\)m. The complementary energy (\(J_{b}\)') was 100 J/m\({}^{2}\). When the composite is homogenous and isotropic, the maximum fiber-bridging stress is identical to the value of \(f_{\text{fs}}\) of the composite. Otherwise, because the fiber-bridging capacity depends on matrix properties and fiber distribution, it is generally observed that the tensile strength is lower than the theoretical maximum fiber-bridging stress. The specimen fractured at the section with the lowest fiber-bridging capacity. Details of a comparison of \(f_{\text{fs}}\) and \(W_{c}\) between the micromechanical analysis and the experimental results are provided in Table 6. The maximum fiber-bridging stress was only 2.3% higher than the value of \(f_{\text{fs}}\) of the composite. Although a relatively high difference was observed between analytical and experimental results for crack width, it seems that the theoretical fiber-bridging curve simulate the tensile behavior of the composite well, especially considering the variation of matrix properties and the fiber distribution. This implies that the fiber-bridging analysis performed in this study is appropriate to investigate the fundamental mechanisms of the strain-hardening and highly ductile behavior of the composite. The matrix fracture energy (\(I_{tip}\)), which is the energy for crack propagation of the matrix at the crack tip, was calculated using Equation (1) [26]. \[J_{tip}\ =\ \frac{{K_{m}}^{2}}{E_{m}} \tag{6}\]  
Here, \(K_{m}\) is the matrix fracture toughness, defined in the previous section. \(E_{m}\) is the elastic modulus of the matrix and was found to be 10.3 GPa. It was obtained in the linear elastic region of the tensile stress and tensile strain curves.