# Mechanical and Microstructural Studies of High Performance Concrete with Condensed Silica Fume
## 3 Results and Discussion
### Flexural Strength and Fracture Properties  
Prismatic specimens with notch in the center-point loading arrangement were employed in accordance with the RILEM TC 89-FMT standard [53]. Two supports were set at a distance of 600 mm. The constant rate of loading throughout the test was 0.05 mm/min. In this experiment, the flexural strength, stress intensity factor, crack mouth opening displacement (CMOD), crack tip opening displacement (CTOD), fracture energy, elastic-plastic failure parameter, and characteristic length of HPC were evaluated. In the specimen mid-span, a notch was cut with a depth and thickness of 50 mm and 3 mm, respectively. Two steel sheets (see Figure 6) were then glued close to the notch to install the gauge. A linear variable displacement transducer (LVDT) installed at the midpoint of 80 \(\times\) 150 \(\times\) 700 mm HPC specimen was used to assess the deflection. As the experiment was conducted on the specimens, the applied load, deflection, and CMOD were automatically recorded by the data acquisition software. Representative load-deflection curves are shown in Figure 7.  
The response of the HPC prisms with notch in the three-point bending test was approximately linear up to the peak load. Then, the curves suddenly dropped as the prisms separated. The flexural strength and fracture properties were calculated based on Equations (1)-(8) to assess the cracking resistance of HPC beam with different CSF substitutions. The following formula was used to determine the flexural strength of the HPC prisms with notch:  
\[f_{t,fl}=\frac{3F_{p}l}{2b(h-a_{0})^{2}}\ (\text{MPa}\,) \tag{1}\]  
Figure 5: Variation of the elastic modulus with CSF replacement. Figure 6: Geometry of notched beam specimens under center-point loading arrangement. The stress intensity factor \(K_{lc}\), representing the field of stress around the tip of the crack, was aimed at evaluating the cracking resistance of HPC prisms with various CSF substitutions on the basis of the following formula [54]:  
\[K_{lc}=\frac{F_{p}}{b\sqrt{h}}f(\alpha)\ (\mathrm{MN/m}^{1.5}), \tag{2}\]  
where \(F_{p}\) is the peak load, \(b\) is the width of the prism, \(h\) is the height of the prism, and \(f(\alpha)\) is a function of the sample geometry defined below:  
\[f(\alpha)=6\sqrt{\alpha}\left\{\frac{1.99-\alpha(1-\alpha)\big{(}2.15-3.93 \alpha+2.7\alpha^{2}\big{)}}{(1+2\alpha)(1-\alpha)^{3/2}}\right\}, \tag{3}\]  
where \(\alpha\) is the relative crack length, \(\alpha=a_{0}/h\), \(a_{0}\) is the notch depth. A higher factor (\(K_{lc}\)) indicates that HPC has greater resistance to crack development and has enhanced fracture assets. The fracture energy \(G_{lc}\), defined as the energy needed to break the specimen in load mode I, is calculated as follows:  
\[G_{lc}=\frac{{K_{lc}}^{2}(1-\
u)}{E_{c}}\ (\mathrm{N/mm}), \tag{4}\]  
where \(\
u\) is the Poisson's ratio, \(E_{c}\) is the elastic modulus. Figure 7: Exemplary test dependences of HPC load-deflection for CSF content of (**a**) 0%; (**b**) 5%; (**c**) 20%; (**d**) 25%. The parameter of elastic-plastic failure (\(J_{lc}\)) was calculated by using the area under the load versus deflection curve until the specimen rupture:  
\[J_{lc}=\frac{A}{2b(h-a_{0})}\ (\mathrm{N/mm}), \tag{5}\]  
where \(A\) is the energy retained by the unit cross-sectional area in the beam sample until the peak load \(F_{p}\) is reached. The crack tip opening displacement (CTOD) is another feature needed to evaluate the fracture toughness in the elastic-plastic region. The CTOD representation is the displacement at the primary tip of crack and the 90\({}^{\circ}\) intersection, and can be calculated as follows:  
\[\mathrm{CTOD}=\frac{6F_{p}a_{0}l}{E_{c}bh^{2}}V_{1}(\alpha)\ (\mathrm{\mu m}), \tag{6}\]  
\[f_{t,fl}=\frac{3F_{p}l}{2b(h-a_{0})^{2}}\ (\text{MPa}\,) \tag{1}\]0  
The HPC brittleness could be determined by the characteristic length \(l_{ch}\). The shorter the characteristic length, the more brittle the HPC is. Conversely, the greater the characteristic length, the more ductile the material is.