# Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models
## 2 Size and Boundary Effect Models
### Bazant's Type 2 Size Effect Law  
Bazant's Type 2 size effect law was originally developed for the size effect study of geometrically similar specimens [13]. The law reads  
\[\sigma_{N}=\hat{B}f_{t}\left(1+\frac{W}{W_{0}}\right)^{-1/2}, \tag{24}\]  
where \(\hat{B}\) is a geometrical dimensionless constant; \(W_{0}\) is a material constant proportional to \(l_{ch}\). \(\hat{B}f_{t}\) and \(W_{0}\) were given as follows in [15]:  
\[\hat{B}f_{t}=\sqrt{\frac{EG_{c}}{g^{\prime}(\alpha)c_{f}}}, \tag{25}\]  
\[W_{0}=\frac{c_{f}g^{\prime}(\alpha)}{g(\alpha)}, \tag{26}\]  
where \(g(\alpha)=K_{l}^{2}(\alpha)W(t/P)^{2}=Y^{2}(\alpha)\pi\alpha\) is defined as the dimensionless energy release function of LEFM; the length parameter \(c_{f}\) is proportional to \(l_{ch}\), which is the effective size of fracture process zone [15]. \(g^{\prime}(\alpha_{0})\) can be written as a function of the geometrical correction factor \(H(\alpha)\) presented in Equation (20):  
\[g^{\prime}(\alpha)=H^{2}(\alpha)\pi. \tag{27}\]  
After the substitution of Equations (7), (25)-(27) into Equation (24), the type 2 size effect law can be finally transformed to a shape similar to the boundary effect model:  
\[\sigma_{N}=\frac{1.12f_{t}\sqrt{a_{\infty}^{\ast}/c_{f}}}{H(\alpha)}\left(1+ \frac{a_{\sigma 3}}{c_{f}}\right)^{-1/2}, \tag{28}\]  
where \(a_{\sigma 3}\) shares the same definition as the one presented in [PERSON]'s size and boundary effect model. This form of the [PERSON]'s Type 2 size effect law can be further written as:  
\[\frac{1}{\sigma_{n4}^{2}}=\frac{c_{f}}{K_{c}^{2}}+\frac{1}{K_{c}^{2}}a_{\sigma 3}, \tag{29}\]with  
\[\sigma_{n4}=\sqrt{\pi}H(\alpha)\sigma_{N}. \tag{30}\]  
Thus, the fracture toughness \(K_{c}\) and the effective size of the fracture process zone \(c_{f}\) can be identified from the size effect or boundary effect fracture test results. To obtain the tensile strength of the material, a ratio \(\gamma_{1}\) (\(\gamma_{1}=c_{f}/l_{ch}\)) between \(c_{f}\) and the material characteristic length \(l_{ch}\) should be assumed. It is reported in [20; 39; 40] that \(\gamma_{1}\) can be 0.28 or 0.29 for concrete material. Since \(l_{ch}=1.12^{2}\pi a_{\infty}^{*}\), another ratio \(\gamma_{2}=c_{f}/a_{\infty}^{*}\) can also be obtained, which is close to 1 when \(\gamma_{1}\approx 0.28\) or 0.29 (\(\gamma_{2}\approx 1.10\) or 1.14), means the effective size of the fracture process zone may be roughly equal to the transition crack size.  
It is interesting to notice that the effective size of the fracture process zone \(c_{f}\) in [PERSON]'s Type 2 size effect law might be able to link to the fictitious crack length \(\Delta a_{fic}\) proposed in [PERSON]' ([PERSON]') boundary effect model (see Section 2.1.2). A reasonable choice of \(\gamma_{1}\) or \(\gamma_{2}\) would provide a better estimation of material tensile strength \(f_{\rm f}\). The theoretical and experimental studies can be done in the future work to find out the approximate relation between \(c_{f}\) and maximum aggregate size \(d_{max}\) or average aggregate size \(d_{av}\).