# Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models
## 2 Size and Boundary Effect Models
### Comparison of Different Equivalent Crack Lengths  
Different size and boundary effect models have adopted different expressions for the calculation of equivalent crack lengths. \(a_{e1}\) (Equation (11)) in [PERSON]'s boundary effect model depends only on the crack-length-to-height ratio \(\alpha\), while \(a_{e2}\) (Equation (18)) proposed in modified [PERSON]'s boundary effect model believes the equivalent crack length depends on both the crack-length-to-height ratio \(\alpha\) and fictitious crack-length-to-height ratio \(\Delta\alpha_{fic}\), thus becomes geometry- and material-related. The expression \(a_{e3}\) (Equation (21)) in [PERSON]'s size and boundary effect model, and [PERSON]'s Type 2 size effect law is identical, which is only geometry-dependent. To better compare the different equivalent crack lengths, a wedge-splitting specimen with \(W=1000\) mm (see Figure 1a) is taken as an example, and its initial crack length varies from 50 mm to 950 mm. The variations of the equivalent crack length for different maximum aggregate size \(d_{max}\) with fixed discrete number \(\beta_{1}\), and variable \(\beta_{1}\) with fixed \(d_{max}\) are plotted respectively in Figures 3 and 4. In different constructions, the maximum aggregate size \(d_{max}\) can be very different. For concrete used in Wudongde and Baihetan super-high arch dams, \(d_{max}=40\) mm, 80 mm or 150 mm for different parts of the dam. Meanwhile for the relatively smaller structures such as piers or beams, the frequently used \(d_{max}\) is only 10 mm or 20 mm. These possible \(d_{max}\) are all considered in this section, and the corresponding \(a_{e2}\) are plotted in Figure 3, together with the material (\(d_{max}\)) independent equivalent crack lengths \(a_{e1}\) and \(a_{e3}\). For the original [PERSON]'s boundary effect model, \(a_{e1}\) reaches its maximum value for any \(d_{max}\) when the crack-length-to-height ratio \(\alpha\) is around 0.2, means for this ratio, the LEFM contributes the most to the failure of the specimen, thus should be recommended as the initial ratio for the fracture test on wedge-splitting specimen. This recommended ratio for [PERSON]'s size and boundary effect model, and [PERSON]'s Type 2 size effect law is 0.3 to 0.4, which is closer to the usual initial crack-length-to-height ratio reported in the references [1; 10]. In terms of \(a_{e2}\) proposed in this study, it is obvious that it is different from \(a_{e1}\), and this difference becomes larger as the increase of \(d_{max}\). Hence, the proposed modified [PERSON]'s boundary effect model would be more appropriate than its original version, due to the fact that the two failure mechanisms are correctly combined even for the fracture analysis of dam concrete with maximum aggregate size reaches 150 mm. However, it should be admitted that for \(d_{max}\) such as 10 mm or 20 mm, the difference in \(a_{e2}\) and \(a_{e1}\) may has neglectable effect on the fracture properties estimated by the original and modified Hu-Guan's boundary effect model. In addition, the increase of \(a_{e2}\) at the tail part can be observed in Figure 3. This trend appears naturally due to its definition. An increase \(a_{e2}\) could maintain the validity of the failure mechanism of LEFM for all the possible crack-length-to-height ratios. Figure 4 shows the equivalent crack length \(a_{e}\) versus initial crack length \(a_{0}\) for different discrete number \(\beta_{1}\) when \(d_{max}=150\) mm. \(a_{e1}\) and \(a_{e3}\) are the same as the ones plotted in Figure 3 since they are independent on the fictitious crack length. \(\beta_{1}\) ranges from 0.25 to 2 is considered, which is a reasonable range for concrete material [23]. As shown in Figure 4, \(a_{e2}\) is strongly affected by the choice of \(\beta_{1}\) when \(d_{max}=150\) mm, thus may provide very different fracture parameters for different boundary effect models.