# Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models
## 3 Model Verification and Comparison
### Tests of Specimens with Different Initial Crack-Length-to-Height Ratios  
#### 3.2.1 [PERSON]'s Experiments  
[PERSON] et al. [31] performed the wedge-splitting tests made of concrete with different initial crack-length-to-height ratios. The maximum aggregate size used was 20 mm. The height and thickness were fixed for all the specimens (\(W=170\) mm, \(t=200\) mm). There were three different initial crack lengths \(a_{0}\), which were 60 mm, 80 mm and 100 mm, respectively. The measured compressive strength \(f_{cu}\) is 47.96 MPa, and the tensile strength \(f_{t}\) was estimated as \(f_{t}=0.24f_{cu}^{3/2}=3.17\) MPa by [PERSON] et al. [31]. Details of the specimen dimensions, test results and equivalent crack lengths are listed in Table 3.  
Figure 7 presents the test results in [31] and the fitted curves of the different size and boundary effect models. It should be noted that the low coefficients of determination (\(R^{2}\)) are simply due to the fluctuation of the test results. As shown in Figure 7, the linear trend of the test results is very clear and therefore, can be used to the prediction of fracture properties. For the original [PERSON]'s boundary effect model (see Figure 7a), it is provided in [23] that \(f_{t}=2.78\) MPa, \(K_{c}=1.73\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=97.65\) mm and \(l_{ch}=384.84\) mm when \(\beta_{1}=1\). The modified [PERSON]'s boundary effect model (see Figure 7b) gives \(f_{t}=2.94\) MPa, \(K_{c}=1.71\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=85.63\) mm and \(l_{ch}=337.45\) mm when \(\beta_{1}=1\). For [PERSON]'s size and boundary effect model, as shown in Figure 7c, one can obtain \(f_{t}=3.08\) MPa, \(K_{c}=1.10\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=32.25\) mm and \(l_{ch}=127.09\) mm. The estimated values for [PERSON]'s Type 2 size effect law (see Figure 7d) are: \(K_{c}=1.10\) MPa\(\sqrt{m}\) and \(c_{f}=32.25\) mm. When \(\gamma_{1}=0.28\), the material tensile strength \(f_{t}=3.24\) MPa can be obtained by Equation (31), and the corresponding \(a_{\infty}^{\star}=29.23\) mm and \(l_{ch}=115.18\) mm.  
The tensile strength calculated by [PERSON]'s size and boundary effect model is again within 5% difference with the estimated tensile strength \(f_{t}=0.24f_{cu}^{3/2}=3.17\) MPa by [PERSON] et al. [31]. With the same discrete number \(\beta_{1}\), the original [PERSON]'s boundary effect model gives a lower tensile strength than its modified one. In terms of fracture toughness, the estimation of [PERSON]'s size and boundary effect model is the same as the [PERSON]'s Type 2 size effect law, and lower than the values given by the original and modified [PERSON]'s boundary effect models. However, under current conditions, it is difficult to decide which model provides the more appropriate fracture toughness. This is because the tested specimen size \(W=170\) mm, which is only 8.5 times bigger than the maximum aggregate size. For such a \(W/d_{max}\) ratio, the failure of the pre-cracked specimen is certainly controlled by the non-LEFM, and the real fracture toughness of the material should be evaluated by a larger specimen or estimated by the size and boundary effect models.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Label**} & \(\mathbf{W}\) & \multirow{2}{*}{\(\mathbf{a_{0}/W}\)} & \multirow{2}{*}{\(\mathbf{t}\)} & \(\mathbf{P_{max}}\) & \(\mathbf{a_{e1}}\) & \(\mathbf{a_{e2}}\) & \(\mathbf{a_{e3}}\) & \(\sigma_{N}\) & \(\sigma_{n2}\) \\  & (mm) & & & (km) & (km) & (mm) & (mm) & (MPa) & (MPa) \\ \hline WS70-1 & 170 & 0.353 &