# 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
#### 3.2.2 [PERSON]'s Experiments  
Another series of wedge-splitting tests with different initial crack-length-to-height ratios were carried out by [PERSON] et al. [32; 33]. The maximum aggregate size was 20 mm. The height and thickness of the specimens were \(W=170\) mm and \(t=200\) mm, respectively. The samples were classified as 5 groups according to the different initial crack-length-to-height ratios ranging from 0.2 to 0.8. The measured compressive strength \(f_{cu}\) was 38.8 MPa, and splitting tensile strength was 3 MPa. Details of the tests are listed in Table 4, the failure loads \(P_{max}\) were the mean values of the different groups provided in [32; 33]. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Label**} & \(\mathbf{W}\) & \multirow{2}{*}{\(\mathbf{a_{0}}\)/\(\mathbf{W}\)} & \(\mathbf{t}\) & \(\mathbf{P_{max}}\) & \(\mathbf{a_{e1}}\) & \(\mathbf{a_{e2}}\) & \(\mathbf{a_{e3}}\) & \(\mathbf{\sigma_{N}}\) & \(\mathbf{\sigma_{n2}}\) \\  & **(mm)** & & **(mm)** & **(kN)** & **(mm)** & **(mm)** & **(mm)** & **(MPa)** & **(MPa)** \\ \hline WS1 & 400 & 0.2 & 200 & 34.20 & 39.22 & 47.46 & 68.95 & 1.710 & 2.672 \\ WS2 & 400 & 0.4 & 200 & 25.20 & 30.25 & 39.51 & 77.94 & 1.260 & 3.675 \\ WS3 & 400 & 0.5 & 200 & 16.00 & 23.67 & 32.77 & 67.03 & 0.800 & 3.400 \\ WS4 & 400 & 0.6 & 200 & 10.80 & 17.92 & 26.95 & 52.63 & 0.540 & 3.578 \\ WS5 & 400 & 0.8 & 200 & 3.20 & 8.79 & 19.09 & 25.94 & 0.160 & 3.800 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Geometry, test results and equivalent crack length of wedge-splitting specimens [32; 33]. Figure 7: Test results in [31] and the corresponding fitted curves of (**a**) [PERSON]’s boundary effect model with \(\beta_{1}=1\), (**b**) modified [PERSON]’s boundary effect model with \(\beta_{1}=1\), (**c**) [PERSON]’s size and boundary effect model and (**d**) Ba2 ant’s Type 2 size effect law. The linear regressions of different models are plotted in Figure 8. When \(\beta_{1}=1\), the estimated materials parameters (see Figure 8a) for original [PERSON]'s boundary effect model are: \(f_{t}=4.83\) MPa, \(K_{c}=1.42\) MPa\(\sqrt{m}\), \(a_{\infty}^{*}=22.0\) mm, \(l_{ch}=86.6\) mm, and for modified [PERSON]'s boundary effect model (see Figure 6b), the extrapolated parameters are: \(f_{t}=6.74\) MPa, \(K_{c}=1.39\) MPa\(\sqrt{m}\), \(a_{\infty}^{*}=10.8\) mm and \(l_{ch}=42.5\) mm. [PERSON]'s size and boundary effect model gives \(f_{t}=3.90\) MPa, \(K_{c}=1.61\) MPa\(\sqrt{m}\), \(a_{\infty}^{*}=43.1\) mm and \(l_{ch}=169.9\) mm (see Figure 6c). [PERSON]'s Type 2 size effect law (see Figure 6d) gives \(K_{c}=1.61\) MPa\(\sqrt{m}\) and \(c_{f}=43.13\) mm. In addition, the tensile strength \(f_{t}=4.10\) MPa is obtained for \(\gamma_{1}=0.28\). For this example, the extrapolated tensile strengths of all the models are larger than the measured splitting tensile strength (3 MPa), this is because the different fracture mechanisms of the splitting and wedge-splitting tests. The latter one's failure mechanism is close to the flexure test. Therefore, the extrapolated tensile strength from the wedge-splitting tests should be close to the tensile strength measured by the flexure test. [PERSON] [11; 12] concluded from plenty of the experimental results that the tensile strength measured by the flexure test is around 35% higher than the one measured by the splitting test. Hence, the extrapolated tensile strengths by the different models are still reasonable. The fracture toughness values given by the [PERSON]'s model and [PERSON]'s model (\(K_{c}=1.61\) MPa\(\sqrt{m}\)) are higher than the values obtained from the [PERSON]'s boundary effect model  
Figure 8: Test results in [32; 33] and the corresponding fitted curves of (**a**) [PERSON]’s boundary effect model with \(\beta_{1}=1\), (**b**) modified [PERSON]’s boundary effect model with \(\beta_{1}=1\), (**c**) [PERSON]’s size and boundary effect model and (**d**) [PERSON]’s Type 2 size effect law. (\(K_{c}=1.42\) MPa\(\sqrt{m}\)) when \(\beta_{1}=1\).