# Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models
## 3 Model Verification and Comparison
### Tests of Geometrically Similar Specimens with Constant Crack-Length-to-Height Ratios
#### 3.1.2 Wittmann's Experiments  
[PERSON] et al. [30] performed the fracture tests on geometrically similar compact tension specimens. The heights of the specimens were 300 mm, 600 mm and 1200 mm, with 6 samples for each specimen size and the maximum aggregate size was 16 mm. The measured compressive strength at 28 days was 42.9 MPa. The sample dimensions, test results and model parameters are listed in Table 2. Only the mean value of the maximum loads of each group (6 samples) has been reported in [30].  
The linear regressions of different size and boundary effect models are plotted in Figure 6. When \(\beta_{1}=1\), the estimated materials parameters (see Figure 6a) for original [PERSON]'s boundary effect model are: \(f_{t}=3.83\) MPa, \(K_{c}=2.10\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=76.2\) mm, \(l_{ch}=300.3\) mm, and for modified [PERSON]'s boundary effect model (see Figure 6b), the extrapolated parameters are: \(f_{t}=4.03\) MPa, \(K_{c}=2.09\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=68.3\) mm and \(l_{ch}=269.3\) mm. [PERSON]'s size and boundary effect model gives \(f_{t}=2.93\) MPa, \(K_{c}=1.87\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=103.3\) mm and \(l_{ch}=407.2\) mm (see Figure 6c). [PERSON]'s Type 2 size effect law (see Figure 6d) gives \(K_{c}=1.87\) MPa\(\sqrt{m}\) and \(c_{f}=103.34\) mm. In addition, the tensile strength \(f_{t}=3.09\) MPa is obtained for \(\gamma_{1}=0.28\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Series**} & \(\begin{array}{c}\boldsymbol{W}\\ \textbf{(mm)}\end{array}\) & \multirow{2}{*}{\(\boldsymbol{a_{0}/W}\)} & \(\boldsymbol{t}\) & \(P_{max}\) & \(a_{e1}\) & \(a_{e2}\) & \(a_{e3}\) & \(\sigma_{N}\) & \(\sigma_{n2}\) \\  & & & **(mm)** & **(kN)** & **(mm)** & **(mm)** & **(mm)** & **(MPa)** & **(MPa)** \\ \hline Small & 300 & 0.5 & 120 & 7.30 & 17.76 & 42.18 & 50.27 & 0.811 & 2.631 \\ Medium & 600 & 0.5 & 120 & 12.60 & 35.51 & 58.10 & 100.54 & 0.700 & 2.736 \\ Large & 1200 & 0.5 & 120 & 20.70 & 71.02 & 92.58 & 201.09 & 0.575 & 2.518 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Geometry, test results and equivalent crack length of compact tension specimens [30].  
The tensile strength can be estimated from the compressive strength by \(f_{t}=0.24f_{ct}^{3/2}=2.94\) MPa [31]. Similar to the estimated results of [PERSON]'s experiments (see Section 3.1.1), the extrapolated tensile strengths of [PERSON]'s model and [PERSON]'s model seem to be more reasonable than the values given by the original and modified [PERSON]'s boundary effect models. In terms of fracture toughness, the difference of the values given by the different models is within 15%, therefore, the model predictions are all acceptable. However, for the largest tested specimen (\(W=1200\) mm), the measured \(K_{c}\) is only 1.521 MPa\(\sqrt{m}\). According to the experimental size effect study [1; 10], for such a relatively huge specimen with ligament length (0.5\(W\)) to maximum aggregate size (\(d_{max}\)) ratio equals 37.5, the measured fracture toughness should be close to the real material fracture toughness already. Therefore, the lower fracture toughness prediction given by [PERSON]'s model and [PERSON]'s model might be more appropriate. As already mentioned in Section 3.1.1, a choice of a smaller \(\beta_{1}\) could lower the prediction of fracture toughness given by the [PERSON]'s model, but at the same time, an unrealistic larger tensile strength would be obtained. This issue of the [PERSON]'s boundary effect model should be further studied.  
Figure 6: Test results in [30] 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.