# 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  
#### 3.1.1 [PERSON]'s Experiments  
[PERSON] et al. [28; 29] carried out a series of experiments with geometrically similar wedge-splitting specimens, to investigate the size effects of concrete with small aggregate size. The height of the specimens ranges from 150 mm to 1200 mm, with the maximum aggregate size of 10 mm. There were 24 specimens in total, which were divided into 7 groups according to their heights, and each group had at least 3 samples. The compressive strength at 60 days was 29.56 MPa. The sample dimensions, test results, equivalent crack lengths and nominal stresses in different boundary effect models are listed in Table 1. It should be noted that the dimensions of different samples in one group were slightly different due to the size deviations of the molds. To obtain simultaneously the tensile strength and fracture toughness, the linear regressions of original [PERSON]'s boundary effect model (\(\Delta a_{fic}=d_{max}\)), modified [PERSON]'s boundary effect model (\(\Delta a_{fic}=d_{max}\)), [PERSON]'s size and boundary effect model, and [PERSON]'s Type 2 size effect law are plotted respectively in Figure 5. The estimated materials parameters (see Figure 5a) for original [PERSON]'s boundary effect model (\(\beta_{1}=1\)) are: \(f_{t}=3.05\) MPa, \(K_{c}=1.25\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=42.9\) mm and \(l_{ch}=168.9\) mm. It should be noticed that the results of [PERSON]'s boundary effect model are slightly different from the results presented in [23], because in this analysis, the more accurate sample dimensions and peak loads are adopted from [28; 29]. For modified [PERSON]'s boundary effect model (see Figure 5b), when \(\beta_{1}=1\) (\(\Delta a_{fic}=d_{max}\)), one can obtain \(f_{t}=3.23\) MPa, \(K_{c}=1.25\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=38.1\) mm and \(l_{ch}=150.0\) mm. For [PERSON]'s size and boundary effect model, \(f_{t}=2.36\) MPa, \(K_{c}=1.16\) MPa\(\sqrt{m}\), \(a_{\infty}^{\star}=61.6\) mm and \(l_{ch}=242.9\) mm can be calculated from the intercept and slope shown in Figure 5c. [PERSON]'s Type 2 size effect law (see Figure 5d) gives \(K_{c}=1.16\) MPa\(\sqrt{m}\) and \(c_{f}=61.6\) mm (Equation (29)). As already introduced previously in Sections 2.3 and 2.5, in order to obtain the tensile strength \(f_{t}\) of the material, a ratio \(\gamma_{1}\) should be assumed, then \(f_{t}\) can be calculated by Equation (31). It is obvious that mathematically \(f_{t}\) would increase monotonically as the increase of \(\gamma_{1}\). For the test results analyzed in this section, \(\gamma_{1}=0.1\) and \(1\) gives \(f_{t}=1.48\) MPa and \(4.69\) MPa, respectively. If \(\gamma_{1}=0.28\) or \(0.29\), the corresponding \(f_{t}=2.48\) MPa _or_ \(2.52\) MPa, which is close to the estimation of Koval-Gao's size and boundary effect model. The transition crack length \(a_{\infty}^{\star}=55.86\) mm and material characteristic length \(l_{ch}=220.13\) mm can also be calculated for the chosen \(\gamma_{1}=0.28\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Label**} & \(\mathbf{W}\) & \multirow{2}{*}{\(a_{0}\)/\(W\)} & \  
The tensile strength was not measured in [29], alternatively, we can estimate it from the measured compressive strength by \(f_{t}=0.24f_{ct}^{3/2}=2.29\) MPa [31], which is only 2.9% difference with the estimation of [PERSON]'s size and boundary effect model. The [PERSON]'s boundary effect model and its modified version still somehow overestimate the material tensile strength with \(\beta_{1}=1\). However, as the increase of \(\Delta a_{fic}\), the estimated \(f_{t}\) would decrease accordingly, as what has done in [23], to test the effect of different \(\Delta a_{fic}\) on the predicted tensile strength.