# Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models
## 2 Size and Boundary Effect Models
### Method for the Determination of Fracture Parameters  
The fracture tests on geometrically similar specimens with constant crack-length-to-height ratios, and specimens with different initial crack-length-to-height ratios are frequently performed to study the dependence of fracture behaviors on specimen sizes and initial crack lengths. The test results  
Figure 4: Equivalent crack length \(a_{e}\) versus initial crack length \(a_{0}\) for different discrete number \(\beta_{1}\) when \(d_{max}=150\) mm.  
Figure 3: Equivalent crack length \(a_{e}\) versus initial crack length \(a_{0}\) for different maximum aggregate size \(d_{max}\) when \(\beta_{1}=1\).  
obtained from these two types of specimens can be used to extrapolate the tensile strength \(f_{t}\) and fracture toughness \(K_{c}\) by the size and boundary effect models.  
The linear forms of original [PERSON]'s boundary effect model (Equation (14)), modified [PERSON]'s boundary effect model (Equation (19)), [PERSON]'s size and boundary effect model (Equation (22)), and [PERSON]'s Type 2 size effect law (Equation (29)) have presented previously in this section. Only the maximum loads and geometrical information of the tested specimens are needed for the linear regression analysis. The nominal stresses in the different linear models can be calculated by Equation (13) (original and modified [PERSON]'s boundary effect model), Equation (23) ([PERSON]'s size and boundary effect model), and Equation (30) ([PERSON]'s Type 2 size effect law). In addition, the different equivalent crack lengths can be calculated by Equation (11) (original [PERSON]'s boundary effect model), Equation (18) (modified [PERSON]'s boundary effect model), and Equation (21) ([PERSON]'s size and boundary effect model and [PERSON]'s Type 2 size effect law). With these two known values from the test results and specimens' geometrical information, one can do the linear regression analysis by the different models. Thus, the intercept and slope obtained from the best linear fit can be used to predict the material properties.  
It should be noted that for the original and modified [PERSON]'s boundary effect models, the extrapolated fracture parameters vary together as the adjustment of the discrete number \(\beta_{1}\). Thus, special attention should be paid to the regression analysis, to ensure the predicted tensile strength and fracture toughness are all acceptable. For [PERSON]'s Type 2 size effect law, as introduced in Section 2.3, only the fracture toughness \(K_{c}\) and the effective size of the fracture process zone \(c_{f}\) can be obtained from the linear fit. To estimate the tensile strength \(f_{t}\) of the material, a ratio \(\gamma_{1}\) (\(\gamma_{1}=c_{f}/l_{ch}\)) between \(c_{f}\) and the material characteristic length \(l_{ch}\) should be assumed. Then \(f_{t}\) can be calculated as follows:  
\[f_{t}=\sqrt{\frac{\gamma_{1}K_{c}^{2}}{c_{f}}}. \tag{31}\]  
[PERSON]'s size and boundary effect model is less flexible than the other two models, because the liner fit is unique for one set of the test results, and one can calculate the tensile strength from the intercept and fracture toughness from the slope, respectively.