# 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
For fracture toughness \(K_{c}\), when the sample size is larger than 600 mm, the nominal fracture toughness \(K_{c}\) tends to be stable, and its value ranges from 0.93 to 1.13 MPa\(\sqrt{m}\), hence, \(K_{c}=1.16\) MPa\(\sqrt{m}\) estimated by the [PERSON]'s model and Bazant model is acceptable. While for the original [PERSON] model and its modified version, the estimated \(K_{c}=1.25\) MPa\(\sqrt{m}\) when \(\Delta a_{fic}=d_{max}\) seems slightly higher, since the largest measured \(K_{c}\) is only 1.09 MPa\(\sqrt{m}\) for the largest specimen with \(W=1200\) mm, whose failure can be assumed as totally LEFM control. Nevertheless, as the variation of the value of \(\Delta a_{fic}\) (or \(\beta_{1}\)), the best estimations of \(f_{t}\) and \(K_{c}\) given by [PERSON]'s boundary effect model can be achieved. It is worthwhile to mention that the adjustment of \(\beta_{1}\) in original and modified [PERSON]'s boundary effect models would alter the predicted \(f_{t}\) and \(K_{c}\) values simultaneously, to be more exact, as the increase of \(\beta_{1}\), the estimated \(f_{t}\) would decrease and \(K_{c}\) would increase. This is simply due to the model assumption about the competition between two failure mechanisms (strength of material and  
Figure 5: Test results in [28,29] 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. LEFM). A larger fictitious crack length gives a lower non-local stress \(\sigma_{n2}\) (see Figure 2a) distributed along the fictitious crack, thus reduces the contribution of strength of material on the specimen failure, and finally leads to the different combination of predicted fracture parameters. This feature may introduce some difficulties to directly adopt the extrapolated fracture parameters to the numerical simulation of a real structure, since one may suspend the veracity of the estimated parameters. Therefore, before using the [PERSON]'s boundary effect model to extrapolate the fracture parameters, it is recommended to fix one parameter by tests or give a narrow range to it in advance. In this way, the other parameter can be more reliable, and a reasonable \(\beta_{1}\) is naturally obtained. In terms of fracture parameters extrapolation, [PERSON]'s Type 2 size effect law is also aided by a length assumption (\(c_{f}=\gamma_{1}l_{ch}\)). However, the different choice of \(\gamma_{1}\) would change the prediction of \(f_{t}\) only. It is reported in [20; 39; 40] that \(\gamma_{1}\) is about 0.28 for concrete material, but as the increase of maximum aggregate size, especially for the dam concrete with \(d_{max}=150\) mm, \(\gamma_{1}\approx 0.28\) should be better verified by more experiential results. The third model, [PERSON]'s size and boundary effect model only provides one set of fracture parameters, which seems to be less flexible than the other models, but its estimated parameters are acceptable for this case.