# Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models
## 4 Conclusions  
The local and non-local versions of boundary effect models developed by [PERSON] et al., the modified [PERSON]'s boundary effect model proposed in this study, the [PERSON]'s size and boundary model, and [PERSON]'s Type 2 size effect law are all able to capture the effects of crack length and sample size on the fracture behavior of wedge-splitting and compact tension specimens. The proposed modified [PERSON]'s boundary effect model provides a more appropriate definition of equivalent crack length. This proposed model can correctly combine the contributions of strength of material and LEFM on the material failure for concrete with any maximum aggregate size. The boundary effect shape of [PERSON]'s Type 2 size effect law shares the same equivalent crack length as [PERSON]'s size and boundary effect model. This improvement could extend the applicability of the model to extrapolate the material parameters by the test results obtained from both the geometrically similar specimens with constant crack-length-to-height ratios and specimens with different initial crack-length-to-height ratios. The applicability of the different size and boundary effect models on the determination of fracture toughness and tensile strength for concrete material are verified and compared by test results reported in the literature. The original and modified [PERSON]'s boundary effect model is more flexible, because the two fracture parameters vary together as the adjustment of the discrete number \(\beta_{1}\). However, there exists the risk that the reasonable fracture parameters cannot be obtained with one single \(\beta_{1}\). 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. Besides, the proposed modified [PERSON]'s boundary effect model would be more appropriate than its original version, due to the fact that the two failure mechanisms are correctly combined even for the fracture analysis of dam concrete with maximum aggregate size reaches 150 mm. Nevertheless, for \(d_{max}\) such as 10 mm or 20 mm, this improvement is neglectable. [PERSON]'s Type 2 size effect law is less flexible than [PERSON]'s boundary effect model, because once the test results are given, the estimated fracture toughness is decided. The variation of the length scale or \(\gamma_{1}\) would change the estimated value of tensile strength only. In terms of [PERSON]'s size and boundary effect model, only one set of fracture parameters can be extrapolated from the test results. However, for the test results adopted in this study, the predictions of the fracture properties given by [PERSON]'s size and boundary effect model are all acceptable. For dam concrete, the specimen size needed to measure the stable tensile strength and fracture toughness can be huge. Hence, it is recommended to use the size and boundary effect to determine the two fracture parameters simultaneously. The fracture tests on at least three different sizes of geometrically similar specimens (e.g., \(W=750\) mm, 1500 mm, 2250 mm with \(\alpha_{0}=0.4\)) and one specimen size with three different initial crack-length-to-height ratios (e.g., \(W=1125\) mm with \(\alpha_{0}=0.2,\ 0.4,\ 0.6\)) are encouraged to be performed, for the purpose of the extrapolation of fracture parameters by the theoretical models. More experimental findings will be reported in the authors' further coming publications. Comprehensive experimental studies on the wedge-splitting specimens made of fully graded concrete are currently undergoing. All the specimens are casted in the same time at the construction sites of Wudongde and Baihetan super-high arch dams. The mechanical and fracture tests are performed simultaneously, to obtain the reliable and comprehensive test results for the dam concrete.