# Experimental Study on Hydraulic Fracturing of High Asphalt Concrete Core Rock-Fill Dam
## 4 Analysis of Model Test Result  
### Strain Rate and Tensile Strength  
In the hydraulic fracture feasibility test provided in Sections 2.4 and 2.5, there was no vertical deformation restriction to the top surface of the specimen. Assuming that all of the recorded deformation came from the middle part of the specimen, the tensile strain rate of the middle asphalt concrete with a lower voids content was obtained to be about \(1.4\times 10^{-5}\)/min from Figure 5. In the model test given in Section 3.2, the top surface of the specimen was restricted to deform vertically by the pressure sensor. According to the recorded vertical deformation from the top dial gauge (not given in this paper for abbreviation) and the recorded force value, the deformation rigidity of the pressure sensor can be estimated to be 625 N/mm. As described in the above section, the hydraulic fracturing process can be divided into two stages. Taking the test with a 3.4% voids content (i.e., Figure 8), the tensile strain rate of the two stages are \(5.7\times 10^{-7}\)/min and \(1.0\times 10^{-3}\)/min based, respectively, on the recorded force value and estimated rigidity; the second strain rate is much larger than that from Figure 5. Here, the average strain rate of \(7.3\times 10^{-6}\)/min of the feasibility test and the first stage model test was selected to make the following analysis. The tensile strength under this train rate was obtained from the following test.  
The tensile specimens of \(220\times 40\times 40\) mm were cut from an asphalt concrete slab compacted by tamping a plate under the specific temperature (less than 130 \({}^{\circ}\)C). The mass of the asphalt mixture was carefully weighed according to the required voids content, selected to be 3.5% \(\pm\) 0.5%. The test was carried out under the temperature of 14.6 \({}^{\circ}\)C on a UTM-100 test device. The displacement was measured using an Linear Variable Differential Transformer (LVDT) displacement sensor. Four tensile rates were selected for the test according to the possible lowest tensile rate of the device; those were 0.01/min, 0.001/min, 0.0005/min, and 0.0001/min. The test results are shown in Table 4.  
Due to the scarcity of Table 4's test data, the tensile strength result of asphalt concrete of the Chenhab [20] is referenced, as shown in Figure 16. According to the result of the LVDT strain rate of Chenhab [20], a function shown in Equation (1) is adopted to fit the test data in Table 4. The function is a built-in function of origin software (a software for drawing curves). It has an upper and a lower limit that is compliant with our test results. It is used to describe the process of something growing from  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Strain Rates/1/min** & **Tensile Strength/MPa** \\ \hline
0.01 & 0.4708 \\
0.001 & 0.3318 \\
0.0005 & 0.1597 \\
0.0001 & 0.1563 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Result of tensile test (voids content of 3.5%).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Voids Content/\%** & **Critical Force/N** & **Critical Time/h** & **Failure Time/h** \\ \hline
3.0 & & & (477 h, no fracture) \\
3.4 & 131 & 77.7 & 104 \\
3.5 & 152 & 0.56 (outlier) & 3.4 \\
4 & 93 & 43.3 & 56.9 \\
5.2 & 0 & 0 & 1.8 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The relationship between the critical fracturing time, the critical fracturing pressure, and the voids content of the specimen.  
one limit to another. The fitting result is given in Equation (1), shown in Figure 17, where \(\sigma_{t}\) is tensile strength (MPa), and \(\dot{\epsilon}\) is tensile strain rate (1/min); the Correlation coefficient (R\({}^{2}\)) is 0.94. According to Equation (1), if the tensile strain rate \(\dot{\epsilon}\) is \(7.3\times 10^{-6}\)/min, the tensile strength \(\sigma_{t}\) will be 0.1563 MPa.  
\[\sigma_{t}=0.1563+\frac{0.31921}{1+10^{2.5(-3.03-\log\dot{\epsilon})}} \tag{1}\]