## 3 Analysis of Test Results
### The Force-Displacement Curves for SBS-Modified Asphalt Mastic for Different Numbers of Cycles  
Based on the test outcomes in Figure 8, it can be determined that the alteration in performance of SBS-modified asphalt mastic within the temperature range of \(-\)20 \({}^{\circ}\)C to 60 \({}^{\circ}\)C is notably significant. Consequently, the SBS-modified asphalt mastic subjected to temperatures ranging from \(-\)20 \({}^{\circ}\)C to 60 \({}^{\circ}\)C was designated as the focal point of this investigation. Utilizing the DENT test findings, a load-displacement curve was graphed, as depicted in Figure 9, facilitating the determination of the critical crack tip opening displacement \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Temperature Range** & **Number of Cycles** & \multicolumn{2}{c}{\(\mathbf{lg(\eta(T))-1/T}\)**} & \multicolumn{2}{c}{**Coefficient of Determination \(R^{2}\)**} & \multicolumn{1}{c}{\(p\)} & \multicolumn{1}{c}{\(E_{\eta}\)/(kJ-mol\({}^{-1}\))**} \\ \hline  & 0 & y = 3923.8x \(-\) 8.67418 & 0.990 & 0.0003 & 75.13 \\  & 3 & y = 3965.9x \(-\) 8.74436 & 0.993 & 0.0001 & 75.94 \\ \(-\)20\(\cdot\)60 \({}^{\circ}\)C & 6 & y = 3978.1x \(-\) 8.73747 & 0.994 & 0.0001 & 76.17 \\  & 9 & y = 4011.8x \(-\) 8.79068 & 0.995 & 0.0001 & 76.81 \\  & 12 & y = 4062.6x \(-\) 8.88664 & 0.994 & 0.0001 & 77.79 \\ \hline  & 0 & y = 3923.8x \(-\) 8.67418 & 0.990 & 0.0003 & 75.13 \\  & 3 & y = 3941.0x \(-\) 8.69152 & 0.996 & 0.0001 & 75.46 \\ \(-\)20\(\cdot\)40 \({}^{\circ}\)C & 6 & y = 3955.1x \(-\) 8.6973 & 0.999 & 0.0001 & 75.73 \\  & 9 & y = 3983.2x \(-\) 8.74042 & 0.999 & 0.0001 & 76.27 \\  & 12 & y = 4005.6x \(-\) 8.77071 & 0.999 & 0.0001 & 76.7 \\ \hline  & 0 & y = 3923.8x \(-\) 8.67418 & 0.990 & 0.0003 & 75.13 \\  & 3 & y = 3945.7x \(-\) 8.69562 & 0.997 & 0.0000 & 75.55 \\
0\(\cdot\)60 \({}^{\circ}\)C & 6 & y = 3964.8x \(-\) 8.7412 & 0.998 & 0.0000 & 75.91 \\  & 9 & y = 3981.6x \(-\) 8.73365 & 1.000 & 0.0000 & 76.24 \\  & 12 & y = 4010.3x \(-\) 8.78018 & 0.999 & 0.0000 & 76.79 \\ \hline  & 0 & y = 3923.8x \(-\) 8.67418 & 0.990 & 0.0003 & 75.13 \\  & 3 & y = 3939.0x \(-\) 8.68776 & 0.997 & 0.0000 & 75.42 \\
0\(\cdot\)40 \({}^{\circ}\)C & 6 & y = 3944.9x \(-\) 8.68674 & 0.999 & 0.0000 & 75.53 \\  & 9 & y = 3964.4x \(-\) 8.71990 & 0.998 & 0.0000 & 75.91 \\  & 12 & y = 3981.5x \(-\) 8.74841 & 0.998 & 0.0000 & 76.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation results of visco-flow activation energy. (CTOD) of the specimen in accordance with Equations (3)-(7). Prior research [32] suggests a strong association between CTOD values and the fatigue properties of asphalt blends, underscoring the capacity of CTOD values to delineate the fracture resistance of asphalt under medium temperatures. The total fracture energy of SBS-modified asphalt mastic is mainly composed of basic fracture energy and plastic deformation energy, which can be determined by the area under the load-displacement curve as follows:  
\[W_{t}=W_{e}+W_{P}=w_{e}\times LB+\beta w_{P}\times BL^{2} \tag{3}\]  
In the above formula, \(W_{t}\) is the total fracture energy, kJ; \(W_{e}\) is basic fracture energy, kJ; \(W_{p}\) is plastic deformation energy, kJ; \(W_{e}\) is fracture ratio basic energy, kJ\(\cdot\)m\({}^{-2}\); \(W_{p}\) is fracture specific plasticity work, kJ\(\cdot\)m\({}^{-3}\); \(B\) is specimen thickness, mm; \(L\) is specimen ligament width, mm; and \(\beta\) is a geometrically defined parameter that characterizes the configuration of the plastic zone. \[w_{t}=W_{t}/BL \tag{4}\]  
Associative Equations (3) and (4) can be obtained as follows:  
\[w_{t}=w_{e}+\beta w_{p}\times L \tag{5}\]  
A linear fit can be made to the \(W_{t}\) fracture ratio fundamental and \(L\) ligament width according to Equation (5), which yields a straight-line intercept of \(W_{e}\) and a slope of \(\beta w_{p}\).