# Evaluation of Low-Temperature Performance of Recycled
## 3 Results and Discussion
### Prediction of Low-Temperature Cracking Temperature
\[\alpha(T)=exp\left[\frac{\Delta E_{a}}{R}\bigg{(}\frac{1}{T}-\frac{1}{T_{0}} \bigg{)}\right] \tag{12}\]  
where \(\alpha(T)\) is the shift factor; \(\Delta E_{a}\) is the surface activation energy; \(R\) is the universal gas constant, taken as \(8.314\) J/K/mol; and \(T_{0}\) is the reference temperature, taken as \(273\) K.  
Then, the temperature stress calculation formula of the asphalt mixture during the cooling process can be expressed as follows:  
\[\sigma(t)=\frac{\dot{\varepsilon}\eta_{1}\eta_{2}}{E_{1}E_{2}\sqrt{\Lambda}} \left[\left(E_{1}-\frac{E_{1}E_{2}}{2\eta_{2}}\Big{(}M+\sqrt{\Lambda}\Big{)} \right)e^{-\frac{M-\sqrt{\Lambda}}{2\eta_{1}E_{2}}t}-\left(E_{1}-\frac{E_{1}E_ {2}}{2\eta_{2}}\Big{(}M-\sqrt{\Lambda}\Big{)}\right)e^{-\frac{M+\sqrt{\Lambda }}{2\eta_{1}E_{2}}t}\right]+\eta_{1}\dot{\varepsilon} \tag{13}\]Putting Formula (13) into the asphalt mixture temperature strain energy formula, the specific expression of the temperature strain energy can be obtained as follows:  
\[\omega_{\epsilon}=\int_{\epsilon(\mathbb{T}_{0})}^{\epsilon(\mathbb{T})}\sigma( \mathbb{t})d\epsilon=\frac{i\eta_{1}\eta_{2}}{E_{1}E_{2}\sqrt{\Lambda}}\int_{0} ^{\epsilon(\mathbb{T})}\Bigg{\{}\Bigg{[}\Big{(}E_{1}-\frac{E_{1}E_{2}}{2\eta _{2}}\Big{(}M+\sqrt{\Lambda}\Big{)}\Big{)}e^{-\frac{M\omega\sqrt{\Lambda}}{E_{ 1}\sqrt{\Lambda}}\frac{L}{L}}-(E_{1}-\frac{E_{1}E_{2}}{2\eta_{2}}\Big{(}M- \sqrt{\Lambda}\Big{)})e^{-\frac{M\omega\sqrt{\Lambda}}{E_{1}\sqrt{\Lambda}} \frac{L}{L}}\Bigg{]}+\frac{E_{1}E_{2}\sqrt{\Lambda}}{\eta_{2}}\Bigg{\}}d\epsilon \tag{14}\]  
To simplify the calculation, Formula (14) is discretized with a time step of 2 s. The integral is then discretized to derive the temperature strain energy calculation presented in Formula (15), as follows:  
\[\omega_{\epsilon}=\frac{i^{2}\eta_{1}\eta_{2}}{1800E_{1}E_{2}\sqrt{\Lambda}} \sum_{t=0}^{J=\frac{\eta_{1}\eta_{2}}{\eta_{2}}}\Bigg{\{}\Bigg{[}\Big{(}E_{1} -\frac{E_{1}E_{2}}{2\eta_{2}}\Big{(}M+\sqrt{\Lambda}\Big{)}\Big{)}e^{-\frac{M \omega\sqrt{\Lambda}}{E_{1}\sqrt{\Lambda}}\frac{L}{L}}-(E_{1}-\frac{E_{1}E_{2 }}{2\eta_{2}}(M-\sqrt{\Lambda}))e^{-\frac{M\omega\sqrt{\Lambda}}{E_{1}\sqrt{ \Lambda}}\frac{L}{L}}\Bigg{]}+\frac{E_{1}E_{2}\sqrt{\Lambda}}{\eta_{2}}\Bigg{\}} \tag{15}\]  
The temperature strain energy of an asphalt mixture refers to the internal energy generated due to the accumulation of temperature shrinkage stress within a unit volume of the mixture as the temperature decreases. This energy gradually increases as the temperature declines. In contrast, the bending strain energy density of the mixtures, derived from the trabecular bending test, decreases progressively as the temperature declines. The opposing trends observed in these two strain energy densities indicate the existence of a critical temperature. When the ambient temperature drops below this critical threshold, resulting in the temperature strain energy density surpassing the bending strain energy density, low-temperature cracking occurs in the asphalt mixture. This critical temperature serves as an estimate of the cracking temperature of the mixtures. However, the accuracy of the bending strain energy test results at \(-30\)\({}^{\circ}\)C is limited due to equipment constraints and other factors. The test was conducted at only \(-20\)\({}^{\circ}\)C, \(-10\)\({}^{\circ}\)C, and \(0\)\({}^{\circ}\)C, with data at \(-30\)\({}^{\circ}\)C obtained through calculations. By applying the time-temperature equivalence principle, viscoelastic parameters at \(-30\)\({}^{\circ}\)C are calculated to ascertain the temperature strain energy density value at that temperature. The comparison between the estimated temperature at the final intersection and the fracture temperature from the TSRST is depicted in Figure 14.  
Upon analysis of the data, it is evident that there is a variation between the estimated temperature and the actual fracture temperature obtained from TSRST experiments.