# Nano-Interaction Mechanism Between Crumb Rubber and Asphal Components: A Molecular Dynamic Study
## 3 Results and Discussion
### Mechanical Properties  
#### 3.3.1 Mechanical Modulus  
In molecular simulations, any system subjected to an external force is under stress, which causes a change in the relative positions of the particles in the system. For isotropic materials, the stress-strain behavior is fully described by the Lame constant alone. At this point, the system stiffness matrix [c] can then be established as a relation between it and the stress-strain utilizing the Lame constant. Then, the bulk modulus (\(K\)) and shear modulus (\(G\)) of each system can be calculated as in Equations (4)-(9) [28; 38]:  
\[K_{H}=\frac{K_{V}+K_{R}}{2} \tag{4}\]  
\[K_{V}=\frac{c_{11}+c_{22}+c_{33}+2(c_{12}+c_{23}+c_{13})}{9} \tag{5}\]  
\[K_{R}=\frac{1}{s_{11}+s_{22}+s_{33}+2(s_{12}+s_{23}+s_{13})} \tag{6}\]  
\[G_{H}=\frac{G_{V}+G_{R}}{2} \tag{7}\]  
\[G_{V}=\frac{c_{11}+c_{22}+c_{33}-(c_{12}+c_{23}+c_{31})+3(c_{44}+c_{55}+c_{66} )}{15} \tag{8}\]  
\[G_{R}=\frac{15}{4(s_{11}+s_{22}+s_{33})-4(s_{12}+s_{23}+s_{31})+3(s_{44}+s_{55} +s_{66})} \tag{9}\]  
where \(K_{V}\), \(K_{R}\), and \(K_{H}\) are the approximate upper limit, lower limit, and mean value of the bulk modulus of the Voigt-Reuss-Hill method, respectively; \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Intercept/(J/cm\({}^{3}\))\({}^{0.5}\)**} & \multicolumn{4}{c}{**Fitting Factor**} \\ \cline{2-5}  & **Saturates** & **Aromatics** & **Resins** & **Asphaltene** \\ \hline
9.698 & 23.238 & 4.783 & \(-\)15.647 & \(-\)1.901 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of model normalization of asphalt \(S_{p}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Intercept/(J/cm\({}^{3}\))\({}^{0.5}\)**} & \multicolumn{4}{c}{**Fitting Factor**} \\ \cline{2-5}  & **Saturates** & **Aromatics** & **Resins** & **Asphaltene** \\ \hline
9.698 & 2.556 & 1.526 & \(-\)2.707 & \(-\)0.756 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Multiple linear regression fitting parameters for asphalt \(S_{p}\). \(G_{\
u}\), \(G_{R}\), and \(G_{H}\) are the approximate upper and lower limits and mean values of the Hill method shear modulus, respectively;  
\(c_{ij}\) (\(i=1\), \(2\ldots 6\); \(j=1\), \(2\ldots 6\)) are the component values in the stiffness matrix; and  
\(S_{ij}\) (\(i=1\), \(2\ldots 6\); \(j=1\), \(2\ldots 6\)) are the values of the components of the flexibility matrix. The mechanical properties of different temperature models were analyzed using the Forcite module. The elastic stiffness matrix (GPa) and flexibility matrix (1/TPa) can be obtained. The following equation shows the stiffness and flexibility matrices calculated for base asphalt at 250 K:  
\[c=\left[ \begin{array}{rrrrr}4.7834&2.2209&2.2503&0.0293&-0.1448&-0.0861\\ 2.2209&5.9683&2.0312&0.0398&0.0446&0.2277\\ 2.2503&2.0312&4.972&0.1067&-0.1689&0.144\\ 0.0293&0.0398&0.1067&1.9277&-0.0438&-0.0484\\ -0.1448&0.0446&-0.1689&-0.0438&1.7134&0.0023\\ -0.0861&0.2277&0.144&-0.0484&0.0023&0.2748\end{array} \right]\]  
\[S=\left[ \begin{array}{rrrrr}304.311&-84.8265&-109.194&9.2015&17.0979&224.4 199\\ -84.8265&222.6233&-47.6112&-5.7702&-17.5548&-186.997\\ -109.194&-47.6112&275.0898&-15.7231&18.9172&-141.911\\ 9.2015&-5.7702&-15.7231&522.5906&12.5895&107.9183\\ 17.0979&-17.5548&18.9172&12.5895&587.7126&7.3104\\ 224.4199&-186.997&-141.911&107.9183&7.3104&3958.218\end{array} \right]\]  
The stiffness and flexibility matrices were carried out using the equations above for joint analysis. The \(K\) and \(G\) of each model can be obtained, respectively. The calculation results are shown in Figure 16.  
Overall, the bulk and shear modulus of base asphalt showed a decreasing trend with increasing temperature. The bulk and shear modulus decreased by 81% and 60% when the temperature increased from 250 K to 400 K, respectively. This is due to asphalt being a temperature-sensitive viscoelastic material.