# Steady-State Creep of Asphalt Concrete
## 3 Results and Discussion
### Defining Relations for Steady-State Creep of Asphalt Concrete at Complex Stressed Condition
Stress intensity:  
\[\sigma_{i}=\frac{1}{\sqrt{2}}\sqrt{\left(\sigma_{xx}-\sigma_{yy}\right)^{2}+ \left(\sigma_{yy}-\sigma_{zz}\right)^{2}+\left(\sigma_{zz}-\sigma_{xx}\right) ^{2}+6\left(\sigma_{xy}{}^{2}+\sigma_{yz}{}^{2}+\sigma_{zx}{}^{2}\right)}. \tag{13}\]  
Strain rate intensity:  
\[\overset{\bullet}{\epsilon}_{i}=\frac{\sqrt{2}}{3}\sqrt{\left(\overset{ \bullet}{\epsilon}_{xx}-\overset{\bullet}{\epsilon}_{yy}\right)^{2}+\left( \overset{\bullet}{\epsilon}_{yy}-\overset{\bullet}{\epsilon}_{zz}\right)^{2} +\left(\overset{\bullet}{\epsilon}_{zz}-\overset{\bullet}{\epsilon}_{xx} \right)^{2}+6\left(\overset{\bullet}{\epsilon}_{xy}^{2}+\overset{\bullet}{ \epsilon}_{yz}^{2}+\overset{\bullet}{\epsilon}_{zx}^{2}\right)}. \tag{14}\]  
Having substituted the Equation (12) into Equation (13), we have  
\[\sigma_{i}=\frac{1}{\sqrt{2}}\sqrt{\psi^{2}\left[\left(\overset{\bullet}{ \epsilon}_{xx}-\overset{\bullet}{\epsilon}_{yy}\right)^{2}+\left(\overset{ \bullet}{\epsilon}_{yy}-\overset{\bullet}{\epsilon}_{zz}\right)^{2}+\left( \overset{\bullet}{\epsilon}_{zz}-\overset{\bullet}{\epsilon}_{xx}\right)^{2} +6\left(\overset{\bullet}{\epsilon}_{xy}^{2}+\overset{\bullet}{\epsilon}_{ yz}^{2}+\overset{\bullet}{\epsilon}_{zx}^{2}\right)\right]}. \tag{15}\]  
Comparison of Equation (15) with Equation (14) shows that  
\[\sigma_{i}=\frac{3}{2}\,\psi\overset{\bullet}{\epsilon}_{i}. \tag{16}\]  
From Equation (16), we obtained  
\[\psi=\frac{2}{3}\,\frac{\sigma_{i}}{\overset{\bullet}{\epsilon}_{i}}. \tag{17}\]  
Then, we write Equation (12) again considering Equation (17):  
\[ \begin{array}{l}\sigma_{xx}-\sigma_{0}=\frac{2}{3}\,\frac{\sigma_{i}}{ \overset{\bullet}{\epsilon}_{i}}\,\overset{\bullet}{\epsilon}_{xx},\quad \sigma_{xy}=\frac{2}{3}\,\frac{\sigma_{i}}{\overset{\bullet}{\epsilon}_{i}} \,\overset{\bullet}{\epsilon}_{xy},\\ \sigma_{yy}-\sigma_{0}=\frac{2}{3}\,\frac{\sigma_{i}}{\overset{\bullet}{ \epsilon}_{i}}\,\overset{\bullet}{\epsilon}_{yy},\quad\sigma_{yz}=\frac{2}{3} \,\frac{\sigma_{i}}{\overset{\bullet}{\epsilon}_{i}}\,\overset{\bullet}{ \epsilon}_{yz},\\ \sigma_{zz}-\sigma_{0}=\frac{2}{3}\,\frac{\sigma_{i}}{\overset{\bullet}{ \epsilon}_{i}}\,\overset{\bullet}{\epsilon}_{zz},\quad\sigma_{zx}=\frac{2}{3} \,\frac{\sigma_{i}}{\overset{\bullet}{\epsilon}_{i}}\,\overset{\bullet}{ \epsilon}_{zx}.\end{array}  \tag{18}\]  
(3) There is a functional dependence between stress intensity and strain rate intensity, which does not depend on the type of stressed condition:  
\[\overset{\bullet}{\epsilon}_{i}=f\,(\sigma_{i}). \tag{19}\]  
For uniaxial tension \(\sigma_{xx}>0\), \(\sigma_{yy}=\sigma_{zz}=\sigma_{xy}=\sigma_{yz}=\sigma_{zx}=0\). Therefore, from Equation (13), we have  
\[\sigma_{i}=\sigma_{xx} \tag{20}\]  
Considering the condition of incompressibility for the material from Equation (11) for uniaxial tension is also true, \(\overset{\bullet}{\epsilon}_{xx}>0\), \(\overset{\bullet}{\epsilon}_{xy}=\overset{\bullet}{\epsilon}_{yz}=\overset{ \bullet}{\epsilon}_{zx}=0\), \(\overset{\bullet}{\epsilon}_{yy}=\overset{\bullet}{\epsilon}_{zz}=-\frac{1} {2}\overset{\bullet}{\epsilon}_{xx}\). Then, from Equation (14), we have  
\[\overset{\bullet}{\epsilon}_{i}=\overset{\bullet}{\epsilon}_{xx}. \tag{21}\]Taking into account Equations (19)-(21) for the considered asphalt concrete, the relation between strain rate intensity and stress intensity will have the following form:  
\[\overset{\bullet}{\epsilon}_{i}=0.6597\cdot\sigma_{i}^{3.9292}, \tag{22}\]  
where \(\overset{\bullet}{\epsilon}_{i}\) is strain rate intensity, \(\%/s\); \(\sigma_{i}\) is stress intensity, MPa. Comparing Equations (19) and (22), we have  
\[f(\sigma_{i})=0.6597\cdot\sigma^{3.9292}. \tag{23}\]  
The previous paper in Reference [22] showed that asphalt concrete is deformed as a plastic material at 20 \({}^{\circ}\)C during cyclic loading under scheme \"loading--deforming under constant load--rest\".