# Steady-State Creep of Asphalt Concrete
## 3 Results and Discussion
### Defining Relations for Steady-State Creep of Asphalt Concrete at Complex Stressed Condition  
In the reality of pavement structure, the points of asphalt concrete pavement during the loading impact of vehicle wheels are in a complex stressed and strained condition [3; 25]. Experimental tests for determining the mechanical characteristics of asphalt concrete, as a rule, are carried out under simple loading and strain schemes (uniaxial tension, uniaxial compression, three-point bending, four-point bending) [2]. Therefore, the integration of these experimental results, which are carried out under simple loading and strain schemes for cases of complex stress and strained condition, is of significant importance for use in practice [24]. By assuming that (\(x\), \(y\), \(z\)) is a Cartesian coordinate system, the stressed condition in point of asphalt concrete pavement is described by stress tensor [26]:  
\[\left( \begin{array}{ccc}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\end{array} \right)=\left( \begin{array}{ ccc}\sigma_{0}&0&0\\ 0&\sigma_{0}&0\\ 0&0&\sigma_{0}\end{array} \right)+\left( \begin{array}{ccc}\sigma_{xx}- \sigma_{0}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}-\sigma_{0}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}-\sigma_{0}\end{array} \right), \tag{9}\]  
where \(\left( \begin{array}{ccc}\sigma_{xx}-\sigma_{0}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}-\sigma_{0}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}-\sigma_{0}\end{array} \right)\) is the deviator of stress; and \(\sigma_{0}=\frac{1}{3}(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})\) is the mean stress. Strained condition in the point is determined by strain rate tensor:  
\[\left( \begin{array}{ccc}\bullet&\epsilon_{xx}&\epsilon_{xy}&\epsilon_{xz}\\ \epsilon_{yx}&\epsilon_{yy}&\epsilon_{yz}\\ \epsilon_{zx}&\epsilon_{zy}&\epsilon_{zz}\end{array} \right)=\left( \begin{array} []{ccc}\bullet&\epsilon_{0}&0&0\\ 0&\bullet&\epsilon_{0}&0\\ 0&0&\bullet&\epsilon_{0}\end{array} \right)+\left( \begin{array}{ccc} \bullet&\epsilon_{xx}-\bullet&\bullet&\bullet_{xy}&\bullet_{xz}\\ \epsilon_{yx}&\epsilon_{yy}-\epsilon_{0}&\bullet_{yz}\\ \epsilon_{zx}&\epsilon_{zy}&\epsilon_{zz}-\epsilon_{0}\end{array} \right), \tag{10}\]  
where \(\left( \begin{array}{ccc}\bullet&\epsilon_{xx}-\bullet&\bullet_{0}&\bullet_{ xy}&\bullet_{xz}\\ \epsilon_{yx}&\epsilon_{yy}-\epsilon_{0}&\bullet_{yz}\\ \epsilon_{zx}&\epsilon_{zy}&\epsilon_{zz}-\epsilon_{0}\end{array} \right)\) is the deviator of strain rates; and \(\epsilon_{0}=\frac{1}{3}(\epsilon_{xx}^{\bullet}+\epsilon_{yy}^{\bullet}+ \epsilon_{zz}^{\bullet})\) is the mean strain. As seen in Figure 13, the strain of asphalt concrete at the start \(\epsilon_{1}\) and final \(\epsilon_{2}\) of the steady-state creep depends on the stress and at a minimum stress of 0.055 MPa, does not exceed 1.2% and 1.9% respectively, i.e., they are very small. To describe the steady-state creep of asphalt concrete, we assumed the following three hypotheses: (1) Material is uncompressible, i.e., the incompressibility condition is true [24]:  
\[\epsilon_{xx}^{\bullet}+\epsilon_{yy}^{\bullet}+\epsilon_{zz}^{\bullet}=0. \tag{11}\]  
Figure 13: Dependence of strains \(\epsilon_{1}\) and \(\epsilon_{2}\) on stress. (2) The deviator of stresses is proportional to deviator of strain rates [24]:  
\[ \begin{array}{l}\sigma_{xx}-\sigma_{0}=\psi\overset{\bullet}{\epsilon}_{xx}, \quad\sigma_{xy}=\psi\overset{\bullet}{\epsilon}_{xy},\\ \sigma_{yy}-\sigma_{0}=\psi\overset{\bullet}{\epsilon}_{yy},\quad\sigma_{yz}= \psi\overset{\bullet}{\epsilon}_{yz},\\ \sigma_{zz}-\sigma_{0}=\psi\overset{\bullet}{\epsilon}_{zz},\quad\sigma_{xx}= \psi\overset{\bullet}{\epsilon}_{zx},\end{array}  \tag{12}\]  
where \(\psi\) is the parameter of proportionality depending on stress and strain rates.