# Impacts of Material Orthotropy on Mechanical Behaviors of Asphal Pavements
## 2 Methodology
### Mathematical Method  
Materials with three mutually orthogonal elastic symmetrical surfaces are defined as orthotropic [32], and are composed of nine independent parameters (\(E_{x},E_{y}\), and \(E_{z}\); \(\mu_{xy},\mu_{zz}\), and \(\mu_{yz}\); and \(G_{yz},\hat{G}_{zx}\), and \(G_{xy}\)). The constitutive equation of orthotropic materials is given by  
\[\left\{\begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \varepsilon_{z}\\ \gamma_{yz}\\ \gamma_{xx}\\ \gamma_{xy}\end{array}\right\}=\left[\begin{array}{cccccc}\frac{1}{E_{x}}& \frac{-\mu_{yx}}{E_{y}}&\frac{-\mu_{yx}}{E_{z}}&0&0&0\\ \frac{-\mu_{yx}}{E_{x}}&\frac{1}{E_{y}}&\frac{-\mu_{xy}}{E_{z}}&0&0&0\\ \frac{-\mu_{xz}}{E_{x}}&\frac{-\mu_{yz}}{E_{y}}&\frac{1}{E_{z}}&0&0&0\\ 0&0&0&\frac{1}{\hat{G}_{yz}}&0&0\\ 0&0&0&0&\frac{1}{\hat{G}_{zx}}&0\\ 0&0&0&0&\frac{1}{\hat{G}_{xy}}\end{array}\right]\left\{\begin{array}{c} \sigma_{x}\\ \sigma_{y}\\ \sigma_{z}\\ \tau_{yz}\\ \tau_{zx}\\ \tau_{xy}\end{array}\right\} \tag{1}\]  
Figure 1: Our research methodology.  
where \(\varepsilon_{x},\varepsilon_{y},\varepsilon_{z},\gamma_{yz},\gamma_{zx}\), and \(\gamma_{xy}\) are the components of strains corresponding to the stresses \(\sigma_{x},\sigma_{y},\sigma_{z},\gamma_{yz},\tau_{zx},\) and \(\tau_{xy}\). \(E_{x},\ E_{y}\), and \(E_{z}\) are the moduli of elasticity in the transverse direction, longitudinal direction, and vertical directions, respectively; \(G_{yz},\)\(G_{zzx},\) and \(G_{xy}\) are the elastic shear modulus transverse, longitudinal, and vertical directions, respectively; \(\mu_{xy},\ \mu_{xz},\) and \(\mu_{yz}\) are the major Poisson's ratios (PR) in the vertical, transverse, and longitudinal directions, respectively; and \(\mu_{yx},\ \mu_{zx},\) and \(\mu_{zy}\) are the minor Poisson's ratios (NU) in the vertical, transverse, and longitudinal directions, respectively.  
The relationship between the parameters of the orthotropic body is given by  
\[\frac{\mu_{yx}}{E_{y}}=\frac{\mu_{xy}}{E_{x}},\ \frac{\mu_{xx}}{E_{z}}=\frac{ \mu_{xz}}{E_{x}},\ \frac{\mu_{zy}}{E_{z}}=\frac{\mu_{yz}}{E_{y}} \tag{2}\]  
The degree of anisotropy represents the ratio of material modulus or Poisson's ratio in any two directions of \(x,y,\) and \(z\). Therefore, \(k_{1}\) is defined as the anisotropy degree between the longitudinal direction and vertical direction. \(k_{2}\) is defined as the anisotropy degree between the transverse direction and longitudinal direction. If \(k_{1}\) and \(k_{2}\) are equal to 1.0, the material is isotropic [26].  
The anisotropy degree for the elastic modulus, shear modulus, and Poisson's ratio are assigned as \(k_{E},k_{G},\) and \(k_{\mu},\) respectively. The anisotropy degree of the elastic modulus, shear modulus, and Poisson's ratio is assumed as equated and changing synchronously [27]; thereby, the specific anisotropy degree is given by  
\[k_{E1}=\frac{E_{y}}{E_{z}},k_{E2}=\frac{E_{x}}{E_{y}} \tag{3}\]  
\[k_{\mu 1}=\frac{\mu_{zx}}{\mu_{yx}},k_{\mu 2}=\frac{\mu_{yz}}{\mu_{xz}} \tag{4}\]  
\[k_{G1}=\frac{G_{yx}}{G_{zx}},k_{G2}=\frac{G_{xz}}{G_{yz}} \tag{5}\]  
Combining Equations (1) and (2), the relationship among orthotropic parameters is obtained as follows:  
\[\mu_{xz}=\frac{\mu_{yz}}{k_{2}},\ \mu_{xy}=\frac{\mu_{yz}}{k_{1}^{2}\cdot k_{2}} \tag{6}\]  
\[G_{xz}=k_{2}\cdot G_{yz},\ G_{xy}=k_{1}\cdot k_{2}\cdot G_{yz} \tag{7}\]  
Therefore, material anisotropy in each layer can be represented by five independent parameters (\(E_{x},\ E_{y},\ E_{z},\ \mu_{yz},\ \text{and}\ G_{yz}\)). Equations (6) and (7) are also applicable to the parameter calculation of transversely isotropic materials. For the transverse isotropy material, \(k_{2}\) in Equations (6) and (7) is equal to 1.0. Therefore, the relationship among transverse isotropic parameters is given by  
\[\mu_{xz}=\mu_{yz}=\mu^{\prime},\ \ \mu_{xy}=\frac{\mu^{\prime}}{k_{1}^{2}} \tag{8}\]  
\[G_{xz}=G_{yz}=G^{\prime},\ G_{xy}=k_{1}\cdot G^{\prime} \tag{9}\]