## 4 Numerical Analysis
### Numerical Analysis of Governing Equations for Carbon Fiber-Heated Pavement
#### 4.1.2 Temperature Stress Transfer Process Governing Equations  
The concrete pavement model was believed to be an elastic material in this numerical model. The thermoelasticity equations, according to elasticity mechanics, contain the equilibrium equation, geometric equation, and intrinsic equation.  
Without taking into account external loads, the equilibrium equation is (4):  
\[\begin{cases}\frac{\partial x_{i}}{\partial x}+\frac{\partial y_{xy}}{\partial y }+\frac{\partial z_{xx}}{\partial z}=0\\ \frac{\partial x_{xy}}{\partial x}+\frac{\partial y_{xy}}{\partial y}+\frac{ \partial z_{yy}}{\partial z}=0\\ \frac{\partial z_{xx}}{\partial x}+\frac{\partial z_{yy}}{\partial y}+\frac{ \partial z_{yy}}{\partial z}=0\end{cases} \tag{4}\]where \(\sigma_{x}\) is the principal stress in the x-direction (MPa), \(\sigma_{y}\) is the principal stress in the \(y\) direction (MPa), \(\sigma_{z}\) is the principal stress in the \(z\) direction (MPa), \(\tau_{yx}\) is the shear stress in the \(yx\) plane (MPa), \(\tau_{xx}\) is the shear stress in the \(zx\) plane (MPa), \(\tau_{xy}\) is the shear stress in the \(xy\) plane (MPa), \(\tau_{zy}\) is the shear stress in the \(zy\) plane (MPa), \(\tau_{xz}\) is the shear stress in the \(xz\) plane (MPa), and \(\tau_{yz}\) is the shear stress in the \(yz\) plane (MPa).  
The geometric equation denotes the relationship between pavement strain and displacement, which may be determined using Equation (5):  
\[\begin{cases}&\varepsilon_{x}=\frac{\partial u_{x}}{\partial x}\\ &\varepsilon_{y}=\frac{\partial u_{y}}{\partial y}\\ &\varepsilon_{z}=\frac{\partial u_{z}}{\partial z}\\ &\gamma_{xy}=\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{y}}{\partial x }\\ &\gamma_{yz}=\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{z}}{\partial y }\\ &\gamma_{zx}=\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{z}}{\partial z }\end{cases} \tag{5}\]  
where \(\varepsilon_{x}\) is the strain in the \(x\) direction (\(\mathrm{\SIUnitSymbolMicro}\epsilon\)), \(\varepsilon_{y}\) is the strain in the \(y\) direction (\(\mathrm{\SIUnitSymbolMicro}\epsilon\)), \(\varepsilon_{z}\) is the strain in the \(z\) direction (\(\mathrm{\SIUnitSymbolMicro}\epsilon\)), \(u_{x}\) is the displacement in the \(x\) direction (\(\mathrm{\SIUnitSymbolMicro}\)), \(u_{y}\) is the displacement in the \(y\) direction (\(\mathrm{\SIUnitSymbolMicro}\)), \(u_{z}\) is the displacement in the \(z\) direction (\(\mathrm{\SIUnitSymbolMicro}\)), \(\gamma_{xy}\) is the shear strain in the \(xy\) plane, \(\gamma_{yz}\) is the shear strain in the \(yz\) plane, and \(\gamma_{zx}\) is the shear strain in the \(zx\) plane.  
The current constitutive equation illustrates the relationship between pavement strain and stress, with the pavement strain containing two components generated by thermal stress and temperature change, which may be determined using Equation (6):  
\[\begin{cases}\varepsilon_{x}=\frac{\sigma_{y}}{E}-\frac{\mu\left(\sigma_{y} +\sigma_{z}\right)}{E}+\alpha\Delta T\\ \varepsilon_{y}=\frac{\sigma_{y}}{E}-\frac{\mu\left(\sigma_{z}+\sigma_{z} \right)}{E}+\alpha\Delta T\\ \varepsilon_{z}=\frac{\sigma_{z}}{E}-\frac{\mu\left(\sigma_{x}+\sigma_{y} \right)}{E}+\alpha\Delta T\\ &\gamma_{xy}=\frac{\tau_{xy}}{E}\\ &\gamma_{yz}=\frac{\tau_{xz}}{G}\\ &\gamma_{zx}=\frac{\tau_{xz}}{G}\end{cases} \tag{6}\]  
where \(E\) is [PERSON]'s modulus (MPa), \(\mu\) is [PERSON]'s ratio without dimension, \(G\) is the shear modulus of elasticity (MPa), \(\alpha\) is the linear expansion coefficient (\(1/\mathrm{K}\)), and \(\Delta T\) is the temperature difference (K). The object of this study was the temperature stress inside the pavement, so the pavement was set up with fixed constraints around and at the bottom.