## 3 The Problem Definition
### Internally Pressurized FGM Pipe under Cyclic Temperature Difference through Thickness  
As a second example, we investigate the responses of FGM pipe with the same boundary and geometrical conditions of the prior section, but instead of cyclic bending moment, the FGM pipe undergoes a cyclic through-thickness temperature gradient, as shown in Figure 2a. The outer surface temperature of the FGM pipe \(T_{out}\) is assumed to be held at 0\({}^{\circ}\)C and the inner wall temperature \(T_{in}\) is assumed to cyclically vary. The cyclic pattern of the applied temperature gradient \(T_{in}\) is presented in Figure 2b. As illustrated in Figure 2b, in the first half cycle, temperature increases linearly up to its maximum value and then in the second half cycle, following the same trend, drops down to 0. As mentioned earlier since the rate-independent plasticity constitutive equations were employed, the applied cyclic thermal loading varies slowly, as dictated by a quasi-static loading.  
Figure 2: (**a**) Physical model of internally pressurized an FGM cylindrical shell subjected to a cyclic temperature gradient through-thickness. (**b**) Cyclic thermal load history pattern applied to the thick-walled FGM cylindrical shell.  
The distributions of temperature through-thickness of FGM pipe can be obtained using the following heat conduction equation and thermal boundary conditions in Equations (18) and (19), respectively [30]:  
\[\begin{array}{l}\
abla\ =\ \frac{1}{r}\frac{\partial}{\partial r}\\ \
abla[k(r)\
abla T(r)]\ =\ 0\\ \frac{1}{r}\frac{\partial}{\partial r}[k(r)\frac{1}{\partial r}\frac{ \partial}{\partial r}T(r)]\ =\ 0\\ \frac{1}{r}k_{0,\
u\
u b}(r)^{m_{1}}\frac{\partial T(r)}{\partial r}+(m_{1}/r )k_{0,\
u\
u b}(r)^{m_{1}}\frac{\partial T(r)}{\partial r}+k_{0,\
u\
u b}(r)^{ m_{1}}\frac{\partial^{2}T(r)}{\partial r^{2}}\ =\ 0\\ \\ T(a)\ =\ T_{int},\ T(b)\ =\ 0\end{array} \tag{19}\]  
Using the corresponding thermal properties of FGM pipe in Equations (13) and (14) and solving the heat conduction equation with considered thermal boundary conditions, the through-thickness variations of temperature becomes as follows:  
\[T(r)\ =\ C_{1}r^{-m_{1}}+C_{2} \tag{20}\]  
where:  
\[C_{1}\ =\ T_{int}/(b^{-m_{1}}-a^{-m_{1}}),\ C_{2}\ =\ (T_{in}+C_{1}a^{-m_{1}}) \tag{21}\]  
In summary, for the considered case, the following set of constitutive equations composed of strain compatibility Equation (22), equilibrium Equation (23), Hooke's law Equation (24) are required to obtain the expression for stress/strain components being expressed as [29]:  
\[\frac{d\varepsilon_{\theta}}{dr}+(1/r)(\varepsilon_{\theta}-\varepsilon_{r}) \ =\ 0, \tag{22}\]  
\[\frac{d\sigma_{r}}{dr}+(1/r)(\sigma_{r}-\sigma_{\theta})\ =\ 0, \tag{23}\]  
\[\begin{array}{l}\varepsilon_{r}\ =\ \frac{1}{\varepsilon}(\sigma_{r}-\
u( \sigma_{\theta}+\varepsilon_{z}))+\ \varepsilon_{r}^{p}+\alpha T(r)+\ \varepsilon_{r}^{r \text{res}},\\ \varepsilon_{\theta}\ =\ \frac{1}{\varepsilon}(\sigma_{\theta}-\
u(\sigma_{r}+ \varepsilon_{z}))+\ \varepsilon_{\theta}^{p}+\alpha T(r)+\ \varepsilon_{\theta}^{r \text{res}},\\ \varepsilon_{z}\ =\ \frac{1}{\varepsilon}(\sigma_{z}-\
u(\sigma_{\theta}+ \sigma_{r}))+\varepsilon_{z}^{p}+\ \varepsilon_{z}^{r\text{res}},\end{array} \tag{24}\]  
where \(\ \varepsilon_{r,\theta,z}^{r\text{res}}\) are the residual strain components. The corresponding mechanical boundary conditions are those given in Equation (17).