## 5 Numerical Results
### Internally Pressurized FGM Pipe under Cyclic Bending Moment
Hence, the processing parameters, used in the manufacturing of FGM composite materials, can be modified to drive the variations of microstructures and subsequently the performances of the material in relation to the given application. This characteristic distinguishes FGM composite materials from monolithic and homogeneous composite materials. The Numerical Results of Internally Pressurized FG Pipe under Cyclic through the Thickness Temperature Gradient  
The 2D case of an infinite pipe with 390 quadratic quadrilateral elements of type CPE8R, as shown in Figure 8, was employed to simulate in ABAQUS finite element code. Moreover, in order to show the through-thickness temperature profiles of the two adopted FG pipes, based on Equations (20) and (21), the variations of dimensionless temperature defined as \(T\)(r)/\(T_{\text{in}}\) versus dimensionless radial direction \(r\)/\(b\) is presented in Figure 9.  
Figure 8: The applied mesh in FE modeling. Figure 3:0 Temperature profiles through-thickness of the FGM pipe with \(b\)/\(a\) = 1.3 for two prescribed FGM profiles including the inner and outer surfaces made of CS 1026. In order to confirm the performances of the proposed computational algorithm under thermal effects, the comparisons against the simulation obtained by ABAQUS/standard are carried out and shown in Figure 10a,b. It should be noted, Figure 10a,b, that for an FGM pipe made with an outer surface containing pure CS 1026, \(T_{y}\) and \(p_{y}\) represent the temperature and pressure causing initial yielding, respectively, equals to 84 \({}^{\circ}\)C and 20.5 MPa. The hysteresis curve of non-dimensional circumferential stress (\(\sigma_{\theta}/p_{y}\)) versus \(\varepsilon_{p}^{\theta}\) and also the variations of absolute values of \(\varepsilon_{p}^{\theta}\) as a function of the number of cycles for the homogeneous pipe with a geometric parameter of \(b/a=1.3\) and the material properties listed in Table 1 are plotted in Figure 10a,b, respectively. According to Figure 10a,b, it is easily realized that the predicted dimensionless circumferential stress-plastic strain curve by the present approach agrees well with those obtained by ABAQUS simulation. Figures 11-13 show the cyclic elastoplastic behavior of the FGM pipe subjected to steady internal pressure and a cyclic through-thickness temperature gradient, as illustrated in Figure 2b, using the corresponding material given in Table 1 and the ratio of outer to inner radii of 1.3. The behaviors of the two different designs of FGM pipes under arbitrary loading conditions, with the pure CS 1026 on the inner and outer surfaces, are well exemplified by their representative hysteresis stress-plastic strain loops in the circumferential direction under the specified loading conditions, as illustrated in Figures 11 and 12. Based on these findings, the extensions of stress regime zones occurring at the inner wall of the FGM pipe are identified and brought together in the interaction diagram in Figure 3:1 The dashed and solid lines in Figure 13 identify the extensions of cyclic plasticity regimes for FGM pipes with the inner and outer surfaces made up of CS 1026, respectively. Note that like the prior [PERSON]'s diagram presented in Figure 7, internal pressure and thermal loading are given in percentages of yield pressure \(p_{y}\) and yield thermal loadings \(T_{y}\) of the FGM pipes with outer surface in CS 1026, respectively. Figure 3:2 Comparison between proposed numerical model and finite element (FE) simulation with \(m_{i}(i\,=\,1\,:\,5)\,=\,0\). (**a**) Hysteresis stress–plastic strain curve in hoop direction, \(p_{in}=0.65p_{y}\) and \(T_{in}=3.6\)\(T_{y}\). (**b**) Evolution of absolute values of circumferential plastic strain as a function of the number of cycles, \(p_{in}=0.25p_{y}\) and \(T_{in}=2.5\)\(T_{y}\). Figure 3:3 Corresponding stress–plastic strain curves in the circumferential direction of FGM tubes with outer surfaces made of CS 1026 under sustained internal pressure plus cyclic thermal loading for different types of elastic–plastic behaviors.