## 5 Numerical Results
### Internally Pressurized FGM Pipe under Cyclic Bending Moment
The third possibility is that the FGM pipe may incur reversed plasticity; in this case, the stress and strain will cycle along a closed loop with a vanishing total plastic strain per cycle. The last cyclic characteristic response of the structure is based on the allowance for ratcheting, with plastic strain rate not only changing the sign but also with the accumulation of plastic strain in each cycle. Figure 5a-c shows the typical stress-plastic strain hysteresis curve in the hoop direction for three kinds of cyclic responses of the FGM pipe with the outer surface made of CS 1026. In addition, the typical hysteresis loops of stress-plastic strain in the hoop direction of FGM pipe with the inner surface made of CS 1026 for three cyclic elastic-plastic responses, as discussed earlier, are depicted in Figure 6a-c Based on four different possible cyclic responses exhibited by the internally pressurized FGM pipe under cyclic bending, the interaction diagram can bring them together in one diagram, such as that shown in Figure 7. It should be noted that for both types of FGM pipe, the internal pressure and cyclic bending moment are normalized with respect to their own yield pressure (\(p_{y}\)) and yield bending moment (\(M_{y}\)), respectively. Note that the dashed and solid lines that determine the boundaries between cyclic stress regimes are representatives' lines of FGM pipe with the inner/outer surfaces made up of CS 1026, respectively. Figure 4: Comparisons of experimental results [32] and present approach; (**a**) axial stress/strain hysteresis curve for pipe made of CS 1026 under axial strain symmetric test. (**b**) Variations of plastic circumferential strain peak at each cycle be for pipe made of CS 1026 under axial symmetric cycling with strain amplitude of 0.5%and constant internal pressure of 46.767 MPa. Figure 5: Representative stress-plastic strain hysteresis loops along the hoop direction of an FGM pipe with outer surfaces made of CS 1026 under sustained internal pressure plus cyclic bending loading for various types of elastic–plastic behaviors. (**a**) Elastic shakedown under \(P_{in}=0.33P_{y}\) and \(M=1.85M_{y}\); (**b**) plastic shakedown under \(P_{in}=0.24P_{y}\) and \(M=2.75M_{y}\); (**c**) ratcheting under \(P_{in}=0.94P_{y}\) and \(M=3.8M_{y}\). Figure 6: _Cont._  
From Figure 7 it is possible to note that for the FGM pipe with its outer surface made up of CS 1026, the reverse plasticity domain is significantly larger than in the case of the FGM pipe with the inner surface made up of CS 1026. This means that changing the FGM properties through the thickness has a significant influence on the size of the ratcheting domain. The deviation between the curves which specify the boundary between the ratcheting and plastic shakedown domains is much more noticeable between 0.2 and 0.4 normalized steady internal pressure. It is also noticed that the curves specifying the elastic and elastic shakedown ranges are almost overlapping in both designs of FGM pipe. From this output, it can be inferred that the variations of material properties are the most critical points within the FGM pipe since they control the ratcheting responses as the root cause of failure. Figure 6: Representative stress–plastic strain hysteresis loops in the hoop direction of FGM pipe with the inner surfaces made of CS 1026 under sustained internal pressure plus cyclic bending loading for different types of elastic–plastic behaviors. (**a**) elastic shakedown under \(P_{in}=0.33P_{y}\) and \(M=1.85M_{y}\); (**b**) plastic shakedown under \(P_{in}=0.24P_{y}\) and \(M=2.75M_{y}\); (**c**) ratcheting under \(P_{in}=0.94P_{y}\) and \(M=3.8M_{y}\). Figure 7: [PERSON]’s diagram for internally pressurized FGM pipe under cyclic bending moment for two different FGM profiles including the inner and outer surfaces made of CS 1026.