## 5 Numerical Results  
### Internally Pressurized FGM Pipe under Cyclic Bending Moment  
In this research, it is assumed that the FGM pipe is made up of carbon steel (CS) grade 1026 with corresponding material properties listed in Table 1. The ratio of outer to inner radii of the FGM pipe is taken to be \(b/a=1.3\). For the selected FGM pipe, two cases are considered. Case one, FGM properties are assumed to continuously change from those of CS 1026, located at the outer surface, to those with FGM grading indexes of \(m_{1}=3\), \(m_{2}=-1.1\), \(m_{3}=3\), \(m_{4}=2\) and \(m_{5}=-1.5\) at the inner surface. Case two, the inner surface of FGM pipe consists of pure CS 1026 and the properties smoothly vary along the radial direction according to Equation (12) with material in-homogeneity parameters of \(m_{1}=-3\), \(m_{2}=1.1\), \(m_{3}=-3\), \(m_{4}=-2\) and \(m_{5}=1.5\) at the inner surface [28]. In order to validate the capabilities of the proposed numerical technique in predicting the elastic-plastic behaviors of metals under cyclic loading conditions, the comparisons were carried out with those given by experiments [32]. The experiments were conducted on a tube made from CS 1026 under loading conditions consisting of: cyclic, symmetric, axial strain-controlled loading as in Figure 4a; and axially strain symmetric cycling with an amplitude of 0.5% combined with a steady internal pressure of 46.767 MPa as in Figure 4b. It is possible to remark that the present approach gives results that are very close to the experimental results, providing validation. It is worth mentioning that the stable hysteresis loop achieved after 12 cycles was chosen and illustrated in Figure 4a. It is worthwhile to emphasize that, in order to calibrate the prediction modeling, the stable hysteresis loop experimentally achieved after 12 cycles was chosen as shown in Figure 4a. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{E_{0}}\) (GPa) & \(\mathbf{\
u}\) & \(\mathbf{\sigma_{y0}}\) (MPa) & \(\mathbf{\alpha_{0}}(1/0^{K})\) & \(\mathbf{C_{0}}^{1,2,3}\) (MPa) & \(\mathbf{\gamma_{0}}^{1,2,3}\) \\ \hline
181.3 & 0.3 & 186.2 & \(12.1\times 10^{-6}\) & 65,103, 39,584, 1675 & 7511, 405.3, 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of material properties CS 1026 used in numerical procedure [32,33]. Figure 3: Flow diagram of the proposed solution algorithm implemented in user subroutine UMAT. Different modes of cyclic behaviors--namely, completely elastic state, elastic shakedown, reverse plasticity (plastic shakedown and ratcheting in the structures)--become evident under various load combinations. In order to analyze the cyclic behaviors of the FGM pipe with the inner and outer surfaces in steel under two types of prescribed loading conditions, the interaction diagrams known as Bree's diagrams are presented in this section. Bree's diagrams have abscissas and ordinates that denote the levels of constant and cyclic loading levels, respectively. These diagrams help illustrate a variety of cyclic responses of structures under different loading conditions. It should be noted that although the analysis can be extremely tedious and lengthy, the results can be plotted in a simple graphical form. Bree diagrams identifying cyclic stress regime limits have been extensively utilized in different engineering sections. It should also be mentioned that the corresponding results in Figures 5-13, are based upon the responses of the inner surface of the FGM pipe for both designated designs. It is of interest to interpret the different cyclic plasticity regimes and illustrate their representative hysteresis loops. The first possibility is to have a pure elastic stress state, with no plastic deformation. The second possibility is to allow for the elastic shakedown involving the plastic zone during the first load cycles, but ultimately shakedown to state with purely elastic behavior with an elastic response to further cycling.