# Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil
## 6 Numerical Results
### Results of Nonlinear Vibration and Post-Buckling for FGM Pipes Transporting Heavy Crude Oil  
In this subsection, the outcomes of the post-buckling and nonlinear vibration behaviors of the FGM pipes transporting heavy crude oil are illustrated. In this regard, an effort is made to show how material properties and geometrical characteristics affect the nonlinear behavior of the present system. Also, as mentioned previously, the first mode shape is \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Present**} & **[PERSON] et al. [36]** & **[PERSON] [37]** & **[PERSON] et al. [38]** \\ \cline{2-6}  & **Discrepancy (\%)** & **Discrepancy (\%)** & **Discrepancy (\%)** \\ \hline
3.14173 & 3.14159 & 0.004 & 3.14159 & 0.004 & 3.14200 & 0.008 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of NDNFs of a homogeneous pipe in the absence of flow. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{\(\mathbf{r_{o}(\mathbf{m})}\)} & \multirow{2}{*}{**\(\mathbf{r_{i}(\mathbf{m})}\)**} & \multirow{2}{*}{**\(\mathbf{L(\mathbf{m})}\)**} & \multirow{2}{*}{**\(\mathbf{N}\)**} & \multirow{2}{*}{**\(\mathbf{P_{0}(\mathbf{N})}\)**} \\ \cline{2-3} \cline{5-5}  & & & & \\ \hline \(P_{0}\) & \(2.01\times 10^{11}\) & \(8166\) & \(3.48\times 10^{11}\) & \(2370\) \\ \(P_{-1}\) & \(0\) & \(0\) & \(0\) & \(0\) \\ \(P_{1}\) & \(3.08\times 10^{-4}\) & \(0\)  
considered in this study. In this context, the first vibration mode shape of the FGM pipes is demonstrated in Figure 3.  
Figure 4 illustrates the nonlinear vibration behavior of FGM pipes transporting heavy crude oil, for two types, at \(T=300\) K. As observed, the vibration curve for Type II shifts to the right over time, indicating that the natural frequency (NF) of Type I is higher than that of Type II. Each graph represents the pipe's vibration response under identical conditions, providing insights into how material gradation influences structural behavior. The nonlinear vibration response observed in Figure 4 clearly highlights amplitude-dependent frequency behavior, a hallmark of nonlinear systems. Specifically, the gradual shift of the response curve over time reflects the nonlinear stiffness behavior of the FGM  
Figure 4: Nonlinear vibration behaviors of FGM pipes transporting heavy crude oil. **(I)** Nonlinear vibration response across all time intervals. **(II)** Nonlinear vibration response during the initial period. **(III)** Nonlinear vibration behavior over extended periods. Figure 3: First mode shape of the FGM pipes. pipe. For Type I pipes (metal-rich inner layers), the vibration response shows a delayed decay and wider oscillation range, indicative of softening behavior, where the effective stiffness decreases with increasing amplitude. In contrast, Type II pipes (ceramic-rich inner layers) tend to maintain a more concentrated response, suggesting hardening characteristics due to the stiffer ceramic composition. This behavior stems from the inclusion of nonlinear strain-displacement relations (via von Karman assumptions) and the interaction of material gradation with the pipe's structural geometry. Such nonlinear dynamics are not captured by linear models and underscore the importance of considering nonlinearities in the analysis of FGM structures, especially when large amplitude vibrations or long-term operational stability are of concern. Figure 5 presents how varying the volume fraction index (\(N\)) affects the nonlinear vibration behavior of FGM pipes transporting heavy crude oil at \(T=300\) K, for two different types. The corresponding effect of \(N\) on the natural frequency (NF) is also presented in Table 5. The results indicate that as \(N\) increases, the NF decreases for Type I pipes but increases for Type II pipes. This trend can be explained using Equations (1a) and (1b): with increasing \(N\), the material properties of Type I pipes tend toward those of metal, while those of Type II pipes approach ceramic behavior.