# 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
These findings highlight that, in addition to offering resistance to corrosion and fatigue, FGM pipes allow for the tuning of vibration characteristics through adjustment of \(N\), enabling control of the NF to fall within a desired operational range. Figure 5 and Table 5 provide additional insights into how the volume fraction index (\(N\)) affects the nonlinear vibration behavior. For Type I pipes, increasing \(N\) results in a material profile closer to that of the base metal, which has lower stiffness compared with ceramics. This shift leads to a reduction in the pipe's NF, increased oscillation amplitudes, and more pronounced nonlinear effects such as frequency shifts and possible modal coupling. Conversely, in Type II pipes, a higher \(N\) increases the influence of ceramics, resulting in stiffer dynamic behavior and elevated natural frequencies. This reveals a critical nonlinear design feature of FGMs; their dynamic characteristics can be tuned by adjusting \(N\), allow  
Figure 5: Impact of \(N\) on the nonlinear vibration behaviors of FGM pipes transporting heavy crude oil: (I) Type I; (II) Type II. \begin{table}
\begin{tabular}{c c c} \hline \(N\) & Type I & Type II \\ \hline
0.2 & 173.14 & 113.46 \\
1 & 140.13 & 136.69 \\
5 & 115.11 & 170.39 \\ \hline \end{tabular}
\end{table} 
Table 5: Impact of \(N\) on the NF of FGM pipes transporting heavy crude oil. ing engineers to either amplify or suppress nonlinear effects based on the performance requirements. The system's sensitivity to material gradation offers unique flexibility in tailoring structural response under dynamic loading conditions. The impact of temperature (\(T\)) on the NF of two types of pipes made of FGM transporting heavy crude oil is presented in Table 6. As observed, an increase in temperature slightly reduces the NF. This indicates that FGM pipes maintain good performance and are well suited for high-temperature applications. It should be explained that the results in Table 6 demonstrate that temperature has a modest yet measurable effect on the NF of both FGM pipe types. As the environmental temperature increases from 250 K to 350 K, the NF shows a slight decreasing trend. This decline in NF is attributed to the temperature-dependent degradation of the elastic modulus, as defined by the nonlinear coefficients in Equation (2). Although the variations appear minor, they are significant for applications involving long-term exposure to thermal loads, such as pipelines in arctic or desert environments, where extreme temperatures may persist. Notably, Type I pipes (metal-rich inner layer) exhibit a more pronounced decline in NF with temperature increase compared with Type II. This suggests that the ceramic-rich outer layer in Type II enhances thermal stability, helping retain stiffness under elevated temperatures. These findings align with the expectation that ceramic materials have superior thermal resistance and less sensitivity to temperature fluctuations. Therefore, when selecting FGM configurations for thermal environments, Type II designs may offer better resilience and sustained mechanical performance. Overall, the inclusion of temperature-dependent material models enables a more realistic simulation of pipe behavior under thermal loads and reinforces the potential of FGMs to serve as reliable materials for thermally demanding oil transport applications. Figure 6 presents how the CFV changes with respect to the volume fraction index (\(N\)) for the FGM pipe, as predicted by both the Euler-Bernoulli and Timoshenko beam models. There are noticeable differences between the curves generated using Euler-Bernoulli and Timoshenko beam theories. This indicates that modeling FGM pipes with the Euler-Bernoulli theory may lack sufficient accuracy. Additionally, it is observed that as the parameter \(N\) increases, the CFV decreases for Type I FGM pipes and increases for Type II.