# Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil
## 2 FGM Pipes Transporting Heavy Crude Oil  
Figure 1 illustrates the geometry of FGM pipes transporting heavy crude oil, characterized by their mean radius (\(r\)), outer radius (\(r_{o}\)), inner radius (\(r_{i}\)), and length (\(L\)); \(\Gamma\) denotes the traveling velocity of the fluid flow.  
In this study, two types of FGM pipes are considered: one with a metal-rich inner surface and ceramic-rich outer surface, and the other with the reverse configuration comprising metal on the outside and ceramic on the inside. These configurations are illustrated in Figure 2.  
Figure 1: Configuration of FGM Pipes Transporting Heavy Crude Oil.  
The mass density (\(\rho\)), Young's modulus (\(E\)), and shear modulus (\(G\)) of the FGM pipes are evaluated based on the material volume fraction.  
\[\begin{array}{rl}\text{Type I }&\to P_{F}(r)=P_{m}+(P_{c}-P_{m})\Big{(}\frac{r-r_{s}}{r _{o}-r_{i}}\Big{)}^{N};\ r_{i}\leq r\leq r_{o}\\ \text{Type II }&\to P_{F}(r)=P_{c}+(P_{m}-P_{c})\Big{(}\frac{r-r_{s}}{r_{o}-r_{i}} \Big{)}^{N};\ r_{i}\leq r\leq r_{o}\end{array} \tag{1}\]  
where \(P\) represents a general material property that varies through the thickness of the FGM pipe. Specifically, this includes [PERSON]'s modulus (\(E\)) and mass density (\(\rho\)); that is, \(P=E\) or \(\rho\), as appropriate. Additionally, the shear modulus (\(G\)), which is also a required material property for the Timoshenko beam formulation, is not independently graded but is instead calculated from the graded [PERSON]'s modulus using the classical isotropic elasticity relation \(G=\frac{E}{2(1+\
u)}\), where \(\
u\) is the [PERSON]'s ratio, assumed to be constant across the material gradient. \(P_{F}\), \(P_{m}\), and \(P_{c}\) denote the material properties corresponding to the FGM, metal, and ceramic constituents, respectively. In this regard, the material coefficient \(P\) is defined as a nonlinear function of temperature, as follows [27; 28]:  
\[P_{k}=P_{0}\Big{(}P_{-1}T^{-1}+1+P_{1}T+P_{2}T^{2}+P_{3}T^{3}\Big{)};\ \ k=m,\ c \tag{2}\]  
where \(P_{i}\ (-1,\ 0,\ 1,\ \dots,\ 3)\) represents the coefficients related to environmental temperature \(T(\text{K})\) and is specific to the material composition being considered.  
As mentioned previously, Timoshenko beam theory is used in this study. In this regard, it should be noted that although Euler-Bernoulli beam theory is adequate for long and slender structures, it neglects shear deformation and rotational inertia. These omissions can lead to an underestimation of critical flow velocities and a misrepresentation of dynamic responses, particularly in cases involving thick-walled pipes or functionally graded materials with steep property gradients. Therefore, Timoshenko theory is employed in this study to provide a more comprehensive representation of mechanical behavior, especially under post-buckling and nonlinear vibration conditions. Consequently, according to [PERSON] beam theory, the displacement field components can be expressed as follows:  
\[\begin{array}{rl}u_{1}=u-\frac{1}{\varphi}(x,t)\\ u_{2}=0\\ u_{3}=w(x,t)\end{array} \tag{3}\]  
where \(u\) and \(w\) represent the displacements of the mid-surface along the \(x\) and \(z\) axes, respectively, while \(\varphi\) corresponds to the rotation angle of the pipe's cross-section. Utilizing Equation (3) and employing the von [PERSON] nonlinear strain assumptions, the strain-displacement relations can be formulated as follows:  
\[\begin{array}{rl}\varepsilon_{xx}=u_{x}-\frac{1}{\varphi}\varphi_{x}+\frac{ 1}{2}w_{x}^{2}\\ \gamma_{xz}=w_{x}-\varphi\end{array} \tag{4}\]  
Figure 2: The distribution materials in FGM pipes.  
Taking into account the pipe's elastic deformation, the corresponding stress-strain relationships can be expressed as follows:  
\[\begin{split}\sigma_{xx}=E(r)\epsilon_{xx}\\ \tau_{xz}=K_{SF}G(r)\gamma_{xz}\end{split} \tag{5}\]  
where \(K_{SF}\) denotes the shear correction factor, which is taken as \(\pi^{2}/12\)[29; 30; 31].