Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil

Abstract

Functionally graded materials (FGMs) have the potential to revolutionize the oil and gas transportation sector, due to their increased strengths and efficiencies as pipelines. Conventional pipelines frequently face serious problems such as extreme weather, pressure changes, corrosion, and stress-induced pipe bursts. By analyzing the mechanical and thermal performance of FGM-based pipes under various operating conditions, this study investigates the possibility of using them as a more reliable substitute. In the current study, the post-buckling and nonlinear vibration behaviors of pipes composed of FGMs transporting heavy crude oil were examined using a Timoshenko beam framework. The material properties of the FGM pipe were observed to change gradually across the thickness, following a power-law distribution, and were influenced by temperature variations. In this regard, two types of FGM pipes are considered: one with a metal-rich inner surface and ceramic-rich outer surface, and the other with a reverse configuration featuring metal on the outside and ceramic on the inside. The nonlinear governing equations (NGEs) describing the system’s nonlinear dynamic response were formulated by considering nonlinear strain terms through the von Kármán assumptions and employing Hamilton’s principle. These equations were then discretized using Galerkin’s method to facilitate the analytical investigation. The Runge–Kutta method was employed to address the nonlinear vibration problem. It is concluded that, compared with pipelines made from conventional materials, those constructed with FGMs exhibit enhanced thermal resistance and improved mechanical strength.

1. Introduction

FGMs have wide use in many fields, including the chemical industry, solar power systems, heat exchangers, nuclear reactors, and combustion engines. These materials occur in nature, such as bones or human skin, providing toughness, elasticity, and tactility [1]. The first FGM was developed in 1984 in Japan resulting from a space plane project [2]. FGM has two isotropic material phases but there are no such limitations on the number of materials to be used; they are merely required to be chemically and spatially compatible. Materials that can be used as FGMs include steel, copper, titanium, aluminium, etc., and the other material can be a type of ceramic material. FGMs are advanced materials compared with other composite materials that are made of two or more phases with continuous and smooth transitions between the composition of materials [1]. FGMs are superior to homogeneous materials with similar characteristics.

FGMs are currently very popular in engineering and technology. They are used in special nuclear components, spacecraft structures, high-temperature coatings, and in the oil and gas industry as pipes for transportation, which is the prime focus of the current study [3]. The most frequently used FGM blend is metal/ceramic composite, as the ceramic has a very good thermal resistance which enhances the thermal properties, while the metallic part has high fracture toughness. The key property is the gradual change in the properties of the material. FGMs are classified into two broad categories: thin FGMs and bulk FGMs. Thin FGMs are used as surface coatings, while bulk FGM are high-volume materials that require more intensive work. They are produced through the fabrication process [1].

Several studies have explored the nonlinear behaviors of pipes and cylindrical shells [4,5,6,7], with some focusing specifically on the vibration behaviors of pipes [8,9,10], while others have concentrated on their buckling behavior [11,12,13].

Although structures composed of FGMs are widely utilized in various industrial applications, their vibration response has not been thoroughly addressed in the previous literature. Nevertheless, a number of researchers have investigated the nonlinear responses of the structures made of FGMs [14,15,16]. Additional studies have focused on the vibration behavior of FGM cylindrical shells [17,18,19] and pipes [20,21], while others have examined their buckling characteristics [22,23,24]. Recently, Foroutan and Torabi [25] conducted a detailed study on nonlinear vibration behaviors of multilayer FGM cylinders with internal FGM spiral stiffeners exposed to thermal environments. Although FGM pipes have been studied extensively for their vibration [14,15,16,17,18,19,20,21] and buckling behaviors [22,23,24], the majority of existing works focus on idealized or simplified conditions, often assuming linear responses, utilizing inviscid or ideal fluid models, or neglecting the influence of temperature-dependent material gradation. For instance, Zhen et al. [20] investigated nonlinear vibrations of FGM pipes with initial curvature, but they did not consider temperature sensitivity. Similarly, Tang and Yang [26] examined the post-buckling behavior of FGM pipes conveying fluid, but their model excluded nonlinear dynamic effects and heavy viscous fluids such as crude oil.

Transporting heavy crude oil presents numerous engineering challenges, primarily due to its high viscosity, which increases internal friction and results in greater energy losses, pressure drops, and thermal degradation of conventional pipelines. Moreover, traditional metal pipes are prone to corrosion, fatigue cracking, and buckling under high flow velocities and fluctuating thermal conditions. These issues significantly limit the service life and reliability of oil transport infrastructure. In contrast, FGMs provide a promising alternative due to their spatially varying composition, which allows the use of optimized combinations of mechanical toughness and thermal resistance. By enabling smooth transitions between metal and ceramic phases, FGMs can reduce interfacial stress concentrations and enhance resistance to thermomechanical loads. Despite their wide application in aerospace and nuclear sectors, their potential in oil transport pipelines remains underexplored, especially with regard to their nonlinear dynamic stability and post-buckling performance. This study aims to fill this gap by investigating the nonlinear vibration and post-buckling behavior of FGM pipes transporting heavy crude oil, thereby assessing their viability as a next-generation solution in the oil and gas industry. Consequently, it presents a unified and comprehensive nonlinear framework that, to the best of our knowledge, is the first to simultaneously examine the nonlinear vibration and post-buckling behaviors of temperature-dependent FGM pipes conveying heavy crude oil. In this context, 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. Also, the material properties of the FGM pipe are temperature-dependent. Therefore, by analyzing the mechanical and thermal performance of FGM-based pipes under various operating conditions, this study investigates the possibility of using them as a more reliable substitute. The novelty of this study lies in its unified treatment of nonlinear vibration and post-buckling behavior of FGM pipes conveying heavy crude oil, accounting for temperature-dependent material properties and two gradation configurations. Unlike previous studies, the model addresses realistic oil transport conditions involving high-viscosity fluids and thermomechanical interactions, offering new insights for pipeline design in harsh environments.

To develop the nonlinear governing equations, we employ Timoshenko beam theory, which accounts for both rotary inertia and transverse shear deformation. These factors are especially significant for thick-walled or moderately thick FGM pipes where high stiffness gradients and thermal effects are present. Nonlinear strain terms are incorporated through von Kármán assumptions, and Hamilton’s principle is applied to derive the dynamic equations. The equations are discretized via Galerkin’s method, and the nonlinear vibration behavior is investigated through the Runge–Kutta method. Furthermore, our validation (Section 6.1) against benchmark studies confirms the accuracy of the proposed formulation. Notably, the identified nonlinear vibration and post-buckling responses directly impact the structural integrity and operational reliability of FGM pipes transporting heavy crude oil in real-world applications.

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); $\mathsf{\Gamma }$ 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.

The mass density ($\rho$), Young’s modulus ($E$), and shear modulus ($G$) of the FGM pipes are evaluated based on the material volume fraction.

$$\begin{gathered} \mathrm{Type}\mathrm{I}\to P_F\left(r\right)=P_m+\left(P_c-P_m\right){\left({\displaystyle \frac{r-r_i}{r_o-r_i}}\right)}^N;r_i\le r\le r_o \\ \mathrm{Type}\mathrm{II}\to P_F\left(r\right)=P_c+\left(P_m-P_c\right){\left({\displaystyle \frac{r-r_i}{r_o-r_i}}\right)}^N;r_i\le r\le r_o \end{gathered}$$

(1)

where $P$ represents a general material property that varies through the thickness of the FGM pipe. Specifically, this includes Young’s modulus ($E$) and mass density ($\rho$); that is, $P=E\mathrm{o}\mathrm{r}\rho$, 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 Young’s modulus using the classical isotropic elasticity relation $G={\displaystyle \frac{E}{2\left(1+\nu \right)}}$, where $\nu$ is the Poisson’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]:

$$P_k=P_0\left(P_{-1}{T}^{-1}+1+P_{1}T+P_2{T}^2+P_3{T}^3\right);k=m,c$$

(2)

where ${\mathrm{P}}_i\left(-1,0,1,\dots ,3\right)$ represents the coefficients related to environmental temperature $T\left(\mathrm{K}\right)$ 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 Timoshenko beam theory, the displacement field components can be expressed as follows:

$$\begin{gathered} u_1=u-\mathcal{z}\phi \left(x,t\right) \\ {u}_2=0 \\ {u}_3=w\left(x,t\right) \end{gathered}$$

(3)

where $u$ and $w$ represent the displacements of the mid-surface along the $x$ and $z$ axes, respectively, while $\phi$ corresponds to the rotation angle of the pipe’s cross-section. Utilizing Equation (3) and employing the von Kármán nonlinear strain assumptions, the strain–displacement relations can be formulated as follows:

$$\begin{gathered} {\epsilon }_{xx}=u_{,x}-\mathcal{z}{\phi }_{,x}+{\displaystyle \frac{1}{2}}{w}_{,x}^2 \\ {\gamma }_{xz}=w_{,x}-\phi \end{gathered}$$

(4)

Taking into account the pipe’s elastic deformation, the corresponding stress–strain relationships can be expressed as follows:

$$\begin{gathered} {\sigma }_{xx}=E\left(r\right){\epsilon }_{xx} \\ {\tau }_{xz}=K_{SF}G\left(r\right){\gamma }_{xz} \end{gathered}$$

(5)

where $K_{SF}$ denotes the shear correction factor, which is taken as ${\pi }^2/12$ [29,30,31].

3. Nonlinear Governing Equations

In order to apply Hamilton’s principle, it is necessary to determine the total kinetic energy ($T_{total}=T_{pipe}+T_{fluid}$) and the strain energy of the FGM pipe transporting fluid ($U$). In this regard, based on Equations (1), (4), and (5), the kinetic energy of pipe and fluid as well as the strain energy of the pipe made of FGM are as follows:

$$T_{pipe}={\displaystyle \frac{1}{2}}{m}_P\underset{0}{\overset{L}{\int }}\left(u_{,t}^2+w_{,t}^2\right)dx+{\displaystyle \frac{1}{2}}\underset{0}{\overset{L}{\int }}\rho \left(r\right)I{\phi }_{,t}^{2}dx$$

(6)

$$T_{fluid}={\displaystyle \frac{1}{2}}{m}_f\underset{0}{\overset{L}{\int }}\left\{{\left(\mathsf{\Gamma }+u_{,t}+\mathsf{\Gamma }{u}_{,x}\right)}^{ 2}+{\left(w_{,t}+\mathsf{\Gamma }{w}_{,x}\right)}^{ 2}\right\}dx$$

(7)

$$U={\displaystyle \frac{1}{2}}\underset{0}{\overset{L}{\int }}\left\{\overline{EI}{\phi }_{,x}^2+\overline{EA}{\left(u_{,x}+{\displaystyle \frac{1}{2}}{w}_{,x}^2\right)}^2+K_{SF}\overline{GA}{\left(w_{,x}-\phi \right)}^2+P_0\left(2u_{,x}+w_{,x}^2\right)\right\}dx$$

(8)

where $I$ represents the moment of inertia associated with the pipe’s cross-section, and $m_P$ and $m_f$ denote the pipe’s and fluid’s mass per unit length, respectively. $\overline{EI}$, $\overline{EA}$, and $\overline{GA}$ are as follows:

$$\overline{EI}=\underset{0}{\overset{2\pi }{\int }}\underset{r_i}{\overset{r_o}{\int }}E\left(r\right)r^2{\left(\mathit{sin}\theta \right)}^{2}rdrd\theta ;\overline{EA}=\underset{0}{\overset{2\pi }{\int }}\underset{r_i}{\overset{r_o}{\int }}E\left(r\right)rdrd\theta ;\overline{GA}=\underset{0}{\overset{2\pi }{\int }}\underset{r_i}{\overset{r_o}{\int }}G\left(r\right)rdrd\theta$$

(9)

According to Hamilton’s principle:

$$\underset{t_1}{\overset{t_2}{\int }}\left(\delta T_{total}-\delta U\right)dt=0$$

(10)

By substituting Equations (7) and (8) into Equation (10), applying integration by parts, and then equating the coefficients of $\delta u$, $\delta w$, and $\delta \phi$ to zero, the resulting partial differential equations of motion are obtained as follows:

$$m_f\left(u_{,tt}+2\Gamma u_{,xt}+{\Gamma }^2{u}_{,xx}\right)+m_P{u}_{,tt}-\overline{EA}{\left(u_{,x}+{\displaystyle \frac{1}{2}}{w}_{,x}^2\right)}_{,x}-{P_0}_{,x}=0$$

(11)

$$m_f\left(w_{,tt}+2\Gamma w_{,xt}+{\Gamma }^2{w}_{,xx}\right)+m_P{w}_{,tt}-{\left(w_{,x}\left[\overline{EA}\left(u_{,x}+{\displaystyle \frac{1}{2}}{w}_{,x}^2\right)+P_0\right]\right)}_{,x}-K_{SF}\overline{GA}\left(w_{,xx}-{\phi }_{,x}\right)=0$$

(12)

$$\overline{EI}{\phi }_{,xx}+K_{SF}\overline{GA}\left(w_{,x}-\phi \right)-\rho \left(r\right)I{\phi }_{,tt}=0$$

(13)

In this regard, for the simply supported FGM pipes, the essential boundary conditions can be defined as follows:

$$u=w={\phi }_{,x}=0atx=0,L$$

(14)

By focusing solely on the transverse motion and assuming limited stretching, Equation (11) simplifies to:

$${\left(u_{,x}+{\displaystyle \frac{1}{2}}{w}_{,x}^2\right)}_{,x}=0$$

(15)

Therefore, using Equation (14) in the integration of Equation (15) results in the following expression for $u$:

$$u={\displaystyle \frac{x}{2L}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx-{\displaystyle \frac{1}{2}}\underset{0}{\overset{x}{\int }}{w}_{,x}^{2}dx$$

(16)

Substituting Equation (16) into Equation (12) yields the following governing equations of motion:

$$\left(m_f+m_P\right)w_{,tt}+2m_f\Gamma w_{,xt}+\left(m_f{\Gamma }^2-P_0\right)w_{,xx}-{\displaystyle \frac{\overline{EA}}{2L}}{w}_{,xx}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx-K_{SF}\overline{GA}\left(w_{,xx}-{\phi }_{,x}\right)=0$$

(17)

$$\overline{EI}{\phi }_{,xx}+K_{SF}\overline{GA}\left(w_{,x}-\phi \right)-\rho \left(r\right)I{\phi }_{,tt}=0$$

(18)

By omitting the rotary inertia component ${\phi }_{,tt}$ and shear deformation term $\left(w_{,x}-\phi \right)$ from Equation (18), and then integrating the simplified form with Equation (17), a reduced formulation can be obtained as follows:

$$\left(m_f+m_P\right)w_{,tt}+2m_f\Gamma w_{,xt}+\left(m_f{\Gamma }^2-P_0\right)w_{,xx}-\overline{EI}{w}_{,xxxx}-{\displaystyle \frac{\overline{EA}}{2L}}{w}_{,xx}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx=0$$

(19)

It is worth noting that Equation (19) corresponds to the same form as that presented by Tang and Yang [26] in their investigation of Euler–Bernoulli pipes, highlighting consistency with their theoretical framework. As explained, the governing equations reduce to the Euler–Bernoulli beam formulation as a special case when shear deformation and rotary inertia are neglected. This reduction is used for comparison purposes to demonstrate the influence of shear effects on the nonlinear behavior of the system.

Equations (17) and (18) are combined to derive an NGE that characterizes the transverse vibration behavior of the pipes made of FGM when transporting fluid, as follows:

$$\begin{array}{c}\left[\overline{EI}\left(1-{\displaystyle \frac{m_f{\Gamma }^2}{K_{SF}\overline{GA}}}+{\displaystyle \frac{P_0}{K_{SF}\overline{GA}}}\right)+{\displaystyle \frac{\overline{EI}\overline{EA}}{2L{K}_{SF}\overline{GA}}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx\right]w_{,xxxx}-2\overline{EI}{\displaystyle \frac{m_f\Gamma }{K_{SF}\overline{GA}}}{w}_{,xxxt}\hfill \\ +\left[-\overline{EI}{\displaystyle \frac{m_f+m_p}{K_{SF}\overline{GA}}}+\rho \left(r\right)I\left({\displaystyle \frac{m_f{\Gamma }^2}{K_{SF}\overline{GA}}}-{\displaystyle \frac{P_0}{K_{SF}\overline{GA}}}-1\right)-\rho \left(r\right)I{\displaystyle \frac{\overline{EA}}{2L{K}_{SF}\overline{GA}}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx\right]w_{,xxtt}\hfill \\ +2\rho \left(r\right)I{\displaystyle \frac{m_f\Gamma }{K_{SF}\overline{GA}}}{w}_{,xttt}+\rho \left(r\right)I{\displaystyle \frac{m_f+m_p}{K_{SF}\overline{GA}}}{w}_{,tttt}+\left[m_f{\Gamma }^2-P_0-{\displaystyle \frac{\overline{EA}}{2L}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx\right]w_{,xx}+\left(m_f+m_p\right)w_{,tt}\hfill \\ +2m_f\Gamma w_{,xt}=0\hfill \end{array}$$

(20)

4. Galerkin Approach

Equation (20) is converted to a nonlinear ordinary differential equation using the Galerkin method [32]. In this study, a first-order Galerkin approximation is applied, where the transverse displacement $w\left(x,t\right)$ is assumed in the following form:

$$w\left(x,t\right)=W\left(t\right)\zeta \left(x\right)$$

(21)

where $\zeta \left(x\right)$ represents the shape function, and for simply supported FGM pipes, the mode shape is assumed to follow $\zeta \left(x\right)=\sin \left({\displaystyle \frac{\pi x}{L}}\right)$. By substituting Equation (21) into Equation (20) and applying the Galerkin approach, the discretized NGE is obtained as follows:

$$\begin{array}{c}{\displaystyle \frac{d^{4}W\left(t\right)}{d{t}^4}}+\left({\displaystyle \frac{2m_f\Gamma }{m_f+m_p}}\left[{\displaystyle \frac{{\int }_0^L{\zeta }^{\prime }\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right]\right){\displaystyle \frac{d^{3}W\left(t\right)}{d{t}^3}}\hfill \\ +\left({\displaystyle \frac{K_{SF}\overline{GA}}{m_f+m_p}}+\left[{\displaystyle \frac{m_f{\Gamma }^2-K_{SF}\overline{GA}-P_0}{m_f+m_p}}-{\displaystyle \frac{\overline{EI}}{\rho \left(r\right)I}}\right]\left\{{\displaystyle \frac{{\int }_0^L{\zeta }^{″}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}\right){\displaystyle \frac{d^{2}W\left(t\right)}{d{t}^2}}\hfill \\ +\left({\displaystyle \frac{2m_f\Gamma K_{SF}\overline{GA}}{\rho \left(r\right)I\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L{\zeta }^{\prime }\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}-{\displaystyle \frac{2m_f\Gamma \overline{EI}}{\rho \left(r\right)I\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L{\zeta }^{‴}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}\right){\displaystyle \frac{dW\left(t\right)}{dt}}\hfill \\ +\left({\displaystyle \frac{\overline{EI}\left(P_0+K_{SF}\overline{GA}-m_f{\Gamma }^2\right)}{\rho \left(r\right)I\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L{\zeta }^{⁗}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}+{\displaystyle \frac{K_{SF}\overline{GA}\left(m_f{\Gamma }^2-P_0\right)}{\rho \left(r\right)I\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L{\zeta }^{″}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}\right)W\left(t\right)\hfill \\ +\left({\displaystyle \frac{{\left(\overline{EI}\right)}^2}{2\rho \left(r\right)IL\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L\left({\int }_0^L{{\zeta }^{\prime }}^2\left(x\right)dx\right){\zeta }^{⁗}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}-{\displaystyle \frac{\overline{EA}{K}_{SF}\overline{GA}}{2\rho \left(r\right)IL\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L\left({\int }_0^L{{\zeta }^{\prime }}^2\left(x\right)dx\right){\zeta }^{″}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}\right)W^3\left(t\right)\hfill \\ -\left({\displaystyle \frac{\overline{EA}}{2L\left(m_f+m_p\right)}}\left\{{\displaystyle \frac{{\int }_0^L\left({\int }_0^L{{\zeta }^{\prime }}^2\left(x\right)dx\right){\zeta }^{″}\left(x\right)\zeta \left(x\right)dx}{{\int }_0^L{\zeta }^2\left(x\right)dx}}\right\}\right)W^2\left(t\right){\displaystyle \frac{d^{2}W\left(t\right)}{d{t}^2}}=0\hfill \end{array}$$

(22)

where ${\left(\right)}^{\prime }={\displaystyle \frac{d\left(\right)}{dx}}$. Therefore, the nonlinear vibration response of FGM pipes transporting heavy crude oil is determined by numerically solving Equation (22) through the application of the fourth-order Runge–Kutta method. This explicit time integration technique is widely used for solving initial value problems due to its balance between computational efficiency and accuracy. In the current analysis, the time step ($\mathsf{\Delta }t$) was chosen based on convergence testing, with a typical value of $\mathsf{\Delta }t=0.001\hspace{0.17em}\mathrm{s}$, ensuring stability and precise tracking of nonlinear dynamic responses over extended time intervals. The fourth-order Runge–Kutta algorithm updates the system state iteratively by evaluating the ordinary differential equation at four intermediate stages within each time step and combining them to compute the solution with fourth-order accuracy. This approach is particularly suitable for capturing the complex vibration behavior observed in FGM pipes under nonlinear excitation, including transient and steady-state oscillations.

5. Post-Buckling Regime: Nonlinear Equation Derivation

FGM pipes with simple supports transporting heavy crude oil may become unstable due to divergence or buckling when the fluid velocity reaches a critical value [33]. To analyze their post-buckling response, the dynamic (time-dependent) effects in Equation (21) are omitted, allowing the equation to be reformulated using only its static terms, as follows:

$$\left[\overline{EI}\left(1-{\displaystyle \frac{m_f{\Gamma }^2}{K_{SF}\overline{GA}}}+{\displaystyle \frac{P_0}{K_{SF}\overline{GA}}}\right)+{\displaystyle \frac{\overline{EI}\overline{EA}}{2L{K}_{SF}\overline{GA}}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx\right]w_{,xxxx}+\left[m_f{\Gamma }^2-P_0-{\displaystyle \frac{\overline{EA}}{2L}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx\right]w_{,xx}=0$$

(23)

In this regard, the integral $\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx$ in Equation (23), for a given mode shape of $w\left(x\right)$, yields a constant value. To simplify the expression, we introduce the constants ${\beta }_1$ and ${\beta }_2$ as follows:

$${\beta }_1={\displaystyle \frac{\overline{EA}}{2L{K}_{SF}\overline{GA}}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx;{\beta }_2={\displaystyle \frac{\overline{EA}}{2L}}\underset{0}{\overset{L}{\int }}{w}_{,x}^{2}dx$$

(24)

Accordingly, Equation (23) can be rewritten as follows:

$$w_{,xxxx}+{\mathsf{\xi }}^2{w}_{,xx}=0$$

(25)

where

$${\mathsf{\xi }}^2={\displaystyle \frac{m_f{\Gamma }^2-P_0-{\beta }_2}{\overline{EI}\left(1-\frac{m_f{\Gamma }^2}{K_{SF}\overline{GA}}+\frac{P_0}{K_{SF}\overline{GA}}+{\beta }_1\right)}}$$

(26)

Equation (26) was previously solved by Nayfeh and Emam [34], who obtained the following solution:

$$w\left(x\right)=\mathsf{\Lambda }\sin \left(\lambda x\right)$$

(27)

where $\mathsf{\Lambda }$ is an unknown constant. Based on Equations (14) and (27), we obtain the following:

$$\sin \left(\lambda L\right)=0\to \lambda ={\displaystyle \frac{m\pi }{L}}\left(m=1,2,3,\dots \right)$$

(28)

Utilizing Equations (24) and (26), $\mathsf{\Lambda }$ can be determined as follows:

$$\mathsf{\Lambda }=\pm 2\sqrt{{\displaystyle \frac{K_{SF}\overline{GA}\left(m_f{\Gamma }^2-P_0\right)+{\xi }^2\overline{EI}\left(m_f{\Gamma }^2-P_0-K_{SF}\overline{GA}\right)}{{\xi }^2\overline{EA}\left({\xi }^2\overline{EI}+K_{SF}\overline{GA}\right)}}}$$

(29)

To derive the post-buckling configuration of the pipes made of FGM, we can insert Equations (28) and (29) into Equation (27), leading to the following closed-form solution:

$$w\left(x\right)=\pm 2\sqrt{{\displaystyle \frac{K_{SF}\overline{GA}\left(m_f{\Gamma }^2-P_0\right)+{\xi }^2\overline{EI}\left(m_f{\Gamma }^2-P_0-K_{SF}\overline{GA}\right)}{{\xi }^2\overline{EA}\left({\xi }^2\overline{EI}+K_{SF}\overline{GA}\right)}}}\sin \left({\displaystyle \frac{m\pi x}{L}}\right)$$

(30)

Therefore, critical flow velocity (CFV) for the pipes made of FGM is determined using Equation (30), as shown below:

$${\Gamma }_{cr}=\sqrt{{\displaystyle \frac{P_0{K}_{SF}\overline{GA}+{\left(\frac{m\pi }{L}\right)}^2\overline{EI}\left(P_0+K_{SF}\overline{GA}\right)}{m_f\left(K_{SF}\overline{GA}+{\left(\frac{m\pi }{L}\right)}^2\overline{EI}\right)}}}$$

(31)

6. Numerical Results

6.1. Modeling Assumptions and Limitations

In the present study, the fluid flow is modeled as one-dimensional and steady, without incorporating detailed fluid–structure interaction (FSI) phenomena such as boundary layer development, viscous shear forces, or transient pressure pulsations. This simplification allows a more tractable analysis of the nonlinear structural dynamics, though it may underrepresent dynamic coupling effects that become significant in highly unsteady flow conditions or near resonance frequencies. Furthermore, while temperature-dependent material properties are considered using a predefined empirical model, the thermal field within the pipeline is assumed to be uniform. This assumption neglects the possibility of radial temperature gradients across the pipe wall, which may arise in practical applications due to internal convective heat transfer or localized thermal inputs. Such gradients can induce thermal stresses and bending moments, particularly in functionally graded materials, as highlighted in prior studies. To address this, key literature has been cited that elaborates on the role of thermal moments and radial temperature variations in FGM structures [35]. Although this approach provides initial insights into the thermal–mechanical behavior of FGMs, we acknowledge that the uniform temperature assumption is a limitation.

In future works or studies to enhance the current model, it can be extended to incorporate radial and axial heat conduction, realistic convective boundary conditions, and comprehensive FSI coupling to improve physical fidelity and predictive capability. Furthermore, the analysis may be expanded to include various boundary condition configurations, and tabulated results can be provided to enhance the clarity, reproducibility, and practical applicability of the findings.

6.2. Validation of the Present Findings

In this sub-section, we first validate the obtained results against previous research findings. Following this, we examine the post-buckling and nonlinear vibration behaviors of the FGM pipes transporting heavy crude oil. In this regard, in

Table 1. Comparison of NDNFs of a homogeneous pipe in the absence of flow.

Present	Zhu et al. [36]		Païdoussis [37]		Ni et al. [38]
	Discrepancy (%)		Discrepancy (%)		Discrepancy (%)
3.14173	3.14159	0.004	3.14159	0.004	3.14200	0.008

, non-dimensional natural frequencies (NDNFs) of a homogeneous pipe in the absence of flow (i.e., when $\mathsf{\Gamma }=0$) are compared with the findings of Zhu et al. [36], Païdoussis [37], and Ni et al. [38]. In addition,

Table 2. Comparison of NDNFs of an FGM beam.

$\mathit{N}$	Present	Huang and Li [39]
		Discrepancy (%)
0.2	0.0889	0.0890	0.11
1	0.0901	0.0902	0.11
5	0.0884	0.0885	0.11

provides a comparison of the NDNFs for the beam made of FGM with those obtained by Huang and Li [39]. These comparisons demonstrate a strong correlation between the present results and the published data, confirming the accuracy of the proposed approach.

To generate the results for the post-buckling and nonlinear vibration behaviors of the pipe made of FGM transporting heavy crude oil, the material properties and structural dimensions are demonstrated in

Table 3. Material parameters of the FGM pipes [27].

	$\mathbf{S}\mathbf{U}\mathbf{S}304$ (Metal)		${\mathit{S}\mathit{i}}_3{\mathit{N}}_4$ (Ceramic)
	$\mathit{E}\left(\mathbf{P}\mathbf{a}\right)$	$\mathit{\rho }\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathit{E}\left(\mathbf{P}\mathbf{a}\right)$	$\mathit{\rho }\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$
$P_0$	2.01 × 1011	8166	3.48 × 1011	2370
$P_{-1}$	0	0	0	0
$P_1$	3.08 × 10−4	0	−3.07 × 10−4	0
$P_2$	−6.53 × 10−7	0	2.16 × 10−7	0
$P_3$	0	0	−8.95 × 10−11	0

and

Table 4. Geometric parameters of FGM pipes.

${\mathit{r}}_{\mathit{o}}$(m)	${\mathit{r}}_{\mathit{i}}\left(\mathbf{m}\right)$	$\mathit{L}$(m)	$\mathit{N}$	${\mathit{P}}_0\left(\mathbf{N}\right)$
0.12	0.1	10	1	−20

. The density (${\rho }_f$) and velocity ($\mathsf{\Gamma }$) of the heavy oil are 940 $\mathrm{K}\mathrm{g}/{\mathrm{m}}^3$ and 0.1 $\mathrm{m}/\mathrm{s}$.

6.3. 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 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 $T=300\mathrm{K}$. 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 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 Kármán 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 $T=300\mathrm{K}$, for two different types. The corresponding effect of $N$ on the natural frequency (NF) is also presented in

Table 5. Impact of $N$ on the NF of FGM pipes transporting heavy crude oil.

N	Type I	Type II
0.2	173.14	113.46
1	140.13	136.69
5	115.11	170.39

. 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. 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 $N$, allowing 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. Impact of $T$ on the NF of FGM pipes transporting heavy crude oil.

T(K)	Type I	Type II
250	140.69	137.21
300	140.13	136.69
350	139.51	136.11

. 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. As previously explained, based on Equations (1a) and (1b), increasing $N$ causes the material properties of Type I pipes to approach those of metal, whereas Type II pipes exhibit behavior closer to ceramics. These results demonstrate that the resistance of FGM pipes, defined by their critical flow velocity, can be effectively tuned by selecting an appropriate FGM type and adjusting the gradient index $N$.

Figure 7 illustrates how the CFV in the pipe made of FGM varies with the length $L$ for two different material types. As observed, increasing the length results in a decrease in the CFV for both types. Additionally, Figure 8 shows the influence of the outer radius ($r_o$) and inner radius ($r_i$) on the critical flow velocity. The results indicate that an increase in $r_o$ leads to a higher critical flow velocity, while an increase in $r_i$ causes it to decrease.

The nonlinear vibration and post-buckling responses explored in this study demonstrate that FGM pipes possess not only enhanced mechanical strength and thermal resistance but also customizable dynamic behavior. Through control of geometrical features (length, radii) and gradation parameters (such as the power-law index nnn), engineers can modulate the degree of nonlinearity in the structural response. These nonlinear characteristics, including stiffness variation with amplitude and frequency shifts, are especially critical in applications involving fluid–structure interaction and thermal environments, such as in oil and gas pipelines.

In real-world systems, such nonlinear dynamics can either be exploited to improve vibration isolation or mitigated to avoid resonance and fatigue failure. Therefore, the ability to predict and tailor nonlinear behaviors becomes a powerful tool in the optimal design of advanced FGM-based pipeline systems.

6.4. Engineering Implications and Model Extensions

The current model provides valuable insights into the structural response of FGM pipelines under thermal and mechanical loads, specifically, when transporting high-viscosity fluids like heavy crude oil. These findings are especially relevant in applications such as oil sands pipelines, geothermal energy transport, and thermally enhanced recovery processes, where elevated temperatures and pressure-induced instabilities are prevalent.

Importantly, the ability to adjust the vibration characteristics and critical flow velocity by tuning the volume fraction index ($N$) and material configuration enables engineers to design pipelines that are more robust against resonant excitation and buckling. For instance, selecting a ceramic-rich outer surface may be advantageous in high-temperature, erosion-prone environments, while a metal-rich inner core improves ductility and strength under flow pressure.

Moreover, the current analytical framework is versatile and can be extended to include the following:

• Geometric imperfections and initial curvature effects;
• Complex foundation models such as nonlinear Winkler or Pasternak foundations;
• Turbulent flow or pulsatile excitation effects;
• Time-varying temperature profiles or transient thermal shocks;
• Multi-mode Galerkin approximations for higher-order dynamic responses.

These extensions would further enhance the model’s applicability in more realistic field conditions and are proposed as future directions.

7. Conclusions

This research explores the post-buckling and nonlinear vibration behavior of functionally graded material (FGM) pipes transporting heavy crude oil, modeled using Timoshenko beam theory. Two distinct FGM configurations are examined, with their material properties considered to be temperature dependent. The nonlinear governing equations (NGEs) are derived by applying Hamilton’s principle, Timoshenko beam theory, and von Kármán’s nonlinear strain–displacement relations. These equations are then discretized via Galerkin’s method. This study highlights the influence of both material properties and geometric parameters on the system’s nonlinear response. The main findings can be outlined as follows:

• By analyzing pipes made from functionally graded materials (FGMs), this study offers valuable insights into how material gradation affects structural behavior. Results reveal that the natural frequency (NF) of Type I pipes is higher than that of Type II.
• As the volume fraction index ($N$) increases, the NF and critical flow velocity (CFV) decrease for Type I pipes, whereas they increase for Type II pipes. This demonstrates that, beyond their inherent resistance to corrosion and fatigue, FGM pipes offer a unique advantage. Their vibration characteristics can be tailored by adjusting the gradient index $N$. This tunability allows engineers to control the NF within a desired operational range. Additionally, the flow resistance—reflected by the critical flow velocity—can be effectively optimized by selecting the suitable FGM type and modifying the value of $N$.
• As $N$ increases, the critical flow velocity decreases for Type I FGM pipes and increases for Type II.
• Increasing the length results in a decrease in the CFV. Additionally, an increase in the outer radius leads to a higher CFV, while an increase in the inner radius causes it to decrease.

The originality of this study lies in its unified treatment of nonlinear vibration and post-buckling behaviors in FGM pipes transporting heavy crude oil, which is a domain where the combined effects of temperature-dependent properties, realistic fluid characteristics, and structural gradation have been largely overlooked. By comparing two distinct FGM configurations under identical thermal and flow conditions and integrating these into a temperature-sensitive nonlinear model, this work offers new perspectives on how to tailor structural performance in thermomechanical environments.

The findings of this study have practical implications in engineering scenarios where systems are subject to complex loading and vibration, such as fluid-conveying pipes operating under high pressure and temperature, aerospace panels exposed to turbulent aerodynamic forces, and advanced structural elements in robotic or mechatronic systems. By tuning geometric and material parameters, engineers can tailor the dynamic behaviors of such systems to mitigate resonance, enhance stability, and optimize performance. Future work could extend this framework to include active vibration control or real-time monitoring strategies for critical infrastructure.