# Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil
## 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 continuousand 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, [PERSON] and [PERSON] [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, [PERSON] et al. [20] investigated nonlinear vibrations of FGM pipes with initial curvature, but they did not consider temperature sensitivity. Similarly, [PERSON] and [PERSON] [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.