Nonlinear Transient Dynamics of Graphene Nanoplatelets Reinforced Pipes Conveying Fluid under Blast Loads and Thermal Environment

: This work aims at investigating the nonlinear transient response of ﬂuid-conveying pipes made of graphene nanoplatelet (GPL)-reinforced composite (GPLRC) under blast loads and in a thermal environment. A modiﬁed Halpin–Tsai model is used to approximate the effective Young’s modulus of the GPLRC pipes conveying ﬂuid; the mass density and Poisson’s ratio are determined by using the Voigt model. A slender Euler–Bernoulli beam is considered for modeling the pipes conveying ﬂuid. The vibration control equation of the GPLRC pipes conveying ﬂuid under blast loads is obtained by using Hamilton’s principle. A set of second-order ordinary differential equations are obtained by using the second-order Galerkin discrete method and are solved by using the adaptive Runge–Kutta method. Numerical experiments show that GPL distribution and temperature; GPL weight fraction; pipe length-to-thickness ratio; ﬂow velocity; and blast load parameters have important effects on the nonlinear transient response of the GPLRC pipes conveying ﬂuid. The numerical results also show that due to the ﬂuid–structure interaction, the vibration amplitudes of the GPLRC pipes conveying ﬂuid decay after the impact of blast loads.


Introduction
As the main structure for oil and gas transportation, fluid-conveying pipes are widely used in aerospace engineering, marine engineering, water conservancy systems, oil and gas transportation systems, and the petroleum energy/chemical/nuclear industries [1][2][3][4][5]. The structure of pipes employed in engineering nuclear power plants are inevitably impacted by nuclear and fuel explosions [6]. Therefore, in order to improve the explosion resistance of fluid-conveying pipes and to minimize the possibility of catastrophic losses, it is necessary to carry out detailed analyses of the nonlinear transient dynamic behaviors of pipes conveying fluid under blast loads.
In recent years, numerous scholars have paid attention to the study of the mechanical behavior of fluid-conveying pipes. Païdoussis and Gregory did some pioneering work [7]. The transverse vibration equation of horizontal cantilever pipes conveying fluid was derived by using the classical Newton's second law. The complex frequencies of the first four modes of the system were calculated and verified by experiments [8]. In addition, they also studied the vibration instability of fluid-conveying pipes under various boundary conditions [9]. Based on Païdoussis's work, Semler and Li used an energy method and Newton method to derive the nonlinear governing equations of cantilever pipes conveying fluid and the pipes conveying fluid supported at both ends, respectively [10]. Up to now, these two dynamic equations have been the most widely used. With the development of research on fluid-conveying pipes, structural and mechanical behavior analyses of pipes conveying fluid have been the meaningful topics in the current research. For example, at the beginning of the 21st century, Lee and Chung deduced a new type of nonlinear coupled motion equation of simply supported pipes with two ends [11], based on the Euler-Bernoulli beam theory and the nonlinear Lagrange strain theory. On the basis of the above research, Wang's research team analyzed the vibration frequencies of elastic axial functionally gradient cantilever pipes [12], micro/nanopipes [13,14], flexible bending pipes [15], and cantilever carbon nanotube pipes [16], and discussed the sensitivity of flow rate to the mechanical behavior of pipes conveying fluid. In addition, they conducted a host of work on the vibration of [17] cantilever pipes conveying fluid under nonlinear elastic boundary. Meanwhile, some scholars have paid attention to the mechanical behavior of high-speed pipes conveying fluid. Tan discussed the parametric resonance of the Timoshenko pipeline with pulsating high-speed fluids [18] and super-harmonic resonance [19]. Ye [20] studied the natural frequencies of curved pipes conveying fluid at supercritical fluid. Zhen [21] analyzed the nonlinear vibrations of functionally graded material pipes conveying fluid with initial curvature under supercritical fluid. Ding [22] studied the nonlinear frequency and forced motion of high-speed pipes conveying fluid. Moreover, regarding mechanical analyses, Selmi [23] explored the post-buckling vibration of functionally graded pipes conveying fluid; Khodabakhsh [24] studied nonlinear vibrations and conducted a buckling analysis of Timoshenko pipes conveying fluid; El-Sayed [25] investigated free vibrations through a stability analysis of multi-span pipes conveying fluid; Askarian [26] performed a stability analysis of viscoelastic pipes conveying fluid with different boundary conditions.
With the development of new composite materials, researchers have begun to pay attention to the mechanical behaviors of composite pipes conveying fluid. For example, Amini [27] and Dehrouyeh-Semnani [28] discussed the stability and the thermal resonance response of functionally graded material (FGM) fluid-conveying pipes. Liang conducted a series of studies on the mechanical behaviors of spinning FGM fluid-conveying pipes considering variable velocity [29,30] and multi-span cases [31]. Dang [32] investigated the nonlinear vibrations and stability of FGM fluid-conveying nanotubes. Ghasemi [33] conducted a buckling analysis of multi-walled carbon nanotubes.
Sedighi [34,35] examined the divergence and flutter instabilities of a hetero-nanotube as well as the vibrational characteristics and critical divergence velocity of a hybridnanotube constructed using carbon (C) and boron nitride (BN) nanotubes in a magnetic and thermal environment.
GPLs have the advantages of low cost, high Young's modulus, and large contact interface [36]. As a reinforcing material, they are uniformly or gradient distributed in a matrix, which can significantly improve the mechanical and electrical properties of composite materials [37]. Research on the mechanical behavior of GPLRC structures is a hot issue at home and abroad. So far, a large number of studies have focused on the mechanical behaviors of GPLRC beams [38], plates [39,40], shells [6], and joined shells [41]. However, to the best of our knowledge, there are few studies in the literature on mechanical analyses of GPLRC pipes conveying fluid, in particular, for the case of the nonlinear transient dynamic response.
In this paper, the nonlinear transient dynamic response of GPLRC pipes conveying fluid under blast loads is studied, taking into account the thermal environment. The Young's modulus of the GPLRC pipes conveying fluid is estimated using a modified Halpin-Tsai model. The model is established by using Hamilton's principle and solved using the Runge-Kutta method. The effects of the GPLs' content and distribution and geometric parameters, on the nonlinear transient response of the GPLRC pipes conveying fluid, are discussed by numerical simulation.

Equations of Motion
A GPLRC pipe conveying fluid as a pinned-pinned supported beam subjected to a blast load was considered. The geometric diagram of the structure with length L, inner radius r i , and outer radius r o is depicted in Figure 1. Three GPL distributions, denoted as GPL-U, GPL-O, and GPL-X, are shown in Figure 2. GPL-U represents a uniform distribution of GPLs; GPL-O represents a distribution of GPLs rich in the mid-plane vicinity; GPL-X represents a distribution of GPLs rich in the inner and outer layers. The latter two types also represent FG distributions of GPLs in the thickness direction of the fluid-conveying pipes, in which dark colors represent a greater number of GPLs within that layer. geometric parameters, on the nonlinear transient response of the GPLRC pipes conveying fluid, are discussed by numerical simulation.

Equations of Motion
A GPLRC pipe conveying fluid as a pinned-pinned supported beam subjected to a blast load was considered. The geometric diagram of the structure with length L, inner radius i r , and outer radius o r is depicted in Figure 1. Three GPL distributions, denoted as GPL-U, GPL-O, and GPL-X, are shown in Figure 2. GPL-U represents a uniform distribution of GPLs; GPL-O represents a distribution of GPLs rich in the mid-plane vicinity; GPL-X represents a distribution of GPLs rich in the inner and outer layers. The latter two types also represent FG distributions of GPLs in the thickness direction of the fluid-conveying pipes, in which dark colors represent a greater number of GPLs within that layer.  The effective material parameters of the GPLRC pipes conveying fluid, such as Poisson ratio c  and Young's modulus c E , should be clearly introduced, and can be referred to in Appendix A. The nonlinear in-plane strain-displacement relations on the basis of the Von Karman's assumption for the moderately large oscillations of the fluid-conveying pipes can be written as: in which  xx is the longitudinal strain; 1 u and 3 u are the longitudinal and lateral components of displacement field, respectively. The prime indicates the derivative with respect to coordinate x. On the basis of the Euler-Bernoulli beam model assumption of pipes conveying fluid, the in-plane components of displacement field can be obtained as follows: geometric parameters, on the nonlinear transient response of the GPLRC pipes conveying fluid, are discussed by numerical simulation.

Equations of Motion
A GPLRC pipe conveying fluid as a pinned-pinned supported beam subjected to a blast load was considered. The geometric diagram of the structure with length L, inner radius i r , and outer radius o r is depicted in Figure 1. Three GPL distributions, denoted as GPL-U, GPL-O, and GPL-X, are shown in Figure 2. GPL-U represents a uniform distribution of GPLs; GPL-O represents a distribution of GPLs rich in the mid-plane vicinity; GPL-X represents a distribution of GPLs rich in the inner and outer layers. The latter two types also represent FG distributions of GPLs in the thickness direction of the fluid-conveying pipes, in which dark colors represent a greater number of GPLs within that layer.  The effective material parameters of the GPLRC pipes conveying fluid, such as Poisson ratio c  and Young's modulus c E , should be clearly introduced, and can be referred to in Appendix A. The nonlinear in-plane strain-displacement relations on the basis of the Von Karman's assumption for the moderately large oscillations of the fluid-conveying pipes can be written as: in which  xx is the longitudinal strain; 1 u and 3 u are the longitudinal and lateral components of displacement field, respectively. The prime indicates the derivative with respect to coordinate x. On the basis of the Euler-Bernoulli beam model assumption of pipes conveying fluid, the in-plane components of displacement field can be obtained as follows: The effective material parameters of the GPLRC pipes conveying fluid, such as Poisson ratio ν c and Young's modulus E c , should be clearly introduced, and can be referred to in Appendix A. The nonlinear in-plane strain-displacement relations on the basis of the Von Karman's assumption for the moderately large oscillations of the fluid-conveying pipes can be written as: in which ε xx is the longitudinal strain; u 1 and u 3 are the longitudinal and lateral components of displacement field, respectively. The prime indicates the derivative with respect to coordinate x. On the basis of the Euler-Bernoulli beam model assumption of pipes conveying fluid, the in-plane components of displacement field can be obtained as follows: where u and w represent the displacement of the middle surface along the x-axis and z-axis, respectively. Therefore, the longitudinal strain as a function of the longitudinal and transverse displacements of the intermediate plane and the coordinates can be expressed as: The temperature variation of the pipes conveying fluid is expressed by ∆T. Therefore, the generalized Hooke's law can be expressed in the following form [28]: Here, σ xx is the stress in the x direction. The thermal stress is expressed as ε T = α(r)∆T, α(r) is the coefficient of thermal expansion.
Through calculation, the potential energy of the fluid-conveying pipes is obtained as: Substituting Equations (3) and (4) into Equation (5) as the potential energy of the fluid-conveying pipes leads to the following expression: In which where A 11 and D 11 are the extensional and the bending stiffness coefficients, respectively; N T and N T 0 are the thermal stress resultants. It is worth noting that the thermal moment is zero due to symmetrical distribution of the material with respect to the neutral axis of the pipes conveying fluid.
The kinetic energy of the GPLRC pipes conveying fluid can be expressed as: In addition, the kinetic energy of fluid can be written as follows: The expression of virtual work caused by fluid lateral motion and viscous damping produced by the GPLRC pipes conveying fluid is: when the GPLRC pipes conveying fluid is impacted by blast load, the virtual work is calculated as follows: In Equations (11)- (14), m, M, c w , p, V are the mass per unit of the GPLRC pipe, the mass per unit of fluid, viscosity coefficient, explosion pressure, and fluid velocity, respectively. The non-linear governing equations of the GPLRC pipes conveying fluid under blast load can be obtained by using Hamilton's principle: Substitution of Equations (6), (11), and (12) into Equation (15) results in the following nonlinearly coupled partial differential equations of motion in the longitudinal and the lateral directions respectively: Furthermore, the following natural and essential boundary conditions are deduced: According to [28], the time-dependent terms (m + M) ..

u and 2MV
. u are negligible, and MV 2 is much smaller than A 11 . Then, we reformulate the longitudinal equation of the governing equation, i.e., Equation (17) as: The left and right ends of the GPLRC pipes conveying fluid are pinned. Therefore, the longitudinal boundary condition can be expressed as: Due to Equations (20) and (21), u can be written as a function of w as follows: Substitution of Equation (22) into Equation (16) as well as Equations (18) and (19) results in the following new governing equation, i.e., Equation (23) and corresponding boundary conditions, i.e., Equations (24) and (25) in terms of the lateral motion, respectively: Assuming that the explosion source has a certain distance from the GPLRC pipes conveying fluid, the impact moment of explosion load can be expressed as the Friedlander equation [6]: In the equation, p 0 is the explosion pressure peak, t p is the explosion action time, γ is the attenuation parameter, and t represents the time.
According to the obtained nonlinear partial differential equation of transverse motion of the GPLRC pipes conveying fluid, it can be discretized into a set of nonlinear coupled ordinary differential equations with finite degrees of freedom by using the Galerkin method. In order to obtain the discrete form of the motion control equation, the following series expansion with a set of orthogonal functions is selected: where ψ n (x) = sin nπx L [42] represents the n-th eigenfunction of the transverse vibration of the GPLRC pipes conveying fluid. In addition, q n (τ) represents the n-th generalized coordinate of transverse motion.
The transverse motion equation of the GPLRC pipes conveying fluid is discretized, and a set of nonlinear coupled ordinary differential equations are obtained by using the Galerkin technique [43,44]: in which In which, i = 1, 2, . . . , N.

Numerical Simulations
In this section, the nonlinear transient dynamic behavior of GPLRC pipes conveying fluid with three distribtuions of GPLs under blast loads is considered. Here, epoxy resin and GPLs are selected as the polymer matrix and reinforcement [45]. Then, the numerical results are obtained by using the adaptive Runge-Kutta method with zero initial value (w 1 , w 2 ) and time step 10 −8 s. The fluid is taken as lead with the following density ρ f = 10, 678 Kg/m 3 and melting point 600.6 K. The reference temperature is considered to be T 0 = 300 K. Therefore, the minimum temperature rise at the inner surface of the GPLRC pipes conveying fluid equals ∆T min = 300.6 K. The following geometric and material property parameters [45] listed in Table 1 remain unchanged, except as specifically stated. Table 1. Material parameters and geometric parameters of pipes.

Parameters
Data Parameters Data  Table 2. There is good agreement between the results obtained by the present study and those by Xu [42] and by Housner [46] as well as the FEM results using software Ansys in Table 3, which demonstrates the reliability of the model and methods employed in this paper. In addition, the same accuracy of fundamental frequencies with the degrees of freedom N = 2, 4, and 6 in Equation (27) (shown in Figure 3) implies that N = 2 can be taken for the next whole simulation. Table 2. Material and geometry parameters used in studies by Xu [42] and Housner [46].   As shown in Figure 4, the vibrations of GPLRC pipes conveying fluid with a uniform distribution of GPLs under the impact of blast load will lead to energy loss over time and the amplitude of the vibrations will have an obvious downward trend. As shown in Figure 4, the vibrations of GPLRC pipes conveying fluid with a uniform distribution of GPLs under the impact of blast load will lead to energy loss over time and the amplitude of the vibrations will have an obvious downward trend. As shown in Figure 4, the vibrations of GPLRC pipes conveying fluid with a uniform distribution of GPLs under the impact of blast load will lead to energy loss over time and the amplitude of the vibrations will have an obvious downward trend.  Figure 5 indicates a higher frequency and lower amplitude of vibrations, because the structural stiffness is increased by the addition of GPLs. However, the amplitude difference of the GPLRC pipes conveying fluid under three distribution modes is not significant. This is because the pipes conveying fluid are slender hollow beam structures with thin thickness. The results imply that in an actual industrial production process, any GPL distribution can be selected for structure production according to convenience. In the follow-up study, the GPL-X distribution is selected for research and discussion.  Figure 5 indicates a higher frequency and lower amplitude of vibrations, because the structural stiffness is increased by the addition of GPLs. However, the amplitude difference of the GPLRC pipes conveying fluid under three distribution modes is not significant. This is because the pipes conveying fluid are slender hollow beam structures with thin thickness. The results imply that in an actual industrial production process, any GPL distribution can be selected for structure production according to convenience. In the follow-up study, the GPL-X distribution is selected for research and discussion.  on the effects of nonlinear transient dynamic behavior of GPLRC pipes conveying fluid. As can be seen from Figure 6, the vibration amplitude of the pipes conveying fluid decreases as the attenuation parameter increases from 0.1 to 2, due to the gradually attenuated blast load. Furthermore, according to Figures 7 and 8, it can be inferred that the vibration amplitude is increased, whether or not the explosion  Figure 6, the vibration amplitude of the pipes conveying fluid decreases as the attenuation parameter increases from 0.1 to 2, due to the gradually attenuated blast load. Furthermore, according to Figures 7 and 8, it can be inferred that the vibration amplitude is increased, whether or not the explosion duration is longer or the peak explosion pressure is greater. However, the transient blast load has little effect on vibration frequency, mainly because the time it acts on the structure is too short. In the following analysis, the blast load parameter (γ = 0.1, t p = 0.25 ms, p = 10 5 N/m 2 ) selection remains unchanged.  on the effects of nonlinear transient dynamic behavior of GPLRC pipes conveying fluid. As can be seen from Figure 6, the vibration amplitude of the pipes conveying fluid decreases as the attenuation parameter increases from 0.1 to 2, due to the gradually attenuated blast load. Furthermore, according to Figures 7 and 8, it can be inferred that the vibration amplitude is increased, whether or not the explosion duration is longer or the peak explosion pressure is greater. However, the transient blast load has little effect on vibration frequency, mainly because the time it acts on the structure is too short. In the following analysis, the blast load parameter     Considering the above temperature difference ∆T = 300.6 K as the basic temperature change unit, we investigated the influence of temperature differences ∆T, 2∆T, 4∆T on vibrations in GPLRC pipes conveying fluid. It can be seen from Figure 9 that the vibration amplitude becomes larger and the vibration frequency of the structure becomes smaller with an increase in temperature difference. This is due to the lower elastic modulus with the higher temperature.  Considering the above temperature difference 300.6K T   as the basic temperature change unit, we investigated the influence of temperature differences ,2 ,4 T T T    on vibrations in GPLRC pipes conveying fluid. It can be seen from Figure 9 that the vibration amplitude becomes larger and the vibration frequency of the structure becomes smaller with an increase in temperature difference. This is due to the lower elastic modulus with the higher temperature.    This is because the addition of GPLs improves the system's stiffness and also enhances the GPL impact resistance of the pipes conveying fluid. Figure 11 shows the effect of the GPL length-to-thickness ratio with a constant GPL thickness. The higher the ratio, the faster the vibration frequency because of the larger contact area between the pipe and GPLs, which effectively increases the stiffness of the structure. Figure 12 shows the time history diagram of the vibrations of the GPLRC pipes conveying fluid at different flow rates. It can be observed that an increase in the flow rate V = (0 m/s, 5 m/s, 20 m/s, 50 m/s) causes a reduction in the vibration frequency of the GPLRC pipes conveying fluid and an increment of vibration amplitude, which can be seen by taking 10-11 s as an example. The main reason is that the increasing fluid velocity can weaken the pipe stiffness.  The vibration frequency of the pipes conveying fluid increases and the amplitude decreases with a change in GPL mass fraction from 0 to 1.2%. This is because the addition of GPLs improves the system's stiffness and also enhances the GPL impact resistance of the pipes conveying fluid.  Figure 11 shows the effect of the GPL length-to-thickness ratio with a constant GPL thickness. The higher the ratio, the faster the vibration frequency because of the larger contact area between the pipe and GPLs, which effectively increases the stiffness of the structure.   Here, we give the inner and outer diameters r i = 0.20 m and r 0 = 0.225 m of the GPLRC pipes conveying fluid as a constant, and study the effect of the structure's lengthto-thickness ratio on its nonlinear transient vibration. As shown in Figure 13, there is an obvious increase in the maximum amplitude of the structure with an increase in the length-to-thickness ratio; the inverse results are obtained for the vibration frequency. This phenomenon indicates that under certain thicknesses within the acceptable accuracy range, appropriately increasing the length-to-thickness ratio of the pipes conveying fluid is more helpful for reducing the vibration frequency of the pipes conveying fluid in the actual production process. Figure 12 shows the time history diagram of the vibrations of the GPLRC pipes conveying fluid at different flow rates. It can be observed that an increase in the flow rate (0 ) m s , 5 m s , 20 m s ,50 m s V  causes a reduction in the vibration frequency of the GPLRC pipes conveying fluid and an increment of vibration amplitude, which can be seen by taking 10-11 s as an example. The main reason is that the increasing fluid velocity can weaken the pipe stiffness. Here, we give the inner and outer diameters 0.20 m and 0.225 m of the GPLRC pipes conveying fluid as a constant, and study the effect of the structure's length-to-thickness ratio on its nonlinear transient vibration. As shown in Figure 13, there is an obvious increase in the maximum amplitude of the structure with an increase in the length-to-thickness ratio; the inverse results are obtained for the vibration frequency. This phenomenon indicates that under certain thicknesses within the acceptable accuracy range, appropriately increasing the length-to-thickness ratio of the pipes conveying fluid

Conclusions and Discussions
In the framework of this study, nonlinear transient dynamic responses of GPLRC pipes conveying fluid subjected to time-varying blast loads were analyzed by using the Galerkin method in conjunction with the adaptive Runge-Kutta method. The polymer matrix and reinforcement were considered to be epoxy resin and GPLs, respectively, and the fluid was regarded to be lead. According to Hamilton's principle, nonlinear partial differential governing equations of pipes conveying fluid were derived. The numerical simulations mainly focused on analyzing the effects of GPL content, fluid velocity, geometry size, and blast load parameters on the vibration frequencies and vibration amplitudes of GPLRC pipes conveying fluid. The accuracy of the model was confirmed by comparing the frequencies with the published results in the literature. Based on the parametric study, the following important conclusions are presented:  GPLs as reinforcement can enhance the vibration frequency and impact resistance of pipes conveying fluid, but their distribution mode has little effect. This also means that in an industrial production process, the most convenient and economical GPL distribution among the three distribution modes can be selected for production.  Transient load parameters have significant influence on vibration amplitude and little influence on vibration frequency.  Vibration frequency has a descending trend with respect to temperature difference, Figure 13. Effects of pipes conveying fluid length-to-thickness ratio on nonlinear transient vibrations.

Conclusions and Discussions
In the framework of this study, nonlinear transient dynamic responses of GPLRC pipes conveying fluid subjected to time-varying blast loads were analyzed by using the Galerkin method in conjunction with the adaptive Runge-Kutta method. The polymer matrix and reinforcement were considered to be epoxy resin and GPLs, respectively, and the fluid was regarded to be lead. According to Hamilton's principle, nonlinear partial differential governing equations of pipes conveying fluid were derived. The numerical simulations mainly focused on analyzing the effects of GPL content, fluid velocity, geometry size, and blast load parameters on the vibration frequencies and vibration amplitudes of GPLRC pipes conveying fluid. The accuracy of the model was confirmed by comparing the frequencies with the published results in the literature. Based on the parametric study, the following important conclusions are presented:

•
GPLs as reinforcement can enhance the vibration frequency and impact resistance of pipes conveying fluid, but their distribution mode has little effect. This also means that in an industrial production process, the most convenient and economical GPL distribution among the three distribution modes can be selected for production.
• Transient load parameters have significant influence on vibration amplitude and little influence on vibration frequency. • Vibration frequency has a descending trend with respect to temperature difference, pipes conveying fluid length-to-thickness ratios, and flow rates; however, the opposite conclusion was obtained for the effects of GPL length-to-thickness ratio and vibration amplitude. Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.