Nonlinear Buckling Analysis of Cylindrical Nanoshells Conveying Nano-Fluid in Hygrothermal Environment

The present work addresses the critical buckling of circular cylindrical nano-shells containing static/dynamic nanofluids under the influence of different thermal fields that can also lead to appear the effect of thermal moisture so-called hygrothermal forces fields. To this end, the classical Sanders theory of cylindrical plates and shells is generalized by utilizing the non-classical nonlocal elasticity theory to derive the modified dynamic equations governing the nanofluid-nanostructure interaction (nano-FSI) problem. Then, the dimensionless obtained equations are analytically solved using the energy method. Herein, the applied nonlinear heat and humidity fields are considered as three types of longitudinal, circumferential, and simultaneously longitudinal-circumferential forces fields. The mentioned cases are examined separately for both high- and room-temperatures modes. The results show a significant effect of nanofluid passing through the nanostructure and its velocity on the critical buckling strain of the nano-systems, especially at high temperatures.

One of the main topics that had a tremendous impact on the life of contemporary man undoubtedly started with the famous statement of the great scientist Richard Feynman, who said there is plenty of room at the bottom. 1 Maybe one can say that one of those rooms is carbon nanotubes (CNTs) discovered by Iijima 2 in 1991. Carbon nanotubes are single-atom thick tubes constructed by wrapping a sheet of graphite made out of hexagonally-arranged atoms of carbon. Radushkevich and Lukyanovich published clear images of carbon tubes with a 50 nm diameter. 3 It was not until the experimental reidentification in 1991 that CNTs attracted considerable curiosity to study their electrical and thermo-mechanical behavior. Experiments illustrate that CNTs have exceptional electrical, 4-7 thermal, 8,9 and mechanical properties. [10][11][12][13] For instance, mechanically, CNTs have a tensile strength that is twenty times that of high-strength steel, 14 and its Young modulus is in the order of a tera-pascal. 15 Electrically, CNTs have demonstrated a high current density carrying capacity of 10 9 Amp/Cm 216 They have high resistance to electro-migrationinduced defeat. 17 Hence, CNTs have a high potential to replace traditional metals such as copper, aluminum, and its alloys whit a current-carrying capacity of 10 6 Amp/Cm 2 in IC interconnect usages. 16,18 In recent years, much attention has been drawn to the development of micro/nano-mechanical and micro/nano-electromechanical systems such as actuators, capacitive sensors, switches, etc. These nanostructured elements for nano-electronic devices may experience high temperatures during production and operation. It leads to thermal expansion, creating residual stress and affecting the device's reliability. 19 So far, there has been a great deal of interest in analyzing and accurately predicting the CNTs' dynamic behavior. For example, one can refer to the investigations conducted in the field of the effect of nano-fluids passing through nanostructures and vibrations and buckling caused by the applied forces like heat. [20][21][22][23][24][25][26][27][28][29][30][31] From the point of view of the solution method, analytical solutions with a large number of terms and conditions are not suitable for solutions for use by engineers and designers. 32 As a result, numerical or approximate methods have always been used to solve problems. However, the classical method for finding an accurate analytical solution is still much important because it serves as a criterion for numerical solutions. Furthermore, precise solutions are essential for developing efficient numerical simulation tools. In addition to experimental experiments, 33,34 which may be very expensive and effortful on the nano-scale, there are three primary approaches for modeling nanostructures: atomic modeling, hybridcontinuous mechanics, and continuum mechanics. Techniques such as classical molecular dynamics (MD), density functional theory (DFT), and tight-binding molecular dynamics (TBMD) are utilized to use the atomic modeling approach. [34][35][36] The combination of hybrid atomistic and continuum mechanics makes feasible the direct incorporation of interatomic potentials into the continuum analysis. It can be achieved by matching the molecular potential energy of a nanostructured material with the mechanical strain energy of the continuum model's representative volume element. 37 Finally, the continuum mechanical approach includes local rod, beam, plate, and shell theories, which are used to analyze nanostructures for macroscale systems. 38,39 The continuum mechanics approach is computationally cheaper than the previous two approaches, and their formulations are relatively simple. These modeling advantages caused to widely use it as an alternative to simulating some phenomena in nano-scale structures such as buckling, 40 wave propagation, 10 and free vibration. 24 Because the continuum mechanics theory is based on a continuous assumption in modeling, confirmation of the obtained results using the existing results through molecular dynamics or experiments is inevitable when the continuum theory is applied to analyzing the nanomaterials.
At the atomic length scale, the material structure is far more significant so that the effects of the small-size scale cannot be relinquished. To improve the constructive relationship of CNT, many researchers 41,42 adopted Eringen's theory of indirect (nonlocal) elasticity and incorporated them into several continuous models. The utilization of nonlocal elasticity relations for beam models like Euler-Bernoulli or Timoshenko models has been demonstrated to be accurate for long CNTs. 38,40 Silvestre et al. 43 examined the use of Sanders' refined thin cylindrical shell theory to model the buckling behavior of CNTs with a small aspect ratio. They showed that Sanders' shell theory can properly produce CNTs' buckling strains and mode shapes that are length-dependent.
The coefficient of thermal expansion (CTE) is the main feature for industries such as nano-electronics. Nanostructures such as nanowires, armchair carbon nanotubes, and nano-plates have shown important effects in thermal environments. Jiang et al. 44 showed that CNTs' CTE is positive for high temperatures and negative for room temperatures. The thermal expansion of carbon nanotubes is fundamentally different from that of other carbon derivatives such as carbon fibers and graphite or diamonds. Experimental observations have presented that 1.0 nm Single-Walled carbon nanotubes z E-mail: a_ghassemi@pmc.iaun.ac.ir; aazam77@yahoo.com have about 800 GPa of tensile stiffness. 45 In addition, it is now well known that the thermal expansion coefficient of CNTs is essentially isotropic. At room temperatures, the thermal conductivity of SWCNTs and MWCNTs are approximately 200 W mK −1 and 3000 W mK −1 , respectively. 46,47 In CNTs, the thermal behaviors are performed by phonons i.e. vibration modes, and thus the effect on the overall properties of CNTs is significant.
On the other hand, in general, structures in manufacture and use are often exposed to high temperatures and humidity. It seems that the changing environmental conditions due to the absorption of moisture and temperature harm the structures' hardness and strength. Increasing humidity and temperature decreases the elastic modulus of the material and induces internal pressures that can affect the stability of the structures. 48 Therefore, it seems that deformations due to the effect of moisture arising from temperature are important in analysis and design. According to the classical theory of plates, Whitney and Ashton 49 investigated the effect of hygro-thermal conditions on bending, buckling, and vibration of laminated plates applying the Ritz method. Sai Ram and Sinha 50 examined the effects of moisture on the free vibration of composite plates using the finite element method. Lee et al. used the classic plate theory to apply the hygrothermal effects on the cylindrical bending of symmetrically angle-ply plates that are subjected to uniform lateral pressure for different boundary conditions. 51 Based on Kirchhoff's plate theory, the hygrothermal bending response of a sector-shaped sigmoid plate with a variable radial thickness has been studied by Mashat and Zenkour. 52 Wang et al. 53 investigated the effect of thermal moisture on dynamic inter-laminar stresses induced by a piezoelectric actuator. A precise solution for the hygrothermal response of nonhomogeneous piezoelectric hollow cylinders (on macro-scale) exposed to mechanical load and electric potential has been yielded by Zenkour. 54 A limited number of articles are related to the effect of humidity conditions on nanostructures. Yao et al. 55 demonstrated the effect of water absorption on the electrical properties of graphene oxide films utilizing experimental measurements. They revealed that in the low moisture rate, the graphene has poor conductivity, while at high humidity, its conductance is increased. In addition, it has been observed that the maximum stress transfer at the CNT boundaries is very different from the change in thermal and hygrothermal coefficients. 56 Moreover, from experimental observations, it can be concluded that for low concentrations of CNT (1.2% by weight), CNTs are well dispersed in the matrix. Thus, CNT nano-composites become more rigid and be similar to the behavior of fiber-reinforced composites. 57 Besides, according to experimental studies, the polymers can allow SWCNT to disperse, whereas it may not happen for carbon fibers, fullerenes, and graphite shells. 58 Resistance analysis has shown that CNTs with low concentrations (less than 12% by weight) have depicted a growth in resistance and thermal expansion.
However, for high CNT concentrations (12% and 23% by weight), the resistivity reduces due to the inner-net between the nanotubes. 59 The main objective of this article is to investigate the influence of temperature change and the effects of a parameter caused by a change in temperature called hygrothermal on the buckling behavior of a nanofluid-nanostructure interaction (nano-FSI) system. For this purpose, Sanders shell theory as a classical shell and plate cylindrical theory is generalized by utilizing Eringen's nonlocal theory as a non-classical elasticity theory. Using Hamilton's principle, the governing equations for a shell cylindrical carbon nano-structure conveying the stationary/flowing nano-fluid are derived in non-dimensional form. Then, the governing relations are analytically solved in order to study the effects of applied thermo-and hygro-thermo-forces as well as the nano-fluid and its velocity passing through the CNT. Herein, variations of the amount of critical buckling strain due to applying the thermal and hygrothermal forces in longitudinal, circumferential, and simultaneous longitudinal-circumferential directions in the room-temperatures and high-temperatures ranges are discussed, separately. In addition, the effects of flowing fluid and its velocity on the buckling behavior of the structure and also its effect on the influence rate of the thermal moisture are investigated.

Mathematical Simulation
As shown in Fig. 1, as a circular cylindrical shell, a cylindrical hollow carbon nanotube with a radius of r, a cylindrical coordinate system (z, θ, x), and a reference point O at the center of one end of the shell is considered. The longitudinal direction of this system is x ϵ [0; L], its circumferential direction as θ ϵ [0; 2π], and its radial direction as z ϵ [−h/2; h/2]. In addition, the displacement coordinates of the desired point on the middle plane of the shell in the longitudinal, circumferential, and radial directions are illustrated with u, v, and w, respectively. In this work, Sanders' shell theory is used to investigate the buckling behavior of the structure. Then, the system's governing equations are presented by generalizing the Sanders theory using the non-classical Eringen theory in order to apply the effects of the size scale of the nanostructure. Furthermore, the effects of externally applied force on the carbon nanoshell and the possible changes in system buckling behavior are investigated utilizing cylindrical coordinates. The external force applied here is considered to be the force caused by the fluid passing through the nanotube, as well as the types of thermal forces and their effects, such as the hygrothermal force at room/high temperatures.
Thermal considerations.-In general, temperature changes occur during the manufacturing and application processes of each structure and system. Temperature changes will be effective in two ways: First, except in exceptional cases, materials expand and contract Figure 1. A schematic of a carbon nanotube and cylindrical shell coordinate system. respectively by heating and cooling. In most cases, these variations are proportional to the temperature changes. For example, the ratio of the change in length of the rod, ΔL, to its initial length, L, depends on the temperature, T. This linear relationship can be expressed mathematically as: [60][61][62][63][64] where α is the coefficient of thermal expansion. This parameter is constant almost in all materials unless a phase change occurs in the material. The above relationship means that there is a constant correlation between thermal strain, ΔL/L, and temperature change, ΔT, from a reference temperature at which there is no thermal stress or thermal strain. The second major effect of temperature changes is related to the hardness and strength of the material. Most materials become softer, more malleable, and looser under the influence of heat. For an orthotropic material such as a composite, there can be up to three different coefficients of thermal expansion and three different thermal strains in each orthogonal direction. In this case, Eq. 1 will have subscripts 1, 2, and 3 for both strain and thermal expansion coefficients. It should be noted that in the initial axes of materials, all thermal effects are only dilated, stretched, or compressed, and there is no thermal effect in shear directions. 65,66 Another important physical phenomenon called thermal humidity or hygrothermal has attracted the attention of materials scientists when studying the polymer matrix of composites. 67 In these researches, it was found that the combination of high temperature and high humidity doubles the destructive effects on the structural performance of the composites. This is due to the fact that the combination of high temperature and humidity in the polymer matrix causes moisture retention and weight gain, which causes lamination of the matrix. Scientists have also shown that in the studied structures, moisture absorption in different layers of the system changes linearly. 67 These changes can be calculated using the following equation: where Δm H is the increase rate of humidity from zero in terms of the weight percentage increase. In addition, β is the coefficient of hygrothermal expansion corresponding to the coefficient of thermal expansion shown in Eq. 1. Therefore, it can be seen that the effects of thermal moisture are mathematically quite similar to the heat effects. Hence, if you want to have answers to the thermo-elastic problem, just add βΔm H to the αΔT term. Experimental methods for obtaining thermal moisture expansion coefficient values are given in Ref. 67. Thus, the thermal and hygrothermal strains must also be considered to achieve governing relations in actual conditions. Therefore, according to the provided descriptions we have: It is important to note that although the thermal and humidity effects are similar, they have scales of varying degrees. For a structure under temperature changes, it takes a few minutes or up to an hour to reach equilibrium at a new temperature. In a similar structure, even if the structure is dry, it will take weeks or months to reach a moisture balance (saturation) by being in an environment with high relative humidity.
Equations 1 and 2 are linear relationships of temperature changes. Herein, these relationships are generalized and then applied in the governing equations to investigate the effects of nonlinear changes in temperature and humidity. For this purpose, it is assumed that the nonlinear temperature changes occur in the direction of the shell's thickness from T o on the outer surface to T i on the inner surface. Therefore, the relationship between thermal changes can be written as follows: 61,68 T T T z h where ΔT = T i −T o , and α p represent the non-negative power index of the temperature variation function. In the above relation, the thermal changes will be nonlinear in the direction of structure thickness by considering 2.
p α ⩾ Furthermore, if zˆis the position of the neutral axis along the z-axis, z ĉ as the distance between an arbitrary plane and the neutral axis is calculated as: 69 Noteworthy, although the percentage of the hygrothermal effects depends on the applied heat and its effectiveness can be estimated by calculating the thermal effects and the available moisture percentage, according to Eq. 4, the following relationship can also be introduced and presented for independent calculation: where . In the following sections, the effects of thermal and hygrothermal fields applied in the longitudinal, circumferential, and simultaneous in both longitudinal and circumferential directions, as well as the coefficients of thermal and humidity expansion appropriate to each case will be examined, separately.
Formulation of the generalized cylindrical shell theory for the Nano-FSI problem.-At first, to achieve the equations of the displacement field, the set of principal equations, i.e. compatibility equations, structural equations, and equilibrium equations, will be examined and used. Then, by considering the boundary conditions and using numerical or analytical methods, the equations can be solved and reach the final answer.
The compatibility equations.-Herein, considering the characteristics of the cylindrical shell, the strain-displacement relations (compatibility equations) will be presented using the classical shell theory of Sanders and then the governing equations will be obtained. This subsection also deals with strains resulting from applied thermal forces.
Taking into account the strain components at a desired point in the shell, the compatibility equations are generally defined as follows according to the coordinates of the cylindrical shell: Sanders developed the theory of thin shells using the principle of virtual work and the Kirchhoff-Love assumptions. Sanders' theory was able to solve the instabilities matter of other classical theories. It should be noted that this theory ignores the effect of lateral shear deformation, 70 so the relation of mid-surface strain-displacements and curvatures are stated as follows: Therefore, according to Eqs. 10-12, the relation of mid-surface strain-displacements and curvatures based on Sanders cylindrical shell theory include thermal and hygrothermal considerations are achieved as follows: The constitutive equations.-It is clear that nanoscale structures such as carbon nanoshells (Fig. 1) are very thin in the z-direction, so there can be a little variation in the components of the stresses in the direction of thickness. As a result, these components' values will be approximately zero. Furthermore, in the case of plate stress, it can be argued that other non-zero components of stress will have little change compared to z; this argument can be summed up by expressing stress in cylindrical coordinates as follows: Nanostructures behave differently from macro-scale behavior due to the effective factor of size and effects such as surface tension, strain gradient, and non-local effect of stress. The results of Eringen's non-local theory are very consistent with the obtained results of molecular dynamics simulations. 71 According to the differential equations governing the theory, Hooke's law for the relations of stress-strain in cylindrical (polar) coordinates is expressed as follows: 72  in which r is the radius measured from the middle area of the cross-section.
The equilibrium equations.-The equations of static equilibrium are the relations between forces/internal momentum and stresses. These equations are expressed as the stress and couple resultants, which are defined as follows using the plates and shells theories: It should be noted that in Sanders' theory, strain-displacement relations are linear functions in the z-direction, so the approximation of z r 1 1 ( + / ) ≈ can be used to simplify the relations, which indicates that N xθ and M xθ are respectively equal to N θx and M θx . Applying Eringen's non-local theory and concerning Eqs. 15a−15b, 16 are expressed as By introducing the following parameters, Then, by discretizing and sorting the relations and also considering the Eqs. 12 and 18, equations of 17 can be written in the following matrix form: Then, taking into account the values of A, B, C, and D in Eq. 18, considering the values of Q ij in Eq. 15b, and discretization of the relations, we have Therefore, the relations of forces and momentum in Eq. 19 can be represented as follows: In Eqs. 21 and 22, C R , D R , and G R , respectively, demonstrate the shell membrane rigidity, the shell flexural rigidity, and the modulus of rigidity (shear modulus), which are defined as follows: Finally, by substituting the strain components of the general obtained compatibility equation (Eq. 13) in Eq. 22, the nonlocal force and momentum components of the cylindrical shell are acquired.
Governing equations.-The Hamiltonian principle is used here to achieve the equations governing the problem, 73 so in the Lagrangian form we have: where U and W are potential energy components that represent the function of strain energy and external forces (the work of external force), respectively, and K is kinetic energy. In Eq. 24, operator δ represents the variation (operator of the changes) and the variable t indicates the time so that the above integration is performed over a time period of t 2 -t 1 . The variations of strain energy, external force work, and kinetic energy in cylindrical (polar) coordinates are stated as follows: In the above relations, the range of variables i and j are defined according to the assumed assumptions. By achieving structural equations in the previous section, at first, the considered kinetic energy of the carbon nanotubes is studied. Next, after determining each of these cases as virtual strain energy, virtual work, and virtual kinetic energy, and then forming the Eq. 24, the final relationship of the governing equations is yielded.
Strain energy.-The general equation for obtaining variations of the virtual kinetic energy is acquired using the following equation: 74 in which Ω depicts the area (i.e. dΩ = dxdθ). Due to the consideration of cylindrical coordinates and the small changes in the direction of the thickness (radial direction), the variables of the above relation in a symmetrical system are considered as i = 1,2,4 and therefore the above relation is discretized as follows: Considering the equations of compatibility and equilibrium and also according to the general definition of stress and couple resultants in the classical theory of plates and shells, the above relation can be rewritten as: External force work.-The external force work variations can be obtained using the following equation: 74 This part of the potential energy, also known as the external force (virtual work), is in two general types of body forces such as weight and surface forces (extensive loads) such as hydrostatic pressure that are applied to the system. These forces can be distinguished and defined as follows: 75 In the above relations, N , nl ε θ and x nl γ θ are determined as nonlinear strains or nonlinear sentences of the main strains in compatibility equations. Based on studies and assumptions of nonlinear elasticity of plates and shells in classical theories of cylindrical shells, 76 after partial integrating, factoring, and simplifying, the body force variations in the form of divided displacements are as The buckling in a cylindrical system under lateral and axial pressures, under conditions where w is constant, v is equal to zero, and the deformations are independent of θ before the occurrence of the buckling, the following relations will be established: In these relations, N xx 0 and N 0 θθ are unknown constant values as internal forces that depend on the amount and amplitude of the applied loads in the axial and lateral directions, respectively. In order to more focus on thermal effects and to simplify, the value of N 0 θθ is considered to be approximately zero and its application in equations is ignored, here.
For calculating the applied virtual work, in addition to the body forces, if it is assumed that the considered shell is subjected to an external force such as the force caused by the passing fluid, the pressure during deformation will always remain perpendicular to the middle surface of the shell. 74 Therefore, the potential energy of the external force due to, for example, the applied hydrostatic pressure, is the product of the pressure on the shell middle surface per unit length, and Eq. 31b must be rewritten as follows: 77 ECS Advances, 2023 2 011002 In Eq. 35, considering the dead load or hydrostatic pressure load the ξ coefficient will take 0 and 1 values, respectively. In this study, the surface forces are assumed to be a dead load, so that the applied pressure is directed centrally. It means that constant pressure on the middle surface of the shell, which has not any deformation will always remain perpendicular and the surface effects can be ignored.
Kinetic energy.-As mentioned before, the kinetic energy variations of the system can be calculated as follows: 74 where u j̇i s the velocity of the object in the direction of j, and the volumetric integral of ρ is related to the object mass. In cases where, in addition to the velocity of the structure, force is applied to it by passing fluid or particle, in order to solve the problem more accurately, it is necessary to pay attention to the effect of this external force on the behavior of the body. Hence, the structure and the fluid passing through it are generally considered as a system. In these problems, which are also known as fluid-structure interaction (FSI), according to the physical principle of the consistency condition at the point of collision of structure and fluid, velocities and accelerations are the same in the direction of displacement. 78 Therefore, to find the total velocity of the system, the velocity of the fluid at the point of impact with the structure can be calculated. In addition, herein, both the structure and the passing fluid are nanoscopic scales, thus the effects of size must be taken into account in the theoretical relations.
To achieve equations of motion, the following relationships are provided using the initial definition of kinetic energy: As can be seen, considering the conditions and assumptions in each problem and integrating Eq. 37, the final obtained equations are consistent with the Navier-Stokes equations. Herein, the components of Eq. 37 including their velocities and variations, are achieved as follows using the definition of the material derivative: By substituting the Eqs. 39a and 39b in the final relation of Eq. 37, simplifying the material derivative in the form of u , and after multiplying and discretizing the parameters, the displacement field equations (displacement variations) caused by the passing fluid are obtained as: It is noteworthy that the dynamic pressure due to the interaction of fluid and structure automatically appears in the above equations of motion using the material derivative. It includes the expressions of m U t , 2 2 (∂ ∂ ) mu U x , 2 2 2 (∂ ∂ ) and mu U t x 2 (∂ ∂ ∂ ) (for U = u,v,w), which represent the internal force due to translational transverse accelerations, the internal forces corresponding to the centripetal or centrifugal accelerations, and the internal force resulting from Coriolis accelerations, respectively.
Moreover, the external pressure means the total pressure exerted by the nanotube on the fluid (load pressure) and the external tension applied to the nanotube by the fluid (thrust pressure) remain constant throughout the nanotube and neutralize each other. On the other hand, it is noticeable that here, the effect of the flowing fluid over the considered nanostructure has been investigated using the cylindrical shell model. Furthermore, it should be noted that the mass in Eq. 37 contains both parts of this system, i.e. structure and fluid. Therefore, by following the described principles, Eq. 37 is rewritten as follows: in which m c is the mass of the structure (carbon nanotubes) and m f is the mass of the passing fluid. However, it is noteworthy that in structural-fluid interaction problems, the velocity at the point of collision is considered and calculated, which are equal in terms of molecular physics. In addition, given that the structure is assumed to be stationary here, all variations related to the structure will be zero relative to spatial variables, and only the terms that contain time derivatives, m u v w , c (̈+̈+) will be included in the final equations. Finally, the obtained equations will be provided by simplifying and assuming that the changes in the thickness direction are negligible, as well as assuming that the fluid velocity only is in the longitudinal direction (u).
To consider the nanoscale effects in fluids, a dimensionless parameter called the velocity correction factor (VCF) is used 80 so that in the resulting equations, wherever there is a fluid velocity parameter, it must be multiplied by the VCF coefficient. In the following, the final equations are presented by applying this parameter. Herein, VCF is defined as the ratio of the average velocity of the fluid flow for a slip boundary condition to the average velocity of the flow for a non-slip boundary condition, as follows: 78 VCF V V C rk n Kn bKn in which, V slip and V no-slip are the flow velocities with and without slip boundary conditions, respectively. Furthermore, b and v σ are the general slip coefficient and the tangential momentum accommodation coefficient. Besides, Cr(Kn) indicates the rarefaction coefficient of fluid (dilution) which is defined as the ratio of dynamic viscosity to total viscosity (bulk) of fluid that is defined as follows using the Polard suggestion relation: 81  Table I.

Solution Procedure
So far, the displacement field equations have been derived using the generalized classical elasticity theory, which includes a set of differential and algebraic relations between stresses, strains, and displacements that describe the considered system.
In this section, first, the final governing equations are presented. Then, by determining the appropriate boundary conditions, the displacement field equations are completed. Finally, the obtained equations are rewritten in dimensionless form and are analytically solved.
Nanofluid-nanostructure interaction equations (Nano-FSI).-In this subsection, by substituting the obtained results of the previous sections for potential energy (strain energy and virtual work) and kinetic energy into the Hamilton principle equation, the general displacement field and the final motion equations governing the problem of carbon nanotube conveying passing nanofluid is obtained based on non-local Sanders shell theory (n-SST). After substituting, utilizing the integration-by-parts method, and classifying, these equations will eventually be obtained as follows:   In the resulting relations, the fourth power of the nabla operator, , 4 ∇ is the biharmonic operator, which is calculated as the square of the Laplacian. This operator can also be shown in cylindrical coordinates as . Boundary conditions.-The boundary conditions determine physical behavior at the boundary of an object. In other words, the boundary loads that physically cause stress, strain, and internal displacement will be considered in the equations. Although the formulation of field equations is specified for each theory, the boundary conditions can be different in each problem. Therefore, it is necessary to determine the appropriate boundary conditions to solve the problem. Here, the Essential type of boundary conditions is considered. In order to analytically solve the equations, the values of where U, V, and W are the movement amplitude, which are the unknown coefficients of the problem; n is the circumferential wave number (meaning the mode number in the mechanical behavior of the structure); ω is the system frequency; t is the time, i is a parameter equal to 1 − and Φ is an axially function that determines the geometric boundary conditions. Considering that this study aims to investigate the effect of external force due to flowing nanofluids and the effect of thermal and hygrothermal forces on the buckling behavior of nanostructure, the simply-supported ends (SS) are used as boundary conditions to simplify the process of the accurate analytical solution according to the considered geometry. Therefore, the axial function of Φ as the characteristic function of the structure is expressed as follows: The buckling behavior of the system is independent of time, so in relations 42, t is considered equal to zero, and then by placing Eqs. 43, 45 is rewritten as follows: Solving the dimensionless governing equations.-Before solving the final equations, the following parameters are defined in order to achieve the general non-dimensional equations: Hence, by setting the coefficient matrix (F ij matrix) equal to zero, the values of the unknown matrix (U mn , V mn , and W mn ) will be obtained.

Results and Discussion
In this section, the buckling behavior of a shell cylindrical carbon nanotube conveying nanofluid is analyzed and discussed under the influence of thermal and hygrothermal forces applied to the system in longitudinal and circumferential directions.
The applied thermal forces are of three types: longitudinal, circumferential, and longitudinal-circumferential thermal forces. The dynamic and physical relations of the system under study are formulated using Sanders' cylindrical shell theory. Then, those are generalized using Eringen's small-scale relations and are presented in dimensionless form. Next, the buckling behavior of carbon nanotubes containing nanofluid is examined under the effects of thermal forces for two modes of high temperatures and room temperatures, separately. Moreover, the effect of the fluid passing through the nanostructure and the effect of its velocity are investigated. Here, the effect of thermal humidity is assumed to be 20% of the applied thermal force. In addition, the nano-water is considered as a fluid to analyze the effect of the presence and velocity of fluid passing through the nano-shell and the critical buckling variations of the nano-system. Material and geometric characteristics of fluid and structure are given in Table II.
It is reported that all the coefficients of thermal expansion for CNT are negative at low and room temperatures and are positive at high temperatures. 44 In the present work, the coefficients of the thermal expansion (CTE) and hygrothermal expansion (CHE) are the same. The values of the thermal expansion coefficients are given in Table III for different thermal fields and conditions. By default, the value of the temperature difference (ΔT) between inside and outside of the structure at low temperatures and high temperatures are considered 30°C, and 230°C, respectively. Besides, the hygrothermal difference (Δm H ) is assumed to be 20% of the heat difference.
An issue that has received less attention in similar articles and research works is that variations of temperature, and consequently thermal humidity, will lead to changes in the density of the structure and fluid under study. Hence, density changes, especially at the nanoscale, can be significant and should be considered. In this subsection, the effect of heat and humidity on the density of nanostructures and nanofluids passing through it, and the effect of these density variations on the natural frequencies of the system are investigated. According to the laws of thermal physics, with volumetric thermal expansion, the change in material density due to temperature variations can be calculated as follows: 78 where m is mass and ρ * is the apparent density under ambient conditions without temperature changes. According to the dependence and direct relationship of thermal humidity to the amount of heat, the following new relationship can be defined by generalizing Eq. 52: From the above relationship, it can be found that the density of the material (both structure and fluid) decreases by increasing thermal and humidity effects.
Validation of the results.-In this subsection, our obtained numerical results of critical buckling strains are compared with those of molecular dynamic (MD) simulation results. To this end, the passing nanofluid is ignored and then the obtained results of MD simulations and nonlocal-Sanders shell theory (n-SST) are listed in Table IV.
According to observations of Silvestre et al., 43 the use of Donnell's theory leads to accurate results only for modes of the high circumferential half-wave number, n. In addition, Pellicano and Amabili 82 revealed that the condition of n 1 2 ⩾ must be satisfied in order to have fairly good accuracy and proposed that n 1 ⩾ is adequate. Herein, in order to compare the results of the assessed modified theory with the results from MD simulations presented in Ref. 43, the value of the circumferential half-wave number, n, and the longitudinal half-wave number, m, is considered equal to 10 and 1, respectively.
Taking into account Table IV, it can be seen that the obtained critical buckling strains are in good agreement with the MD simulation results. To achieve the best result in terms of the agreement with the molecular dynamics outlets the different values of the internal characteristic length parameter (e 0 a) of the nanotubes are investigated and the closest responses to the molecular simulation results are provided in Table IV. Moreover, it can be found that the available models are unable to show the correct trend in critical axial buckling strains of carbon nanotubes with a specific length if the correct value of the parameter of e 0 a is not considered.
Effect of thermal and hygrothermal forces at high-temperatures.-In this subsection, the effect of thermal and hygro-thermal forces at high temperatures on the buckling behavior of the nanostructure-nanofluid system is examined. Figure 2 shows the changes in the amount of critical dimensionless buckling created in the cylindrical shell carbon nanotube due to variations of thermal strength in three modes: longitudinal, circumferential, and simultaneous application of longitudinal and circumferential thermal forces at high temperatures. Critical buckling values are obtained here in the presence of motionless fluid inside the structure as well as without considering the effects of thermal moisture. As can be seen in Fig. 2, the amount of dimensionless critical buckling that occurred in the structure decreases due to the application of longitudinal thermal force at high temperatures, i.e. increasing temperature from 50 to 1000 causes to reaches the amount of dimensionless buckling strain from −2.071e10 to −4.143e11. In addition, the amount of critical buckling decreases from −2.862e9 to −5.724e10 due to applying the circumferential thermal force. The dimensionless critical buckling force decreases from −2.357e10 to −4.715e11 when both longitudinal and circumferential thermal forces are applied simultaneously at high temperatures. Interestingly, in all three cases, the reduction in critical buckling force with an increase in temperature from 50 to 1000 is approximately 444%. It indicates the predictability of these thermal forces' effect on the behavior of the system. It can also be found from Fig. 2 that due to the application of circumferential, longitudinal thermal forces, and simultaneous applying the longitudinal-circumferential forces, the less critical buckling value is obtained, respectively. It is noteworthy that the critical strain value due to the exertion of longitudinal thermo-load relative to this value due to the application of circumferential thermo-load at any point in temperature (in the high-temperature range) shows a reduction of about 96.7%. In addition, the amount of critical strain due to the simultaneous application of longitudinal and circumferential thermal loads compared with the critical buckling strain value obtained by applying longitudinal or lateral thermal load at any point of temperature is decreased by 13.8% and 123.9%, respectively; This constant trend can be used in similar research. Figure 3 illustrates the amount of critical buckling load variations relative to temperature changes at high degrees for the three applying modes of longitudinal thermal force, circumferential thermal force, and simultaneous application of longitudinal and lateral thermal force. Here, in addition to thermal forces, the effects of thermal humidity (the so-called hygrothermal) have also been applied. As can be observed, the process of decreasing the critical buckling strain is started by increasing the temperature, which also causes to increase in the effect of thermal moisture. Herein, it is assumed that the effect of hygro-thermal force (HTF) is 20% of the applied thermal force (TF).
The critical buckling loads of cylindrical carbon nanoshells under the influence of high-temperature with and without applying longitudinal (LHTF), circumferential (CHTF), and longitudinal-circumferential (LCHTF) hygrothermal force effects, which are shown in Figs. 2 and 3, are simultaneously displayed in Fig. 4. As can be seen and compared in Fig. 4, with the addition of the effect of hygrothermal, which is a natural and indisputable effect on physical phenomena, the reduction rate of the critical buckling load is the same as the assumed value of 20%. Therefore, it can be found that increasing or decreasing the effect of hygrothermal has the opposite and equal relation with decreasing or increasing the critical load of buckling at high temperatures, respectively. Furthermore, due to the constant change value in the amount of critical buckling of the structure, only by calculating, for example, the amount of critical buckling load caused by longitudinal thermal force at any temperature point, the amount of critical buckling caused by applying the circumferential or simultaneous longitudinal-circumferential thermal load can be achieved with or without hygro-thermal effect.
Effect of thermal and hygrothermal forces at room-temperatures.-In this subsection, the effect of thermal and hygrothermal forces at room temperatures on the buckling behavior of the CNT conveying nano-fluid is investigated. Figure 5 shows the amount of the variations of critical buckling strain of shell cylindrical carbon nanotubes against the application of the longitudinal, circumferential thermal forces as well as the simultaneous applied longitudinal and circumferential thermal forces at room temperature. In this figure, it is demonstrated that at room  temperature, if the temperature is less than zero, the value of the critical buckling load will be less than zero, and as the temperature rises, this value will gradually increase. As can be seen, the rate of the changes is the highest when the longitudinal and circumferential thermal forces are simultaneously applied and the lower variations rate is observed when the lateral thermal forces are utilized. For example, at the temperature of −10, the critical buckling load for longitudinal, circumferential, and longitudinal−circumferential thermal forces is −6.026e9, −8.586e8, and −6.884e9, respectively, and at the temperature of 50, they are 2.953e10, 4.207e9, and 3.373e10, respectively. It is significant that in all three cases, the enhancement rate in the critical load value of the buckling strain due to the increase in temperature from −10 to 50 is 175%. As can be seen, shortly after zero degree temperature (at room temperature) with the application of each type of thermal force (LTF, CTF, and LCTF), the critical buckling load is approximately  equal to zero. It can also be found that by applying lateral thermal load at temperatures below 1, at any point in temperature, the dimensionless critical buckling load increases about 47.5% and 54.1%, respectively, relative to longitudinal load, and the simultaneous application of longitudinal-lateral load. Moreover, by applying CTF the dimensionless value of critical buckling strain at any point in temperatures above 1 decreases approximately 90.1% and 118% relative to the applied LTF and LCTF.
Changes in the dimensionless critical buckling strain of cylindrical nanoshells due to applying the longitudinal, circumferential, and longitudinal-circumferential thermal forces at room temperature are demonstrated in Fig. 6, taking into account the effects of hygrothermal effects. The process of increasing the critical dimensionless buckling in all three types of thermal force is similar to Fig. 5. Hence, the amount of critical buckling load is raised by increasing temperature. Exerting the effect of thermal humidity (20% of the  applied thermal force) for this structure at room temperature causes to increase in the critical load values. As can be seen, again at the dimensionless temperature of 1, the critical load value in all three types of applied thermal forces reaches zero It is also observed that, for example, at the temperature of −10, the amount of critical buckling load for longitudinal, circumferential, and longitudinalcircumferential thermal forces is −7.231e9, −1.03e9, and −8.261e9, respectively, and at the temperature of 50, the values are respectively equal to 3.543e10, 5.049e9, and 4.048e10. Therefore, it is determined that by considering the effects of hygrothermal force, the amount of dimensionless critical load at each temperature point theoretically decreases by 20% before the dimensionless temperature of 1 and increases by 20% after that temperature.
A comparison between the critical non-dimensional buckling load values of the system respectively against the applied nondimensional longitudinal, lateral, and longitudinal-lateral thermal forces at room temperature with and without considering the hygrothermal effects, is displayed in Fig. 7. As mentioned earlier, for temperatures less than 1, the effect of thermal moisture causes to reduce the critical buckling strain values, and in thermal magnitude greater than 1, the effect of hygro-thermal increases the critical load value of buckling strain. As can be seen in Fig. 7, the highest ratio of the hygrothermal effect is related to applying the longitudinalcircumferential (LCTF), longitudinal (LTF), and circumferential (CTF) thermal forces, respectively.
Comparison of the hygrothermal effects at high-temperatures and room-temperatures.-In this subsection, the effects of hygrothermal from 10% to 80% of the amount of thermal force are investigated. Herein, the effectiveness of the hygrothermal parameter in each type of studied thermal force (LTF, CTF, and LCTF) and also in the two modes of room-temperature and high-temperature are discussed. From Fig. 8, it can be seen that at room temperature, an increase in the effect of hygrothermal cause to increase the critical buckling value of the structure, while this effect is reversed at high temperatures, it means the effect of critical buckling load is decreased by enhancing the hygrothermal effects.
Furthermore, it can be reported from this scrutiny that, for example, by increasing the effect of thermal humidity from 10% to 80%, in all types of studied thermal forces, as well as in both room temperature and high temperature, increase or decrease the amount of critical buckling load is 38.89%. It is noticeable that the rate of increase or decrease in the critical buckling load value due to increasing the percentage of hygro-thermal in all cases is the same. That is, for example, with every 10 percent increase in the effect of thermal humidity parameter to 20,30,40,50,60,70, and 80 percent, in all considered cases (LTF, CTF, and LCTF), the critical loads of buckling respectively 8.34%, 7.7%, 7.13%, 6.66%, 6.25%, 5.88%, and 5.55% are increased (at room-temperatures) or decreased (at high-temperatures).
The effect of passing fluid velocity on the critical buckling load.-In this subsection, the effect of the passing nano-fluid velocity throughout the shell cylindrical carbon nanotube on the critical buckling of the structure under the influence of longitudinal, circumferential, and longitudinal-circumferential thermal forces with and without the hygrothermal effects is discussed.
Firstly, Fig. 9 illustrates that the critical buckling load value of the nanostructure significantly decreases as the flowing nanofluid velocity increases. Second, like the previous results, it can be seen that in all cases of high-temperatures conditions, when the hygrothermal effects are also considered the critical buckling value of the structure is less than when the only effects of thermal force on the nanotube are applied. The effect of hygrothermal is considered to be 20%, here.
As shown in Fig. 9, it is noteworthy that, for example, by applying lateral thermal force without and with considering the effect of thermal moisture, the amount of non-dimensional buckling load at zero fluid velocity equal to −2.862e9 and −3.435e9 and for the dimensionless fluid velocity of 100 are −5.095e11 and −5.1e11, respectively. As indicated in this figure, at zero velocity, the effect of the hygrothermal parameter causes a 20% reduction in the critical load of the buckling. However, this effect is reduced at higher speeds of the passing fluid. For example, at a non-dimensional velocity of 100, the thermal humidity causes a decrease of 0.1% in the critical buckling load. This reduction in the case where the longitudinal thermal forces (LTF) and longitudinal-circumferential thermal forces (LCTF) apply at zero fluid velocity is equal to 20% and at a dimensionless speed of 100 is approximately 0.85%. Therefore, it can be concluded that increasing the velocity of the fluid passing through the carbon nanotube leads to reducing the hygrothermal effects on the amount of critical buckling load. Figure 10 shows the rate of change in the critical dimensionless buckling strain of the carbon nanotube at room temperature with Figure 9. The dimensionless critical buckling strain changes of the shell cylindrical carbon nanotubes exposed to longitudinal, circumferential, longitudinalcircumferential thermal forces at high-temperatures relative to velocity variations of the nanofluid passing through the structure with and without hygrothermal effects. respect to the changes in nanofluid velocity passing through it so that the system is exposed to longitudinal, circumferential, and longitudinal-lateral thermal forces with and without the effect of thermal moisture. It is again observed that by enhancing the flowing nanofluid velocity, the critical buckling value of the structure decreases, significantly. It can also be seen that for room-temperatures mode (unlike high-temperatures mode), considering the effect of thermal humidity, the critical buckling value of the structure is more than when only the effects of thermal force are applied to the system. In Fig. 10, by applying circumferential thermal force without and with (20%) considering the effect of thermal moisture, the amount of non-dimensional buckling load at zero fluid velocity is equal to 4.293e9 and 5.152e9, respectively, and it is equal to −5.023e11 and −5.014e11 at the dimensionless velocity of 100. These values, if the longitudinal thermal force is applied to the system, are equal to 3.013e10 and 3.615e10 at zero velocity, and equal to −4.765e11 and −4.704e11, at the velocity of 100, respectively. In addition, the dimensionless critical buckling loads due to applying the longitudinal-circumferential thermal force without and with the hygrothermal effects are 3.442e10, 4.13e10, −4.722e11, and −4.653e11 for the fluid velocities of 0 and 100, respectively.
As can be observed, at zero speed, the effect of the thermal moisture parameter on all three types of applied thermal forces leads to a 20% enhancement of the critical buckling load. However, at higher fluid velocities, this effect is reduced so that, for example, at a dimensionless speed of 100, the hygrothermal effect causes to 0.18%, 1.3%, and 1.46% reduction in the critical buckling load respectively related to the application of circumferential (CTF), longitudinal (LTF) and longitudinal-circumferential thermal forces (LCTF). Therefore, it can be revealed that in room-temperatures mode, increasing the velocity of the fluid passing through the carbon nanotube reduces the effect of hygrothermal on the critical buckling load value.
It should be noted that at the dimensionless fluid velocity between 10 and 30 (for each type of thermal force with and without applying the effect of thermal moisture), the points can be observed in which the critical buckling value of the structure has reached zero, that is, in this range of fluid flow velocity, the structure will be more stable.

Conclusions
In this paper, an attempt was made to investigate the effect of thermal force and hygrothermal resulting from the applied heat force on the critical buckling value of a system in the form of nanofluidnanostructure interactions (Nano-FSI). Hence, a cylindrical carbon nano-shell containing static/dynamic nanofluid was considered. The applied thermal forces are considered in longitudinal, circumferential, and simultaneous longitudinal-circumferential directions. Herein, the effect of thermal humidity is assumed by 20% of the applied thermal force. The dynamic and physical relations of the system under study have been formulated using Sanders' cylindrical shell theory so that it was generalized utilizing the small-scale relations of nonlocal elasticity theory and then the resulting equations were formed in dimensionless form.
All of the above-mentioned were examined separately for both high-temperatures and room-temperatures modes. Moreover, the effect of the fluid passing through the nanostructure and its velocity on the critical buckling load and the hygrothermal effects were also investigated. From the obtained results the following conclusions are noticeable: • Due to applying the thermal forces at high-temperatures, the amount of dimensionless critical buckling strain of the system has a decreasing trend. So that raises the temperature and application of circumferential, longitudinal, and longitudinal-circumferential thermal forces respectively leads to obtaining a less critical buckling value.
• At high temperatures, by applying longitudinal thermal force, lateral thermal force, and also simultaneous application of longitudinal-circumferential thermal force, the process of reducing the amount of critical buckling strain is maintained by increasing the temperature, which also causes to increase in the effect of thermal moisture. It can also be realized that increasing or decreasing the Figure 10. The dimensionless critical buckling strain changes of the shell cylindrical carbon nanotubes exposed to longitudinal, circumferential, longitudinalcircumferential thermal forces at room-temperatures relative to velocity variations of the nanofluid passing through the structure with and without hygrothermal effects. hygrothermal effects has the equality and opposite ratio to decreasing or increasing the amount of critical buckling load at hightemperatures, respectively.
• It can be reported that at room temperatures, if the temperature is less than zero, the critical buckling load will also be less than zero, and this value will gradually increase as the temperature rises. It is also observed that in room-temperatures mode, by applying each type of thermal force (LTF, CTF, and LCTF), the critical buckling load value of zero can be achieved, which means that the system reaches a stable state.
• By applying longitudinal, lateral, and longitudinal-lateral thermal forces at room-temperatures and considering the effects of thermal humidity, it can be seen that the critical load of the buckling increases as temperature rises. It can also be stated that before reaching the stability point, the hygrothermal effect will reduce the amount of critical buckling strain and after that, it leads to an increase in the critical buckling load value. At room-temperatures mode, the highest impact ratio of thermal humidity effect belongs to applying the longitudinal-circumferential (LCTF), longitudinal (LTF), and circumferential (CTF) thermal forces, respectively.
• It was concluded that in room temperatures and high temperatures modes, increasing the hygrothermal effects causes an increase and decrease in the amount of critical buckling of the structure, respectively. It can also be seen that by increasing the hygrothermal effects on all types of thermal forces under consideration, at both room temperatures and high temperatures, the rate of increase or decrease of the critical buckling load is the same.
• By investigating the effect of the velocity of the nanofluid passing through the shell cylindrical carbon nanotube on the critical buckling of the nano-FSI system, it is observed that by increasing the nanofluid flow velocity, the critical buckling load is significantly reduced in all cases. It was also concluded that at both high-and low/room-temperatures, enhancing the velocity of fluid flowing through the carbon nanotube causes to reduce the effect of hygrothermal on the critical buckling load value.