On the Solution of Thermal Buckling Problem of Moderately Thick Laminated Conical Shells Containing Carbon Nanotube Originating Layers

This study presents the solution for the thermal buckling problem of moderately thick laminated conical shells consisting of carbon nanotube (CNT) originating layers. It is assumed that the laminated truncated-conical shell is subjected to uniform temperature rise. The Donnell-type shell theory is used to derive the governing equations, and the Galerkin method is used to find the expression for the buckling temperature in the framework of shear deformation theories (STs). Different transverse shear stress functions, such as the parabolic transverse shear stress (Par-TSS), cosine-hyperbolic shear stress (Cos-Hyp-TSS), and uniform shear stress (U-TSS) functions are used in the analysis part. After validation of the formulation with respect to the existing literature, several parametric studies are carried out to investigate the influences of CNT patterns, number and arrangement of the layers on the uniform buckling temperature (UBT) using various transverse shear stress functions, and classical shell theory (CT).


Introduction
Laminated anisotropic shells have been the subject of much research because they are used as the main bearing elements of engineering structures used in modern aerospace and rocket technology, shipbuilding, energy and chemical engineering, and other fields. The widespread use of composites as structural elements, which can best meet the demands of harsh working conditions, has revealed the need for the application of new theories and methods in the mechanics of laminated composites and structural mechanics based on them. These factors have led to the development of various refined theories instead of the classical shell theory, which neglects the effect of transverse shear deformations in the calculation of laminated anisotropic plates and shells. The formation and development of these theories are summarized in monographs published in different periods [1][2][3][4][5][6]. Since laminated homogeneous shells made of traditional composites are used in thermal environments, their thermal buckling behavior has always been the focus of attention for researchers [7][8][9][10][11][12][13].
where θ 1 = θ sin γ the cone-1 a and 2 a being the coordinates of the points where this axis intersects the small and large bases, respectively. The mid-surface 0 z  is located at the interface of the layers for even values of N , while the mid-surface for the odd values of N is located in the mid-surface of the middle lamina (Figure 1b). The displacement components of the mid-surface along the , S  and z axes are designated by , u v and w, respectively. The mid-surface rotations of the normals about  and S axes are denoted by 1  and 2  , respectively. The stress resultants are given by  , as in [48,49]

Material Properties of Nanocomposite Layers
The effective material properties and thermal expansion coefficients of nanocomposite layer th k are given by [43][44][45][46][47]: and

Material Properties of Nanocomposite Layers
The effective material properties and thermal expansion coefficients of nanocomposite layer kth are given by [43][44][45][46][47]: and α (k) m are the Young and shear moduli, Poisson's ratio, the thermal expansion coefficient in the lamina kth, θcn are the similar elastic and thermal properties for CNTs in the lamina kth, and η (k) i (i = 1, 2, 3) is the efficiency parameter for the lamina kth. The following equality is satisfied for the volume fraction of CNTs and lamina: V The distribution of volume fractions for CNTs across the thickness of the layer kth is modeled as the U-, V-, O-and X-shaped elements (See, Figure 2): The distribution of volume fractions of CNTs across the thickness of the layer are illustrated in Figure 2

Basic Relations and Equations
In this section, the basic relations and equations of moderately thick laminated orthotropic conical shells consisting of CNT originating layers are reviewed. In the presence of a temperature field, the constitutive relations for the CNT originating layer th k in the framework of STs can be determined by a generalization of Hooke's law as follows [26,47]: 11 12 21 22 66 , , and The distribution of volume fractions of CNTs across the thickness of the layer are illustrated in Figure 2

Basic Relations and Equations
In this section, the basic relations and equations of moderately thick laminated orthotropic conical shells consisting of CNT originating layers are reviewed. In the presence of a temperature field, the constitutive relations for the CNT originating layer kth in the framework of STs can be determined by a generalization of Hooke's law as follows [26,47]: (5) and σ (k) where σ θz are the stresses in the kth layer, e S , e θ , γ Sθ , γ Sz , γ θz are the strains, and Q (k) ijz , (i, j = 1, 2, 6) is the plane stress-reduced stiffnesses defined in terms of engineering constants in the material axes of the lamina kth. It is given by: Sθ (Z). (7) in which σ where ∆T = T − T 0 is the uniform temperature rise from the reference temperature (T 0 = 300 K) at which the cone is free of thermal stresses.
The stresses σ (k) Sz and σ (k) θz in the lamina kth are expressed by ϕ 1 and ϕ 2 as follows [1,47]: where f  Using (8), (5) and (6), the strains (e S , e θ , γ Sθ ) can be expressed as those of their midsurface (e 0S , e 0θ γ 0Sθ ) as follows: where J The in-plane forces (T S , T θ , T Sθ ), moments (M S , M θ , M Sθ ), and transverse shear forces (Q S , Q θ ) for laminated nanocomposite truncated conical shells composed of CNT originating layers are obtained from the following integrals [1][2][3][4][5][6]48]: The resultants for the thermal forces and moments are [26]: The basic equations for a truncated conical shell, based on the STs, are expressed as [48]: where T 0 S , T 0 θ , and T 0 Sθ are the membrane forces for the condition with zero initial moments. Since the temperature is constant in the longitudinal and circumferential directions of the laminated conical shell, and varies only in the thickness direction, the prebuckling deformation can be expressed by the following equations: Thus, the prebuckling thermal force T 0 S is defined in [49]: where Γ T is the thermal parameter. When the temperature changes uniformly throughout the thickness of laminated nanocomposite conical shells, the thermal parameter Γ T is defined as Using the relationships (5), (6) (10), (12) and (13), the governing Equation (14) is transformed into the following form: where L ij (i, j = 1, 2, . . . , 4) is a differential operator, and is given in Appendix A.
The set within Equation (19) is the set of basic equations of laminated conical shells with CNT-patterned layers based on STs.

Solution Procedure
The two end edges of the laminated truncated conical shell are assumed to be simply supported, and to be restrained against expansion longitudinally, while temperature is increased steadily, so that the boundary conditions are ζ = −ζ 0 and ζ = 0 [26,48,49]: Here, the following denotations are introduced for convenience: ζ = ln S a 2 and ζ 0 = ln a 1 a 2 . The solution for (19) is defined as [47]: β m = mπ ζ 0 and β n = n sin γ , wherein (m, n) is the buckling temperature mode, p is the unknown parameter that is defined based on the minimum condition of the buckling temperature, and C i (i = 1, 2, . . . , 4) represents unknown coefficients.
By substituting approximation Equation (21) into the set within Equation (19), and then applying the Galerkin method to the resulting equations, one obtains: where in which l ij (i, j = 1, 2, . . . , 4) and l T are given in Appendix B. From Equations (17) and (22), the following expression is found for the uniform buckling temperature of laminated truncated conical shells composed of CNT originating layers: Considering the problem within the framework of the CT (that is, considering only the relationships in (5)-the governing equations of laminated conical shells with CNT originating layers) one obtains: Similarly, substituting the first two approximation functions from (21) into (25), and then applying the Galerkin method to the resulting equations, the following expression for the uniform buckling temperature for CNT shaped laminated conical shells based on the CT is obtained [41]: where l j (j = 1, 2, . . . , ) is the parameter depending on the CNT-shaped laminated conical shell characteristics based on the CT, and l T = l T a 4 2 is the thermal parameter (both are presented in Appendix C).

Comparative Studies
To check the accuracy of the expressions obtained for the uniform buckling temperature, a comparison is made with the results of the single-layer homogeneous isotropic truncated conical shell, which is presented in Ref. [50] (see Table 1). The data used in the comparison are taken from Ref. [50], and are as follows: Y To compare the results of Ref. [50], the expression (28) was multiplied by α m are considered in the comparison. It is seen that the magnitudes of UBT (α

Thermal Buckling Analysis
In this subsection, thermal buckling analyses are presented for laminated conical shells consisting of CNT originating layers under uniform temperature rise. The properties of the nanocomposite composed of CNT-reinforced polymethyl methacrylate (PMMA) are given in Table 2 (see, Shen [51]). Table 2. Properties of nanocomposites and efficiency parameters in the layers.

Thermo-Mechanical Properties of PMMA in the Layer
The Efficiency Parameters in the Layer The geometrical properties of CNT a cn = 9.26 nm, r cn = 0.68 nm, h cn = 0.067 nm, ν cn Sθ = 0.175 The transverse shear stress functions are defined as: Par-TSS functions, or f [5,51]. The following definition applies here: f The uniform buckling temperatures of laminated nanocomposite truncated conical shells within ST and CT are found by minimizing Equations (24) and (27) versus m, n, and p. The lowest values of buckling temperature for laminated nanocomposite cones within ST and CT are achieved at approximately p = 2.1, and the number of longitudinal waves is equal to one for all cases. The cross-section types of laminated conical shells, as well as the cross section of the (0 • )-single-layer conical shells patterned by CNTs, which are used in the comparison, are shown in Figure 3. In this subsection, the percentages are obtained from the following expressions: types of laminated conical shells, as well as the cross section of the (0°)-single-layer conical shells patterned by CNTs, which are used in the comparison, are shown in Figure 3. In this subsection, the percentages are obtained from the following expressions: 100%, 100%, 100%      (Table 3). When comparing all laminated and (0 • )-single-layer conical shells for the U-TSS function, the biggest differences between UBT values are found in the U-, V-, and O-shaped (0 • /90 • /0 • /90 • )-array conical shell when γ = 10 • (which are the values of 44.93%, 30.81%, and 83.36%). While in the X-pattern, it occurs when the (90 • /0 • /90 • )-array conical shell obtains a value of (−30.84%) at γ = 30 • ( Table 4). As can be seen from Tables 3 and 4, the influence of the arrangement and number of layers on the buckling temperature is reduced when using U-TSS compared to Par-TSS (or Cos-Hyp-TSS) functions in U-, V-and O-shaped conical shells. This effect is more pronounced in nanocomposite conical shells with an X-shaped pattern.

T UBT /10 3 (n cr )for Par-and Cos-Hyp-TSS Functions
The variations of the magnitudes of UBT for (0 • )-single-layer and laminated truncated conical shells composed of U-, V-, O-and X -originating layers within ST and CT (versus r 1 /h) are tabulated in Table 5 and

Conclusions
The thermal buckling of laminated truncated conical shells composed of CNT originating layers within STs is studied. The modified Donnell type shell theory is applied to derive the basic equations, and then the Galerkin method is applied to the basic equations to find a new expression for the UBT of laminated truncated conical shells composed of CNT originating layers (within ST and CT). Four types of single-walled carbon nanotube distributions across the thickness of the layers are considered (namely uniform and functionally graded). The Par-, Cos-Hyp-and U-transverse shear stress functions are used in the analysis. The influences of change in CNT models, and the arrangement and number of the layers on the UBT using different shear stress functions, are examined.
Numerical analyses revealed the following generalizations:   Figure 4). Also, the use of laminated conical shells reduces the effects of shear stresses on the UBT compared to (0 • )-single-layer shells. Concerning the laminated cones starting with the array starting 0 • , the effect of shear stresses on the buckling temperature shows a significant decrease in laminated cones starting with the array starting 90 • . For example: while the value of effect of shear stresses on the magnitudes of UBT is 33.84% in the (0 • /90 • /0 • /90 • )-array shells consisting of V-shaped layers, it is a value of 15.57% in the (90 • /0 • /0 • /90 • )-array shells ( Figure 5). Depending on the increase in r 1 /h ratio, the effect of arrangement and number of layers on the UBT shows significant changes compared to the (0)-single-layer shell. The most change occurs in the UBT of the (0 • /90 • /0 • /90 • )-array conical shell consisting of X-shaped layers when compared to the (0 • )-single-layer conical shell. For instance, the difference between buckling temperatures is 22.31% when r 1 /h = 20, while this difference is (−2.39%) when r 1 /h = 35 ( Figure 6).

Conclusions
The thermal buckling of laminated truncated conical shells composed of CNT originating layers within STs is studied. The modified Donnell type shell theory is applied to derive the basic equations, and then the Galerkin method is applied to the basic equations to find a new expression for the UBT of laminated truncated conical shells composed of CNT originating layers (within ST and CT). Four types of single-walled carbon nanotube distributions across the thickness of the layers are considered (namely uniform and functionally graded). The Par-, Cos-Hyp-and U-transverse shear stress functions are used in the analysis. The influences of change in CNT models, and the arrangement and number of the layers on the UBT using different shear stress functions, are examined.
Numerical analyses revealed the following generalizations: Since laminated heterogeneous nanocomposite conical shells, reinforced with carbon nanotubes with robust heat resistance and high strength, are frequently used in modern aerospace and rocket technology, shipbuilding, energy and chemical engineering, and other fields exposed to very high temperatures, the results obtained in this research on their thermal buckling behavior should be considered during design.

Conflicts of Interest:
The authors declare no potential conflicts of interest with respect to the research, authorship and publication of this article.