Large deflection analysis of nanocomposite cylindrical shell subjected to internal pressure


 In this work, large deflection behavior of a functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shell under internal pressure is studied. The composite cylindrical shell reinforced along the longitudinal direction and made from a polymeric matrix. Based on first-order shear deformation shell theory (FSDT) and von Kármán geometrical nonlinearity, the set of governing equations are derived. Using the dynamic relaxation (DR) technique, these systems of equations are solved for various boundary conditions and the roles of volume fraction of CNTs, CNTs distributions and geometrical ratios are examined on the responses.

Functionally graded (FG) materials can be utilized to tackle the problems of laminated composites' delamination since the initial advent of it in the 1980s (by Japanese researchers). In other words, these continuous spatial gradient materials have been tailored to amend composite defects [3]. FG-CNTRC can be named a novel sub-branch of the composite which is fabricated and brought into play in many engineering performances in recent years. These innovative materials are the outcome of reinforcing the nanofibers with carbon in the ultra-small scale, distributed smoothly and continuously in a polymeric matrix [4]. Shen [5] can be called one of the first researcher who analyzed the mechanical features of FGCNTRC then published an article in 2009 which examined the large deflection of FG-CNTRC in thermal environment. The static examination of an ultrasmall size composite beam strengthen by a SWNT via Airy stress-function approach was studied by Vodenitcharova and Zhang [6] both experimentally and theoretically. In another work, Shen [7] perused post-buckling of reinforced nanocomposite cylindrical shells exposed to axial compression in thermal surroundings. Seidel and his colleague Lagoud [8] scrutinized micromechanical elastic properties of reinforced structures by CNTs for recognizing the mechanical properties of nanotubes. With the aid of molecular dynamic simulation, Zhang and Shen [9] carried out a research on CNTs mechanical properties including temperature-dependent properties. Vibrational of FG carbon nanotube-reinforced cylindrical panels via Eshelby-Mori-Tanaka technique and quadrature method was investigated in Ref. [10]. The researchers of Ref. [11] investigated deflection analyzes of FG-CNTRC rectangular shell with simply supported edges exposed to thermo-mechanical loads on the basis of the 3D theory of elasticity. Moradi et al. [12] conduct a research on the dynamic investigation of the reinforced nanocomposite cylinders subjected to an impact load with the aid of a mesh-free technique. The authors of Ref. [13] studied both statistic and free vibration analyzes of single-walled carbon nanotubes composite shells considering the variable thickness on the basis of FSDT via the finite element procedure. Kwon et al. [14] can be called pioneers of fabricating FG-CNTRC. Later, many researches were conducted about the static and dynamic examination of FG-CNTRC, which in the following some of them are discussed. In Ref. [15] FG-CNTRC plates were studied analytically and in Ref. [16] the nonlinear vibration response of FG-CNTRC beams were implemented. Shen et al. [17] carried out the nonlinear dynamic investigation of FG-CNTRC rectangular plates exposed to blast loads using Reddy's higher-order shear deformation theory (HSDT) by means of the weak form quadrature element approach. Jiao and his coworkers [18] analyzed the buckling of thin rectangular FG-CNTRC plate subjected to compression loads by means of the differential quadrature technique (DQM) with the work equivalent technique. Ansari et al. [19] perused the free vibration of diverse forms of FG-CNTRC plates using HSDT with the aid of the generalized differential quadrature method (GDQM). The modified FSDT was implemented by Mellouli [20] to conduct free vibration of FG-CNTRC shell by means of a method named mesh free radial point interpolation.
Civaleka and Jalaeib [21] scrutinized the shear buckling of the FG-CNTRC asymmetric plate with various boundary conditions. Large deflection analysis of the FG-CNTRC annular variable thickness plate on the Pasternak elastic foundation was implemented by Keleshteri and his colleagues [22] using the third-order shear deformation model. Zhong and his assistants [23] studied the vibration of FG-CNTRC circular/annular as well as sector plates utilizing FSDT with the aid of the semi-analytical method. The dynamic response of aero-thermoelastic FG-CNTRC panels was investigated in Ref. [24]. Hajlaoui and Chebbi [25] analyzed the buckling of FG-CNTRC shells on the basis of modified first-order enhanced solid-shell element formulation.
Nguyena et al. [26] perused the nonlinear postbuckling of FG-CNTRC shells based on nonuniform rational B-Spline basis functions and FSDT by means of modified Riks numerical operators. The nonlinear buckling of FG-CNTRC cylindrical shells was studied in Ref. [27]. Zhou and Song [28] investigated the nonlinear deflection analysis of FG-CNTRC plates based on the Strain-Rotation (S-R) decomposition using the 3-D element-free Galerkin method. Soni et al. [29] investigated the bending of FG-CNTRC plates on the basis of inverse hyperbolicshear deformation principle by means of Navier method and they inferred that the analysis of FG-CNTRC depends markedly on some items including loading conditions, the volume fraction of CNT, sort of distribution of CNT and span-thickness ratio. Golmakani and Zeighami [30] perused the nonlinear thermo-elastic of the static examination of FG-CNTRC plates on elastic foundations with the aid of FSDT and by means of dynamic relaxation method.
According to this literature and available contexts, the large deflection of nanocomposite cylindrical shell reinforced by carbon nanotubes in the longitudinal direction under internal pressure has not been published on the basis of FSDT via DR technique. The material properties of FG-CNTRC shells are assumed to vary smoothly along thickness and gained through a micromechanical model. Eventually, the roles of effective factors including volume fraction of CNTs, boundary conditions, thickness-to-radius and length-to-radius ratios and CNTs distributions on the responses are examined.

2-Mathematical formulation and theoretical frameworks
In this part, the theoretical frameworks on the properties of carbon nanotube materials are discussed and the equations that set their distribution are expressed initially. Then, the governing equations based on axial symmetry and strain-displacement are expressed.
The distributions of the nanotubes in the longitudinal direction of the cylindrical shell with uniform direction (UD) and functional gradients including FG-O, FG-X and FG-V are illustrated in Fig 1. More specifically, in the longitudinal direction, 1 and 2 signify that the fibers are in (longitudinal axis) and (or θ) directions, respectively.
Utilizing principle of minimum energy, the total potential energy is the total of strain energy and the potential energy from external loads. Hence, the equation (11) is applied to gain the equations of equilibrium: where on the basis of the principle of minimum energy, the variations of Eq. (11) are as follows: By inserting Eq. (5) in Eq. (12), the Eq. (13) is gained: Hence, the governing equations of the cylindrical shell under internal pressure are extracted by the aid of the principle of minimum energy. The equations of equilibrium can be written on the basis of FSDT and utilizing Eq. (7), as [13,33]: The governing relations would be written in terms of the displacement as the set of equation (15) A set of boundary conditions should be applied in the above equations to obtain the results. In this article, the boundary conditions are as follows: (a) SS boundary condition: (b) CC boundary condition: (c) SC boundary condition: (d) CS boundary condition:

Solution procedure
In this context, the DR technique with the central finite-difference procedure is utilized to solve nonlinear problems including the nonlinear differential relationships of the CNTRC shell [34][35][36].
DR is an iterative approach which its basic aim is to gain a steady-state solution with the aid of converting the static problem into a dynamic one [37][38][39]. For solving the shell equations by means of this technique, damping and fictitious inertia terms are added to the right sides of the shell equations of equilibrium. Therefore, they would convert from the boundary value problem to an initial value format. Subsequently, the equations of equilibrium can be expressed as: (20) where in the equation (20)  at node i in the th n iteration is to apply the Gershgörin principle. By means of this theory, the equation (21) needs to be satisfied in order to assure the convergence of iterations [38]: In equation (21), the symbols n and  signify the th n iteration step and an increment of fictitious time, respectively. The element ij k of the stiffness matrix can be obtained by computing equation (28) as follows: where ,, expresses a rough solution vector. Furthermore, internal forces i P is the left side of the equilibrium relationships (Eqs. (15)) and for instance, it is expressed on the basis of stress and moment resultants in Eq. (21). Here, the instant critical damping factor n i c for node i at the n th iteration can be achieved as follows [39]: Various values of c for diverse nodes in each direction can be considered for each element of diagonal fictitious damping matrices C ( N N  ) in a discrete system connected through N nodes to achieve the form used in the DR method as follows: Replacing the velocity and the acceleration based on Eq. (21) with the equivalent central finitedifference equations, the converting process is achieved. Therefore, the equations of equilibrium can be written into an initial value format as equation (25) (for more details refer to [39]): After integrating the velocities of each time step, the displacements can be computed by: Utilizing the above equations together with the certain boundary conditions in their finitedifference formats create the set of equations for the sequential DR technique [35].

4-Results
In this section, numeric outcomes of the bending examination of the FG-CNTRC cylindrical shell in the longitudinal direction exposed to internal pressure are discussed. The CNTRC is supposed to be consisted of the Poly methyl methacrylate (referred to as PMMA) with CNT which fibers aligned in the longitudinal direction of the shell.
In order to verify the accuracy of the numeric outcomes of the mechanical analysis, two examples are considered and comparative studies with similar articles (in this field) or Abaqus finite element software [40] are conducted. Also, parametric studies have been shown to examine effective items such as the distribution of carbon nanotubes, the thickness-to-radius, length -toradius, boundary conditions, and the volume fraction of CNTs. Due to the unavailability of a reliable published article in the present subject, the results were first compared with the same article for the isotropic state or with those gained from the Abaqus finite element software [40]. In the following, two examples of them are discussed.

Example 1:
As can be seen in figure 3, in order to verify the accuracy of the linear bending examination of the isotropic cylindrical shell, the results of the present study are obtained with those published by Ref. [41] in CC boundary condition. The research is concerned with FGM and the comparison is made for = 0 (which represents pure metal). It can be monitored that the maximum values achieved by the present procedure are roughly similar to those of Ref. [41].  The maximum radial dimensionless deflection results of the linear bending obtained by the DR method are compared with those of results gained by Abaqus finite element software [40] and gathered in tables 2. and 3. It can be deduced that the maximum value of non-dimension deflection obtained from the DR technique is almost similar to those of results obtained from Abaqus finite element software [40]. Also, it can be comprehended that the maximum values of the deflection increase in FG state under different loads are more than those of values with uniform distribution.
Furthermore, the maximum dimensionless deflections in the two states of FG-X and FG-V are very close to each other. Table 2. Comparison between the maximum dimensionless deflections obtained from linear analysis and Abaqus software [40] for CC boundary condition with / = 1.5 and * = 0.12.

UD FG-V FG-X
[40] Present study [40] Present study [        More specifically, that of difference in the CC boundary condition is less than SS condition. Also, it can be deduced that for both boundary conditions and both length-toradius ratios, the highest percentage difference between linear and nonlinear dimensionless deflection is related to FG-X distribution and the lowest that of difference is in FG-O mode. It should be noted that as the length-to-radius ratio increases, the difference of non-dimensional deflections between the two linear and nonlinear states decreases.   Table 7. Comparison between /ℎ obtained from linear and nonlinear examinations for CC boundary condition with V CN * = 0.17 and L/r = 1.5.     boundary condition. Moreover, in / > 3 , the rate of non-dimension maximum deflection growth will reduce significantly and will be less than 1% , for all distributions.  Table 9. Non-dimension maximum deflection in terms of increasing the ratio of length-to-radius for FG-X distribution (V CN * = 0.17, = 50 , ℎ = 1 ) and CC boundary condition.  Table 10. Non-dimension maximum deflection in terms of increasing the ratio of length-to-radius for FG-V distribution (V CN * = 0.17, = 50 , ℎ = 1 ) and CC boundary condition. 5 Figure 10 reveals the non-dimension stress resultant versus the dimensionless length for axialsymmetric shell with = 50 , = 100 , ℎ = 2 and under internal pressure = 10 at CC and SS boundary conditions. As can be seen, the change in the value of membrane force for uniform and FG distributions along the shell is roughly constant. Also, for both boundary conditions, the FG-X distribution will have the highest non-dimension stress resultant but the UD has the lowest value.    For all distributions, with the increase of non-dimensional thickness (ℎ/ ) from 0.03 to 0.04, the decrease rate of non-dimensional deflection is about 3 times more than the increase of ℎ/ from 0.04 to 0.05.
 As the length-to-radius ratio increases, the difference of non-dimensional deflections between the two linear and nonlinear analyzes decreases.
 By the increase of the length-to-radius ratio the maximum dimensionless deflection goes up.
 With increasing the ratio of length-to-radius and rising the volume fraction (for all four distributions and both SS and CC boundary conditions), the growth of maximum dimensionless deflection reduces by approximately 2.8 times.
 For both CC and SS boundary conditions, the FG-X and UD distributions have the highest and the lowest value of stress resultants, respectively.
 By increasing the ratio of length-to-radius for uniform, X, V and O distributions, the highest drop in non-dimensional maximum deflection is in the volume fraction of 0.12 with SS boundary condition, but the lowest that of drop is in the volume fraction of 0.28 with CC boundary condition.
 The change in the value of membrane force for uniform and FG distributions along the shell is roughly constant.
 In / > 3 , the rate of non-dimensional maximum deflection growth will reduce significantly and will be less than 1% , for all distributions.
 The highest maximum dimensionless moment versus dimensionless length is in the FG-V distribution for both SS and CC boundary conditions.

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