Buckling Behavior of SWCNTs and MWCNTs Resting on Elastic Foundations Using an Optimization Technique

This paper aims to investigate the buckling behavior of multi-walled carbon nanotubes (MWCNTs) and single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium using the nonlocal cylindrical shell theory. The SWCNT is treated as a cylindrical shell and the MWCNT is considered as multiple SWCNTs nested inside one another; they interact with each other via van der Waals interactions. The interaction between the matrix and the outer wall is modeled as a foundation using Winkler, Pasternak, and Kerr models. An optimization technique is developed to estimate the nonlocal critical buckling load of SWCNT and MWCNT. Furthermore, analytical formulas are proposed to describe the buckling behavior of SWCNTs embedded in an elastic medium without taking into account the effect of the nonlocal parameter. In the proposed formulas, van der Waals interactions between adjacent tubes and the effect of terms involving tube radii differences are taken into account, although they are generally neglected in expressions published in the literature. The effects of the number of layers, the nonlocal parameter, and the elastic foundation parameters are investigated. Moreover, the effects of different parameters on the stability behavior of the carbon nanotubes are also discussed.


INTRODUCTION
Carbon nanotubes (CNTs) were discovered by Iijima [1]. Since that time, many scientific research activities have been carried out on CNTs. In these research works, the study of the mechanical behavior of CNTs is based on two main types of modeling: atomistic modeling [2][3][4] and continuum mechanics modeling [5][6][7]. However, atomistic modeling is very expensive, which limits the computational capacity. To give an idea on the problem of the atomistic modeling in molecular dynamics (MD) simulation [8], we refer readers to the work of Liew et al. [9] which shows that the calculation of buckling behavior for SWCNTs needs 36 hours using a single CPU SGI origin 2000 system and 2000 atoms. In addition, the computation time increases considerably with increasing number of atoms or number of layers in a MWCNT. The analysis of a SWCNT and MWCNT was conducted by several continuum models [10][11][12][13][14][15][16]. One of the most important studies is the excel-lent mechanical resilience of CNTs embedded in a matrix. The researchers continue to study this kind of problems until now, as evidenced by works [5,7,[17][18][19][20][21][22][23][24][25][26][27][28]. In the following, we present some examples of scientific research for each foundation model. For the Winkler foundation model based on a single parameter, the stability analysis of cantilevered curved microtubules in axons was presented by Shariati et al. [29]; the impacts of covering MAP Tau proteins were taken into account using the Winkler elastic medium. Chaabane et al. [30] studied the mechanical behavior of functionally graded material (FGM) beams using a model based on the hyperbolic shear deformation theory. The authors also discussed the effect of several parameters such as the Winkler spring constant, fundamental frequency, normal and shear stresses.
Here are some other recent research papers [13,31,32] that study the effect of the Winkler elastic medium on CNTs. Using the Pasternak foundation model based on two parameters, Chikr et al. [33] studied the buckling behavior of FGM sandwich plates with several boundary conditions using a refined trigonometric shear deformation theory. Refrafi et al. [34] analyzed the buckling and hygrothermal behavior of a FGM sandwich plate embedded in an elastic foundation. Bousahla et al. [35] studied the buckling and vibration of composite beams reinforced by SWCNTs and embedded in an elastic foundation. Bellal et al. [36] studied the buckling behavior of a single layer graphene sheet seated on a visco-Pasternak medium. Tounsi et al. [37] analyzed the static effects of nonlinear hygrothermo-mechanical loading on advanced FGM ceramic-metal plates. Boukhlif et al. [38] presented a dynamic investigation of FGM plates seated on an elastic foundation. Some works focus on the study of structures resting on viscoelastic foundations [39][40][41]. Boulefrakh et al. [39] studied the FGM plates embedded in visco-Pasternak foundations using a quasi three-dimensional (quasi 3D) model of hyperbolic shear deformation. Malikan et al. [40] [42,43]. For the Kerr foundation model based on three parameters, Kaddari et al. [44] discussed the mechanical behavior of functionally graded porous plates embedded in elastic foundations. Timesli [45] analyzed the buckling of double-walled carbon nanotubes (DWCNTs) seated on Winkler, Pasternak, and Kerr elastic foundations using the nonlocal Donnell shell theory. Addou et al. [46] investigated the dynamic behavior of FGM plates with porosities using different elastic foundation models.
There are several applications of CNTs in nanocomposite structures. Arshid et al. [47] analyzed the vibration of FGM microplates with porosities embedded in polymeric nanocomposite patches using an innovative plate theory to take into account the hygrothermal effect. Bendenia et al. [48] presented studies on defections, stresses and free vibration of FGM reinforced by CNTs and embedded in an elastic foundation of a Pasternak-type model. Al-Furjan et al. [49] performed frequency simulations of viscoelastic multi-phase reinforced fully symmetric systems resting on an elastic foundation. Bourada et al. [50] analyzed the dynamics and stability of a SWCNT-reinforced concrete beam on an elastic foundation using an integral first-order shear deformation beam theory. Al-Furjan et al. [51] investigated the characteristics of propagated waves in a sandwich doubly curved nanocomposite panel and presented a parametric study of the effects of the CNT weight fraction. Timesli [6] analyzed the stability of a concrete cylindrical shell reinforced by SWCNTs and embedded in an elastic foundation using the Donnell cylindrical shell theory.
A continuum model of a CNT based on the Donnell shell theory [6,52,53] was chosen in this work to describe the mechanical behavior of CNTs. The use of this type of model requires the use of equivalent parameters of Young's modulus and shell thickness. To determine these equivalent parameters of a CNT, molecular dynamic simulations and a continuum mechanics shell model can be used as indicated in the literature [54]. The choice of the Donnell theory [52] of circular cylindrical shells is based on several considerations: (i) it is based on the simplifying shallow-shell hypothesis, (ii) it is practically accurate and relatively simple and that is why it has been widely used, (iii) the introduction of a stress function is the most used form of the Donnell shell theory, this reduces the problem to two equations involving only the radial displacement w and the stress function. This theory gives accurate results when we take into consideration the following hypotheses: large aspect ratio of length/radius ≥ 10, (thickness/radius) 2 << 1 and 1/n 2 << 1, where n is the circumferential half wavenumber.
In the present paper, the relations of the classical Donnell shell theory [6,52] are used for the buckling analysis of SWCNT and MWCNT embedded in an elastic foundation taking into account the small-scale effects. The Winkler, Pasternak, and Kerr foundations are used to determine the effect of the matrix on the outermost tube. New explicit analytical formulas are developed for the local critical buckling load of embedded SWCNTs resting on the Winkler, Pasternak, and Kerr foundations. Moreover, an optimization technique is used to study the nonlocal critical buckling load of the SWCNT and MWCNT using the Winkler, Pasternak and Kerr models for the surrounding elastic medium, where we take into account the effects of the internal small length scale and the van der Waals interactions. The results of MD simulation [9] without the effect of the small length scale are used to validate the proposed MWCNT model. One of the developments of this research in the study of the buckling behavior of a MWCNT embedded within an elastic foundation, compared with the open literature, is the use of the Kerr foundation. In this context, my previous work [45] on DWCNTs em-bedded in an elastic foundation is generalized for a MWCNT and a SWCNT.

CONTINUUM MODELING OF A CNT EMBEDDED IN AN ELASTIC MEDIUM BASED ON CYLINDRICAL SHELL THEORY
Consider a thin cylindrical shell of wall thickness h, radius R, and length L. This elastic structure is considered as a homogeneous isotopic material with the following properties: Young's modulus E and Poisson's ratio . The CNT, represented by the thin cylindrical shell, is embedded in an elastic foundation. Using the thin cylindrical shell and the foundation model [6,45,55,56], the equilibrium equation is given as follows: where N x and N θ are the normal forces, N xθ is the internal shear force, M x and M θ are the bending moments, M xθ is the twisting moment, w is the transverse displacement of the middle surface, p is the external axial pressure, and f is the contact pressure between the CNT and the elastic foundation, which can be written by the following relations [57][58][59]: where K W is the lower spring modulus, K G is the intermediate shear layer modulus, and K C is the upper spring modulus; these last parameters are also called the Winkler modulus, Pasternak modulus, and Kerr modulus, respectively. The nonlocal elasticity theory developed by Eringen [60,61] is used to obtain a nonlocal Hooke's law which takes into account the small scale effect in the buckling behavior, as shown in [6]. This theory was successfully used in several works, such as [6,45,[62][63][64][65][66][67]. On the other hand, we can relate the membrane forces with the stress function Φ as follows: , .
  The possible existence of adjacent equilibrium configurations is studied using the adjacent equilibrium criterion [68]. Assuming that the pre-buckling and post-buckling quantities are represented by the indices 0 and b respectively and the second-order terms in the index b are neglected, we obtain the following equation: ) is the bending stiffness of the shell, e 0 is the nonlocal parameter, and ρ = 1/R is the curvature. The compatibility condition satisfied by the stress function ( , ) x   is given in [45,56]: If we neglect the shear membrane forces N xθ0 = 0, the circumferential membrane force is N θ0 = F and the axial compression is N xθ0 = P. System (2), (3) is written in the following form: (1 ( ) ) 0, ( ) where λ = P/(Eh) is the load parameter (or the buckling load) of CNTs.

A CONTINUUM APPROACH OF MULTIPLE SHELLS FOR A MWCNT
Consider N tubes of radius R 1 , R 2 , ..., and R N . The tubes have the same geometric and material characteristics: length L, thickness h, Young's modulus E, and Poisson's ratio ν. Each tube is modeled as an individual cylindrical shell. MWCNTs are represented by these tubes interacting through the van der Waals forces. A schematic of a MWCNT embedded in an elastic medium is shown in Fig. 1. The transverse displacement w j (x, θ) and the corresponding stress functions ( , ) j x   are solutions of the following equilibrium equations: where μ = e 0 a, j = 1, 2, ..., N, ρ j = 1/R j represents the curvature radius, 2 is the bi-Laplacian operator, j  represents the stress function, F j are the forces by the length unit in the circumferential direction, and c jk represents the van der Waals coefficients.

Critical Buckling Load of a MWCNT Embedded in an Elastic Medium
We can assume that the solution of problem (11) is written in the following form [45,56,69]: 1, 2, ..., , j N  where A j and a j express arbitrary complex constants, n and m represent respectively the circumferential and axial half wavenumbers. In this paper, it is interesting to consider DWCNTs consisting of N tubes. After the substitution of solution (7) in system (6), the first equation in this system gives: The second equation in system (7) gives: As shown in Eq. (10), we can calculate the constants a j : With the expression of constants a j in Eq. (11), system (8), (9) can be written as follows: where β j = q j /p expresses the aspect ratios with β j+1 = R j /R j+1 β j and η j = (1 + μ 2 p 2 (1 + β j 2 )). System (12), (13) leads to the following homogeneous matrix system: The  (15) with δ ji being the Kronecker symbol.
The coupled system of N equations (14) and N unknowns (A 1 , A 2 , ..., A N ) has a nonzero solution if its determinant is zero: Solving Eq. (16) gives the buckling loads λ for fixed values of aspect ratios β 1 , β 2 , ..., and β N . The critical buckling load designated by λ cr represents the small-est value; it is given as a function of the critical axial wave number p cr . The number p cr can be computed through the numerical minimization of λ with respect to p. In this work, we use the proposed algorithm as shown in Fig. 2, which gives the value of p cr quickly and automatically. The principle of this algorithm is explained in detail elsewhere [45,56,[70][71][72].
At the beginning this algorithm shows the decrease of the relative error (error (p i n ) = |(λ cr (n-1)λ cr n )/λ cr (n-1) |) with the increase of the parameter p. When p continues to increase, the algorithm calculates an incorrect result, which leads to a larger relative error.
The implementation of the proposed algorithm in Fig. 2 is based on the following routine:  where d denotes the accuracy of p cr to the mth decimal place, ε is related to the starting point of the test scope and gets a value between 0 and 1.

Critical Buckling Load of a SWCNT Embedded in an Elastic Medium
The nonlocal critical buckling load for a SWCNT is simplified as follows: For the local buckling (e 0 = 0) of a SWCNT embedded in the Kerr elastic medium under axial compression, the local critical buckling load is For a fixed aspect ratio β, we obtain the critical buckling load λ cr by minimizing the buckling load λ(β, p) with respect to the axial wave number p: Equation (20) leads to the following polynomial of degree 6 in p: We substitute Λ = p 2 , so Eq. (21) can be written as a polynomial of degree 3 in Λ: (1 ) , We divide Eq. (22) by b 1 and substitute Λ = Xb 2 /(3b 1 ) to obtain the following reduced form: The critical axial wave number p cr is therefore given by For an elastic foundation based on the Pasternak model, the local critical buckling load of SWCNTs under axial compression is written as follows: By minimizing the above expression with respect to the axial wave number p, we obtain the following polynomial of degree 4 in p: which allows us to calculate the critical axial wave number p cr as follows: As a result, we find explicit analytical formulas of the critical axial wave number p cr for the Winkler, Pasternak, and Kerr media. Subsequently, the critical buckling load λ cr of SWCNTs is determined according to p cr as follows: λ cr = λ (p = p cr ).

Validation of Explicit Analytical Formulas for the Local Critical Buckling Load of a SWCNT
This numerical analysis is performed with the following data: h = 0.066 nm, a = 0.142 nm, E = 5500 GPa, ν = 0.34, β = 0.6, e 0 = 0, and different values of elastic foundation parameters K W (nN/nm 3 ), K G (nN/nm 3 ), and K C (nN/nm 3 ) [4,8,9]. In the numerical analysis, the critical axial wave numbers p cr of a SWCNT embedded in the Winkler, Pasternak, and Kerr media are obtained by the proposed analytical formulas (26) and (29). The critical values of p cr for the fixed aspect ratio β allow us to compute the local critical buckling loads as follows: λ cr = λ(p = p cr ) using Eq. (19). For the numerical solution, λ cr is obtained by the minimization procedure as shown in algorithm (17).
In Fig. 3a, the critical wave number p cr in the axial direction versus radius R is plotted for the absence and presence of the Winkler elastic medium. The value of p cr decreases with increasing radius R. It increases in the presence of the Winkler elastic me-dium compared to that without the medium, and it is not affected by the Pasternak modulus K G as shown in Eq. (29). Figure 3b presents the critical buckling load λ cr of the SWCNT versus radius R. The critical load decreases with increasing radius R. It increases in the presence of the Winkler and Pasternak media compared to that without the medium. These results also show that λ cr of the Winkler model is less important than that of the Pasternak. Note that there is an agreement between the numerical results obtained by minimization and the analytical ones.
The critical wave number p cr of a SWCNT embedded in the Kerr elastic foundation is given by Eq. (26). Figures 4a and 4b present the variations of p cr and λ cr with respect to radius R. λ cr decreases with increasing radius R and increases in the presence of the Kerr elastic medium compared to that without the medium.

Study of the Nonlocal Effect on the Critical Buckling Load of a SWCNT Embedded in an Elastic Medium
In the following numerical tests, we study the nonlocal effect using Eq. (19). To show the smallscale effects on the critical buckling load λ cr of   The radius R = 0.34 nm is retained in the rest of the paper. We notice that λ cr of SWCNTs increases with increasing aspect ratio β for the three types of the elastic medium as shown in Fig. 7.
To illustrate the influence of the elastic matrix on the critical buckling load λ cr , the values of λ cr versus elastic foundation parameters K W , K G and K C are plotted in Fig. 8. It is very clear from Fig. 8a that λ cr rises most quickly with the increase in K W for the Winkler (K G = 0 and K C = 0) and Pasternak (K C = 0) elastic media. The increase of λ cr becomes gradually slower when the value of K W exceeds a value close to 10 4 . A reverse phenomenon is observed for the Kerr elastic medium where λ cr descends most quickly with increasing K W . The decrease of λ cr becomes gradually slower when the value of K W exceeds a value close to 10 3 . For the variation of λ cr versus shear modulus K G as shown in Fig. 8b, λ cr increases with increasing K G for the Pasternak and Kerr elastic media, but it rises  most quickly for the Pasternak medium. In the case of λ cr versus Kerr modulus K C as shown in Fig. 8c, λ cr rises most quickly with increasing K C , and the increase of λ cr becomes gradually slower when the value of K C exceeds a value close to 2 × 10 3 .

Numerical Analysis for the Critical Buckling Load of MWCNT Embedded in an Elastic Medium
We consider a MWCNT embedded in an elastic medium with the same data from [4,8,9]. The innermost radius is R 1 = 0.34 nm and the outermost radius is R N . The MWCNT is subjected to the combined action of van der Waals interaction and axial compression. The initial interlayer spacing between the two adjacent layers is assumed to be δR = 0.34 nm. In these numerical examples, the aspect ratios of length to radius L/R N = 10, h = 0.066 nm, a = 0.142 nm, E = 5500 GPa, ν = 0.34, β = 0.6 and van der Waals coefficients c jk are given by 12 where m jk E is an elliptic integral,  is the depth of the Lennard-Jones potential with = 2.968 meV, and  is a parameter determined by the equilibrium distance with  = 0.3407 nm.
To validate the proposed model, we compare its results with the available literature data. We consider a MWCNT without an elastic medium and without the small scale effect (e 0 = 0). The critical buckling loads λ cr of the MWCNT are computed by the present model as shown in Fig. 9. The results of the present model are compared with those obtained by Liew et al. [9]. We also compare the present model results with the molecular dynamics simulation results presented by Liew et al. [8] obtained for double-, tripleand four-walled CNTs. From Fig. 9, we clearly observe a good agreement with the literature data [8,9].
The proposed model also takes into account the internal small scale effect as shown by the following tests. Figure 10 shows that an increase in the number of layers N leads to a decrease in λ cr . It can be seen that the difference between the critical buckling loads of different values of e 0 is quite large for small values of N (or small outermost radius R N ), which indicates that the small scale effect plays an important role for the critical buckling loads when the number of layers is small (or the outermost radius R N is small). We can also conclude that as the outermost radius increases, the difference between these results becomes very small and their values tend towards the local critical buckling loads. This implies that the effect of the nonlocal parameter is very small for MWCNTs with large radii and it can be neglected.

CONCLUSIONS
An efficient methodology based on the continuum cylindrical shell theory was presented for the buckling analysis of SWCNTs and MWCNTs embedded  in an elastic foundation. To treat the interaction between the CNT and the elastic medium, the medium is modeled as the Winkler, Pasternak and Kerr elastic models. In the case of a SWCNT embedded in an elastic medium, analytical formulas of the local critical buckling load are developed. In the case of the nonlocal buckling analysis of SWCNTs and MWCNTs resting on elastic foundations, an optimization technique is developed to obtain the nonlocal critical buckling load. Several numerical studies were conducted to show the effects of many parameters such as the nonlocal parameter and the elastic foundation parameters. The following results were obtained: -The critical buckling load λ cr decreases with the increase in the number of MWCNT layers for a fixed innermost radius.
-The effect of the nonlocal parameter is very small for MWCNTs with large radii and it can be neglected.
-The effect of the nonlocal parameter is also very small for SWCNTs with large radii, in particular in the case of the Kerr elastic medium.
-The critical buckling load λ cr increases with increasing aspect ratio β for the three types of elastic media.
-The parameters of the Winkler, Pasternak and Kerr foundations increase λ cr of SWCNTs and MWCNTs.
-SWCNT and MWCNT resting on the Pasternak foundation have the highest critical buckling load values.
-The effect of the Winkler foundation parameter is less significant than that of the Pasternak foundation parameter.
-The critical buckling load λ cr increases with increasing elastic foundation parameters except in the case of λ cr versus Winkler modulus for a CNT embedded in the Kerr medium, which leads to a decrease in λ cr with increasing Winkler modulus.
-The increase in the number of layers increases the rigidity of a MWCNT, and a MWCNT without a medium is more susceptible to axial buckling than that with a medium.
The analysis of the buckling response performed in this paper will be extended by considering a cylindrical shell model for nonlocal buckling behavior of a CNT embedded in an elastic foundation under coupled effects of the magnetic field, the temperature change, and the number of walls.

CONFLICTS OF INTEREST
The author declares to have no conflicts of interest.