Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations

Author’s Accepted Manuscript Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations Fahimeh Mehralian, Yaghoub Tadi Beni, Mehran Karimi Zeverdejani www.elsevier.com/locate/physb

PII: DOI: Reference:

S0921-4526(17)30358-7 http://dx.doi.org/10.1016/j.physb.2017.06.058 PHYSB310036

To appear in: Physica B: Physics of Condensed Matter Received date: 15 May 2017 Revised date: 21 June 2017 Accepted date: 21 June 2017 Cite this article as: Fahimeh Mehralian, Yaghoub Tadi Beni and Mehran Karimi Zeverdejani, Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.06.058 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations Fahimeh Mehralian1, Yaghoub Tadi Beni 2*, Mehran Karimi Zeverdejani 1

1

Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran Faculty of Engineering, Shahrekord University, Shahrekord, Iran * Tel/Fax: +98-38-32324438. [email protected] 2

Abstract: The present paper is concerned with the applicability of nonlocal strain gradient theory for axial buckling analysis of nanotubes. The first order shear deformation theory with the von Kármán geometrical nonlinearity is utilized to establish theoretical formulations. The governing equations and boundary conditions are derived using the minimum potential energy principle. As main purpose of this study, the small length scale parameters are calibrated for the axial buckling problem of carbon nanotubes (CNTs) using molecular dynamics (MDs) simulations. Further the influences of different geometrical and material parameters, such as length and thickness ratio as well as small length scale parameters on the buckling response of nanotubes are studied. It is indicated that the effect of small length scale parameters on the critical buckling load becomes more prominent by increasing thickness and decreasing length ratio. Moreover, the calibrated small length scale parameters presented herein would be useful for the purpose of applying the nonlocal strain gradient theory for the analysis of nanotubes. The calibrated nonlocal strain gradient theory presented herein should be useful for researchers who are using the nonlocal strain gradient shell theories for analysis of micro/nanotubes.

Keyword: Calibration, Molecular dynamics simulations, Nonlocal strain gradient theory, Shell model, Axial buckling. 1

1. Introduction Nowadays, nanotechnology has found tremendous amount of interest due to its fundamental role in different fields. Among nanostructures, carbon nanotubes (CNTs) as leading candidate which has been emerged for nanodevice applications, attracted the attention of a lot of researchers [1-5]. According to arrangement of carbon atoms, CNTs are divide to three types namely chiral, zigzag and armchair CNT [6,7]. Due to the great technical importance, the stability of carbon nanotubes is one of the important engineers concerns. Since CNTs have found a wide range of applications in nanostructures, their buckling analysis subjected to different type of loadings is of primary importance in designs of nanodevices [8]. In stability investigations, according to the type of loading and geometrical conditions, various behaviors have been observed, such as bifurcation, snap-through and limit load [9,10]. The buckling analysis of cantilever single-walled carbon nanotubes (SWCNTs) was carried out by Akgöz et al. [11]. Buckling of piezoelectric nanotubes subjected to lateral pressure was examined by Mehralian et al. and the influences of different parameters on buckling pressure were illustrated [12]. Bending, vibration and buckling were studied by Reddy and the influence of different parameters was presented on deflection, buckling load and natural frequency [13]. Accurately analytical approach, due to difficulty of experiments at nanoscale and time consuming of molecular dynamics simulation, attracted a lot of attention [14-20]. The challenge of developing more accurate model has been met by a lot of researchers [21-23]. Also, considerable interest has been devoted toward higher order continuum theories due to a major drawback in classical continuum theory, which are not capable to consider size effect [24-29]. Many researchers have studied the mechanical behavior of nano-systems using nonlocal theory [30,31]. By using size dependent nonlocal theory and Euler-Bernoulli beam model, Pradhan et al. investigated the effects of temperature gradient and boundary conditions on critical buckling load of CNTs [32]. Free vibration of SWCNTs was examined by Ansari et al. based on 2

nonlocal beam theories and the results obtained compared with those achieved by MD simulation [33]. Nonlinear vibration and stability of a fluid-conveying nanotube coupled with a smart piezoelectric polymeric beam was investigated on the basis of nonlocal theory by Atabakhshian et al. and the influences of different material and geometrical parameters were illustrated [34]. Based on nonlocal theory, in the particular point, the stress is assumed to be dependent on the strains at all of the body points in specific domain [35]. Moreover, in recent years, some studies are conducted on nano-systems based on strain gradient theory [36-38]. Using Euler-Bernoulli beam model, Akgöz et al. evaluated the buckling of isotropic CNTs based on couple stress theory which is a form of strain gradient theory [39]. Static bending of single walled carbon nanotubes embedded in an elastic medium was studied based on modified strain gradient theory by Akgöz et al. and the effect of various parameters on deflection were discussed [40].According to strain gradient theory, the strain energy is function of strain and strain gradients. Also, nonlocal theory has softening effect on structures; in contrast to, strain gradient theory which has stiffening influence [41,42]. Scientists have shown that the nonlocal theory cannot provide accurate results for some cases such as bending problem, and for this problem, the results obtained by nonlocal theory are identical to results of classical continuum theory [43,44]. Thus, in recent years, by combining the nonlocal elasticity and strain gradient theory, the nonlocal strain gradient theory (NSGT), proposed by Lim et al., considers higher order stress gradients and strain gradient nonlocality simultaneously. Also this theory consists of both nonlocal effects of the strain field and first gradient strain field [45]. Lately, many studies are carried out based on this theory. Li et al. examined buckling of various beam models using NSGT [46]. Simsek based on nonlocal strain gradient theory studied nonlinear vibration of functionally graded beams [47]. As well as, Li et al. investigated wave propagation of FG beams according to nonlocal strain gradient model [48]. Nonlinear vibration of CNTs conveying viscous nanoflow and resting on elastic foundation was examined by Mohammadimehr et al. based on the nonlocal higher order stress theory [49]. Free vibration of nanotubes was studied on the basis of nonlocal strain gradient theory by

3

Mehralian et al. and two small length scale parameters were calibrated by the means of MD simulation [50]. For the sake of predicting buckling behavior of nanotubes more accurately, molecular dynamics simulation (MDs) is strongly recommended [51,52]. Jakobson et al. studied buckling behavior of CNT under axial load using molecular dynamics simulation [53]. Buehler et al. investigated the impact of aspect ratio on the critical axial load of CNT using MD simulation [54]. Buckling of armchair and zigzag SWCNTs under axial compression, torsion and external pressure was studied by Zhang et al. based on MD simulation [55]. Torsional buckling of SWCNTs filled with hydrogen or silicon atoms was investigated by Wang et al. based on MD simulation [56]. Also, using MD, buckling of defective SWCNTs and DWCNTs under axial compressive loads was examined by Hao et al. [57]. Cylindrical shell is the most suitable geometrical model for simulation of carbon nanotube. This model according to thickness to radius ratio have been divided into three types, namely classical model, first order shear deformation model (FSDT), and higher order shear deformation model (HSDT) [58]. Scientists revealed that for cylindrical shell by low aspect ratio (L/R) size effect is significant [59]. In recent years, analysis of nanoshells has been one of the most popular topics in literature [60,61]. Based on nonlocal elasticity, torsional buckling of protein microtubule investigated by Shen using higher order shell model [62]. Sahmani et al. studied size dependent buckling of shells based on the first order shear deformation shell model using nonlinear model [63]. Mehralian et al. investigated buckling of size dependent nanoshell under combined loadings on the basis of first order shear deformation shell model [64]. Gholami et al. studied buckling of nanoshell, based on first order shear deformation shell model [65]. Motivated by this considerations, in the present study, for the first time, the stability response of cylindrical nanoshell is studied based on the nonlocal strain gradient theory using first order shear deformation theory with the von Kármán geometrical nonlinearity. The equilibrium equations and boundary conditions are derived based on the minimum potential energy principle. Afterward, using adjacent equilibrium criterion, the stability equations are obtained. In results section, first, the amount of 4

NSGT scale parameters are calibrated for particular case of CNT using MD simulation. Next, variation of critical axial buckling loads versus different geometrical and material parameters are explored. 2. Buckling analysis based on nonlocal strain gradient theory The equilibrium equations are obtained based on the minimum potential energy principle. Accordingly, variation of energy function (Π) must be set equal to zero. Energy function is defined as difference between strain energy (Us) and work done by external works (Wexw):

   U s  We  0

(1)

Based on the nonlocal strain gradient theory, proposed by Lim et al [45], the strain energy is function of strain εij, nonlocal stress σij, strain gradient εij,m and higher order nonlocal stress σ(1)ij components, as below:

Us 

1     (1)ij  ij  dV 2 V ij ij

(2)

Besides, the work done on nanotube by external force is obtained as follows:  p U   Rdxd 2 R x  x

We    

(3)

which p is axial compression. By substituting equations (2) and (3) into equation (1), following equation is obtained:     tij  ij dV    (1)ij  ij V

A

L 2  p  U  Rd dz    (1)ij  ij dxdz      Rdxd  0 0 0 2 R x  A x  

(4)

Based on the nonlocal strain gradient theory, the stress field, tij takes into account not only the nonlocal stress field but also the strain gradient stress field, also that is [45]:

tij   ij   (1)ij

(5)

According to nonlocal strain gradient theory, the constitutive equation is defined as [45]: 1   e a 2 2  t  C   l 2C 2 i jkl kl i jkl 0 kl   ij

(6)

5

which (e0a) is the nonlocal parameter and (l) is the material length scale parameter. Also, Cijkl is the elasticity tensor and defined as follows:

Cijkl

E  E  0 0 0  1   2 1   2   E  E 0 0 0  1   2 1   2    E  0  0 0 0   2 1      E  0 0 0 0    2 1      E   0 0 0 0  2 1     

(7)

In the present work, nanotube is modeled as FSDT cylindrical shell. For FSDT cylindrical shell model components of displacement field are defined as:

u  x, , z   U  x,   z x  x,  v  x, , z   V  x,   z   x, 

(8)

w  x, , z   W  x,  where, U, V and W are middle surface displacements along axial, circumferential and radial directions, respectively. As well as, ψx and ψθ represent the rotation around θ and x axis, respectively (see Fig. 1). Furthermore, z corresponds to distance of each point from middle surface along radial direction.

6

Fig.1: FSDT cylindrical shell model. For FSDT cylindrical shell model based on von Kármán geometrical nonlinearity, components of strain tensor are obtained as follow:

 xx

 x U 1  W    z x 2  x  x



1  V 1  W  z      W   2    R   R   2 R   

2

2

1  V

1 U

W W

 1 

  

 x   x      z   2  x R   x    R  x   1 W

(9)

  z      2 R  R 

1

V

W

  xz   x  2 x 

1

Consequently, using the principle of minimum potential energy and variational approach, the equilibrium equations and boundary conditions of cylindrical nanoshell are obtained as:

U :

N xx 1 N x  0 x R 

(10-1)

V :

N x 1 N Qz   0 x R  R

(10-2)

W :

Qxz 1 Qz   W N x W    N xx  x R  x  x R 

 1    R  

W  N xx W  R   N x x 

7

 N  R  0 

(10-3)

 x :

M xx 1 M x   Qxz  0 x R 

(10-4)

  :

M x 1 M   Q z  0 x R 

(10-5)

The boundary conditions on x = 0, L are presented in Appendix A. Besides, the stress resultants are introduced as follows:  N xx , N (1) xx  h 2  t xx ,  (1) xx    (1)  (1)   N x , N x    t x ,  x dz  N , N (1)   h 2 t ,  (1)     

(11-1)

h

(1) 2  Qzx , Q (1) zx   t zx ,  zx   dz     (1) (1)  Qz , Q z   h 2 t z ,  z 

(11-2)

h

 M xx , M (1) xx  2  txx ,  (1) xx     zdz  (1)  (1)   M  , M    h 2 t ,   

(11-3)

Finally, by combining equations (7), (10) and (11), the equilibrium equations in terms of displacements are obtained as:  Eh  2U  U : 1  2   1   2 x 2  

 Eh Eh   2V Eh W Eh  2U       0  R 1   2  2 R 1     x R 1   2  x 2 R 2 1     2   

 W  Eh   2U  2V  Eh Eh Eh       k    R 2 1   2   2  R 2 1   2  2 R 2 1       R 1   2  2 R 1     x     

 V : 1   2  

(12-1)

Eh

Eh  2V Eh V 1   k    2 2 2 1    x 2 1     R R 

  x  2W Eh k  2 x  2 1     x

 W : 1   2  

   0  

  1   Eh 1  2W  Eh k  2   R  2  1   2  2 1     R 

  V Eh Eh Eh W    2 k 2  2  2 R 1       1   2 R 2  R 1        W N x W  1   2    N xx  x R   x 

W    1   N xx W   R   R   N x x    0   

8

(12-2)

 1 U     R x 

(12-3)



  2    2 x Eh3 Eh3 Eh3        2 2 2   24 R 1    12 R 1     x 24 R 1     Eh W     k  x    0 2 1     x  

  2 x  2 12 1   2   x

 x : 1   2  

Eh3

(12-4)



  2  2  Eh3 Eh3 Eh3  2  Eh3 x      24 R 1    12 R 1   2   x 24 1    x 2 12 R 2 1   2   2  

  : 1   2  

Eh 1 W V     k       0 2 1     R  R  

(12-5)

In bifurcation analysis, it is necessary to study prebuckling deformations first. However, the prebuckling equilibrium path of initially curved structures is more complicated than flat structures. By the way, the symmetrical case of equilibrium equations with boundary conditions should be solved to trace the primary path [66,67]. In linear membrane approach, as most simple case of prebuckling solution, the von Kármán nonlinear terms as well as bending moments and curvatures are ignored. Therefore, the axial compressive force of the shell subjected to axial load is obtained as: N xx 0 

3.

P , N 0  0, N x 0  0 2 R

(13)

Solution procedure

The critical buckling load is determined using the stability equations derived based on the adjacent equilibrium criterion. According to this criterion, which is based on the perturbation technique, the components of displacements on the primary equilibrium path are perturbed infinitesimally to establish an adjacent equilibrium position [66,67]. Assume that the equilibrium state of the cylindrical shell under load is defined in terms of U0, V0, W0, ψx0, ψθ0 and the displacement components of a neighboring state of stable equilibrium differ by U1, V1, W1, ψx1 and ψθ1 from the equilibrium state. Therefore, the displacement components of a neighboring state are as below:

9

U  U 0  U1 V  V0  V1 W  W0  W1

(14)

 x   x 0   x1      0   1 Upon substitution of Eq. (14) in Eq. (12), the terms in resulting equations with subscript 0 drop out due to satisfying equilibrium conditions and nonlinear terms of incremental values are ignored because they are small compared to linear terms. Also, the remaining terms form the stability equations are as follows:  Eh  2U  Eh  2U1  Eh   2V1 Eh W1 Eh 1    U : 1  2       0 1   2 x 2  R 1   2  2 R 1     x R 1   2  x 2 R 2 1     2     

 W  Eh   2U  2V1  Eh Eh Eh 1   2  k 1   2 2 2 2  R 1    2 R 1       R 1    2 R 1     x  R 1         

 V : 1   2  

(15-1)

Eh

2

2

V  Eh  2V1 Eh 1   k   1  12    0 2 1    x 2 2 1     R R 

(15-2)



  x1  2W1   1   1 1  2W1  Eh  1 U1  Eh Eh k   k  2    x 2  2 1     R  R  2  1   2  R x   2 1     x

 W : 1   2  

  V Eh Eh Eh W1   1  2 k 2 2 2  2  2 R 1    R 1      1   R  

(15-3)

  W1 N x 0 W1  1   N 0 W1 W1    1   2    N xx 0    R   R   N x x    0  x  x R       



  2   2 x1    2 x1 Eh3 Eh3 Eh3 1      2  2 2 2 12 1     x   24 R 1    12 R 1     x 24 R 1     W   Eh   k  x1  1    0 2 1     x  

 x : 1   2  

Eh3

2

(15-4)



  2  2  1 Eh3 Eh3 Eh3  2  1 Eh3 x1     2 2  24 R 1    12 R 1   2   x 24 1    x 12 R 2 1   2    

  : 1   2  

Eh 1 W1 V1     k   1   0 2 1     R  R  

And, the boundary conditions on x = 0, L are given in Appendix A.

10

(15-5)

In order to solve the stability equations of simply supported cylindrical nanoshell, the following approximate solutions, are utilized:  m x  U 1  x ,   U 1 cos   cos  n   L   m x  V 1  x ,  V 1 sin   sin  n   L 

 m x W 1  x ,  W 1 sin   L

  cos  n  

 m x   cos  n   L   m x    1  x,     1 sin   sin  n   L 

(16)

 x1  x,    x1 cos 

In the above equation, ̅ , ̅ , ̅ , ̅

and ̅

are constant coefficients. As well as, m and n represent

the longitudinal and circumferential mode numbers, respectively.

Thus, substituting approximate

solutions (16) into stability equations (Eq. (15)), yields:

 K 55 U1

V1 W1  x1   1  0 T

(17)

In order to find the smallest buckling load corresponding to axial and circumferential wave numbers, m and n, the determinant of coefficient matrix should be set equal to zero.

4. Buckling analysis based on molecular dynamics simulation In order to explore the buckling behavior of CNTs, molecular dynamics simulations (MDs) are carried out using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [68]. The interatomic pair potential between carbon atoms in CNT is described using the adaptive intermolecular reactive empirical bond order (AIREBO) potential [69]. AIREBO includes three sub-components, REBO, Lennard-Jones and torsional potentials and may be expressed as [69]: ∑∑[

∑ ∑

11

]

(18)

where in Eq. (18),

,

,

are the potential for pairwise covalent bonding interactions of

atoms based on the second generation reactive empirical bond order potentials of Brenner, the potential for the pairwise sum of Lennard - Jones interactions and the potential for the torsion interactions depending on the dihedral angles of the system, respectively. In the following, each CNT is partitioned into three regions depending on the boundary conditions [70]. The upper region is restrained at all directions such that only displacement in the z-direction is allowed (see Fig. 2). The middle region is free to evolve in time with no additional constraints and the lower region is held fixed. The upper region is 5 Å long. The length of the lower region changes in such a manner that the length of middle region is kept fixed for a particular length of LT to ensure uniformity. The simulation is conducted by adopting the so-called constant Number of atoms, Volume and Temperature ensemble (NVT ensemble). The Nose - Hoover thermostat is applied to middle part in the simulation so that the temperature is maintained at the desired ranges. The velocity-Verlet integrator algorithm is implemented to integrate the Newton equations of motion, with 1 fs as the time step for integration. A random velocity is adopted to reach the room temperature (300 K). The simulation commences by relaxing CNTs at room temperature (300 K) for 5 ps. The initial atomic structure of CNT is optimized by conjugate gradient algorithm with an energy convergence norm for energy minimization in order to remove the existing residual stresses of CNT. After the system reaches the equilibrium state, the axial compression is accomplished by prescribing a displacement of 0.01 Å along the axial direction to the atoms in upper region of CNT, in such a manner that in each displacement increment upper region remains circular and perpendicular to the deformed axis, as shown in Fig. 2. Afterward the structure is allowed to relax for 5 ps. Consequently, the critical buckling load and critical strain are determined by summing up the load in the z-direction for all atoms in middle region. Such displacement controlled loading is extensively utilized in literature to simulate the axial buckling of CNTs in MD.

12

Fig. 2: MD simulation setup.

5.

Results and discussions

5.1 MD calibration of μ and η With the purpose of applying the nonlocal strain gradient shell theory for the buckling analysis of nanotubes, it is necessary to provide the values of small length scale parameters. In this section, μ = (e0a)2 and η = l2 are calibrated using MD results, as a benchmark of good accuracy, for simply supported (10,10) armchair CNT as illustrated in Fig. 2. The critical buckling load of a (10,10) armchair CNT obtained by the present model, nonlocal model, strain gradient model and MD simulation are indicated for various length ratios in Fig. 3. As observed, there is more suitable agreement between the results of nonlocal strain gradient model with MD results in comparison to the nonlocal and strain gradient model. Moreover, from this figure one can see that the results are reasonable because the critical buckling load decreases with increasing length ratio. Notice that the following material and geometrical parameters are considered for calibrating quantitatively μ and η parameters [71]: E = 1.06 TPa, υ = 0.19, h = 0.34 nm, d = 1.356 nm.

13

According to surface elastic constants [6] and MD results, it can be seen that μ and η are not constant values but they are rather dependent on length ratio, such that the range of μ = (e0a)2 and η = l2 values are found to be (1)2 to (1.5)2 nm2 and (0.4)2 to (0.9)2 nm2, respectively, for different length ratios. In the following, the buckling analysis of nanotubes using the nonlocal strain gradient model, as a better substitution of MD simulation, is evaluated in the calibration range of small length scale parameters for various material and geometrical parameters. In what follows, the radius is taken to be 2 nm.

Fig. 3: Comparison between MD results and other theories.

5.2 Influence of small length scale parameters on the critical buckling load Fig. 4 plots the critical buckling load as a function of small length scale parameters. It is shown that the critical buckling load at a certain scale factor (η) decreases by increasing the nonlocal parameter (μ) which reveals the softening effect of nonlocal parameter. However, by increasing scale factor at a certain nonlocal parameter, the critical buckling load will increase and it means that the effective stiffness of nanotube becomes larger with increasing scale factor. These phenomena indicate that by using nonlocal

14

strain gradient theory, the nanotube exerts the softening and stiffening behavior by increasing the nonlocal parameter and scale factor, respectively. Further from this figure one can see that the influence of small length scale parameters is significant by increase in scale factor and decrease in nonlocal parameter. In order to see the effects of small length scale parameters for different geometrical parameter, Table 1 is presented. As is clear, the trends of the critical buckling load variation versus small length scale parameters for all thickness ratios are similar to Fig. 4. Moreover, from this table it can be seen that the influence of small length scale parameters diminishes as thickness ratio decreases.

Fig. 4: Effect of small length scale parameters on the critical buckling load (L = 10R, h = 0.34 nm).

Table 1: Variation of critical buckling load (nN) against small length scale parameters (nm2). μ

h/R=0.05 h/R=0.1 h/R=0.2 2 2 2 2 2 2 η = 0.5 η = 0.7 η = 0.9 η = 0.5 η = 0.7 η = 0.9 η = 0.5 η = 0.72 η = 0.92 23.144 27.676 33.717 154.667 184.947 209.903 372.225 395.186 425.801 17.154 20.512 24.99 114.636 137.078 167.002 324.044 344.033 370.685 14.176 16.951 20.652 94.736 113.282 138.011 293.239 311.327 335.446 2

12 1.32 1.52

15

5.3

Influence of length ratio on the critical buckling load

The critical buckling load versus length ratio for different scale factor is presented graphically in Fig. 5. The nonlocal parameter is considered to be (1)2 nm2. As depicted in this figure, the critical buckling load decreases by increasing length ratio. This effect diminishes as the scale factor decreases. In other words, the effects of length ratio on the critical buckling load with greater scale factor are more than those of ones with small scale factor. Besides, according to this figure, as the length ratio increases the critical buckling loads are not considerably affected by the scale factor. In order to see the effects of length ratio more clearly, the critical buckling loads of nanotubes with η = (0.9)2 nm2 against length ratio for several nonlocal parameters are graphed in Fig.6. As is clear, the trends of the critical buckling load variation versus length ratios for various nonlocal parameter are similar to Fig. 5. Such that at different nonlocal parameter, as length ratio increases the critical buckling load decreases. Moreover, from this figure it can be seen that the influence of nonlocal parameter diminishes as the length ratio increases. Also, according to Figs. 5 and 6, at high length ratio the results of the present model approach to those of classical ones which shows the capability of classical model to predict the buckling response of large-scale structures. Additionally, Figs. 7 and 8 are presented to find better insights on the influence of scale factor and nonlocal parameter on the critical buckling load. As the benefits of these figures, the variations of critical buckling load are illustrated continuously versus scale factor and nonlocal parameter, respectively.

16

Fig. 5: Influence of length ratio for different scale factor on the critical buckling load (μ = (1)2 nm2, h = 0.34 nm).

Fig. 6: Influence of length ratio for different nonlocal parameter on the critical buckling load (η = (0.9)2 nm2, h = 0.34 nm).

17

Fig. 7: Effect of length ratio versus scale factor on the critical buckling load (μ = (1)2 nm2, h = 0.34 nm).

Fig. 8: Effect of length ratio versus nonlocal parameter on the critical buckling load (η = (0.9)2 nm2, h = 0.34 nm).

5.4 Influence of thickness ratio on the critical buckling load 18

Fig. 9 indicates the plot of the critical buckling load versus thickness ratio for different scale factor. Herein the value of nonlocal parameter is (1)2 nm2. It can be seen that since increasing thickness ratio ascends the stiffness of nanotube, the increase in thickness ratio contributes to the higher critical buckling load for various values of scale factor; besides, this effect is intensified when the scale factor goes up. Further it is also found that the effect of scale factor on the critical buckling load becomes more pronounced by increasing thickness ratio. This is regarded as evidence that the scale factor makes nanotube stiffer. In order to have a deeper insight into the influence of thickness ratio, Fig. 10 is illustrated for various nonlocal parameter. The scale factor is assumed to be (0.9)2 nm2. According to this figure one can see that the critical buckling load experiences similar variation with respect to the thickness ratio for various nonlocal parameter and are sensitive to thickness ratio especially when the nonlocal parameter is smaller. This phenomenon is attributed to the intrinsic softening effect of nonlocal parameter. In addition, when the thickness ratio is larger, the critical buckling load is more sensitive to nonlocal parameter because a small change in nonlocal parameter causes a significant change in the critical buckling load. In order to show clearly the influence of scale factor and nonlocal parameter on the critical buckling load, Figs. 11 and 12 are depicted additionally. Effects of scale factor and nonlocal parameter are illustrated continuously on the critical buckling load in Figs 11 and 12. It should be noted that, the influence of the transverse shear deformation is significant when thick and short nanotubes are investigated and the first order shear deformation theory is capable to consider this effect.

19

Fig. 9: Influence of thickness ratio for different scale factor on the critical buckling load (μ = (1)2 nm2).

Fig. 10: Influence of thickness ratio for different nonlocal parameter on the critical buckling load (η = (0.9)2 nm2).

20

Fig. 11: Effect of thickness ratio versus scale factor on the critical buckling load (μ = (1)2 nm2).

Fig. 12: Effect of thickness ratio versus nonlocal parameter on the critical buckling load (η = (0.9)2 nm2).

6. Conclusion

21

In this paper, the new size dependent shell model is presented for the bucking analysis of cylindrical nanoshell. Equilibrium equations are obtained based on nonlocal strain gradient theory using first order shear deformation shell model, and then, using adjacent equilibrium criterion, the stability equations are obtained. By comparing MD results with those of different continuum theories, it was observed that results obtained from NSGT is in excellent agreement with MD results. As case study, the buckling response of simply supported cylindrical nanoshell was investigated and the influence of different parameters on critical buckling load was studied. Also, it was found that: -

The nanotube exerts the softening and stiffening behavior by increasing the nonlocal parameter and scale factor, respectively.

-

The influence of geometrical parameters becomes significant more prominent by increase in scale factor and decrease in nonlocal parameter.

-

It was revealed that influence of small length scale parameters is considerable by increase in thickness ratio and decrease in length ratio.

Appendix A The boundary conditions on x = 0, L are as below: U  0 U 0 x

V  0 V 0 x

N xx 

or

or

1 N x(1) 0 R 

N xx(1)  0

N x 

or

or

(A-1)

(A-2)

1 N (1) Q(1)z  0 R  R

(A-3)

N x(1)  0

(A-4)

22

W  0

Qxz 

or

N(1) 1 Q(1)z W N x W   N xx  R R  x R 

1   1 W  1   1 W  1 Qz  2 0  N   N x    R   x  R  R   1

 W 0 x

Qxz(1)  N xx1

or

 x  0

M xx 

or

 x 0 x

M x 

or

  0 x

(A-6)

1 M x(1)  Q xz(1)  0 R 

(A-7)

M xx(1)  0

or

   0

1 W N x W  0 x R 

(A-8)

1 M (1)  Q(1)z  0 R 

(A-9)

M x(1)  0

or

(A-5)

(A-10)

The stability boundary conditions on x = 0, L are as:  U1  0  U1 0 x

 V1  0

1 N x(1) 1 0 R 

N xx1 

or

N xx(1)1  0

or

N x 1 

or

(A-12)

1 N(1)1 Q(1)z1  0 R  R

 V1 0 x

or

N x(1) 1  0

 W1  0

or

Qxz1  

 W1 0 x

 x1  0

(A-11)

or

or

(A-14)

N(1)1 1 Q(1)z1 W1 N x 0 W1   N xx 0  R R  x R 

1   1 W1  1  N 0    R  R 2  

Qxz(1)1  N xx10

M xx1 

(A-13)

  1 W1  1 Qz 1 N   x 0 x  R   0   1

1 W1 N x0 W1  0 x R 

(A-15)

(A-16)

1 M x(1) 1  Qxz(1)1  0 R 

(A-17)

23

 x1 0 x

  1  0   1 0 x

or

or

or

M xx(1)1  0

M x 1 

(A-18)

1 M(1)1  Q(1)z1  0 R 

(A-19)

M x(1) 1  0

(A-20)

Conflict of Interest: The authors declare that they have no conflict of interest.

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