Nonlinear three-dimensional stability characteristics of geometrically imperfect nanoshells under axial compression and surface residual stress

Muhammad Atif Shahzad; Babak Safaei; Saeid Sahmani; Mohammed Salem Basingab; Abdul Zubar Hameed

doi:10.1515/ntrev-2022-0551

Article Open Access

Nonlinear three-dimensional stability characteristics of geometrically imperfect nanoshells under axial compression and surface residual stress

Muhammad Atif Shahzad , Babak Safaei , Saeid Sahmani , Mohammed Salem Basingab and Abdul Zubar Hameed

Published/Copyright: June 9, 2023

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From the journal Nanotechnology Reviews Volume 12 Issue 1

Abstract

Through reduction of thickness value in nanostructures, the features of surface elasticity become more prominent due to having a high surface-to-volume ratio. The main aim of this research work was to examine the surface residual stress effect on the three-dimensional nonlinear stability characteristics of geometrically perfect and imperfect cylindrical shells at nanoscale under axial compression. To do so, an unconventional three-dimensional shell model was established via combination of the three-dimensional shell formulations and the Gurtin–Murdoch theory of elasticity. The silicon material is selected as a case study, which is the most utilized material in the design of micro-electromechanically systems. Then, the moving Kriging meshfree approach was applied to take numerically into account the surface free energy effects and the initial geometrical imperfection in the three-dimensional nonlinear stability curves. Accordingly, the considered cylindrical shell domain was discretized via a set of nodes together using the quadratic polynomial type of basis shape functions and an appropriate correlation function. It was found that the surface stress effects lead to an increase the critical axial buckling load of a perfect silicon nanoshell about 82.4 % for the shell thickness of 2 nm , about 32.4 % for the shell thickness of 5 nm , about 15.8 % for the shell thickness of 10 nm , and about 7.5 % for the shell thickness of 20 nm . These enhancements in the value of the critical axial buckling load for a geometrically imperfect silicon nanoshell become about 92.9 % for the shell thickness of 2 nm , about 36.5 % for the shell thickness of 5 nm , about 17.7 % for the shell thickness of 10 nm , and about 8.8 % for the shell thickness of 20 nm .

Keywords: nanostructures; 3D elasticity; nonlinear buckling; surface residual stress; meshfree numerical technique

1 Introduction

Recently, many research works have demonstrated that for nanoscaled structures [1], relevant electrical [2], mechanical, and other characteristics were significantly changed, and some fascinating responses could be observed [3,4,5,6]. For example, it has been experimentally revealed that both fatigue lifetime and yield strength of copper nanofilm extremely depended on the thickness of the film and were increased by the decrease of the thickness of the film [7]. Research on nanomechanics needs both molecular and atomic modeling and continuum scale modeling, which was especially true for mechanical characteristics, which depended on phenomena at all possible length scales [8,9,10,11]. Sahmani and Aghdam [12] developed a beam model nonlocal strain gradient-based to investigate the size-dependent frequency response of an axially loaded drug delivery system. Kim et al. [13] employed MCST to explore size-dependent effects on the static and dynamic behavior of porous functionally graded (FG) microplate. Gao et al. [14] used NSGT together with the surface elasticity theory to establish the size-dependent model for postbuckling response of FG grapheme-reinforced composite curved nanobeam. Fan et al. [15,16] constructed a NURBS-based model to study the nonlinear size-dependent buckling, postbuckling, and vibration behaviors of FG micro/nanoplate including porosity by considering nonlocal parameters. Yuan et al. [17] anticipated the NSGT-based frequency stability behavior of FG truncated conical microshell embedded in nonlinear Winkler–Pasternak medium. Tang et al. [18] constructed a quasi-3D plate model to analyze the nonlinear free vibration characteristics of FG composite elliptical shape microplates based on MCST. Yang et al. [19] explored the unconventional flexural features of FG composite microplate by incorporating MCST into the isogeometric finite element model. Sahmani and Safaei [20] proposed a size‐dependent microplate model for the stability behavior of cylindrical graphene nanoplatelet reinforcement using MSGT and the third-order shear flexible shell. Tang and Qing [21] proposed an integral nonlocal strain gradient model to anticipate elastic buckling and vibration response of the FG Timoshenko beam.

To incorporate atomistic characteristics into the classical continuum framework to consider size effects, modified continuum models have been developed and applied [22,23]. More recently, Thai et al. [24,25] developed nonlocal strain gradient and modified strain gradient plate models for mechanical analysis of sandwich and FG composite plates at nanoscale and microscale via the meshfree approach. Yue et al. [26,27] introduced quasi-3D agglomerated beam and plate models on the basis of, respectively, nonlocal strain gradient and nonlocal couple stress elasticity theories to analyze their nonlinear dynamic stability responses. Yang et al. [28] established the nonlocal strain gradient-based arch formulations for the nonlinear thermomechanical in-plane buckling analysis of sandwich microsize arches. The influence of surface free energy, also called surface stress effect, is an important size effect that may greatly affect the mechanical characteristics of nanostructures. Surface free energy effect is exerted only on atom layers on or near the surface and does not exist in the bulk of material. Therefore, in nanoscale structures with comparatively high surface-volume ratios, surface free energy effects play important roles.

By using rigorous mathematical modeling, Gurtin and Ian Murdoch [29,30] applied continuum mechanics concepts to develop a theoretical framework to consider surface free energy effects, where the surface was modeled as a zero-thickness mathematical layer completely bonded to its underlying bulk. Moreover, surface layer was considered to have its own characteristics that differed from those of the bulk. Gurtin–Murdoch elasticity theory has extensively been utilized in nanomechanics to explain nanoscaled size-dependent behavior. For example, Sharma et al. [31,32] explored size-dependent elastic fields of ellipsoidal and spherical nanoinclusions using the surface elasticity theory considering residual and strain-dependent surface stresses. Lim and He [33] developed a size-dependent model for the prediction of geometrically nonlinear responses of thin elastic films with nanoscaled thicknesses by surface elasticity theory-based continuum method. Li et al. [34] investigated the surface effect on stress concentration around spherical cavities in linearly isotropic elastic media according to surface elasticity theory. Tian and Rajapakse [35,36] predicted size-dependent elastic fields due to nanoscale elliptical and circular defects in an isotropic matrix using the Gurtin–Murdoch elasticity theory. Lü et al. [37] developed a general theory for nanoscaled FG films taking into account surface free energy influences as film surface layers were modeled by surface elasticity continuum theory. Gordeliy et al. [38] solved a 2D transient and uncoupled thermoelastic problem of an infinite medium with a circular nanoscaled cavity using Gurtin–Murdoch elasticity theory. Mogilevskaya et al. [39] investigated the influences of the tension and elasticity of surface on unidirectional nanoscaled fiber-reinforced composite transverse overall behaviors. Fu et al. [40] established a modified nanoscaled beam continuum model considering surface elasticity to explain surface energy effects on critical axial forces of buckling, postbuckling, and linear free vibration frequency of nanobeams. Ansari and Sahmani [41] used analytically solved the buckling and bending responses of nanobeams using surface elasticity theory according to various beam theories. Also, Ansari and Sahmani [42] discussed rectangular nanoscaled plate free vibration responses taking into account surface free energy effects. Wang [43] studied the postbuckling responses of supported nanobeams with the internal flowing fluid having two surface layers using a nonlinear theoretical model. Ansari et al. [44,45] used the Gurtin–Murdoch elasticity theory to predict the effects of surface stress on the postbuckling characteristics of nanobeams modeled via Timoshenko and Euler-Bernoulli beam theories. Kiani [46] established a surface elasticity theory-based model to explore surface effects on the instability and free transverse vibrations of current-carrying nanowires immersed in longitudinal magnetic fields. Gao et al. [47] analyzed nanowire buckling in elastomeric substrates taking into account surface stress effects. Sahmani et al. [48] applied surface elasticity theory to evaluate free vibration responses of postbuckled third-order shear deformable nanobeams. Sahmani et al. [49] applied the Gurtin–Murdoch elasticity theory to introduce a nonclassical beam model to investigate the nonlinear forced vibrations of nanobeams with surface effects. Liang et al. [50] established a theoretical model for the prediction of surface effect on piezoelectric nanowire postbuckling behaviors taking into account surface piezoelectricity, surface elasticity, and residual surface stresses. Sahmani et al. [51] used surface elasticity theory to evaluate the free vibrations of postbuckled FG third-order shear deformable nanobeams. Sahmani et al. [52] predicted nonlinear postbuckling behaviors of circular nanoplates under surface energy effects with residual tension and elasticity of surface. Sahmani et al. [53] anticipated the nonlinear postbuckling behavior of axially loaded cylindrical nanoshells based on the surface elasticity theory. Li et al. [54], Sarafraz et al. [55], and Sahmani et al. [56] analyzed the mode interactions in the nonlinear primary resonance of graded porous cylindrical nanoshells in the presence of surface stress effects. Sahmani and Safaei [57] examined the surface stress effect on the large-amplitude free vibrations of composite conical nanoshells having in-plane heterogeneity. Tong et al. [58] extracted the critical buckling loads of nanoplates on the basis of the nonisotropic surface stress theory. Wang et al. [59] and Sahmani et al. [60] predicted the surface stress effect on the quasi-3D nonlinear free vibrations and bending of arbitrary-shaped nanoplates having nonuniform thickness. Fan et al. [61] examined the quasi-3D thermal postbuckling behavior of porous composite nanoplates having a central cutout on the basis of the surface theory of elasticity. Recently, Yang et al. [62] anticipated the effect of surface stress on the nonlinear thermomechanical in-plane stability behavior of FG laminated curved beams at a nanoscale.

In reality, the geometry of a structure may have some imperfections due to some inaccuracies in the manufacturing process, especially at a very small scale. Therefore, it is necessary to analyze the effect of an initial geometrical imperfection in a stability solution of nanoshells. On the other hand, to design efficiently nano-electromechanical systems, it is necessary to predict accurately the size-dependent mechanical responses of nanostructures utilized in the fabrication of them. In this regard, the surface residual stress together with the surface elasticity play essential roles in mechanical characteristics of structures at the nanoscale. So, the main goal of this research was to formulate for the first time, the three-dimensional surface elastic-based shell model to analyze the three-dimensional nonlinear axial buckling and postbuckling behaviors of cylindrical shells at the nanoscale in the presence and absence of initial imperfection under surface free energy effects. To do so, the Gurtin–Murdoch elasticity theory along with the nonlinear Green–Lagrange strain tensor was employed. Afterward, the MKM approach was applied to take numerically into account the surface free energy effects and the initial geometrical imperfection in the three-dimensional nonlinear stability curves. Accordingly, the considered cylindrical shell domain was discretized via a set of nodes together using the quadratic polynomial type of basis shape functions and an appropriate correlation function.

2 Preliminaries

As can be seen in Figure 1, a cylindrical nanoshell with thickness h , mid-surface radius R , and length L was considered. Nanoshells contain three parts including a bulk part and two thin inner and outer surface layers. For the bulk part, material characteristics included Young’s modulus E and Poisson’s ratio ν . The two surface layers were assumed to have the surface residual tension τ s and surface Lame constants of λ s and μ s .

Figure 1

Schematic view of a cylindrical shear deformable nanoshell with surface layers.

Within the framework of a three-dimensional theory of elasticity, the associated displacement field for any arbitrary point within the nanoshell can be adopted as follows:

(1) U = U x U y U z = u ( x , y , z ) v ( x , y , z ) w ( x , y , z ) + w * ( x , y , z ) ,

where u , v , and w are the displacement variables along x , y , and z axes, respectively. Also, w * stands for the initial geometrical imperfection, which is assumed as a coefficient of the lateral deflection associated with the initial buckling mode obtained by the solution process.

Now, the strain-displacement equations including the initial geometrical imperfection based on the nonlinear Green–Lagrange strain tensor [63] can be written as follows:

ε xx = ∂ u ∂ x + 1 2 ∂ u ∂ x 2 + ∂ v ∂ x 2 + ∂ w ∂ x 2 + ∂ w ∂ x ∂ w * ∂ x ,

ε yy = ∂ v ∂ y + 1 2 ∂ u ∂ y 2 + ∂ v ∂ y 2 + ∂ w ∂ y 2 + ∂ w ∂ y ∂ w * ∂ y ,

(2) ε zz = ∂ w ∂ z + 1 2 ∂ u ∂ z 2 + ∂ v ∂ z 2 + ∂ w ∂ z 2 + ∂ w ∂ z ∂ w * ∂ z ,

γ xy = ∂ u ∂ y + ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x ,

γ xz = ∂ u ∂ z + ∂ w ∂ x + ∂ u ∂ x ∂ u ∂ z + ∂ v ∂ x ∂ v ∂ z + ∂ w ∂ x ∂ w ∂ z + ∂ w ∂ x ∂ w * ∂ z + ∂ w ∂ z ∂ w * ∂ x ,

γ yz = ∂ v ∂ z + ∂ w ∂ y + ∂ u ∂ y ∂ u ∂ z + ∂ v ∂ y ∂ v ∂ z + ∂ w ∂ y ∂ w ∂ z + ∂ w ∂ y ∂ w * ∂ z + ∂ w ∂ z ∂ w * ∂ y .

Accordingly, the three-dimensional stress–strain constitutive equations can be expressed in the following form:

(3) σ xx σ yy σ zz τ xy τ xz τ yz = Q 11 Q 12 Q 13 0 0 0 Q 12 Q 22 Q 23 0 0 0 Q 13 Q 23 Q 33 0 0 0 0 0 0 Q 66 0 0 0 0 0 0 Q 55 0 0 0 0 0 0 Q 44 ε xx ε yy ε zz γ xy γ xz γ yz ,

where

Q 11 = Q 22 = Q 33 = ( 1 − ν ) E ( 1 − 2 ν ) ( 1 + ν ) ,

(4) Q 12 = Q 13 = Q 23 = ν E ( 1 − 2 ν ) ( 1 + ν ) ,

Q 44 = Q 55 = Q 66 = E 2 ( 1 + ν ) .

In addition, on the basis of the Gurtin–Murdoch continuum elasticity theory, the surface stress components can be extracted as follows [29,30]:

(5a) σ α β S = τ s δ α β + ( τ s + λ s ) ε ll δ α β + 2 ( μ s − τ s ) ε α β + τ s U α , β S α , β = x , y ,

(5b) σ α z S = τ s U z , α S ,

where λ s and μ s are the surface Lame constants associated with the surface elasticity, and τ s denotes the surface tension.

As a consequence, one will have

σ xx S ± = ( λ s + 2 μ 2 − τ s ) ε xx + ( λ s + τ s ) ( ε yy + ε zz ) + τ s 1 + ∂ U x ∂ x = ( λ s + 2 μ 2 ) ∂ u ∂ x + 1 2 ∂ u ∂ x 2 + ∂ v ∂ x 2 + ∂ w ∂ x 2 + ∂ w ∂ x ∂ w * ∂ x + ( λ s + τ s ) ∂ v ∂ y + ∂ w ∂ z + ∂ w ∂ y ∂ w * ∂ y + ∂ w ∂ z ∂ w * ∂ z + 1 2 ∂ u ∂ y 2 + ∂ u ∂ z 2 + ∂ v ∂ y 2 + ∂ v ∂ z 2 + ∂ w ∂ y 2 + ∂ w ∂ z 2 − τ s 2 ∂ u ∂ x 2 + ∂ v ∂ x 2 + ∂ w ∂ x 2 + 2 ∂ w ∂ x ∂ w * ∂ x + τ s

σ yy S ± = ( λ s + 2 μ 2 − τ s ) ε yy + ( λ s + τ s ) ( ε xx + ε zz ) + τ s 1 + ∂ U y ∂ y = ( λ s + 2 μ 2 ) ∂ v ∂ y + 1 2 ∂ u ∂ y 2 + ∂ v ∂ y 2 + ∂ w ∂ y 2 + ∂ w ∂ y ∂ w * ∂ y + ( λ s + τ s ) ∂ u ∂ x + ∂ w ∂ z + ∂ w ∂ x ∂ w * ∂ x + ∂ w ∂ z ∂ w * ∂ z + 1 2 ∂ u ∂ x 2 + ∂ u ∂ z 2 + ∂ v ∂ x 2 + ∂ v ∂ z 2 + ∂ w ∂ x 2 + ∂ w ∂ z 2 − τ s 2 ∂ u ∂ y 2 + ∂ v ∂ y 2 + ∂ w ∂ y 2 + 2 ∂ w ∂ y ∂ w * ∂ y + τ s

τ xy S ± = ( μ s − τ s ) γ xy + τ s ∂ U x ∂ y = μ s ∂ u ∂ y + ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x − τ s ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x

(6) τ yx S ± = ( μ s − τ s ) γ xy + τ s ∂ U y ∂ x = μ s ∂ u ∂ y + ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x − τ s ∂ u ∂ y + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x ,

τ xz S ± = τ s ∂ U z ∂ x = τ s ∂ w ∂ x + ∂ w * ∂ x ,

τ yz S ± = τ s ∂ U y ∂ z = τ s ∂ w ∂ y + ∂ w * ∂ y .

Consequently, the three-dimensional surface stress–strain relationships can be written in a matrix form as follows:

(7) σ xx S ± σ yy S ± τ xy S ± τ yx S ± τ xz S ± τ yz S ± = Q 11 S − τ s Q 12 S Q 13 S 0 0 0 Q 21 S Q 22 S − τ s Q 23 S 0 0 0 0 0 0 Q 66 S 0 0 0 0 0 Q 66 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ε xx ε yy ε zz γ xy γ xz γ yz + τ s 1 + ∂ U x ∂ x τ s 1 + ∂ U y ∂ y τ s ∂ U x ∂ y τ s ∂ U y ∂ x τ s ∂ U z ∂ x τ s ∂ U z ∂ y ,

where

(8) Q 11 S = Q 22 S = λ s + 2 μ 2 , Q 12 S = Q 21 S = Q 13 S = Q 23 S = λ s + τ s , Q 66 S = μ s − τ s .

In this regard, the conventional and surface elastic-based parts of the variation of the strain energy for a nanoshell on the basis of the three-dimensional theory of elasticity can be written as follows:

(9a) δ Π C = ∫ S ∫ − h 2 h 2 { σ xx δ ε xx + σ yy δ ε yy + σ zz δ ε zz + τ xy δ γ xy + τ yz δ γ yz + τ xz δ γ xz } d z d S ,

(9b) δ Π NC = ∫ S + { σ xx S δ ε xx + σ yy S δ ε yy + τ xy S δ γ xy + τ yx S δ γ xy + τ xz S δ γ xz + τ yz S δ γ yz } d S + + ∫ S − { σ xx S δ ε xx + σ yy S δ ε yy + τ xy S δ γ xy + τ yx S δ γ xy + τ xz S δ γ xz + τ yz S δ γ yz } d S − .

Also, the virtual work caused by the axial compressive load P can be written as follows:

(10) δ Π W = 1 2 ∫ S P ∂ ( w + w * ) ∂ x 2 d S .

On the basis of the virtual work principle, and through substituting equations (3) and (6) to equations (9a) and (9b), one will have

(11) ∫ S δ ( P b T ) ξ b P b + δ ( P s T ) ξ s P s − P δ ∂ ( w + w * ) ∂ x 2 d S + ∮ S { δ ( S b 1 T ) ζ b 1 S b 1 + δ ( S b 2 T ) ζ b 2 S b 2 } = 0 ,

where

P b = ∂ u ∂ x + 1 2 ∂ u ∂ x 2 + ∂ v ∂ x 2 + ∂ w ∂ x 2 + ∂ w ∂ x ∂ w * ∂ x ∂ v ∂ y + 1 2 ∂ u ∂ y 2 + ∂ v ∂ y 2 + ∂ w ∂ y 2 + ∂ w ∂ y ∂ w * ∂ y ∂ w ∂ z + 1 2 ∂ u ∂ z 2 + ∂ v ∂ z 2 + ∂ w ∂ z 2 + ∂ w ∂ z ∂ w * ∂ z T , ξ b = ∫ − h 2 h 2 Q 11 Q 12 Q 13 Q 12 Q 22 Q 23 Q 13 Q 23 Q 33 d z ,

P s = ∂ u ∂ z + ∂ w ∂ x + ∂ u ∂ x ∂ u ∂ z + ∂ v ∂ x ∂ v ∂ z + ∂ w ∂ x ∂ w ∂ z + ∂ w ∂ x ∂ w * ∂ z + ∂ w ∂ z ∂ w * ∂ x ∂ v ∂ z + ∂ w ∂ y + ∂ u ∂ y ∂ u ∂ z + ∂ v ∂ y ∂ v ∂ z + ∂ w ∂ y ∂ w ∂ z + ∂ w ∂ y ∂ w * ∂ z + ∂ w ∂ z ∂ w * ∂ y ∂ u ∂ y + ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x T , ξ s = ∫ − h 2 h 2 Q 44 0 0 0 Q 55 0 0 0 Q 66 d z ,

S b 1 = Z 11 Z 12 0 Z 21 Z 22 0 0 0 Z 61 0 0 Z 61 0 0 0 0 0 0 T ,

Z 11 = ∂ u ∂ x + 1 2 ∂ u ∂ x 2 + ∂ v ∂ x 2 + ∂ w ∂ x 2 + ∂ w ∂ x ∂ w * ∂ x ,

Z 12 = ∂ v ∂ y + ∂ w ∂ z + ∂ w ∂ y ∂ w * ∂ y + ∂ w ∂ z ∂ w * ∂ z + 1 2 ∂ u ∂ y 2 + ∂ u ∂ z 2 + ∂ v ∂ y 2 + ∂ v ∂ z 2 + ∂ w ∂ y 2 + ∂ w ∂ z 2

(12) Z 21 = ∂ v ∂ y + 1 2 ∂ u ∂ y 2 + ∂ v ∂ y 2 + ∂ w ∂ y 2 + ∂ w ∂ y ∂ w * ∂ y ,

Z 22 = ∂ u ∂ x + ∂ w ∂ z + ∂ w ∂ x ∂ w * ∂ x + ∂ w ∂ z ∂ w * ∂ z + 1 2 ∂ u ∂ x 2 + ∂ u ∂ z 2 + ∂ v ∂ x 2 + ∂ v ∂ z 2 + ∂ w ∂ x 2 + ∂ w ∂ z 2 ,

Z 61 = ∂ u ∂ y + ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x ,

ζ b 1 = Q 11 S − τ s Q 12 S Q 13 S 0 0 0 Q 21 S Q 22 S − τ s Q 23 S 0 0 0 0 0 0 Q 66 S 0 0 0 0 0 Q 66 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S − S + ,

S b 2 = 1 − 1 2 ∂ u ∂ x 2 + ∂ v ∂ x 2 + ∂ w ∂ x 2 + 2 ∂ w ∂ x ∂ w * ∂ x 1 − 1 2 ∂ u ∂ y 2 + ∂ v ∂ y 2 + ∂ w ∂ y 2 + 2 ∂ w ∂ y ∂ w * ∂ y 1 − 1 2 ∂ u ∂ z 2 + ∂ v ∂ z 2 + ∂ w ∂ z 2 + 2 ∂ w ∂ z ∂ w * ∂ z ∂ v ∂ x + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x ∂ u ∂ y + ∂ u ∂ x ∂ u ∂ y + ∂ v ∂ x ∂ v ∂ y + ∂ w ∂ x ∂ w ∂ y + ∂ w ∂ x ∂ w * ∂ y + ∂ w ∂ y ∂ w * ∂ x ∂ w ∂ x + ∂ w * ∂ x ∂ w ∂ y + ∂ w * ∂ y T , ζ b 2 = τ s τ s τ s − τ s − τ s τ s τ s .

3 Three-dimensional MKM solving process

Via employing the moving Kriging meshfree numerical solving technique, one can formulate a structure having an arbitrary geometry with the aid of correct stabilized dispersion of nodes having the capability to impose the required boundary conditions accurately via the genuine Lagrange multiplier. In this regard, the MKM approach has an excellent proficiency to utilize in predicting different nonlinear mechanical characteristics of arbitrary-shaped structures at various scales [64,65,66,67,68,69,70,71,72]. As illustrated in Figure 2a, the three-dimensional solution territory of O containing n nodes is employed to discretize the domain of the problem together with a three-dimensional support territory of O s incorporating a random point. On the other hand, the necessary integration prouder for the considered nonlinear problem is carried out via using an appropriate nodal integration strategy as represented by Figure 2b.

Figure 2

(a) Representation of the selected support domain using in numerical solving process and (b) moving Kriging nodes together with the relevant integration regions.

To disperse nodes within the nanoshell effectively, the Chebychev dispersion pattern is put to use as shown in Figure 3. Accordingly, through applying the moving Kriging-based interpolation function to the constructed three-dimensional shell model, the correlated nodal displacement vector is derived as follows [73,74]:

(13) U h ( x ) = ∑ i = 1 n K i ( x ) U i ,

where U denotes an undisclosed parameter relevant to the displacement of the ith node, and K refers to the moving Kriging kind of shape function.

Figure 3

Schematic depiction of the employed three-dimensional Chebyshev scheme of node dispersion for the MKM model of a cylindrical shell.

As a consequence, the discretized form relevant to the vector of three-dimensional displacement for a cylindrical nanoshell can be read as follows:

(14) U h ( x , y , z ) = ∑ i = 1 m × n × q K i ( x , y , z ) 0 0 0 K i ( x , y , z ) 0 0 0 K i ( x , y , z ) u i v i w i = ∑ i = 1 m × n × q W i ( x , y , z ) T i ,

where T i exhibits the freedom degree associated with the ith node.

Therefore, the discretized forms of the classical and surface elastic-based strain tensor can be expressed as follows:

P b = P b L + P b NL = ∑ i = 1 m × n × q Z ∼ L b i ( x , y , z ) T i + 1 2 ∑ i = 1 m × n × q Z ∼ NL b i ( x , y , z ) T i ,

P s = P s L + P s NL = ∑ i = 1 m × n × q Z ∼ L s i ( x , y , z ) T i + ∑ i = 1 m × n × q Z ∼ NL s i ( x , y , z ) T i ,

(15) S b 1 = S b 1 L + S b 1 NL = ∑ i = 1 m × n × q f L b 1 i ( x , y , z ) T i + ∑ i = 1 m × n × q f NL b 1 i ( x , y , z ) T i ,

S b 2 = S b 2 L + S b 2 NL = ∑ i = 1 m × n × q f L b 2 i ( x , y , z ) T i + ∑ i = 1 m × n × q f NL b 2 i ( x , y , z ) T i .

where Z ∼ L b i and Z ∼ L s i are the discretized matrices of moving Kriging kind of shape function related to the linear parts of the classical strain tensor, and Z ∼ NL b i and Z ∼ NL s i denote the discretized matrices of moving Kriging kind of shape function associated with the nonlinear parts of the classical strain tensor. In addition, f L b 1 i and f L b 2 i stand for the discretized matrices of moving Kriging kind of shape function related to the linear parts of the surface elastic-based strain tensor, and f NL b 1 i and f NL b 1 i stand for the discretized matrices of moving Kriging kind of shape function related to the nonlinear parts of the surface elastic-based strain tensor.

Consequently, the surface elastic-based nonlinear stability problem can be expressed based on the MKM-based discretization strategy in the following form:

(16) ∑ i = 1 m × n × q ( K ( T i ) − p cr K G ) T i = 0 ,

where p cr designates the critical bifurcation load, and K ( T i ) signifies the three-dimensional overall stiffness matrix of the considered cylindrical nanoshells comprising the both linear and nonlinear parts. Moreover, K G stands for the three-dimensional geometric stiffness matrix.

4 Numerical results and discussion

In this section, the surface elastic-based three-dimensional postbuckling equilibrium paths of imperfect and perfect cylindrical nanoshells under surface stress effects and axial compression were presented. The material characteristics of nanoshells made of silicon are summarized in Table 1. In addition, in all previous numerical findings, nanoshell edge supports were assumed to be clamped. Also, the geometrical parameters of cylindrical nanoshells are considered as follows: L 2 / Rh = 200 , R / h = 50 . In addition, the following dimensionless parameters are used to present the numerical results as follows:

Table 1

Material properties of a cylindrical nanoshell made of silicon [75,76]

E ( GPa )	210
ν	0.24
μ s ( N / m )	−2.774
λ s ( N / m )	−4.488
τ s ( N / m )	0.6048

Dimensionless axial compressive load: p = P / ( 4 π E h 2 ) .

Dimensionless axial shortening: δ = ∆ x R / ( 2 Lh ) .

Dimensionless lateral deflection: W = w / h .

The validity of the proposed shell formulations are checked at the first step. In this regard, by ignoring the expressions relevant to the surface continuum elasticity, the critical axial buckling loads of graded inhomogeneous cylindrical shells at macroscale are extracted corresponding to various gradient indexes of material properties and are compared with those presented by Huang and Han [77] via employing the Galerkin technique. As tabulated in Table 2, a very good agreement is found which represents the accuracy of the proposed three-dimensional MKM shell model.

Table 2

Comparison study on the critical axial buckling loads of graded inhomogeneous cylindrical shells at macroscale ( L / R = 2 , R / h = 200 )

Material property gradient index	Critical buckling load (MPa)
Material property gradient index	Present work	Ref. [77]
0.2	431.603	441.053
1	374.189	382.955
5	329.866	335.512

In another comparison study, in Table 3, the surface elastic-based critical buckling stresses of axially compressed simply supported isotropic piezoelectric nanoshells having various radiuses are obtained corresponding to two different piezoelectric materials and are compared with those reported by Sun et al. [78] using the method of separation of variables. A very good agreement is achieved again, which indicates the validity of the developed three-dimensional surface elastic-based shell model.

Table 3

Comparison study on the critical buckling stresses (MPa) of axially compressed isotropic piezoelectric nanoshells including the surface elastic effects ( h = 1 nm , L = 5 R , ν = 0.3 )

R ( nm )	Ref. [71]		Present model
R ( nm )	PZT-5A	PZT-7A	PZT-5A	PZT-7A
10	3685.8	4400.8	3681.22	4397.73
20	1889.2	2267.8	1887.94	2265.21
30	1269.8	1526.9	1268.19	1525.82
40	956.2	1150.8	955.41	1149.29
50	766.8	923.3	765.97	921.83
60	640.0	770.9	639.18	769.51
70	549.2	661.7	548.48	660.85
80	481.0	579.6	480.51	578.34
90	427.8	515.6	427.49	514.33
100	385.3	464.4	385.06	463.47

Figure 4 shows the dimensionless three-dimensional postbuckling load-deflection paths of silicon cylindrical nanoshells in the presence and absence of initial imperfection using the classical and surface elastic-based MKM shell models. Nanoshells were supposed to be made of silicon, which is the most utilized material in design of micro-electromechanically systems. It is observed that for the both imperfect and perfect nanoshells, the gap between the classical and surface elastic-based load-deflection postbuckling curves enhances for lower shell thicknesses. This meant that the surface stress effect was more significant for three-dimensional cylindrical nanoshells with smaller thicknesses. This fact comes from a higher surface to volume ratio through reduction of the shell thickness. In this regard, the critical axial buckling load of a perfect silicon nanoshell increases about 82.4 % for the shell thickness of 2 nm , about 32.4 % for the shell thickness of 5 nm , about 15.8 % for the shell thickness of 10 nm , about 7.5 % for the shell thickness of 20 nm , and about 2.5 % for the shell thickness of 50 nm . These enhancements in the value of the critical axial buckling load for a geometrically imperfect silicon nanoshell become about 92.9 % for the shell thickness of 2 nm , about 36.5 % for the shell thickness of 5 nm , about 17.7 % for the shell thickness of 10 nm , about 8.4 % for the shell thickness of 20 nm , and about 2.8 % for the shell thickness of 50 nm . In addition, the effects of surface elasticity get more considerable for a larger lateral deflections after occurring the snap-through phenomenon within the postbuckling territory.

Figure 4

Classical and surface elastic-based load-deflection three-dimensional postbuckling paths of silicon nanoshells with and without initial geometrical imperfection.

Figure 5 depicts the classical and surface elastic-based three-dimensional postbuckling load-end shortening paths of imperfect and perfect cylindrical nanoshells made of silicon with different thicknesses. Considering the surface stress effects reveals that the both critical buckling load and critical end-shortening of nanoshells were increased as well as the slope of prebuckling part of the classical load-end shortening curve is a bit higher than that of the surface elastic-based one. It is due to the tensile character of the surface residual stress for the silicon material, which results in a restraint against the applied compressive load. Accordingly, by taking the surface stress effects, the critical axial shortening of a perfect silicon nanoshell enhances about 91.3 % for the shell thickness of 2 nm , about 35.0 % for the shell thickness of 5 nm , about 16.9 % for the shell thickness of 10 nm , about 7.9 % for the shell thickness of 20 nm , and about 2.7 % for the shell thickness of 50 nm . These increments in the value of the critical axial shortening for a geometrically imperfect silicon nanoshell become about 101.2 % for the shell thickness of 2 nm , about 38.7 % for the shell thickness of 5 nm , about 18.7 % for the shell thickness of 10 nm , about 8.8 % for the shell thickness of 20 nm , and about 2.7 % for the shell thickness of 50 nm .

Figure 5

Classical and surface elastic-based load-shortening three-dimensional postbuckling paths of silicon nanoshells with and without initial geometrical imperfection.

Figure 6 shows the surface elastic-based three-dimensional postbuckling load-deflection paths of cylindrical nanoshells in the presence and absence of initial imperfection made of materials having various tensional and compressive surface residual stresses. It is found that a tensional surface residual stress causes to enhance the nonlinear stability of an axially compressed nanoshell, resulting in higher critical buckling and minimum postbuckling loads as well as a wider postbuckling territory. On the other hand, a compressive surface residual stress makes opposite features. In this regard, the tensional surface residual stresses equal to + 0.8 and + 0.4 N / m result in an enhancement in the value of critical axial buckling load about, respectively, 46.8 and 23.9 % for a perfect nanoshell and 53.5 and 26.8 % for an imperfect nanoshell having the shell thickness of 5 nm . Also, the compressive surface residual stresses equal to − 0.8 and − 0.4 N / m cause a reduction in the value of critical axial buckling load about, respectively, 48.0 and 24.0 % for a perfect nanoshell and 53.6 and 26.8 % for an imperfect nanoshell having the shell thickness of 5 nm .

$Figure 6 Influence of positive and negative surface residual stress on load-deflection three-dimensional postbuckling paths of nanoshells with and without initial geometrical imperfection ( λ s + 2 μ s = 0 , h = 5 nm {\lambda }^{s}+2{\mu }^{s}=0,{h}=5{\rm{nm}} ).$

Figure 6

Influence of positive and negative surface residual stress on load-deflection three-dimensional postbuckling paths of nanoshells with and without initial geometrical imperfection ( λ s + 2 μ s = 0 , h = 5 nm ).

Figure 7 represents the surface elastic-based three-dimensional postbuckling load-shortening paths of imperfect and perfect cylindrical nanoshells made of materials having various tensional and compressive surface residual stresses. It is demonstrated that a tensional surface residual stress leads to increase the critical axial shortening as well as the shortening at the minimum postbuckling point. However, a compressive surface residual stress plays an opposite role. Accordingly, the tensional surface residual stresses equal to + 0.8 and + 0.4 N / m make an increment in the value of critical axial shortening about, respectively, 48.5 and 24.2 % for a perfect nanoshell and 53.4 and 26.7 % for an imperfect nanoshell having the shell thickness of 5 nm . Also, the compressive surface residual stresses equal to − 0.8 and − 0.4 N / m cause a reduction in the value of critical axial buckling load about, respectively, 48.6 and 24.2 % for a perfect nanoshell and 53.5 and 26.7 % for an imperfect nanoshell having the shell thickness of 5 nm .

$Figure 7 Influence of positive and negative surface residual stress on load-shortening three-dimensional postbuckling paths of nanoshells with and without initial geometrical imperfection ( λ s + 2 μ s = 0 , h = 5 nm {\lambda }^{s}+2{\mu }^{s}=0,h=5{\rm{nm}} ).$

Figure 7

Influence of positive and negative surface residual stress on load-shortening three-dimensional postbuckling paths of nanoshells with and without initial geometrical imperfection ( λ s + 2 μ s = 0 , h = 5 nm ).

Figures 8 and 9 display the variations of the surface elastic-based to classical three-dimensional buckling load ratio with shell thickness for various surface elastic constants and surface residual stress, respectively. It is seen that the surface elasticity could decrease or increase the critical buckling load of three-dimensional nanoshells depending on the signs of the surface residual stress as well as the surface elastic constants. It is revealed that positive values of the surface residual stress or surface elastic constant increase the nonlinear critical axial buckling load, while negative values result in decrease of the nonlinear critical axial buckling load compared to those of the classical shell formulations. In addition, it was witnessed again that the increase in the shell thickness leads to weakening of the effect of surface elastic constant and surface residual stress, which result in convergence of the variation curves.

$Figure 8 Variation of the three-dimensional critical buckling load ratio with nanoshell thickness corresponding to various surface elastic constants ( τ s = 0 {\tau }^{s}=0 ).$

Figure 8

Variation of the three-dimensional critical buckling load ratio with nanoshell thickness corresponding to various surface elastic constants ( τ s = 0 ).

$Figure 9 Variation of the three-dimensional critical buckling load ratio with nanoshell thickness corresponding to various surface residual stresses ( λ s + 2 μ s = 0 {\lambda }^{s}+2{\mu }^{s}=0 ).$

Figure 9

Variation of the three-dimensional critical buckling load ratio with nanoshell thickness corresponding to various surface residual stresses ( λ s + 2 μ s = 0 ).

5 Conclusion

In this research work, the three-dimensional nonlinear buckling and postbuckling characteristics of cylindrical nanoshells in the presence and absence of initial imperfection were studied under surface stress effects. To do so, surface elastic-based shell formulations were established by the combination of three-dimensional shell model with the Gurtin–Murdoch continuum mechanics using the nonlinear Green-Lagrange strain tensor. Then, the MKM approach was applied to take numerically into account the surface free energy effects and the initial geometrical imperfection in the three-dimensional nonlinear stability curves. Accordingly, the considered cylindrical shell domain was discretized via a set of nodes together with using quadratic polynomial type of basis shape functions and an appropriate correlation function.

It was revealed that the surface stress effect was more significant for three-dimensional cylindrical nanoshells with smaller thicknesses. Therefore, the both critical buckling load and critical end-shortening of nanoshells are increased, as well as the slope of prebuckling part of the classical load-end shortening curve is a bit higher than that of the surface elastic-based one. Accordingly, by taking the surface stress effects, the critical axial shortening of a perfect silicon nanoshell enhances about 91.3 % for the shell thickness of 2 nm , about 35.0 % for the shell thickness of 5 nm , about 16.9 % for the shell thickness of 10 nm , about 7.9 % for the shell thickness of 20 nm , and about 2.7 % for the shell thickness of 50 nm . These increments in the value of the critical axial shortening for a geometrically imperfect nanoshell become about 101.2 % for the shell thickness of 2 nm , about 38.7 % for the shell thickness of 5 nm , about 18.7 % for the shell thickness of 10 nm , about 8.8 % for the shell thickness of 20 nm , and about 2.7 % for the shell thickness of 50 nm .

Furthermore, it was found that a tensional surface residual stress leads to increase the critical axial shortening as well as the shortening at the minimum postbuckling point. However, a compressive surface residual stress plays an opposite role. Accordingly, the tensional surface residual stresses equal to + 0.8 and + 0.4 N / m make an increment in the value of critical axial shortening about, respectively, 48.5 and 24.2 % for a perfect nanoshell, 53.4 and 26.7 % for an imperfect nanoshell having the shell thickness of 5 nm . Also, the compressive surface residual stresses equal to − 0.8 and − 0.4 N / m cause a reduction in the value of critical axial buckling load about, respectively, 48.6 and 24.2 % for a perfect nanoshell and 53.5 and 26.7 % for an imperfect nanoshell having the shell thickness of 5 nm .

Acknowledgments

The authors extend their appreciation to the Deanship for Research and Innovation, Ministry of Education in Saudi Arabia, for funding this research work through the project number “IFPRC-047-135-2020” and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

Funding information: This research work was funded by the Deanship for Research and Innovation, Ministry of Education in Saudi Arabia, through the project number “IFPRC-047-135-2020” and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Guo H, Żur KK, Ouyang X. New insights into the nonlinear stability of nanocomposite cylindrical panels under aero-thermal loads. Compos Struct. 2023;303:116231.10.1016/j.compstruct.2022.116231Search in Google Scholar

[2] Żur KK, Arefi M, Kim J, Reddy JN. Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear deformation theory. Compos Part B Eng. 2020;182:107601.10.1016/j.compositesb.2019.107601Search in Google Scholar

[3] Yao X, Han Q. Torsional buckling and postbuckling equilibrium path of double-walled carbon nanotubes. Compos Sci Technol. 2008;68(1):113–20.10.1016/j.compscitech.2007.05.025Search in Google Scholar

[4] Barati A, Hadi A, Nejad MZ, Noroozi R. On vibration of bi-directional functionally graded nanobeams under magnetic field. Mech Based Des Struct Mach. 2022 Feb;50(2):468–85.10.1080/15397734.2020.1719507Search in Google Scholar

[5] Alshenawy R, Sahmani S, Safaei B, Elmoghazy Y, Al-Alwan A, Al Nuwairan M. Three-dimensional nonlinear stability analysis of axial-thermal-electrical loaded FG piezoelectric microshells via MKM strain gradient formulations. Appl Math Comput. 2023 Feb;439:127623.10.1016/j.amc.2022.127623Search in Google Scholar

[6] Civalek Ö, Uzun B, Yaylı MÖ. An effective analytical method for buckling solutions of a restrained FGM nonlocal beam. Comput Appl Math. 2022;41(2):1–20.10.1007/s40314-022-01761-1Search in Google Scholar

[7] Zhang GP, Sun KH, Zhang B, Gong J, Sun C, Wang ZG. Tensile and fatigue strength of ultrathin copper films. Mater Sci Eng A. 2008;483:387–90.10.1016/j.msea.2007.02.132Search in Google Scholar

[8] Abouelregal AE, Ersoy H, Civalek Ö. Solution of moore–gibson–thompson equation of an unbounded medium with a cylindrical hole. Mathematics. 2021 Jun 30;9(13):1536.10.3390/math9131536Search in Google Scholar

[9] Kouris D, Gao H. Nanomechanics of surfaces and interfaces. J Appl Mech. 2002;69(4):405–6.10.1115/1.1469005Search in Google Scholar

[10] Sofiyev A, Aksogan O, Schnack E, Avcar M. The stability of a three-layered composite conical shell containing a FGM layer subjected to external pressure. Mech Adv Mater Struct. 2008;15(6–7):461–6.10.1080/15376490802138492Search in Google Scholar

[11] Sofiyev AH, Yücel K, Avcar M, Zerin Z. The dynamic stability of orthotropic cylindrical shells with non-homogenous material properties under axial compressive load varying as a parabolic function of time. J Reinf Plast Compos. 2016 Aug;25(18):1877–86.10.1177/0731684406069914Search in Google Scholar

[12] Sahmani S, Aghdam MM. Nonlocal strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules. Math Biosci. 2018;295:24–35.10.1016/j.mbs.2017.11.002Search in Google Scholar PubMed

[13] Kim J, Żur KK, Reddy JN. Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates. Compos Struct. 2019 Feb 1;209:879–88.10.1016/j.compstruct.2018.11.023Search in Google Scholar

[14] Gao Y, Xiao WS, Zhu H. Snap-buckling of functionally graded multilayer graphene platelet-reinforced composite curved nanobeams with geometrical imperfections. Eur J Mech-A/Solids. 2020 Jul;82:103993.10.1016/j.euromechsol.2020.103993Search in Google Scholar

[15] Fan F, Sahmani S, Safaei B. Isogeometric nonlinear oscillations of nonlocal strain gradient PFGM micro/nano-plates via NURBS-based formulation. Compos Struct. 2021 Jan 1;255:112969.10.1016/j.compstruct.2020.112969Search in Google Scholar

[16] Fan F, Safaei B, Sahmani S. Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis. Thin-Walled Struct. 2021 Feb 1;159:107231.10.1016/j.tws.2020.107231Search in Google Scholar

[17] Yuan Y, Zhao X, Zhao Y, Sahmani S, Safaei B. Dynamic stability of nonlocal strain gradient FGM truncated conical microshells integrated with magnetostrictive facesheets resting on a nonlinear viscoelastic foundation. Thin-Walled Struct. 2021 Feb 1;159:107249.10.1016/j.tws.2020.107249Search in Google Scholar

[18] Tang P, Sun Y, Sahmani S, Madyira DM. Isogeometric small-scale-dependent nonlinear oscillations of quasi-3D FG inhomogeneous arbitrary-shaped microplates with variable thickness. J Braz Soc Mech Sci Eng. 2021;43(7):1–16.10.1007/s40430-021-03057-7Search in Google Scholar

[19] Yang Z, Lu H, Sahmani S, Safaei B. Isogeometric couple stress continuum-based linear and nonlinear flexural responses of functionally graded composite microplates with variable thickness. Arch Civ Mech Eng. 2021;21(3):1–19.10.1007/s43452-021-00264-wSearch in Google Scholar

[20] Sahmani S, Safaei B. Microstructural-dependent nonlinear stability analysis of random checkerboard reinforced composite micropanels via moving Kriging meshfree approach. Eur Phys J Plus. 2021;136(8):1–31.10.1140/epjp/s13360-021-01706-3Search in Google Scholar

[21] Tang Y, Qing H. Elastic buckling and free vibration analysis of functionally graded Timoshenko beam with nonlocal strain gradient integral model. Appl Math Model. 2021 Aug;96:657–77.10.1016/j.apm.2021.03.040Search in Google Scholar

[22] Jin Q, Ren Y. Nonlinear size-dependent bending and forced vibration of internal flow-inducing pre- and post-buckled FG nanotubes. Commun Nonlinear Sci Numer Simul. 2022;104:106044.10.1016/j.cnsns.2021.106044Search in Google Scholar

[23] Zuo D, Safaei B, Sahmani S, Ma G. Nonlinear free vibrations of porous composite microplates incorporating various microstructural-dependent strain gradient tensors. Appl Math Mech. 2022;43(6):825–44.10.1007/s10483-022-2851-7Search in Google Scholar

[24] Thai CH, Nguyen-Xuan H, Nguyen LB, Phung-Van P. A modified strain gradient meshfree approach for functionally graded microplates. Eng Comput. 2022;38(Suppl 5):4545–67.10.1007/s00366-021-01493-6Search in Google Scholar

[25] Thai CH, Ferreira AJM, Nguyen-Xuan H, Nguyen LB, Phung-Van P. A nonlocal strain gradient analysis of laminated composites and sandwich nanoplates using meshfree approach. Eng Comput. 2023;39:5–21.10.1007/s00366-021-01501-9Search in Google Scholar

[26] Yue X-G, Sahmani S, Luo H, Safaei B. Nonlocal strain gradient-based quasi-3D nonlinear dynamical stability behavior of agglomerated nanocomposite microbeams. Arch Civ Mech Eng. 2023;23(1):21.10.1007/s43452-022-00548-9Search in Google Scholar

[27] Yue X-G, Sahmani S, Safaei B. Nonlocal couple stress-based quasi-3D nonlinear dynamics of agglomerated CNT-reinforced micro/nano-plates before and after bifurcation phenomenon. Phys Scr. 2023;98(3):035710.10.1088/1402-4896/acb858Search in Google Scholar

[28] Yang Z, Hurdoganoglu D, Sahmani S, Nuhu AA, Safaei B. Nonlocal strain gradient-based nonlinear in-plane thermomechanical stability of FG multilayer micro/nano-arches. Arch Civ Mech Eng. 2023;23(2):90.10.1007/s43452-023-00623-9Search in Google Scholar

[29] Gurtin ME, Ian Murdoch A. A continuum theory of elastic material surfaces. Arch Ration Mech Anal. 1975;57(4):291–323.10.1007/BF00261375Search in Google Scholar

[30] Gurtin ME, Ian Murdoch A. Surface stress in solids. Int J Solids Struct. 1978 Jan 1;14(6):431–40.10.1016/0020-7683(78)90008-2Search in Google Scholar

[31] Sharma P, Ganti S, Bhate N. Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett. 2003;82(4):535–7.10.1063/1.1539929Search in Google Scholar

[32] Sharma P, Wheeler L. Size-dependent elastic state of ellipsoidal nano-inclusions incorporating surface∕interface tension. J Appl Mech. 2007;74(3):447–54.10.1115/1.2338052Search in Google Scholar

[33] Lim CW, He L. Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int J Mech Sci. 2004;46(11):1715–26.10.1016/j.ijmecsci.2004.09.003Search in Google Scholar

[34] Li ZR, Lim CW, He LH. Stress concentration around a nano-scale spherical cavity in elastic media: effect of surface stress. Eur J Mech. 2006;25(2):260–70.10.1016/j.euromechsol.2005.09.005Search in Google Scholar

[35] Tian L, Rajapakse R. Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity. J Appl Mech. 2007;74(3):568–74.10.1115/1.2424242Search in Google Scholar

[36] Tian L, Rajapakse R. Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity. Int J Solids Struct. 2007;44(24):7988–8005.10.1016/j.ijsolstr.2007.05.019Search in Google Scholar

[37] Lü CF, Chen WQ, Lim CW. Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies. Compos Sci Technol. 2009;69(7–8):1124–30.10.1016/j.compscitech.2009.02.005Search in Google Scholar

[38] Gordeliy E, Mogilevskaya SG, Crouch SL. Transient thermal stresses in a medium with a circular cavity with surface effects. Int J Solids Struct. 2009;46(9):1834–48.10.1016/j.ijsolstr.2008.12.014Search in Google Scholar

[39] Mogilevskaya SG, Crouch SL, La Grotta A, Stolarski HK. The effects of surface elasticity and surface tension on the transverse overall elastic behavior of unidirectional nano-composites. Compos Sci Technol. 2010;70(3):427–34.10.1016/j.compscitech.2009.11.012Search in Google Scholar

[40] Fu Y, Zhang J, Jiang Y. Influences of the surface energies on the nonlinear static and dynamic behaviors of nanobeams. Phys E Low-Dimension Syst Nanostructures. 2010;42(9):2268–73.10.1016/j.physe.2010.05.001Search in Google Scholar

[41] Ansari R, Sahmani S. Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories. Int J Eng Sci. 2011;49(11):1244–55.10.1016/j.ijengsci.2011.01.007Search in Google Scholar

[42] Ansari R, Sahmani S. Surface stress effects on the free vibration behavior of nanoplates. Int J Eng Sci. 2011;49(11):1204–15.10.1016/j.ijengsci.2011.06.005Search in Google Scholar

[43] Wang L. Surface effect on buckling configuration of nanobeams containing internal flowing fluid: A nonlinear analysis. Phys E Low-Dimension Syst Nanostructures. 2012;44(4):808–12.10.1016/j.physe.2011.12.006Search in Google Scholar

[44] Ansari R, Mohammadi V, Faghih Shojaei M, Gholami R, Sahmani S. Postbuckling characteristics of nanobeams based on the surface elasticity theory. Compos Part B Eng. 2013;55:240–6.10.1016/j.compositesb.2013.05.040Search in Google Scholar

[45] Ansari R, Mohammadi V, Faghih Shojaei M, Gholami R, Sahmani S. Postbuckling analysis of Timoshenko nanobeams including surface stress effect. Int J Eng Sci. 2014;75:1–10.10.1016/j.ijengsci.2013.10.002Search in Google Scholar

[46] Kiani K. Surface effect on free transverse vibrations and dynamic instability of current-carrying nanowires in the presence of a longitudinal magnetic field. Phys Lett A. 2014;378(26–27):1834–40.10.1016/j.physleta.2014.04.039Search in Google Scholar

[47] Gao F, Cheng Q, Luo J. Mechanics of nanowire buckling on elastomeric substrates with consideration of surface stress effects. Phys E Low-Dimension Syst Nanostructures. 2014;64:72–7.10.1016/j.physe.2014.07.006Search in Google Scholar

[48] Sahmani S, Bahrami M, Ansari R. Surface energy effects on the free vibration characteristics of postbuckled third-order shear deformable nanobeams. Compos Struct. 2014 Sep 1;116(1):552–61.10.1016/j.compstruct.2014.05.035Search in Google Scholar

[49] Sahmani S, Bahrami M, Aghdam MM, Ansari R. Surface effects on the nonlinear forced vibration response of third-order shear deformable nanobeams. Compos Struct. 2014 Dec 1;118(1):149–58.10.1016/j.compstruct.2014.07.026Search in Google Scholar

[50] Liang X, Hu S, Shen S. Surface effects on the post-buckling of piezoelectric nanowires. Phys E Low-Dimension Syst Nanostruct. 2015;69:61–4.10.1016/j.physe.2015.01.019Search in Google Scholar

[51] Sahmani S, Aghdam MM, Bahrami M. On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects. Compos Struct. 2015 Mar 1;121:377–85.10.1016/j.compstruct.2014.11.033Search in Google Scholar

[52] Sahmani S, Bahrami M, Aghdam MM, Ansari R. Postbuckling behavior of circular higher-order shear deformable nanoplates including surface energy effects. Appl Math Model. 2015;39(13):3678–89.10.1016/j.apm.2014.12.002Search in Google Scholar

[53] Sahmani S, Bahrami M, Aghdam MM. Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to torsional load. Compos Part B Eng. 2016 Jan 1;84:140–54.10.1016/j.compositesb.2015.08.076Search in Google Scholar

[54] Li Q, Xie B, Sahmani S, Safaei B. Surface stress effect on the nonlinear free vibrations of functionally graded composite nanoshells in the presence of modal interaction. J Braz Soc Mech Sci Eng. 2020;42(5):237.10.1007/s40430-020-02317-2Search in Google Scholar

[55] Sarafraz A, Sahmani S, Aghdam MM. Nonlinear primary resonance analysis of nanoshells including vibrational mode interactions based on the surface elasticity theory. Appl Math Mech. 2019;41(2):233–60.10.1007/s10483-020-2564-5Search in Google Scholar

[56] Sahmani S, Fattahi AM, Ahmed NA. Surface elastic shell model for nonlinear primary resonant dynamics of FG porous nanoshells incorporating modal interactions. Int J Mech Sci. 2020 Jan 1;165:105203.10.1016/j.ijmecsci.2019.105203Search in Google Scholar

[57] Sahmani S, Safaei B. Large-amplitude oscillations of composite conical nanoshells with in-plane heterogeneity including surface stress effect. Appl Math Model. 2021;89:1792–813.10.1016/j.apm.2020.08.039Search in Google Scholar

[58] Tong LH, Lin F, Xiang Y, Shen H-S, Lim CW. Buckling analysis of nanoplates based on a generic third-order plate theory with shear-dependent non-isotropic surface stresses. Compos Struct. 2021;265:113708.10.1016/j.compstruct.2021.113708Search in Google Scholar

[59] Wang P, Yuan P, Sahmani S, Safaei B. Surface stress size dependency in nonlinear free oscillations of FGM quasi-3D nanoplates having arbitrary shapes with variable thickness using IGA. Thin-Walled Struct. 2021;166:10810.10.1016/j.tws.2021.108101Search in Google Scholar

[60] Sahmani S, Safaei B, Aldakheel F. Surface elastic-based nonlinear bending analysis of functionally graded nanoplates with variable thickness. Eur Phys J Plus. 2021;136(6):1–28.10.1140/epjp/s13360-021-01667-7Search in Google Scholar

[61] Fan F, Cai X, Sahmani S, Safaei B. Isogeometric thermal postbuckling analysis of porous FGM quasi-3D nanoplates having cutouts with different shapes based upon surface stress elasticity. Compos Struct. 2021;262:113604.10.1016/j.compstruct.2021.113604Search in Google Scholar

[62] Yang Z, Hurdoganoglu D, Sahmani S, Safaei B, Liu A. Surface stress size dependency in nonlinear thermomechanical in-plane stability characteristics of FG laminated curved nanobeams. Eng Struct. 2023;284:115957.10.1016/j.engstruct.2023.115957Search in Google Scholar

[63] Greenberg JL, Goodstein JR. Theodore von Kármán and applied mathematics in America. Science. 1983;222(4630):1300–4.10.1126/science.222.4630.1300Search in Google Scholar PubMed

[64] Thai CH, Nguyen TN, Rabczuk T, Nguyen-Xuan H. An improved moving Kriging meshfree method for plate analysis using a refined plate theory. Comput Struct. 2016;176:34–49.10.1016/j.compstruc.2016.07.009Search in Google Scholar

[65] Thai CH, Ferreira AJM, Lee J, Nguyen-Xuan H. An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on modified couple stress theory and moving Kriging-based meshfree method. Int J Mech Sci. 2018 Jul 1;142–143:322–38.10.1016/j.ijmecsci.2018.04.040Search in Google Scholar

[66] Zarei A, Khosravifard A. A meshfree method for static and buckling analysis of shear deformable composite laminates considering continuity of interlaminar transverse shearing stresses. Compos Struct. 2019;209:206–18.10.1016/j.compstruct.2018.10.077Search in Google Scholar

[67] Thai CH, Tran TD, Phung-Van P. A size-dependent moving Kriging meshfree model for deformation and free vibration analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Eng Anal Bound Elem. 2020 Jun;115:52–63.10.1016/j.enganabound.2020.02.008Search in Google Scholar

[68] Thai CH, Ferreira AJM, Phung-Van P. Free vibration analysis of functionally graded anisotropic microplates using modified strain gradient theory. Eng Anal Bound Elem. 2020;117:284–98.10.1016/j.enganabound.2020.05.003Search in Google Scholar

[69] Thai CH, Phung-Van P. A meshfree approach using naturally stabilized nodal integration for multilayer FG GPLRC complicated plate structures. Eng Anal Bound Elem. 2020 Aug 1;117:346–58.10.1016/j.enganabound.2020.04.001Search in Google Scholar

[70] Watts G, Kumar R, Patel SN, Singh S. Dynamic instability of trapezoidal composite plates under non-uniform compression using moving kriging based meshfree method. Thin-Walled Struct. 2021;164:107766.10.1016/j.tws.2021.107766Search in Google Scholar

[71] Liu H, Sahmani S, Safaei B. Nonlinear buckling mode transition analysis in nonlocal couple stress-based stability of FG piezoelectric nanoshells under thermo-electromechanical load. Mech Adv Mater Struct. 2022 May;1–21.10.1080/15376494.2022.2073620Search in Google Scholar

[72] Hou D, Wang L, Yan J. Vibration analysis of a cylindrical shell by using strain gradient theory via a moving Kriging interpolation-based meshfree method. Thin-Walled Struct. 2023;184:110466.10.1016/j.tws.2022.110466Search in Google Scholar

[73] Gu L. Moving kriging interpolation and element‐free Galerkin method. Int J Numer Methods Eng. 2003;56(1):1–11.10.1002/nme.553Search in Google Scholar

[74] Thai CH, Do VNV, Nguyen-Xuan H. An improved moving kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates. Eng Anal Bound Elem. 2016 Mar 1;64:122–36.10.1016/j.enganabound.2015.12.003Search in Google Scholar

[75] Miller RE, Shenoy VB. Size-dependent elastic properties of nanosized structural elements. Nanotechnology. 2000;11(3):139–47.10.1088/0957-4484/11/3/301Search in Google Scholar

[76] Zhu R, Pan E, Chung PW, Cai X, Liew KM, Buldum A. Atomistic calculation of elastic moduli in strained silicon. Semicond Sci Technol. 2006;21(7):906–11.10.1088/0268-1242/21/7/014Search in Google Scholar

[77] Huang H, Han Q. Buckling of imperfect functionally graded cylindrical shells under axial compression. Eur J Mech. 2008;27(6):1026–36.10.1016/j.euromechsol.2008.01.004Search in Google Scholar

[78] Sun J, Wang Z, Zhou Z, Xu X, Lim CW. Surface effects on the buckling behaviors of piezoelectric cylindrical nanoshells using nonlocal continuum model. Appl Math Model. 2018;59:341–56.10.1016/j.apm.2018.01.032Search in Google Scholar

Received: 2023-01-16

Revised: 2023-03-22

Accepted: 2023-04-14

Published Online: 2023-06-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ntrev-2022-0551

Keywords for this article

nanostructures; 3D elasticity; nonlinear buckling; surface residual stress; meshfree numerical technique

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