1 Introduction

Due to continuous evolution of nano electro mechanical systems (NEMS), structures at the scale of microns and sub-microns are viewed as the chief components which cause to promote greatly the advancement of several nanosystems and nanodevices ^[1-7]. According to experimental studies, it has been demonstrably found that the mechanical response of nanostructures is affected by some size effects. Because the classical continuum theory is a scale independent theory, it does not have the capability to consider this inherent size-dependent behavior of structures at nanoscale.

In this respect, some unconventional continuum theories have been developed and utilized to accommodate the size-dependency observed in nanoscaled structures ^[8-26]. More recently, Rabinson and Adali^[27] evaluated the buckling loads of carbon nanotubes under the combined concentrated and triangularly distributed axial compression loading condition based on the nonlocal Euler-Bernoulli beam model. Radić and Jeremić^[28] studied the thermal buckling behavior of double-layered graphene sheets using nonlocal continuum elasticity within the framework of a new first-order shear deformation theory. Wang et al.^[29] obtained the boundary conditions related to a microplate modeled by a strain gradient elasticity theory. Sahmani et al.^[30] analyzed the size dependency of surface stress in the axial postbuckling of piezoelectric nanoshells on the basis of the Gurtin-Murdoch elasticity theory. Akgoz and Civalek^[31] predicted the static bending behavior of carbon nanotubes on the elastic foundation based on the modified strain gradient elasticity theory. Mirsalehi et al.^[32] used the spline finite strip technique to anticipate the mechanical stability and free vibration response of microplates made of functionally graded material (FGM) on the basis of the strain gradient elasticity theory.

As it has been observed in the previous investigations, in comparison with the classical continuum theory, the nonlocal elasticity of Eringen^[33] generally results in a softening influence, while the use of strain gradient elasticity^[34] leads to a stiffening effect. Askes and Aifantis^[35] used the nonlocal elasticity and strain gradient elasticity to study the wave dispersion in carbon nanotubes. Subsequently, Lim et al.^[36] proposed a new size-dependent elasticity theory, namely, a nonlocal strain gradient theory which includes both the softening and stiffening influences to describe the size dependency in a more accurate way. Afterwards, some studies have been carried out to adopt the nonlocal strain gradient elasticity theory in order to predict the size-dependent mechanical characteristics of nanostructures. Li et al.^[37] performed an analysis for longitudinal vibrations of nanorods on the basis of nonlocal strain gradient theory. They also used this theory to find the size-dependent natural frequencies of FGM Timoshenko nanobeams^[38]. Tang et al.^[39] investigated the viscoelastic wave propagation in embedded carbon nanotubes with the nonlocal strain gradient theory of elasticity. Li and Hu^[40] analyzed the small scale effects on the postbuckling behavior of FGM nanobeams based on the nonlocal strain gradient elasticity theory. Xu et al.^[41] analyzed the nonlinear bending and buckling features of nanoscaled Euler-Bernoulli beams within the framework of the nonlocal strain gradient theory. Sahmani and Aghdam ^[42-43] developed a size-dependent shell model based on the nonlocal strain gradient theory of elasticity for axial and radial nonlinear instability of multilayer nanoshells. Li et al.^[44] used the nonlocal strain gradient elasticity theory to predict the size-dependent bending, buckling and vibration of axially FGM nanobeams. Sahmani and Aghdam^[45] explored the nonlinear instability of magneto-electro-elastic composite nanoshells with the theory of nonlocal strain gradient. Zhu and Li ^[46-47] proposed a size-dependent integral elasticity model to analyze the tension and longitudinal dynamics of a nanorod based on the nonlocal strain gradient theory of elasticity. Sahmani and Aghdam^[48] investigated the nonlinear vibration response of the pre-and post-buckled multilayer functionally graded nanobeams based on the nonlocal strain gradient elasticity theory.

In the present study, based on the nonlocal strain gradient elasticity theory, both the nonlocal and length scale parameters are taken into account to anticipate more accurately the size dependency in the nonlinear buckling and postbuckling response of FGM nanoshells under the axial compressive load. A refined shell theory with exponential distribution of shear deformation in conjunction with von Karman kinematic nonlinearity is put to use. In accordance with the Mori-Tanaka technique, the gradual change in the material properties along the shell thickness is considered. On the basis of perturbation-based boundary layer solution methodology, asymptotic solutions for the independent variables of problem in conjunction with explicit expressions associated with the size-dependent nonlinear equilibrium curves are obtained for axially loaded nonlocal strain gradient FGM nanoshells.

2 Nonlocal strain gradient FGM shell model

In Fig. 1, schematic representation of an FGM nanoshell with the length L, the radius of midplane R, the thickness h and the attached coordinate system is displayed. As seen, the outer (z=-h/2) and inner (z=h/2) free surfaces of the nanoshell are metal rich and ceramic rich, respectively. It is assumed that the material properties of the functionally graded nanoshell vary continuously along the shell thickness direction. According to the Mori-Tanaka homogenization scheme^[49], the material properties including the bulk modulus and shear modulus vary along the shell thickness as below,

(1a)

(1b)

where λ, μ, and V represent the bulk modulus, shear modulus, and volume fraction, respectively. Also, the subscripts e, m, and c denote, respectively, effective, metal, and ceramic. Additionally, one will have

(2)

The volume fraction corresponding to the ceramic phase can be expressed as

(3)

in which k stands for the material property gradient index. As a consequence, Young's modulus and Poisson's ratio of an FGM nanoshell can be written, respectively, as

(4)

(5)

On the basis of a refined shell theory incorporating an exponential distribution of the shear deformation, the components of displacement field along different coordinate directions can be given as^[50]

(6a)

(6b)

(6c)

where u, v, and w in order denote the mid-plane displacements along x-, y -, and z-directions. Moreover, ψ_x and ψ_y are the rotations of the mid-plane normal about the y-and x-directions, respectively.

According to the von Karman kinematics nonlinearity, the strain components of nanoshells in terms of displacement field can be written as

(7a)

(7b)

in which ε_ij⁰, κ_ij⁽¹⁾, and κ_ij⁽²⁾(i, j=x, y) represent, respectively, the mid-plane strain components, the first-order curvature components, and the higher-order curvature components.

It has been observed that small scale effects may cause a softening or stiffening influence before. Motivated by this fact, Lim et al.^[36] proposed a new unconventional continuum theory, namely, a nonlocal strain gradient elasticity theory, which contains both nonlocal and strain gradient size effects simultaneously. As a result, the total nonlocal strain gradient stress tensor Λ can be expressed as below^[36],

(8)

where σ and σ^* in order denote the stress and higher-order stress tensors which can be expressed as follows:

(9a)

(9b)

in which C is the stiffness matrix, ρ₁ and ρ₂ are, respectively, the principal attenuation kernel function including the nonlocality and the additional kernel function associated with the nonlocality effect of the first-order strain gradient field, X and X' in order represent a point and any point else in the body, and l stands for the internal length scale parameter. Also, ∇ is the gradient symbol. Following the method of Eringen and assuming that ρ₁=ρ₂=ρ, the constitutive relationship corresponding to the total nonlocal strain gradient stress tensor of a two-dimensional material can be obtained as

(10)

where e₀θ represents the nonlocal parameter in such a way that θ is an internal characteristic constant and e₀ is a constant related to the selected material. In addition, ∇² denotes the Laplacian operator. As a result, the nonlocal strain gradient constitutive relations for an FGM nanoshell can be rewritten as

(11)

in which

In accordance with the nonlocal strain gradient exponential shear deformation shell model, the total strain energy of an FGM nanoshell can be given as

(12)

where the stress resultants are in the following forms:

(13a)

(13b)

(13c)

(13d)

in which

(14a)

(14b)

and

(15a)

(15b)

(15c)

(15d)

(15e)

By applying the virtual work's principle to the total strain energy of the FGM exponential shear deformable nanoshell, the non-classical governing differential equations are derived as

(16a)

(16b)

(16c)

(16d)

(16e)

With the aim of satisfaction of the first two differential equations at hand, the Airy stress function f(x, y) is introduced as below,

(17)

Additionally, for a perfect shell-type structure, the compatibility equation relevant to the mid-plane strain components can be read as

(18)

Therefore, through inserting Eq. (17) into the inverse of Eq. (13) and then using Eqs. (16) and (18), the nonlocal strain gradient governing equations can be presented as functions of the displacement field as follows:

(19a)

(19b)

(19c)

(19d)

where the parameters φ_i (i=1, 2, …, 23) are introduced in Appendix A.

Also, the edge supports at the left and right ends of the FGM nanoshells are assumed to be clamped. As a result, one will have w=0, at x=0, L. On the other hand, the equilibrium in the x-direction can be expressed as

(20)

The periodicity condition relevant to a closed shell-type structure reads

(21)

which can be rewritten as

(22)

Furthermore, the unit end-shortening related to the movable boundary conditions at the left and right ends of an FGM exponential shear deformable nanoshell can be given as

(23)

3 Solving process 3.1 Boundary layer theory of nonlocal strain gradient shell buckling

Firstly, the following dimensionless parameters are put to use in order to obtain the asymptotic solutions of the problem in a more general framework:

(24)

in which A₀₀=(λ_m+2μ_m)h. As a consequence, the nonlinear nonlocal strain gradient governing differential equations can be deduced in boundary layer-type forms as below,

(25a)

(25b)

(25c)

(25d)

The clamped boundary conditions at the left and right ends of nanoshells take the dimensionless form as W=0, at X=0, π.

Moreover, the dimensionless load-equilibrium relationship along the x-direction takes the following form:

(26)

In a similar way, the periodicity condition and the unit end-shortening of an FGM exponential shear deformable nanoshell in dimensionless forms can be rewritten, respectively, as follows:

(27)

(28)

3.2 Perturbation-based solution methodology

The small perturbation parameter ε has been used to construct the boundary layer-type nonlocal strain gradient governing equations (25) before. Now, a two-stepped perturbation technique ^[51-57] is employed, based on the fact that the independent variables are summarized via the summations of the regular and boundary layer solutions as follows:

(29a)

(29b)

(29c)

(29d)

where the accent character -stands for the regular solution, and the accent characters ~ and represent the boundary layer solutions associated with the left (X=0) and right (X=π) ends of an FGM nanoshell, respectively.

Therefore, each part of the solutions can be altered to the perturbation expansions as below,

(30)

in which ξ and ς denote the boundary layer variables as follows:

(31)

Afterwards, in order to extract the sets of perturbation equations corresponding to both regular and boundary layer solutions, Eqs. (29) and (30) are inserted in the nonlocal strain gradient governing equations (25), and then the expressions with the similar order of ε are collected. A tolerance limit smaller than 0.001 is considered to determine the maximum order of ε associated with the convergence of the solution methodology.

An initial buckling mode shape for the FGM exponential shear deformable nanoshell is assumed to continue the procedure as

(32)

Now, some mathematical calculations are carried out to obtain the asymptotic solutions corresponding to each independent variable of the problem which is presented in Appendix A. Subsequently, substitution of them into Eqs. (26) and (28) and then rearranging with respect to the second perturbation parameter (A₁₁⁽²⁾ε) yield explicit expressions for the nonlocal strain gradient load-deflection and load-shortening stability paths, respectively, as follows:

(33)

(34)

where the parameters given in the above equations are introduced in Appendix B. Thereafter, it is supposed that the dimensionless coordinates of the point relevant to the maximum deflection of an FGM nanoshell are (X, Y)=(π/(2m), π/(2n)). As a result, one will have

(35)

where w_m represents the maximum deflection of an FGM nanoshell, and the symbols S₁ and S₂ are presented in Appendix B.

4 Numerical results and discussion

With the aid of the developed nonlocal strain gradient FGM shell model, the nonlinear axial instability characteristics of exponential shear deformable FGM nanoshells are anticipated in this section. The material properties of ceramic phase (silicon) and metal phase (aluminum) of FGM nanoshells are tabulated in Table 1. Moreover, in all of the preceding results, the geometric parameters of nanoshells are selected in such a way that R/h=50 and L/R=2.

In Fig. 2, the nonlocal strain gradient load-deflection responses of FGM nanoshells are depicted corresponding to various values of the nonlocal parameter and internal strain gradient length scale parameter. It is observed that the strain gradient size effect causes to increase the critical axial buckling load, while the nonlocality leads to reduce it. Also, the internal strain gradient length scale parameter decreases the minimum postbuckling load, but increases the associated maximum deflection. However, this pattern is vice versa for the nonlocality size dependency. Furthermore, it can be observed that the influence of the nonlocal type of small scale effect on the axial instability characteristics of nanoshells is more than that of the strain gradient one.

Figure 3 displays the nonlocal strain gradient load-shortening responses of axially loaded FGM nanoshells with different values of nonlocal and internal strain gradient length scale parameters. It is revealed that strain gradient size dependency causes to increase the width of snap-through phenomenon related to the postbuckling regime, while the nonlocality size effect has an opposite influence. Additionally, the both types of small scale effect have no influence on the slope of prebuckling part of the load-shortening stability path. However, by taking the internal strain gradient length scale parameter into account, the shortening of FGM nanoshell at the critical point increases. However, with consideration of the nonlocal parameter, it reduces.

In Figs. 4 and 5, the buckling mode shapes of a nonlocal strain gradient FGM nanoshell at the postbuckling domain and in the vicinity of the critical buckling point are illustrated corresponding to various nonlocal parameters and internal strain gradient length scale parameters, respectively. It is found that the nonlocality size dependency leads to decrease the pick of deflection in the buckling mode shape of the nanoshell, while the strain gradient size effect causes to increase it. Furthermore, as supposed in the solving process, it can be seen that for all values of nonlocal and internal strain gradient length scale parameters, the maximum deflection of the nanoshell occurs at a point with the dimensionless coordinate of X=π/(2m), where m=2.

In order to predict how the shell thickness affects the significance of size-dependent behavior of axially loaded FGM nanoshells, the critical axial buckling load ratio is defined as the buckling load obtained with the local shell model divided by the one obtained with the nonlocal strain gradient shell model.

Figure 6 shows the variation of critical axial buckling load ratio with the small scale parameter (nonlocal or internal length scale parameter) for FGM nanoshells with different shell thicknesses. It is demonstrated that, with the increase in the shell thickness, the slope of the variation reduces, which indicates the reduction of size dependency behavior of thicker nanoshells. Moreover, it is indicated that the significance of nonlocality size effect on the critical axial buckling load of an FGM nanoshell is higher than that of the strain gradient small scale effect. Here, it is seen that the first one for e₀θ=4 nm is equal to 8.26% reduction in the buckling load, while the second one for l=4 nm is equal to 6.34% increment in the buckling load.

Plotted in Figs. 7 and 8 are, respectively, the nonlocal strain gradient load-deflection and load-shortening stability paths of FGM nanoshells with various values of material property gradient index and different nonlocal and internal strain gradient length scale parameters. It can be observed that, by increasing the value of material property gradient index (moving from the ceramic rich nanoshell to the metal rich one), the influences of both nonlocality and strain gradient small scale effects on the axial instability characteristics of FGM nanoshells reduce, especially their influences on the minimum postbuckling load and the associated maximum deflection. This pattern is more considerable for the strain gradient size effect than the nonlocal one.

5 Concluding remarks

The primary objective of the present study is to analyze more accurately the size dependency in the nonlinear buckling and postbuckling behaviors of FGM nanoshells. Therefore, the nonlocal strain gradient elasticity theory incorporating simultaneously both the stiffening and softening influences of size effects is implemented within the framework of a refined exponential shear deformation shell theory to construct a comprehensive size-dependent shell model. By using a two-stepped perturbation technique in conjunction with a boundary layer theory of shell buckling, explicit expressions for nonlocal strain gradient stability paths are proposed.

It is displayed that the strain gradient size effect causes to increase the critical axial buckling load, while the nonlocality leads to reduce it. Also, the internal strain gradient length scale parameter decreases the minimum postbuckling load, but increases the associated maximum deflection. In addition, it is found that strain gradient size dependency causes to increase the width of snap-through phenomenon related to the postbuckling regime, while the nonlocality size effect has an opposite influence. Also, the both types of small scale effect have no influence on the slope of prebuckling part of the load-shortening stability path. Moreover, it is indicated that the nonlocality size dependency leads to decrease the pick of deflection in the buckling mode shape of nanoshell, while the strain gradient size effect causes to increase it.

It is revealed that both nonlocal and strain gradient size dependencies reduce in the axial instability behavior of thicker nanoshells. Furthermore, it is demonstrated that the significance of nonlocality size effect on the critical axial buckling load of an FGM nanoshell is higher than that of the strain gradient small scale effect. Moreover, it is seen that, by increasing the value of material property gradient index (moving from the ceramic rich nanoshell to the metal rich one), the influences of both the nonlocality and strain gradient small scale effects on the axial instability characteristics of FGM nanoshells reduce.

Appendix A

(A1)

It should be noted that the parameters φ_i (i=1, 2, …, 23) are the dimensionless forms.

The solutions in asymptotic forms corresponding to each of independent variables are extracted as below,

(A2)

(A3)

(A4)

(A5)

in which

(A6)

Appendix B

(B1)

(B2)

(B3)

(B4)

(B5)

(B6)

in which

(B7)

where U_i (i=0, 1, …, 9) are constant parameters extracted by the perturbation sets of equations,

(B8)

(B9)