A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

In this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.


Introduction
Functionally graded (FG) materials are an advanced class of composite structures with material properties changing continuously in one or more directions. In recent years, with the rapid development of nanotechnology, FG nanoscale and microscale structures with potential applications in nanoelectromechanical systems (NEMS) and microelectromechanical systems (MEMS) have attracted a great deal of attention. Consequently, investigating the size-dependent mechanical behaviors of FG nano-/micro-structures is of great importance for better understanding and designing those small-scaled systems. Since the experimental study in nanoscale may be technically difficult and financially expensive, and molecular dynamics simulation is timeconsuming for analyzing large size system, the continuum mechanics approach, as an alternative way to model the mechanical response of nano-/micro-structures, has been widely used. It is well-known that classical continuum mechanics is size-independent and cannot capture the size effects in small scale. In this regard, several non-classical continuum theories have been developed to assess the remarkable size effects on the mechanical characteristics of nano-/micro-structures, such as nonlocal elasticity theory [1][2] , strain gradient elasticity theory [3][4][5][6] , and surface elasticity theory [7][8] .
Based on the nonlocal elasticity theory and strain gradient elasticity theory, a large number of works on the size-dependent analysis of nano-/micro-structures have been carried out in the last two decades (see review articles [9]- [11]), these works have shown that the nonlocal theory captures only the stiffness-softening effect and the strain gradient theory captures only the stiffness-hardening effect. Recently, the nonlocal elasticity theory and strain gradient elasticity theory have been combined into a single theory, namely, the nonlocal strain gradient theory (NSGT) [12] . NSGT takes not only the nonlocal stress field but also the strain gradient stress field into account, which is capable of describing both stiffness-softening effect and stiffnesshardening effect. Based upon the NSGT, numerous works have been performed to investigate the size-dependent linear or nonlinear bending, buckling, vibration, and wave propagation of FG small-scaled beams [13][14][15][16][17][18][19][20][21][22][23][24] , plates [25][26][27][28][29][30][31][32] , and shells [33][34][35][36][37] . To mention a few, She et al. [38] analyzed the wave propagation of porous FG nanotubes with the help of NSGT and a refined beam model. Sobhy and Zenkour [39] investigated the buckling and vibration behaviors of a double-layered FG porous nanoplate via NSGT in conjunction with a quasi-3D refined theory. Moreover, by using NSGT and first-order shear deformation shell theory, the free and forced vibrations of porous FG cylindrical nanoshells were studied by Barati [40] and Faleh et al. [41] , respectively. In another work, Ma et al. [42] investigated the wave propagation characteristics in magneto-electro-elastic nanoshells within the framework of NSGT.
On the other hand, it is known that for a solid with large surface area to bulk volume ratio, the atom arrangement and material properties of the surface are different from those in the bulk part, which makes the mechanical behavior of the solid become unusual compared with conventional structures. To capture the surface effects, Gurtin and Murdoch [7][8] proposed the surface elasticity theory, which defines the surface as a membrane without thickness, and suggests that the surface has different material properties and constitutive relations than the bulk part. Based on the surface elasticity theory, a large amount of research has focused on the analysis of surface effects on the mechanical response of FG nanostructures [43][44][45][46][47][48][49][50][51][52] . For example, Zhu et al. [53] investigated surface energy effects on the torsional buckling of FG cylindrical nanoshells covered with piezoelectric nano-layers based on the surface elasticity theory. Norouzzadeh and Ansari [54] analyzed the size-dependent vibration characteristics of FG rectangular and circular nanoplates in the framework of nonlocal elasticity and surface elasticity theories. In a recent work, Attia and Abdel-Rahman [55] studied the simultaneous effects of the microstructure rotation and surface energy on the vibration of FG viscoelastic nanobeams by using the modified couple stress theory and surface elasticity theory.
From the above-mentioned, we can find that NSGT and surface elasticity theory describe the remarkable size effects at small scale in totally different ways. Therefore, there is a scientific need to explore the combined effects of nonlocal stress, strain gradient, and surface energy by using NSGT in conjunction with the surface elasticity theory. Recently, the combined effect of nonlocal stress, strain gradient, and surface energy on the mechanical behaviors of homogenous nanoplates and nanoshells have been addressed by some researchers [56][57][58] . However, to the best of the authors' knowledge, no works have been concerned with combining NSGT with surface elasticity theory to assess the size effects in FG nanoshells so far. In this regard, the primary objective of the present work is to develop a nonlocal strain gradient shell model including surface effects for the size-dependent analysis of FG cylindrical nanoshells. To achieve this goal, firstly, NSGT in conjunction with surface elasticity theory will be applied to establish the governing equations in Section 2. Afterwards, closed-form solutions for the free vibration problem will be formulated in Section 3. Then, a comparative study is performed in Section 4 to examine the validity of the proposed model and the accuracy of the analytical method. Next, the effects of various parameters, such as nonlocal parameter, material length scale parameter, radius-to-thickness ratio, length-to-radius ratio, power-law index, and surface energy on the vibration response of FG cylindrical nanoshells are investigated in Section 5. Finally, the main conclusions of the present work are summarized in Section 6.
2 Nonlocal strain gradient shell model incorporating surface effects As depicted in Fig. 1, an FG cylindrical nanoshell of length L, thickness h, and radius R is considered. The coordinate system (x, θ, z) is established in the mid-plane of the nanoshell, and the x-, θ-, and z-axes are taken along the axial, circumferential, and radical directions, respectively. It is assumed that the FG cylindrical nanoshell is made of a mixture of ceramics and metals, and the material properties vary continuously from metals at the inner surface (z = −h/2) to ceramics at the outer surface (z = h/2) along the thickness direction according to a power-law distribution. Thus, the effective Young's modulus E(z), effective Poisson's ratio µ(z), and effective mass density ρ(z) of the FG cylindrical nanoshell can be written as follows [59] : where ξ is the power-law index (0 ξ < ∞). The subscripts "c" and "m" stand for the ceramic and metal constituents, respectively.

Kinematics
Based on the FSDT, the displacement field of a cylindrical shell is given by where u(x, θ, t), v(x, θ, t), and w(x, θ, t) are the displacements in axial, circumferential, and radical directions, respectively. ϕ x (x, θ, t) and ϕ θ (x, θ, t) represent the rotations about θ-and x-axes, respectively. Accordingly, the strain field can be written as

Constitutive relations
In the current work, constitutive relations of the FG cylindrical nanoshell including surface effects are established within the framework of NSGT. To account for surface effects, the surface elasticity theory proposed by Gurtin and Murdoch [7][8] is applied. According to the surface elasticity theory, the FG cylindrical nanoshell is assumed to be composed of a bulk part and two thin surface layers (inner surface z = −h/2 and outer surface z = h/2). The two surface layers are treated as zero thickness films, and perfectly adhere to the underlying bulk material without slipping. In doing this, the constitutive relations of the two surface layers are introduced as where λ s± and µ s± denote the surface Lamé constants, τ s± represents the surface residual stress, and δ αβ is the Kronecker delta. Note that the superscript s± stand for the outer surface and inner surface, respectively. In surface elasticity theory, the surface balance conditions cannot be satisfied due to the fact that the stress component σ zz is usually neglected in classical continuum mechanics. To improve this weakness, following Lu et al. [60] , it is assumed that σ zz varies linearly through the thickness and satisfies the surface balance conditions, and therefore σ zz is expressed as By substituting Eq. (5) into Eq. (6), σ zz can be rewritten as in which ρ s+ and ρ s− are the surface mass densities of the outer surface and inner surface, respectively.
By taking σ zz into account, the constitutive relations of the bulk part in surface elasticity theory are given by [60] On the other hand, the constitutive relation with the framework of NSGT is written as [12] 1 where ∂θ 2 is the Laplacian operator in cylindrical coordinate system, t ij is the total stress, C ijkl is the elastic modulus, and ε kl is the strain. (ea) is the nonlocal parameter introduced to describe the nonlocal effect, and l is the material length scale parameter involved to capture the strain gradient effect. It is seen that the remarkable nonlocal effect and strain gradient effect in small scale can be captured by the NSGT simultaneously.
By applying Eq. (9) to both the bulk part and two surface layers of the FG cylindrical nanoshell, and substituting Eqs. (3) and (4) into the stress components, the constitutive equations based on the NSGT including surface effects can be obtained as follows:

Variational formulation
The equilibrium equations for the vibration of FG cylindrical nanoshell will be formulated using Hamilton's principle. According to the Hamilton's principle, one can get that where δU , δK, and δW are the first variation of strain energy, kinetic energy, and work done by external forces, respectively. Due to the surface effects, the strain energy should include both the bulk part and two surface layers. Thus, we have in which the resultant forces and bending moments are defined as where κ is the shear correction factor. Similarly, the kinetic energy should also consider the bulk part as well as the two surface layers, which leads to where The first variation of the work done by external forces can be written as where f x , f θ , and f z are the distributed axial, circumferential, and radical forces, respectively. Substituting Eqs. (22), (26), and (28) into Eq. (21) and integrating by parts, the equilibrium equations can be obtained as δv : δw : The associated boundary conditions are given as follows: in which (n x , n θ ) are the directional cosines of the outward unit normal to the boundary of the mid-plane.

Closed-form solutions for natural frequencies
In this section, closed-form solutions for natural frequencies of the FG cylindrical nanoshells under various boundary conditions are obtained through an analytical approach. Three typical types of boundary condition are considered, which are simply supported-simply supported (SS-SS), clamped-clamped (C-C), and clamped-simply supported (C-SS). In order to solve the governing equations, the displacement components are expressed as the following double Fourier series form [40][41] : where U mn , V mn , W mn , Φ xmn , and Φ θmn are unknown constants, m and n are two positive integers denoting the wave numbers in the axial and circumferential directions, respectively, and X m (x) is the axial modal function. The explicit forms of X m (x) for different boundary conditions are given as follows [62][63][64][65] : For the SS-SS boundary condition, For the C-C and C-SS boundary conditions, integrating along the length, we can obtain that where K 5×5 and M 5×5 are the stiffness matrix and the mass matrix, respectively. The elements of the two matrices are given in Appendix B. By setting the determinant of the coefficient matrix of Eq. (66) to be zero, the natural frequencies of FG cylindrical nanoshells can be obtained.

Comparative study
Before carrying out the numerical analyses, the validity of the present model and the accuracy of the analytical method are examined by comparing the degenerated results with those available in the literature. In Table 1, natural frequencies of an FG macroscale cylindrical shell with SS-SS boundary condition predicted by the present model (by ignoring all the size effects) are compared with those obtained by Loy et al. [59] . For comparison purpose, the inner surface of the FG cylindrical shell is assumed to be made of stainless steel with material properties: E m = 207.788 GPa, µ m = 0.317 756, and ρ m = 8 166 kg·m −3 , and the outer surface of the FG cylindrical shell is assumed to be made of Nickel with material properties: E c = 205.098 GPa, µ c = 0.31, and ρ c = 8 900 kg·m −3 . It is seen from Table 1 that the present results are in good agreement with the results of Loy et al. [59] for different circumferential wave numbers, power-law indices, and thickness-to-radius ratios.
Ref. [ To examine the accuracy of the present analytical method for more complex boundary conditions, Table 2 compares the dimensionless natural frequencies of a homogenous macroscale cylindrical shell, under SS-SS, C-SS, and C-C boundary conditions, obtained by the present analytical method with those obtained by Loy et al. [66] using generalized differential quadrature (GDQ) method and Zhang et al. [67] through wave propagation approach. One can find that for the three types of boundary conditions considered, the present analytical solutions agree well with the results obtained by the GDQ method and the wave propagation method.
As the third example, a comparative study is performed to verify the accuracy of the present model in capturing size effects. For this objective, Table 3 tabulates the dimensionless natural frequencies of a homogenous nanoscale cylindrical shell with SS-SS and C-C boundary conditions predicted by the present work and Ghorbani et al. [58] . Results are presented for different values of nonlocal parameter and material length scale parameter with and without surface effects. Material properties are taken as: E = 210 GPa, ρ = 2 331 kg·m −3 , µ = 0.24, λ s = −4.488 N·m −1 , µ s = −2.774 N·m −1 , τ s = 0.605 N·m −1 and ρ s = 3.17 × 10 −7 kg·m −3 . It can be seen from Table 3 that the present results have a good consistency with those of Ghorbani et al. [58] for both SS-SS and C-C boundary conditions. From the above comparisons, the reliability of the present model and analytical method are well-validated.

Results and discussion
In this section, based on the proposed size-dependent shell model, the effects of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, lengthto-radius ratio, and surface effects on the free vibration behavior of FG cylindrical nanoshells with different boundary conditions (SS-SS, C-SS, and C-C) are investigated. The ceramic and metal constituents are selected as Silicon and Aluminum, respectively. Material properties of Silicon and Aluminum are given in Table 4. In the numerical analyses, the thickness of the FG cylindrical nanoshell is considered as h = 1 nm, axial wave number is fixed at m = 1, and the shear correction factor is taken as κ = 5/6. Table 4 Material properties of silicon and aluminum [55,68] Silicon To illustrate the numerical results and to highlight the size effects, the dimensionless natural frequency ̟ mn and frequency ratio are defined as Frequency ratio = ̟ mn predicted by NSGT with/without surface effects ̟ mn predicted by classical continuum theory .
Note that the results of the classical continuum theory can be acquired from the present model by setting the two scale parameters and all the surface elastic constants to be zero.

Benchmark results
Tables 5-7 illustrate the dimensionless natural frequencies of an FG cylindrical nanoshell based on the NSGT without surface effects for SS-SS, C-SS, and C-C boundary conditions, respectively. The geometric parameters are taken as: R/h = 10 and L/R = 5. Results are presented for different values of normalized nonlocal parameter ((ea)/h), normalized material length scale parameter (l/h), circumferential wave number (n) and power-law index (ξ). Note that when ξ = 0, the nanoshell is made of pure silicon, and the inner and outer surface have the same surface material properties. It can be seen from Tables 5-7 that an increase in the nonlocal parameter or power-law index leads to a decrease in the natural frequency; conversely, an increase in the strain gradient parameter results in an increase in the natural frequency. Moreover, with the increase in the circumferential wave number, the natural frequencies first decrease and then increase. In particular, the fundamental frequency always occurs at n = 2. On the other hand, we can find that for the three types of boundary condition considered, the natural frequencies associated with the C-SS boundary condition are higher than those of the SS-SS boundary condition, but lower than those of C-C boundary condition, which indicates that stiffer boundary condition produces higher natural frequencies. The corresponding dimensionless natural frequencies of the FG cylindrical nanoshell based on NSGT with surface effects for different boundary conditions are tabulated in Tables 8-10. Comparing the results with surface effects with those without surface effects, it can be found that surface effects may affect the circumferential wave number at which the fundamental frequency occurs. For instance, when ignoring surface effects, the fundamental frequency of the SS-SS FG cylindrical nanoshell with ξ = 1 and (ea) = 0 occurs at n = 2. However, by taking surface effects into account, the fundamental frequency occurs at n = 1. Moreover, the results tabulated in Tables 5-10 can be used as benchmarks for prospective researchers to compare their results.

Coupling effect of nonlocal parameter and material length scale parameter
To explore the coupling effect of nonlocal parameter and material length scale parameter on the vibrational behavior of FG cylindrical nanoshell under different boundary conditions, Fig.  2 plots the variations of the frequency ratio as a function of the scale parameter ratio (l/(ea)) for SS-SS, C-SS, and C-C boundary conditions, when R/h = 10, L/R = 2, ξ = 1, and n = 3. Surface effects are neglected in this example. It is seen that, for SS-SS boundary condition (see Fig. 2(a)), when the nonlocal parameter is smaller than the material length scale parameter (l/(ea) < 1), the FG cylindrical nanoshell exhibits a stiffness-softening behavior, leading to the natural frequencies predicted by NSGT always lower than those of classical continuum theory. When the nonlocal parameter is larger than the material length scale parameter (l/(ea) > 1),           the FG cylindrical nanoshell exerts a stiffness-hardening behavior, resulting in the natural frequencies of NSGT always higher than those of classical continuum theory. In particular, when the two scale parameters are equal (l/(ea) = 1), the results of NSGT reduce to those of classical continuum theory. These observations have been reported by many researchers [13,14,[69][70] .
Nevertheless, for C-SS and C-C boundary conditions (see Figs. 2(b) and 2(c)), it is interesting to see that, when l/(ea) < 1, the results of NSGT can be lower or higher than or even equal to those of classical continuum theory by taking a specific scale parameter ratio, which is quite different than the case of SS-SS boundary condition. More interestingly, when l/(ea) = 1, unlike the SS-SS FG cylindrical nanoshells whose natural frequencies obtained by NSGT are identical to those of classical continuum theory, the natural frequencies of C-SS and C-C FG cylindrical nanoshells evaluated using NSGT are higher than the results of classical continuum theory. In other words, under C-SS and C-C boundary conditions, the predictions of NSGT cannot reduce to those of classical continuum theory when the nonlocal parameter equals to the material length scale parameter. From this observations, we can conclude that the coupling effect of nonlocal stress and strain gradient on the vibrational behavior of FG nanoshells depends not only on the relative magnitude of the two scale parameters but also on the boundary condition. Therefore, it is of great importance to take different boundary conditions into account when studying the mechanical behaviors of nanostructures with the framework of NSGT.

Effect of power-law index
The effect of power-law index on the frequency ratio of FG cylindrical nanoshell under different boundary conditions is shown in Figs. 3 and 4 for various normalized nonlocal parameters and material length scale parameters, respectively. Parameters R/h = 10, L/R = 5 and n = 2 are taken in this example. It is observed from Figs. 3 and 4 that when surface effects are not included, for different values of scale parameters, the frequency ratios are nearly unchanged as the power-law index increases from 0 to 10. This implies that nonlocal effect and strain gradient effect on the vibration of FG cylindrical nanoshells are not sensitive to the change in the power-law index. However, for the case of including surface effects, it can be seen that increasing the power-law index leads to an increase in the frequency ratio. In other words, surface effects on the vibration behavior of FG cylindrical nanoshell becomes more and more important as the increase in the power-law index. In addition, comparing the results of SS-SS, C-SS and C-C boundary conditions, it is clearly seen that surface effects play a more important role in the vibration of FG cylindrical shells with SS-SS boundary condition.

Effect of radius-to-thickness ratio
The effect of radius-to-thickness ratio on the frequency ratio of FG cylindrical nanoshell with SS-SS, C-SS, and C-C boundary conditions is depicted in Figs. 5 and 6 for different values of normalized nonlocal parameter and material length scale parameter, respectively. Results are obtained with L/R = 10, ξ = 1 and n = 2. It is seen that when surface effects are not considered, the frequency ratios associated with different nonlocal parameters or material length scale parameters all approach to 1 as the radius-to-thickness ratio increases from 5 to 25, which means that nonlocal effect and strain gradient effect gradually decrease with the increase in radius-to-thickness ratio. When accounting for surface effects, one can observe that the gaps between the frequency ratios with surface effects and those without surface effects are getting wider as the radius-to-thickness ratio increases, revealing that increasing the radiusto-thickness ratio will increase surface effects, and this trend is more remarkable for SS-SS boundary condition. Besides, it can be concluded from Figs. 5 and 6 that, at higher values of radius-to-thickness ratio, surface effects play a dominant role in the vibration of FG cylindrical nanoshells, and nonlocal and strain gradient effects are less important in this situation. 5.5 Effect of length-to-radius ratio Figures 7 and 8 illustrate the variations of frequency ratio with respect to the length-toradius ratio for various normalized nonlocal parameters and material length scale parameters, respectively. Same as Figs. 5 and 6, results are presented for SS-SS, C-SS, and C-C FG cylindrical nanoshells with or without surface effects. Parameters R/h = 5, ξ = 1 and n = 1 are selected in this example. It is observed that without considering surface effects, the frequency ratios corresponding to different scale parameters gradually reduce with the increase in lengthto-radius ratio for all three boundary conditions, which implies that increasing the length-toradius ratio results in the decrease in nonlocal effect and strain gradient effect on the vibrational behavior of FG cylindrical nanoshell, but compared with surface effects, nonlocal effect and strain gradient effects are not sensitive to the change in length-to-radius ratio. Under the influence of surface effects, it can be seen from Figs. 7 and 8 that as the length-to-radius ratio increases, the frequency ratios with surface effects are first lower than then higher than those without surface effects. That is to say, surface effects tend to decrease the natural frequencies of short nanoshells, but increase the natural frequencies of long nanoshells. Moreover, it is found that, at higher values of length-to-radius ratio, surface effects play a more important role than nonlocal effect and strain gradient effect in the vibration of FG cylindrical nanoshells, and similar to the case of Figs. 5 and 6, this phenomenon is more notable for SS-SS boundary condition.

Conclusions
In the present work, a size-dependent FG cylindrical shell model is developed within the framework of NSGT and surface elasticity theory. By adopting the proposed model, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-tothickness ratio, length-to-radius ratio and surface effects on the free vibration of FG cylindrical nanoshells under SS-SS, C-SS and C-C boundary conditions are investigated. From the numerical results, the following main conclusions can be drawn: (i) The coupling effect of nonlocal parameter and material length scale parameter on the vibration of FG cylindrical nanoshells depends not only on the relative magnitude of the two scale parameters but also on the boundary condition. In particular, when the two scale parameters are equal, the predictions of NSGT reduce to those of the classical continuum theory for SS-SS boundary condition. However, the results of NSGT are higher than those of the classical continuum theory for C-SS and C-C boundary conditions.
(ii) Nonlocal effect and strain gradient effect on the vibration of FG cylindrical nanoshells are not sensitive to the change in power-law index, while surface effects on the vibration of FG cylindrical nanoshells becomes more and more important with the increase in the power-law index.
(iii) Increasing the radius-to-thickness ratio or length-to-radius ratio will decrease nonlocal effect and strain gradient effect on the vibration of FG cylindrical nanoshells, whereas increase surface effects on the vibration of FG cylindrical nanoshells.
(iv) Surface effects play a more important role than nonlocal effect and strain gradient effect in the vibration of FG cylindrical nanoshells with higher values of radius-to-thickness ratio and length-to-radius ratio, and this situation is more significant for SS-SS boundary condition.