Home Technology A size-dependent 3D solution of functionally graded shallow nanoshells
Article Open Access

A size-dependent 3D solution of functionally graded shallow nanoshells

  • Joao Carlos Monge , Jose Luis Mantari EMAIL logo , Melchor Nicolas Llosa and Miguel Angel Hinostroza
Published/Copyright: November 28, 2023
Download Article (PDF)
Curved and Layered Structures
From the journal Curved and Layered Structures Volume 10 Issue 1

Abstract

An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen’s nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid for spherical and cylindrical shells, and rectangular plates. The stresses and displacements are assumed in terms of the Navier method which is applicable for simply supported structures. The derivatives in terms of thickness are approximated by the differential quadrature method (DQM). The thickness domain is discretized by the Chebyshev–Gauss–Lobatto grid distribution. Lagrange interpolation polynomials are considered as the basis function for DQM. The correct free surface boundary condition for out-of-plane stresses is considered. Several problems of isotropic and functionally graded shells subjected to different types of loads are analyzed. The results are compared with other three-dimensional solutions and higher-order theories. It is important to emphasize that the radii of curvature are crucial at nanoscale, so it should be considered in the design of nanodevices.

Keywords: nanoshell; functionally graded material; Eringen’s nonlocal elasticity theory; equilibrium equations

1 Introduction

Nanotechnology is applied in sensors, actuators, micro- and nano-electromechanical system, and atomic force microscope, and its components use different small-scale structures such as beams, plates, and shells [1]. The nanostructures can be studied by three different methods: quantum mechanics approach by solving the Schrödinger equations, molecular dynamics in which the interaction between atoms and molecules is computer simulated, and non-classical continuum mechanics (NCCM) [2]. The first two methods described incurred a higher computational cost compared to NCCM, so researchers have made valiant efforts and modifications to classical mechanics theories to incorporate the small-scale effect of the material and structure. NCCM can be classified into three different theories which are: strain gradient family, microcontinuum, and nonlocal elastic theories [1]. The strain gradient family can be divided into modified strain gradient theory [3,4] and modified couple stress [5,6]. Microcontinuum is classified into micropolar [7,8], microstretch [9], and micromorphic [10]. Nonlocal elastic theories were developed by Eringen [11,12,13] and were originally formulated in terms of an integral form and then reformulated to a differential form by introducing a Laplacian operator. Scientists have used the differential form of nonlocal theories due to its simplicity and easy extension for beams, plates, and shells. An extensive review of nonlocal theories for shells, plates, beam, frames, and rods was provided by Shariati et al. [14].

A survey on the topic reveals that several researchers have made a great contribution to the field of nonlocal theories. For example, Ansari and Rouhi [15] employed a nonlocal Flugge shell model to study the stability characteristics of single-walled carbon nanotubes under the action of axial load. Ritz method with different hyperbolic functions was utilized for evaluating different boundary conditions. Kananipour [16] investigated the static analysis of rectangular nanoplates subjected to discontinuous loads and different boundary conditions by differential quadrature method (DQM) in the framework of Kirchhoff and Mindlin plate theories. Srividhya et al. [17] developed a nonlocal nonlinear analysis of functionally graded plates subjected to static loads. The model was based on a higher order shear deformation theory (HSDT) and Von Karman nonlinear strains. Phung-Van et al. [18] employed a polynomial HSDT with five unknowns to study the free vibration and bending analysis of functionally graded carbon nano-reinforced composite nanoplates. Isogeometric analysis was employed to approximate the derivatives of the general equations from a weak-form perspective.

Thai et al. [19] proposed a simple beam theory with one unknown to study the static response and free vibration of isotropic nanobeams. Analytical solutions were derived, and more accurate results than Euler–Bernoulli were obtained. Results were comparable with Timoshenko's beam theory. Jape and Sayyad [20] applied a hyperbolic nonlocal theory for the study of functionally graded nanobeams under uniform loads. The solutions are derived in terms of Navier’s approach. Karami et al. [21] implemented the nonlocal strain gradient elasticity theory for the study of wave dispersion in anisotropic doubly-curved nanoshells employing a modified HSDT with 7 unknowns. Arefi and Zenkour [22] evaluated the free vibration analysis of sandwich nanoplates resting on Visco-Pasternak’s foundation with an elastic nano core and two piezo-electromagnetic face sheets as sensors and actors. Mematollahi et al. [23] employed the classic plate theory for analyzing the free vibration of thin nanoplates involved in a thermal environment. The governing equations were obtained in terms of a higher-order nonlocal strain gradient theory and Navier’ closed-form solution was employed. Arefi [24] studied the nonlocal electro-elastic bending of a doubly curved nanoshell resting on the Winkler–Pasternak foundation and subjected to transverse loading and voltage. The electric potential was assumed as a linear and cosine function combination, and the displacement field was approximated as the first-order shear deformation theory.

Ansari et al. [25] implement the Donnel shell theory for evaluating the free vibration of double-walled carbon nanotubes using a nonlocal elastic shell model. Van der Waals interactions between the inner and outer nanotubes were considered. The numerical solutions were given in terms of radial interpolation approximation in the framework of the DQM. Sahmani and Fattahi [26] utilized an HSDT for the study of the nonlocal nonlinear buckling and postbuckling response of temperature-dependent functionally graded carbon nanotube-reinforced composite shells under a combination of axial compression load and through-thickness heat conduction. She et al. [27] studied the free vibration of porous nanotubes which were assumed as functionally graded and vary continuously along the radial direction. The effects of nonlocal, strain gradient parameter, temperature variations, porosity volume fraction, and material variation were discussed.

Şimşek and Yurtcu [28] examined bending and buckling of a functionally graded nanobeam based on the Timoshenko and Euler–Bernoulli beam theory. The effect of nonlocal parameters, aspect ratio, and material constitution were discussed. Karami et al. [29] presented a simple model based on an HSDT for evaluating the static, stability, and free vibration of functionally graded carbon nanotubes-reinforced composite nanoplates resting on Winkler–Pasternak foundation. Thai et al. [30] employed the HSDT developed by Touratier [31] for studying bending, buckling, and free vibration of simply supported isotropic nanoplates. Ansari et al. [32] incorporated the classical Mindlin plate theory for the free vibration of single-layered graphene sheets. The boundary conditions were considered as simply supported and clamped, and the semi-analytical DQM was used for calculating the natural frequencies. The results were matched with molecular dynamic simulations to derive the approximate values of nonlocal parameters. Janghorban [33] employed an HSDT for investigating the characteristics and length parameters of the bending of functionally graded rectangular plates. Shahrbabaki [34] employed trigonometric series approximation functions in conjunction with the Galerkin approach to analyze 3D natural frequencies of nanoplates with different boundary conditions.

Curved structures are modeled by shell theories which can be derived in a two-dimensional (2D) and three-dimensional (3D) perspective. The 2D theories are developed in terms of unknown variables which are related to the midsurface of the shell panel and expansion functions toward the thickness direction of the shell. The 3D solutions [35,36,37,38,39,40,41,42] can be derived in terms of finite elements which are implemented in commercial software; however, for simple geometries, limited boundary conditions, and orthotropic laminated structures some semi-analytical and analytical methods can be formulated. The governing equations for the 3D bending of shallow shell structures have variable coefficients which depend on the thickness direction. So, the solution requires different numerical methods such as finite difference [43], radial basis functions [44,45], sampling surfaces [46,47], discrete singular convolution [48,49], and DQM [50]. The last method described was formulated by Bellman and Casti [51], and several applications can be read in the review by Bert and Malik [52].

In this article, an available non-local semi-analytical 3D solution for bending of simply supported shallow nanoshells subjected to mechanical load is presented. The constitutive equations are based on Eringen’s nonlocal theory. The equilibrium equations written in curvilinear coordinates are employed for the solutions. The system of nine equations is solved by the use of Navier-type solutions for stresses and displacements along with DQM for discretizing the thickness direction in terms of Chebyshev–Gauss–Lobatto grid distribution. Lagrange polynomials are employed as the basis function for DQM. The method is validated with other solutions and cases provided in the literature. Remarkably, it is found that the radii of curvature in crucial at the nanoscale, so it should be considered in the design of nanodevices.

2 Three-dimensional solution for nanoshells

An innovative procedure to obtain a semi-analytical three-dimensional solution for the bending analysis of simply supported shallow nanoshells subjected to mechanical pressure is presented in this article for the first time. The nonlocal effect for stresses is included in the study of the constitutive equations. The presented model is only valid for spherical, cylindrical panels, and rectangular plates. A shell panel is described in terms of orthogonal curvilinear coordinates ( α , β , z ) , and the representation for a spherical shell panel is shown in Figure 1. The length of the shell is denoted as “a,” and the width is represented as “b,” The radii of curvature “R α ” and “R β ” are considered as constant among the midsurface “Ω k ” The global thickness of the shell is denoted as “ h .”

Figure 1 The representation of a doubly curved shell.
Figure 1

The representation of a doubly curved shell.

2.1 Linear strains relations, constitutive equations, and equilibrium equations

The linear strain relations for shallow shell panels were reported and derived in classical textbooks [53,54,55,56], and they are described in what follows:

(1) ε α α ε β β γ α β γ α z γ β z ε z z = 1 H α ∂ ∂ α 0 1 H α R α 0 1 H β ∂ ∂ β 1 H β R β 1 H β ∂ ∂ β 1 H α ∂ ∂ α 0 ∂ ∂ z − 1 H α R α 0 1 H α ∂ ∂ α 0 ∂ ∂ z − 1 H β R β 1 H β ∂ ∂ β 0 0 ∂ ∂ z u v w .

The metric parameters are related to the radii of curvature, and the thickness coordinate of the panel and are written as

(2) H α = 1 + z R α H β = 1 + z R β .

The metric coefficients “ H α ” and “ H β ” establish the type of geometry to be considered. If one of the radii of curvature is considered infinite, the structure has a cylindrical geometry. A rectangular plate is prescribed when both radii of curvatures are infinite. So, these general equations can be applied to spherical, cylindrical, and rectangular plates by degenerating the radii of curvature.

The constitutive equations are used for calculating the stresses, which are related to the linear strain as follows:

(3) ( 1 − μ ∇ 2 ) σ α α σ β β τ α β τ α z τ β z σ z z = C 11 C 12 0 0 0 C 13 C 12 C 22 0 0 0 C 23 0 0 C 66 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 44 0 C 13 C 23 0 0 0 C 33 ε α α ε β β γ α β γ α z γ β z ε z z = σ α α NL σ α α NL τ α β NL τ α z NL τ β z NL σ z z NL .

The parameter μ is considered as a nanometric factor, and its numerical value is considered as 0 ≤ μ ≤ 4 n m 2 [57]. The nonlocal stresses are represented as [ σ α α NL , σ β β NL , σ zz NL , τ α β NL , τ α z NL , τ β z NL ] . The Laplacian operator ∇ 2 is written in terms of the midsurface coordinates and is expressed as:

(4) ∇ 2 = 1 H α 2 ∂ 2 ∂ α 2 + 1 H β 2 ∂ 2 ∂ β 2 .

The stiffness coefficients C ij can be calculated in the following manner:

(5) C 11 = C 22 = C 33 = ( 1 − ν ) E ( 1 + ν ) ( 1 − 2 ν ) C 12 = C 13 = C 23 = ν E ( 1 + ν ) ( 1 − 2 ν ) C 44 = C 55 = C 66 = E 2 ( 1 + ν ) .

The equilibrium equations are developed in orthogonal curvilinear ( α , β , z ) and are valid for doubly curved shell panels with constant curvature. Several applications can be found in refs. [58–61]. The equilibrium equations are reported as follows:

(6) H β ∂ σ α α ∂ α + H α ∂ τ α β ∂ β + H α H β ∂ τ α z ∂ z + 2 H β R α + H α R β τ α z = 0 H β ∂ τ α β ∂ α + H α ∂ σ β β ∂ β + H α H β ∂ τ β z ∂ z + 2 H α R β + H β R α τ β z = 0 H β ∂ τ α z ∂ α + H α ∂ τ β z ∂ β + H α H β ∂ σ zz ∂ z − H β R α σ α α − H α R β σ β β + H β R α + H α R β σ zz = 0 .

The equilibrium equations are derived from the Principle of Virtual Displacement and the complete derivation can be seen in ref. [58].

2.2 3D bending solutions of nanoshells

Closed-form summations of trigonometric harmonics are employed for solving the partial differential equations for the midsurface coordinates “ α ” and “ β . ” The stresses and displacements are written in terms of the Fourier series:

(7) { w , σ α α , σ β β , σ zz } = ∑ m = 1 ∞ ∑ n = 1 ∞ { w ( z ) , σ α α ( z ) , σ β β ( z ) , σ zz ( z ) } sin ( α ̄ α ) sin ( β ̄ β ) { u , τ α z } = ∑ m = 1 ∞ ∑ n = 1 ∞ { u ( z ) , τ α z ( z ) } cos ( α ̄ α ) sin ( β ̄ β ) { v , τ β z } = ∑ m = 1 ∞ ∑ n = 1 ∞ { v ( z ) , τ β z ( z ) } sin ( α ̄ α ) cos ( β ̄ β ) { τ α β } = ∑ m = 1 ∞ ∑ n = 1 ∞ { τ α β ( z ) } cos ( α ̄ α ) cos ( β ̄ β ) α ̄ = r π a , β ̄ = s π b .

These harmonic summations can only be applied for shells with constant radii of curvature and for panels in which edge boundary conditions are considered as simply supported. The wave numbers are denoted as “ r ” and “ s . ” The amplitudes for stresses are named “ σ α α ( z ) , σ β β ( z ) , σ zz ( z ) , τ α z ( z ) , τ β z ( z ) , τ α β ( z ) ” and for displacement “ u ( z ) , v ( z ) , w ( z ) . " Fourier series are replaced in the equilibrium and constitute equations, and the following equations are formed:

(8) 1 + μ α ̄ 2 H α 2 + μ β ̄ 2 H β 2 σ α α − C 13 ∂ w ∂ z − C 11 H α R α + C 12 H β R β w + α ̄ C 11 H α u + β ̄ C 12 H β = 0 1 + μ α ̄ 2 H α 2 + μ β ̄ 2 H β 2 σ β β − C 23 ∂ w ∂ z − C 12 H α R α + C 22 H β R β w + α ̄ C 12 H α u + β ̄ C 22 H β = 0 1 + μ α ̄ 2 H α 2 + μ β ̄ 2 H β 2 σ zz − C 33 ∂ w ∂ z − C 13 H α R α + C 23 H β R β w + α ̄ C 13 H α u + β ̄ C 23 H β = 0 1 + μ α ̄ 2 H α 2 + μ β ̄ 2 H β 2 τ α β − C 66 β ̄ H β u − C 66 α ̄ H α v = 0 1 + μ α ̄ 2 H α 2 + μ β ̄ 2 H β 2 τ α z − C 55 ∂ u ∂ z + C 55 H α R α u − α ̄ C 55 H α w = 0 1 + μ α ̄ 2 H α 2 + μ β ̄ 2 H β 2 τ β z − C 44 ∂ v ∂ z + C 44 H β R β v − β ̄ C 44 H β w = 0 ∂ τ α z ∂ z + 2 H α R α + 1 H β R β τ α z − β ̄ H β τ α β + α ̄ H α σ α α = 0 ∂ τ β z ∂ z + 1 H α R α + 2 H β R β τ α z − α ̄ H α τ α β + β ̄ H β σ β β = 0 ∂ σ zz ∂ z + 1 H α R α + 1 H β R β σ zz − α ̄ H α τ α z − β ̄ H β τ β z − σ α α H α R α − σ β β H β R β = 0 .

The coefficients of the system of nine differential equations are considered as variable due to the metric coefficients and depend on the thickness coordinates. No analytical solutions can be derived because of the thickness dependence coefficients. So, a semi-analytical solution is reported based on the so-called DQM. This numerical procedure permits calculation of the derivative of a stress or displacement by a linear sum of each variable (stress or displacement) evaluated in a certain grid point distribution. The mathematical expression for DQM is reported as follows:

(9) d f d z k z k = z i = ∑ j = 1 T a ij f ( z j ) , i = 1 , 2 , . . . , T .

The weighted coefficients are denoted as a ij . There are different manners of calculating a ij , and several procedures were reported in the survey by Tornabene et al. [62]. The basis function employed is the Lagrange interpolation polynomials (LIP) [63] and are reported as follows:

(10) L ( z j ) = ∏ i = 1 , i ≠ j I T ( z j − z i ) .

The principal advantage of LIP is that an arbitrary grid distribution can be applied. A recursive formula is applied for the weighted coefficients [63], and its formulation for the first- and second-order derivative can be expressed as:

(11) a ij ( 1 ) = L ( 1 ) ( z j ) ( z i − z j ) L ( z j ) , i , j = 1 , 2 , . . . , I T , i ≠ j a ij ( 2 ) = 2 a ii ( 1 ) a ij ( 1 ) − a ij ( 1 ) ( z i − z j ) , i , j = 1 , 2 , . . . , I T , i ≠ j a ii ( n ) = − ∑ j = 1 , j ≠ i I T a ij ( n ) , i , j = 1 , 2 , . . . , I T , i = j , n = 1 , 2 .

The thickness of the shell panel is discretized in terms of Chebyshev–Gauss–Lobatto grid distribution:

(12) z i = ( − h ) 2 cos i − 1 I T − 1 π , i = 1 , 2 , . . . , I T .

The traction conditions of the out-plane stress for the bottom (b) and top (t) are expressed as:

(13) σ zz ( t ) = P mn , σ zz ( b ) = 0 τ α z ( t ) = τ α z ( b ) = 0 τ β z ( t ) = τ β z ( b ) = 0 .

It is assumed that the shell panel is subjected to a transverse load at the top of the panel, and it can be written in terms of harmonics:

(14) p z s k = ∑ r , s P rs k sin r π a α sin s π b β .

The coefficient of the mechanical load for a uniformly distributed load is expressed as:

(15) P rs k = 16 P z π 2 rs , for { r , s } = 1 , 3 , 5 , 7 , . . . 0 , for { r , s } = 2 , 4 , 6 , 8 , . . .

The implementation of weighted coefficients, the traction boundary conditions, and governing equations for shallow nanoshells subjected to a mechanical load are developed in MATLAB code.

3 Results

The target of this article is to perform the static analysis of nanoshells with constant curvature based on Eringen’s nonlocal theory. The shallow shell panels are subjected to bisinusoidal and uniformly distributed loads. The first case problem was reported in the article by Wu and Li [57], who studied an isotropic nanoplate subjected to a bisinusoidal load. The second case problem was reported by Nguyen et al. [64], and it is about functionally graded nanoplates subjected to uniformly distributed load. The authors modeled the properties by the Mori and Tanaka scheme [65]. Overall, these problems can be used for extending the case to doubly curved panels and cylindrical panels.

The transverse displacement w and normal nonlocal stresses σ α α NL , σ β β NL , σ zz NL are evaluated at ( a / 2 , b / 2 , z ) . The in-plane shear stress τ α β NL is calculated at ( 0,0 , z ) . The in-plane displacement u and shear stress τ α z NL are estimated at ( 0 , b / 2 , z ) . The displacement v and out-of-plane stress τ β z NL are assessed at ( a / 2,0 , z ) .

3.1 Isotropic shells subjected to a bisinusoidal load

A simply supported isotropic plate is subjected to a bisinusoidal load. The plate is considered as a single-layered graphene sheet whose properties are E = 1.02 TPa , ν = 0.16 and the thickness of the plate is h = 0.34 nm , the panel key ratios are length-to-thickness ratio ( a / h ) and length aspect ratios ( b / a ) . The normalizations of stresses and displacements are given by the following formula:

(16) ( u ̄ , w ̄ ) = uE h 2 P a 3 , − 100 wE h 3 P a 4 , ( σ ̄ α α NL , τ ̄ α β NL , τ ̄ α z NL ) = σ α α NL h 2 P a 2 , τ α β NL h 2 P a 2 , τ α z NL h Pa

The numerical normalized results for displacements ( u ̄ , w ̄ ) and nonlocal stresses ( σ ̄ α α NL , τ ̄ α β NL , τ ̄ α z NL ) are presented in Table 1. In this case, the thickness of the plate is discretized into I T = 9 points. It is demonstrated that the results were the same as the 3D referential solution proposed by Wu and Li [57] employing the perturbation method. Then, the problem is extended to spherical and cylindrical panels by considering the radius-to-size ( R / a ) ratio. A thickness ratio of a / h = 10 is considered. For a cylindrical panel, the radius R β is equal to infinite. Figures 2 and 3 show the relationship between normalized deformation u ̄ and non-local parameters for a spherical and cylindrical panel, respectively. In the case of normalized deformation of shells, the same relation can be seen in Figures 4 and 5. The results demonstrated as the nonlocal parameter increased, the normalized deformation also increased. On the other hand, as the ratio ( R / a ) decreases, the normalized deformation also decreases. Interestingly, it is important to remark that the radii of curvature play a crucial role at the nanoscale and must be considered in future research activities for the proper design of nanodevices.

Table 1

Results of an isotropic plate subjected to bisinusoidal load

a / h b / a Theory μ = 0 μ = 0.5 μ = 1 μ = 1.5 μ = 2
10 1 u ̄ ( h / 2 ) [57] 0.0472 0.0874 0.1277 0.168 0.2083
u ̄ ( h / 2 ) [Present] 0.0472 0.0874 0.1277 0.168 0.2083
w ̄ ( 0 ) [57] 3.1331 5.8081 8.483 11.158 13.833
w ̄ ( 0 ) [Present] 3.1331 5.8081 8.483 11.158 13.833
σ ̄ α α NL ( − h / 2 ) [57] 0.1774 0.3289 0.4804 0.6319 −0.7833
σ ̄ α α NL ( − h / 2 ) [Present] 0.1774 0.3289 0.4804 0.6319 0.7833
τ ̄ α β NL ( − h / 2 ) [57] −0.1285 −0.2382 −0.3479 −0.4575 −0.5672
τ ̄ α β NL ( − h / 2 ) [Present] −0.1285 −0.2382 −0.3479 −0.4575 −0.5672
τ ̄ α z NL ( 0 ) [57] −0.2383 −0.4418 −0.6453 −0.8488 −1.0523
τ ̄ α z NL ( 0 ) [Present] −0.2383 −0.4418 −0.6453 −0.8488 −1.0523
2 u ̄ ( h / 2 ) [57] 0.1207 0.1851 0.2495 0.314 0.3784
u ̄ ( h / 2 ) [Present] 0.1207 0.1851 0.2495 0.314 0.3784
w ̄ ( 0 ) [57] 7.8941 12.1065 16.3189 20.5313 24.7436
w ̄ ( 0 ) [Present] 7.8941 12.1065 16.3189 20.5313 24.7436
σ ̄ α α NL ( − h / 2 ) [57] 0.4063 0.6231 0.8398 1.0566 1.2734
σ ̄ α α NL ( − h / 2 ) [Present] 0.4063 0.6231 0.8398 1.0566 1.2734
τ ̄ α β NL ( − h / 2 ) [57] −0.1641 −0.2516 −0.3392 −0.4267 −0.5143
τ ̄ α β NL ( − h / 2 ) [Present] −0.1641 −0.2516 −0.3392 −0.4267 −0.5143
τ ̄ α z NL ( 0 ) [57] −0.3816 −0.5852 −0.7888 −0.9924 −1.196
τ ̄ α z NL ( 0 ) [Present] −0.3816 −0.5852 −0.7888 −0.9924 −1.196
20 1 u ̄ ( h / 2 ) [57] 0.0471 0.0572 0.0673 0.0773 0.0874
u ̄ ( h / 2 ) [Present] 0.0471 0.0572 0.0673 0.0773 0.0874
w ̄ ( 0 ) [57] 3.0341 3.6817 4.3293 4.9769 5.6245
w ̄ ( 0 ) [Present] 3.0341 3.6817 4.3293 4.9769 5.6245
σ ̄ α α NL ( − h / 2 ) [57] 0.1766 0.2143 0.252 0.2897 0.3274
σ ̄ α α NL ( − h / 2 ) [Present] 0.1766 0.2143 0.252 0.2897 0.3274
τ ̄ α β NL ( − h / 2 ) [57] −0.1279 −0.1552 −0.1825 −0.2098 −0.237
τ ̄ α β NL ( − h / 2 ) [Present] −0.1279 −0.1552 −0.1825 −0.2098 −0.237
τ ̄ α z NL ( 0 ) [57] −0.2386 −0.2896 −0.3405 −0.3914 −0.4424
τ ̄ α z NL ( 0 ) [Present] −0.2386 −0.2896 −0.3405 −0.3914 −0.4424
2 u ̄ ( h / 2 ) [57] 0.1207 0.1368 0.1529 0.169 0.1851
u ̄ ( h / 2 ) [Present] 0.1207 0.1368 0.1529 0.169 0.1851
w ̄ ( 0 ) [57] 7.7354 8.7674 9.7993 10.8312 11.8631
w ̄ ( 0 ) [Presented] 7.7354 8.7674 9.7993 10.8312 11.8631
σ ̄ α α NL ( − h / 2 ) [57] 0.4051 0.4591 0.5131 0.5672 0.6212
σ ̄ α α NL ( − h / 2 ) [Present] 0.4051 0.4591 0.5131 0.5672 0.6212
τ ̄ α β NL ( − h / 2 ) [57] −0.1636 −0.1854 −0.2072 −0.229 −0.2509
τ ̄ α β NL ( − h / 2 ) [Present] −0.1636 −0.1854 −0.2072 −0.229 −0.2509
τ ̄ α z NL ( 0 ) [57] −0.3819 −0.4328 −0.4838 −0.5347 −0.5856
τ ̄ α z NL ( 0 ) [Present] −0.3819 −0.4328 −0.4838 −0.5347 −0.5856
Figure 2 Normalized displacement u ̄ \bar{u} vs nonlocal factor for an isotropic spherical shell.
Figure 2

Normalized displacement u ̄ vs nonlocal factor for an isotropic spherical shell.

Figure 3 Normalized displacement u ̄ \bar{u} vs nonlocal factor for an isotropic cylindrical shell.
Figure 3

Normalized displacement u ̄ vs nonlocal factor for an isotropic cylindrical shell.

Figure 4 Normalized displacement w ̄ \bar{w} vs nonlocal factor for an isotropic spherical shell.
Figure 4

Normalized displacement w ̄ vs nonlocal factor for an isotropic spherical shell.

Figure 5 Normalized displacement w ̄ \bar{w} vs nonlocal factor for an isotropic cylindrical shell.
Figure 5

Normalized displacement w ̄ vs nonlocal factor for an isotropic cylindrical shell.

3.2 Functionally graded plate subjected to uniformly distributed load

A simply supported plate is subjected to a uniformly distributed load. The functionally graded mechanical properties are modeled by the Mori–Tanaka model as follows:

(17) K e − K m K c − K m = V c 1 + V m K c − K m K m + 4 / 3 G m G e − G m G c − G m = V c 1 + V m G c − G m G m + f 1 f 1 = G m ( 9 K m + 8 G m ) 6 ( K m + 2 G m ) ,

where K and G are defined as the Bulk and shear modulus. The subscript “m” stands for metallic composition, and “c” is for the ceramic constitution. The volume fractions are considered as a function of the thickness direction and are considered by the following relation:

(18) V c = 1 2 + z h p , V m = 1 − V c .

The exponent of the ceramic volume fraction is given as “p.” The elastic modulus and Poisson ratio for computing the stiffness coefficients (Eq. (5)) are calculated as:

(19) E = 9 KG 3 K + G , ν = 3 K − 2 G 2 ( 3 K + G ) .

The properties of the metallic constitution are ( E m = 70 GPa , υ m = 0.30 ) [ Al ] and for ceramic are ( E c = 380 GPa , υ c = 0.30 ) [ A l 2 O 3 ] . The geometry of the plate is considered as a = b = 10 nm . The load is considered uniform distributed (Eq. (15)) and r = s = 31 are considered as the index for the summation of harmonics. For this numerical example, the panel is discretized with I T = 19 . Phung-Van et al. [18] solve this problem by an HSDT and an isogeometric analysis. Nguyen et al. [64] employed an HSDT with four unknowns and isogeometric technique. The normalized transverse displacement is presented:

(20) w ̄ = 100 E m h 3 12 ( 1 − υ m 2 ) P a 4 w ( a / 2 , b / 2 , 0 ) .

The numerical results for transverse displacement are reported in Table 2. It is seen that for thin plates ( a / h = 50 ) , the results are in total concordance with the higher order theories. A slightly numerical difference is evidenced for thick ( a / h = 5 ) and moderated thick plates ( a / h = 10 ) between the presented model and 2D theories. In thick nanoplates, the capability of the 3D solution is evidenced.

Table 2

Results of an isotropic plate subjected to uniform distributed load

p Model a / h = 5 a / h = 10 a / h = 50
μ = 0 μ = 1 μ = 4 μ = 0 μ = 1 μ = 4 μ = 0 μ = 1 μ = 4
0 Nguyen et al. [64] 0.0902 0.1059 0.1529 0.0787 0.0928 0.1351 0.075 0.0886 0.1294
Phung-Van et al. [18] 0.0903 0.1060 0.153 0.0787 0.0928 0.1351 0.075 0.0886 0.1294
Present 0.0885 0.1039 0.1502 0.0783 0.0923 0.1344 0.075 0.0886 0.1293
1 Nguyen et al. [64] 0.2254 0.2646 0.3821 0.1977 0.2331 0.3393 0.1888 0.223 0.3256
Phung-Van et al. [18] 0.2254 0.2646 0.3821 0.1977 0.2331 0.3394 0.1888 0.223 0.3256
Present 0.2243 0.2634 0.3806 0.1974 0.2328 0.339 0.1888 0.223 0.3256
2 Nguyen et al. [64] 0.2723 0.3194 0.4606 0.2344 0.2763 0.4021 0.2222 0.2625 0.3834
Phung-Van et al. [18] 0.2721 0.3192 0.4603 0.2344 0.2763 0.4021 0.2222 0.2625 0.3834
Present 0.2714 0.3185 0.4596 0.2343 0.2762 0.4019 0.2222 0.2625 0.3833
5 Nguyen et al. [64] 0.3247 0.3804 0.5477 0.2730 0.3217 0.4678 0.2564 0.3028 0.4422
Phung-Van et al. [18] 0.3246 0.3803 0.5474 0.2729 0.3217 0.4678 0.2564 0.3028 0.4422
Present 0.3227 0.3782 0.5449 0.2726 0.3212 0.4672 0.2564 0.3028 0.4422
10 Nguyen et al. [64] 0.3620 0.4243 0.6109 0.3052 0.3597 0.5231 0.2869 0.339 0.495
Phung-Van et al. [18] 0.3625 0.4247 0.6114 0.3053 0.3598 0.5233 0.2870 0.339 0.495
Present 0.3594 0.4213 0.6071 0.3047 0.3591 0.5223 0.2870 0.339 0.495

A moderated thick spherical shell ( a / h = 10 ) is considered for parametric analysis. Figure 6 shows the normalized centered transverse displacement as a function of a / R , considering a nonlocal parameter μ = 1 n m 2 . It is seen that as the exponent “ p ” increases, the normalized displacement also increases; as the ratio a / R increases, the normalized displacement decreases. A nonlinear relationship is evidenced, and the radii of curvature are crucial at nanoscale. Figure 7 shows the relationship between the transverse centered displacement and the nonlocal parameter “ μ , ” considering a ratio of R / a = 10 . It is demonstrated a linear correlation for the material exponent parameters considered. As the nonlocal parameter increases, the displacement increases. This pattern is kept for different radius-to-length ratios.

Figure 6 Normalized displacement w ̄ \bar{w} vs ratio a / R a/R for a functionally graded spherical shell.
Figure 6

Normalized displacement w ̄ vs ratio a / R for a functionally graded spherical shell.

Figure 7 Normalized displacement w ̄ \bar{w} vs nonlocal factor for a functionally graded spherical shell.
Figure 7

Normalized displacement w ̄ vs nonlocal factor for a functionally graded spherical shell.

Overall, it is important to remark that is extremally important the influence of the radii of curvature on the bending response of nanoshells. Suppressively, as far as the author is concerned, there is no contribution in this field. Consequently, these problems can be considered as the referential solution for other numerical models.

Moreover, the present technique could be used to solve other problems such as clamped nanoplates by the use of the boundary discontinuous method. Currently, this method can be extended to more sophisticated structures such as functionally graded carbon-nanotubes-reinforced laminated composites [66] or graphene nanoplatelet reinforced composites [67]. Additionally, multilayered structures and complex shell structures can also be faced.

4 Conclusions

A smart semi-analytical solution for the bending of simply supported spherical, cylindrical, and rectangular nanostructures is presented. The nonlocal scale is considered by Eringen’s nonlocal theory. The equilibrium equations are written in terms of curvilinear coordinates. The stresses and mechanical displacements are expressed in closed-form summations of harmonic for the midsurface of the plate and are discretized with the classic Chebyshev–Gauss–Lobatto. The governing equations are solved semi-analytical by the DQM, and Lagrange Interpolation polynomials are used as basis functions. The results are compared with 3D theories and 2D theories provided in the literature for isotropic and functionally graded rectangular plates subjected to bisinusoidal and uniform distributed load. Since remarkable results were obtained for plates, the case problems were extended for cylindrical and spherical panels. Remarkably, the influence of the radii of curvature on the bending response of nanoshells is significant, and it should be a strong topic of research since it is crucial in the design of nanodevices. Nanoshell problems are solved in a closed form in this article, so they can be considered benchmark nanoshell solutions for future works. Finally, the method can be extended for multilayered shells and for different functionally reinforced constitutive such as carbon nanotubes and graphene nanoplatelets.

tel: +51 964358966

  1. Funding information: This article was written in the context of the project: “Desarrollo de un algoritmo autónomo y óptimo de mecánica computacional para un análisis de estructuras complejas impresa con tecnología 3D, utilizando inteligencia artificial y algoritmos genéticos” founded by CONCYTEC under the contract number N°060-2021-PROCIENCIA. The authors of this manuscript appreciate the financial support from the Peruvian Government.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors have no conflicts of interest to declare.

References

[1] Thai HT, Vo T, Nguyen TK, Seung-Eock K. A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos Struct. 2017;177:196–219.10.1016/j.compstruct.2017.06.040Search in Google Scholar

[2] Khoram MM, Hosseimi M, Shishesaz M. A concise review of nano-plates. J Comput Appl Mech. 2019;50(2):420–9.Search in Google Scholar

[3] Thai CT, Ferreira AJM, Phung-Van P. Size dependent free vibration analysis of multilayered GPLRC microplates based on modified strain gradient theory. Compos B: Eng. 2019;169:174–88.10.1016/j.compositesb.2019.02.048Search in Google Scholar

[4] Hajilak EZ, Pourghader J, Hashemabadi D, Bagh SF, Habibi M. Multilayer GPLRC composite cylindrical nanoshell using modified strain gradient theory. Mech Based Des Struct Mach. 2019;47(5):521–45.10.1080/15397734.2019.1566743Search in Google Scholar

[5] Asghari M, Rahaeifard M, Kahrobaiyan MH, Ahmadian MT. The modified couple stress functionally graded Timoshenko beam formulation. Mater Des. 2011;32:1435–43.10.1016/j.matdes.2010.08.046Search in Google Scholar

[6] Zare Jouneghani F, Babamoradi H, Dimitri R, Tornabene F. A modified couple stress elasticity for non-uniform composite laminated beams based on the Ritz formulation. Molecules. 2020;25(6):1404.10.3390/molecules25061404Search in Google Scholar PubMed PubMed Central

[7] Carrera E, Zozulya VV. Carrera unified formulation for the micropolar plates. Mech Adv Mater Struct. 2011;29(22):3163–86.10.1080/15376494.2021.1889726Search in Google Scholar

[8] Karttunen AT, Reddy JN, Romanoff J. Micropolar modeling approach for periodic sandwich beams. Compos Struct. 2018;185:656–64.10.1016/j.compstruct.2017.11.064Search in Google Scholar

[9] Liu X, Hu G. Inclusion of microstretch continuum. Int J Eng Sci. 2004;42:849–60.10.1016/j.ijengsci.2003.07.011Search in Google Scholar

[10] Eringen AC. Compatibility conditions of the theory of micromorphic elastic solids. J Math Mech. 1969;19(6):473–81.10.1512/iumj.1970.19.19044Search in Google Scholar

[11] Eringen AC. Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci. 1972;10(5):425–35.10.1016/0020-7225(72)90050-XSearch in Google Scholar

[12] Eringen AC, Edelen DGB. On nonlocal elasticity. Int J Eng Sci. 1972;10:233–48.10.1016/0020-7225(72)90039-0Search in Google Scholar

[13] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys. 1983;54:4703–10.10.1063/1.332803Search in Google Scholar

[14] Shariati M, Shishehsaz M, Sahbafar H, Pourabdy M, Hosseini M. A review on stress-driven nonlocal elasticity theory. J Comput Appl Mech. 2021;52(3):535–52.Search in Google Scholar

[15] Ansari R, Rouhi H. Nonlocal Flugge shell model for the axial buckling of single-walled Carbon nanotubes: An analytical approach. Int J Nano Dimens. 2015;6(5):453–62.Search in Google Scholar

[16] Kananipour H. Static analysis of nanoplates based on nonlocal based on the nonlocal Kirchhoff and Mindlin plate theories using DQM. Lat Am J Solids Struct. 2014;11(10):1709–20.10.1590/S1679-78252014001000001Search in Google Scholar

[17] Srividhya S, Raghu P, Rajagopal A, Reddy JN. Nonlocal nonlinear analysis of functionally graded plates using third-order shear deformation theory. Int J Eng Sci. 2008;125:1–22.10.1016/j.ijengsci.2017.12.006Search in Google Scholar

[18] Phung-Van P, Lieu QX, Nguyen-Xuan H, Abdel Wahab M. Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos Struct. 2017;166:120–35.10.1016/j.compstruct.2017.01.049Search in Google Scholar

[19] Thai S, Thai HT, Vo TP, Patel VI. A simple shear deformation theory for nonlocal beams. Compos Struct. 2018;183:262–70.10.1016/j.compstruct.2017.03.022Search in Google Scholar

[20] Jape AS, Sayyad AS. Bending of functionally graded nanobeams using hyperbolic nonlocal theory. IOP Conf Ser: Mater Sci Eng. 2022;1236(1):012088.10.1088/1757-899X/1236/1/012008Search in Google Scholar

[21] Karami B, Janghorban M, Tounsi A. Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory. Thin Wall Struct. 2018;129:251–64.10.1016/j.tws.2018.02.025Search in Google Scholar

[22] Arefi M, Zenkour AM. Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation. Phys B: Condens Matter. 2017;521:188–97.10.1016/j.physb.2017.06.066Search in Google Scholar

[23] Mematollahi MS, Mohammadi H, Nematollahi MA. Thermal vibration analysis of nanoplates based on the higher-order nonlocal strain gradient theory by an analytical approach. Superlattices Microstruct. 2017;111:944–59.10.1016/j.spmi.2017.07.055Search in Google Scholar

[24] Arefi M. Analysis of doubly curved piezoelectric nano shell: Nonlocal electro-elastic bending solution. Eur J Mech A/Solids. 2018;70:226–37.10.1016/j.euromechsol.2018.02.012Search in Google Scholar

[25] Ansari R, Rouhi H, Arash B. Vibrational analysis of Double-Walled carbon nanotubes based on nonlocal Donnell shell theory via a new numerical approach. Iran J Sci Technol B. 2013;37:91–105.Search in Google Scholar

[26] Sahmani S, Fattahi AM. Nonlocal size dependency in nonlinear instability of axially loaded exponential shear deformable FG-CNT reinforced nanoshells under heat conduction. Eur Phys J Plus. 2017;132:231.10.1140/epjp/i2017-11497-5Search in Google Scholar

[27] She GL, Ren YR, Yuan FG, Xiao WS. On vibrations of porous nanotubes. Int J Eng Sci. 2018;125:23–35.10.1016/j.ijengsci.2017.12.009Search in Google Scholar

[28] Şimşek M, Yurtcu HH. Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct. 2013;97:378–96.10.1016/j.compstruct.2012.10.038Search in Google Scholar

[29] Karami B, Shahsavari D, Janghorban M. A compressive analytical study on functionally graded carbon nanotubes-reinforced composite plates. Aerosp Sci Technol. 2018;82–83:499–512.10.1016/j.ast.2018.10.001Search in Google Scholar

[30] Thai HT, Vo TP, Nguyen TK, Lee J. A nonlocal sinusoidal plate model for micro/nanoscale plates. Proc Inst Mech Eng C J Mech Eng Sci. 2014;228(14):2652–60.10.1177/0954406214521391Search in Google Scholar

[31] Touratier M. An efficient standard plate theory. Int J Eng Sci. 1991;29:901–16.10.1016/0020-7225(91)90165-YSearch in Google Scholar

[32] Ansari R, Sahmani S, Arash B. Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A. 2010;375:53–62.10.1016/j.physleta.2010.10.028Search in Google Scholar

[33] Janghorban M. Static analysis of functionally graded rectangular nanoplates based on nonlocal third order shear deformation theory. Int J Eng Appl Sci. 2016;8(2):87–100.10.24107/ijeas.252147Search in Google Scholar

[34] Shahrbabaki EA. On three-dimensional nonlocal elasticity: Free vibrations of rectangular nanoplates. Eur J Mech A/Solids. 2018;71:122–33.10.1016/j.euromechsol.2018.03.004Search in Google Scholar

[35] Kulikov GM, Plotnikova SV. A method of solving three-dimensional problems of elasticity for laminated composite plates. Mech Compos Mater. 2012;48(1):15–26.10.1007/s11029-012-9248-8Search in Google Scholar

[36] Kulikov GM, Plotnikova SV. Exact geometry four-node solid-shell element for stress analysis of functionally graded shell structures via advanced SaS formulation. Mech Adv Mater Struct. 2018;27(12):948–64.10.1080/15376494.2018.1502380Search in Google Scholar

[37] Alibeigloo A, Emtehani A. Static and free vibration analysis of carbon nanotube-reinforced composite plate using differential quadrature method. Meccanica. 2015;50:61–76.10.1007/s11012-014-0050-7Search in Google Scholar

[38] Alibeigloo A, Liew KM. Elasticity solution of free vibration and bending behavior of functionally graded carbon nanotube-reinforced composite beam with thin piezoelectric layers using differential quadrature method. Int J Appl Mech. 2015;7(1):1550002.10.1142/S1758825115400025Search in Google Scholar

[39] Brischetto S. Exact three-dimensional static analysis of single- and multi-layered plates and shells. Compos B: Eng. 2017;119:230–52.10.1016/j.compositesb.2017.03.010Search in Google Scholar

[40] Brischetto S. A 3D layer-wise model for the correct imposition of transverse shear/normal load conditions in FGM shells. Int J Mech Sci. 2018;136:50–66.10.1016/j.ijmecsci.2017.12.013Search in Google Scholar

[41] Wu CP, Lu YC. A modified Pagano method for the 3D dynamic response of functionally graded magneto-electro-elastic plates. Compos Struct. 2009;90:363–72.10.1016/j.compstruct.2009.03.022Search in Google Scholar

[42] Wu CP, Tsai YH. Static behavior of functionally graded magneto-electro-elastic shell under electric displacement and magnetic flux. Int J Eng Sci. 2007;45:744–69.10.1016/j.ijengsci.2007.05.002Search in Google Scholar

[43] Demirbaș MD. Thermal stress analysis of functionally graded plates with temperature-dependent material properties using theory of elasticity. Compos B: Eng. 2017;131:100–24.10.1016/j.compositesb.2017.08.005Search in Google Scholar

[44] Wu CP, Chiu KH, Jiang RY. A meshless collocation method for the coupled analysis of functionally graded piezo-thermo-elastic shells and plates under thermal loads. Int J Eng Sci. 2012;56:29–48.10.1016/j.ijengsci.2012.03.001Search in Google Scholar

[45] Ferreira AJM, Roque CMC, Carrera E, Cinefra M, Polit O. Bending and Vibration of laminated plates by a layerwise formulation and collocation with radial basis functions. Mech Adv Mater Struct. 2013;20(8):624–37.10.1080/15376494.2011.643282Search in Google Scholar

[46] Kulikov GM, Plotnikova SV. Strong sampling surfaces formulation for layered shells. Int J Solids Struct. 2017;121:75–85.10.1016/j.ijsolstr.2017.05.017Search in Google Scholar

[47] Kulikov GM, Plotnikova SV. Strong sampling surfaces formulation for laminated composites plates. Compos Struct. 2017;172:73–82.10.1016/j.compstruct.2017.03.061Search in Google Scholar

[48] Wei GW. Discrete singular convolution for beam analysis. Eng Struct. 2001;23(9):1045–53.10.1016/S0141-0296(01)00016-5Search in Google Scholar

[49] Civalek Ö, Gürses M. Discrete singular convolution for free vibration analysis annular membranes. Math Comput Appl. 2009;14(2):131–8.10.3390/mca14020131Search in Google Scholar

[50] Wang X. Differential quadrature and differential quadrature based element methods. Waltham, UK: Butterworth-Heinemann; 2015.10.1016/B978-0-12-803081-3.00002-4Search in Google Scholar

[51] Bellman R, Casti J. Differential quadrature and long-term integration. J Math Anal Appl. 1971;34(2):235–8.10.1016/0022-247X(71)90110-7Search in Google Scholar

[52] Bert CW, Malik M. Differential quadrature method in computational mechanics: A review. Appl Mech Rev. 1996;49(1):1–28.10.1115/1.3101882Search in Google Scholar

[53] Flügge W. Stresses in shells. Berlin: Springer-Verlag; 1960.10.1007/978-3-662-29731-5Search in Google Scholar

[54] Leissa AW. Vibrations of shells. Washington: NASA Sp. 288; 1973.Search in Google Scholar

[55] Gould PL. Analysis of shells and plates. New York (NY), USA: Springer-Verlag; 1988.10.1007/978-1-4612-3764-8Search in Google Scholar

[56] Soedel W. Vibrations of shells and plates. New York (NY), USA: Marcell Dekker; 2004.10.4324/9780203026304Search in Google Scholar

[57] Wu CP, Li WC. Three-dimensional static analysis of nanoplates and graphene sheets by using Eringen’ s nonlocal elasticity theory and the perturbation method. Comput Mater Contin. 2016;52(2):73–103.Search in Google Scholar

[58] Monge JC, Mantari JL. 3D elasticity numerical solution for the static behavior of FGM shells. Eng Struct. 2020;110159.10.1016/j.engstruct.2019.110159Search in Google Scholar

[59] Monge JC, Mantari JL, Arciniega RA. Computational semi-analytical method for 3D elasticity bending solution of laminated composite and sandwich doubly-curved shells. Eng Struct. 2020;221:110938.10.1016/j.engstruct.2020.110938Search in Google Scholar

[60] Brischetto S, Tornabene F. Advanced GDQ models and 3D stress recovery in multilayered plates, spherical and double-curved panels subjected to transverse shear loads. Compos B: Eng. 2018;146:244–69.10.1016/j.compositesb.2018.04.019Search in Google Scholar

[61] Tornabene F, Brischetto S. 3D capability of refined GDQ models for the bending analysis of composite and sandwich plates, spherical and doubly curved shells. Thin Wall Struct. 2018;129:94–124.10.1016/j.tws.2018.03.021Search in Google Scholar

[62] Tornabene F, Fantuzzi N, Ubertini F, Viola E. Strong formulation finite element based on differential quadrature: A survey. Appl Mech Rev. 2015;67(2):1–55.10.1115/1.4028859Search in Google Scholar

[63] Shu C. Differential Quadrature and its application in engineering. London, UK: Springer; 2000.10.1007/978-1-4471-0407-0Search in Google Scholar

[64] Nguyen NT, Hui D, Lee J, Nguyen-Xuan N. An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng. 2015;297:191–218.10.1016/j.cma.2015.07.021Search in Google Scholar

[65] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 1973;21:571–4.10.1016/0001-6160(73)90064-3Search in Google Scholar

[66] Tornabene F, Fantuzzi N, Bacciocchi M, Viola E. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos B: Eng. 2016;89:187–218.10.1016/j.compositesb.2015.11.016Search in Google Scholar

[67] Arefi M, Kiani Moghaddam S, Mohammad-Rezaei Bidgoli E, Kiani M, Civalek O. Analysis of graphene nanoplatelet reinforced cylindrical shell subjected to thermo-mechanical loads. Compos Struct. 2021;255:112924.10.1016/j.compstruct.2020.112924Search in Google Scholar
Received: 2023-06-09
Accepted: 2023-08-27
Published Online: 2023-11-28

© 2023 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Research Articles
  2. Investigation of differential shrinkage stresses in a revolution shell structure due to the evolving parameters of concrete
  3. Multiphysics analysis for fluid–structure interaction of blood biological flow inside three-dimensional artery
  4. MD-based study on the deformation process of engineered Ni–Al core–shell nanowires: Toward an understanding underlying deformation mechanisms
  5. Experimental measurement and numerical predictions of thickness variation and transverse stresses in a concrete ring
  6. Studying the effect of embedded length strength of concrete and diameter of anchor on shear performance between old and new concrete
  7. Evaluation of static responses for layered composite arches
  8. Nonlocal state-space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam
  9. Numerical study of the FRP-concrete bond behavior under thermal variations
  10. Parametric study of retrofitted reinforced concrete columns with steel cages and predicting load distribution and compressive stress in columns using machine learning algorithms
  11. Application of soft computing in estimating primary crack spacing of reinforced concrete structures
  12. Identification of crack location in metallic biomaterial cantilever beam subjected to moving load base on central difference approximation
  13. Numerical investigations of two vibrating cylinders in uniform flow using overset mesh
  14. Performance analysis on the structure of the bracket mounting for hybrid converter kit: Finite-element approach
  15. A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation
  16. Strength analysis of marine biaxial warp-knitted glass fabrics as composite laminations for ship material
  17. Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity
  18. Structural function analysis of shear walls in sustainable assembled buildings under finite element model
  19. In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation
  20. Optimization of structural parameters and numerical simulation of stress field of composite crucible based on the indirect coupling method
  21. Numerical study on crushing damage and energy absorption of multi-cell glass fibre-reinforced composite panel: Application to the crash absorber design of tsunami lifeboat
  22. Stripped and layered fabrication of minimal surface tectonics using parametric algorithms
  23. A methodological approach for detecting multiple faults in wind turbine blades based on vibration signals and machine learning
  24. Influence of the selection of different construction materials on the stress–strain state of the track
  25. A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells
  26. Comparative study of shell element formulations as NLFE parameters to forecast structural crashworthiness
  27. A size-dependent 3D solution of functionally graded shallow nanoshells
  28. Special Issue: The 2nd Thematic Symposium - Integrity of Mechanical Structure and Material - Part I
  29. Correlation between lamina directions and the mechanical characteristics of laminated bamboo composite for ship structure
  30. Reliability-based assessment of ship hull girder ultimate strength
  31. Finite element method on topology optimization applied to laminate composite of fuselage structure
  32. Dynamic response of high-speed craft bottom panels subjected to slamming loadings
  33. Effect of pitting corrosion position to the strength of ship bottom plate in grounding incident
  34. Antiballistic material, testing, and procedures of curved-layered objects: A systematic review and current milestone
  35. Thin-walled cylindrical shells in engineering designs and critical infrastructures: A systematic review based on the loading response
  36. Laminar Rayleigh–Benard convection in a closed square field with meshless radial basis function method
  37. Determination of cryogenic temperature loads for finite-element model of LNG bunkering ship under LNG release accident
  38. Roundness and slenderness effects on the dynamic characteristics of spar-type floating offshore wind turbine
https://doi.org/10.1515/cls-2022-0215
Keywords for this article
nanoshell; functionally graded material; Eringen’s nonlocal elasticity theory; equilibrium equations
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. Investigation of differential shrinkage stresses in a revolution shell structure due to the evolving parameters of concrete
  3. Multiphysics analysis for fluid–structure interaction of blood biological flow inside three-dimensional artery
  4. MD-based study on the deformation process of engineered Ni–Al core–shell nanowires: Toward an understanding underlying deformation mechanisms
  5. Experimental measurement and numerical predictions of thickness variation and transverse stresses in a concrete ring
  6. Studying the effect of embedded length strength of concrete and diameter of anchor on shear performance between old and new concrete
  7. Evaluation of static responses for layered composite arches
  8. Nonlocal state-space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam
  9. Numerical study of the FRP-concrete bond behavior under thermal variations
  10. Parametric study of retrofitted reinforced concrete columns with steel cages and predicting load distribution and compressive stress in columns using machine learning algorithms
  11. Application of soft computing in estimating primary crack spacing of reinforced concrete structures
  12. Identification of crack location in metallic biomaterial cantilever beam subjected to moving load base on central difference approximation
  13. Numerical investigations of two vibrating cylinders in uniform flow using overset mesh
  14. Performance analysis on the structure of the bracket mounting for hybrid converter kit: Finite-element approach
  15. A new finite-element procedure for vibration analysis of FGP sandwich plates resting on Kerr foundation
  16. Strength analysis of marine biaxial warp-knitted glass fabrics as composite laminations for ship material
  17. Analysis of a thick cylindrical FGM pressure vessel with variable parameters using thermoelasticity
  18. Structural function analysis of shear walls in sustainable assembled buildings under finite element model
  19. In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation
  20. Optimization of structural parameters and numerical simulation of stress field of composite crucible based on the indirect coupling method
  21. Numerical study on crushing damage and energy absorption of multi-cell glass fibre-reinforced composite panel: Application to the crash absorber design of tsunami lifeboat
  22. Stripped and layered fabrication of minimal surface tectonics using parametric algorithms
  23. A methodological approach for detecting multiple faults in wind turbine blades based on vibration signals and machine learning
  24. Influence of the selection of different construction materials on the stress–strain state of the track
  25. A coupled hygro-elastic 3D model for steady-state analysis of functionally graded plates and shells
  26. Comparative study of shell element formulations as NLFE parameters to forecast structural crashworthiness
  27. A size-dependent 3D solution of functionally graded shallow nanoshells
  28. Special Issue: The 2nd Thematic Symposium - Integrity of Mechanical Structure and Material - Part I
  29. Correlation between lamina directions and the mechanical characteristics of laminated bamboo composite for ship structure
  30. Reliability-based assessment of ship hull girder ultimate strength
  31. Finite element method on topology optimization applied to laminate composite of fuselage structure
  32. Dynamic response of high-speed craft bottom panels subjected to slamming loadings
  33. Effect of pitting corrosion position to the strength of ship bottom plate in grounding incident
  34. Antiballistic material, testing, and procedures of curved-layered objects: A systematic review and current milestone
  35. Thin-walled cylindrical shells in engineering designs and critical infrastructures: A systematic review based on the loading response
  36. Laminar Rayleigh–Benard convection in a closed square field with meshless radial basis function method
  37. Determination of cryogenic temperature loads for finite-element model of LNG bunkering ship under LNG release accident
  38. Roundness and slenderness effects on the dynamic characteristics of spar-type floating offshore wind turbine
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 20.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/cls-2022-0215/html
Scroll to top button