Analysis of FGM micro cylindrical shell with variable thickness using Cooper Naghdi model: Bending and buckling solutions

doi:10.1016/j.mechrescom.2021.103739

Mechanics Research Communications

Volume 115, July 2021, 103739

https://doi.org/10.1016/j.mechrescom.2021.103739 Get rights and content

•
Bending and buckling analysis of micro cylindrical shell is investigated in this research
•
Differential quadrature method is used for static analysis of micro cylindrical shell with variable thickness
•
Cooper Naghdi model is used to displacement fields
•
Functionally graded material together with MCST are employed in this work

Abstract

Functionally graded materials (FGMs) due to high thermal and mechanical resistance have a special applications in the aerospace industry, nuclear reactors and electronic and magnetism industries. In the present study, the bending and buckling analysis of cylindrical micro shell with variable thickness made of FGMs based on modified couple stress theory (MCST) are presented. In this theory, the effect of material length scale parameter is considered. Power law distribution is used to distribute FGMs and the mechanical properties of micro shell vary in thickness direction. The displacement field follows the Cooper-Naghdi model and the governing equations of equilibrium are obtained analytically using the energy method. Nonlinear equations of cylindrical micro shell with variable thickness for clamped-clamped boundary conditions are solved numerically by differentially quadrature method (DQM). The effect of various mechanical parameters, micro shell dimensional proportions, material length scale parameter, also the effect of the distribution of FGMs on deflection and critical buckling load of the cylindrical micro shell with variable thickness have been investigated. A brief summary of the results of the present study will show that increasing the material length scale parameter increases the critical buckling load, and decrease the deflection of the cylindrical micro shell with variable thickness. Increasing the power distribution of FGMs increases critical buckling load, which is vice versa for the deflection of micro shell.

Introduction

Functionally graded materials (FGMs) are a new kind of composite in which the hardness and temperature resistance increase by steady change of the materials. Cylindrical shells especially FGMs are important and useful parts of various industries, which have been developed for use in high temperature environments and have a wide range of applications in different fields of engineering from aerospace, civil, mechanical, marine vessels to pressurized water reactors, and naval engineering practices in recent years. Due to gradual variation in the microstructure of these materials, which causes the continuous and smooth variation of properties, the problem of mismatch between the properties of bonded materials at the interface that could cause to debonding in high temperature, would vanish. Many of FGMs are composed of a ceramic and a metal and can take the advantage of the desirable properties as heat and corrosion resistance of ceramics and high tensile strength, toughness and bonding capability of metals. Also, metals are desirable for their good toughness, mechanical strength, machinability, ductility and impact tolerance, whereas their resistance to erosion, creep and high temperature is not high enough, particularly at elevated temperatures they may be affected by corrosion and oxidation. Because of good resistance of ceramics to erosion, creep, heat, corrosion and oxidation, beside their brittle nature, metals and ceramics usually complement each other perfectly. Over the past two decades, there has been a growing interest in analysis of structures made of functionally graded materials. Javaheri and Eslami and [1] derived stability and balance equations for a rectangular plate made of functionally graded materials under thermal loading according to the classic plate theory. They investigated buckling of plate for four thermal loading conditions and compared the results with other articles. Bangal et al. [2] used the finite element formulation to study thermal buckling as well as vibration behavior of a semi-cone made of functionally graded materials in a high temperature environment based on first order shear deformation theory (FSDT). Mehralian et al. [3] developed the buckling of a cylindrical nano shell made of piezoelectric functionally graded material based on the modified couple stress theory. Golami et al. [4] used the FSDT based on modified strain gradient theory to analyze longitudinal buckling of a FGM cylindrical micro shell. Due to complexity and time consuming nature of atomic simulation of an atomic system in large scale, continuum mechanics has attracted a lot of attentions. Since classic theory is unable to take the size effect into account, non-classic theories including Eringen's non-local theory [5], Mindlin's couple stress theory [6], Toupin [7] and Koiter [8] have become more important. Yung et al. [9] presented the modified couple stress theory (MCST) which was used in numerous researches that will be discussed later. Ghorbanpour Arani et al. [10] investigated the effect of CNT volume fraction on the magneto-thermo-electro-mechanical behavior of smart nanocomposite cylinder. Mehralian et al. [11] considered the buckling of anisotropic piezoelectric cylindrical shells under axial compression and lateral pressure according to the MCST and FSDT. Zenkour [12] investigated the bending of a curved plate with varying radial thickness using small parameter method and presented the results for various thickness distributions.

By reviewing the literature, it can be seen that up to date there are no researches about bending and buckling analysis of functionally graded material (FGM) micro cylindrical shell with variable thickness based on Cooper Naghdi model using differential quadrature method (DQM). Also, the effect of material length scale parameter is considered. Power law distribution is used to distribute FGMs and the mechanical properties of micro shell vary in thickness direction. The displacement field follows the Cooper-Naghdi model and the governing equations of equilibrium are obtained analytically using the energy method. Nonlinear equations of cylindrical micro shell with variable thickness for clamped-clamped boundary conditions are solved numerically by differentially quadrature method (DQM).

Section snippets

Geometry and Simulation

Fig. 1 shows a schematic view of FGM micro sandwich cylindrical shell with variable thickness which are Length, thickness and mean radius of the FGM micro cylindrical shell are L, hand R respectively. Fig. 2 illustrates a schematic diagram of micro cylindrical shell for (a) transverse (q₀ ) and axial buckling loads ( $N_{x 0}$ ) (b) clamped-clamped boundary conditions at two ends. Fig. 3 depicts a cross-sectional distribution of micro cylindrical shell along the thickness that this formula is defined

Navier's method

Using defined displacement fields in Eq. (7), by employing Navier's method based on the governing equations of the micro cylindrical shell, and boundary conditions, the middle surface displacements and rotations in the longitudinal, circumferential and thickness directions of the micro cylindrical shell (u_x(x,θ), v_θ(x,θ), w(x, θ), β_x(x,θ) and β_θ(x,θ)) are considered as follows: $\begin{matrix} u_{x} (x, θ) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} u_{0 m n} \cos (\frac{m π x}{L}) \cos (n θ) \\ v_{θ} (x, θ) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} v_{0 m n} \sin (\frac{m π x}{L}) \sin (n θ) \\ w (x, θ) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} w_{0 m n} \sin (\frac{m π x}{L}) \cos (n θ) \\ β_{x} ( \end{matrix}$

DQ method

The DQM has been presented for the first time [16] in the framework of solving differential equations (DQ). This method refers to the quadrature method in deriving the partial derivatives of a function as follows: $\begin{matrix} \frac{d^{r} f}{d x^{r}} = \sum_{n = 1}^{N} A_{i j}^{(r)} f_{j} i, j = 1, 2, 3, . . . . . . . . . . . ., N, \\ r = 1, 2, . . . . . . . . . . . . . . ., N - 1 \end{matrix}$ where $A_{i j}^{(r)}$ and N are the weighting coefficient of nth derivative and the number of grid point, respectively [17]. $A^{x} = A_{i j}^{(1)} = {\begin{matrix} \frac{\prod_{\frac{m = 1}{m \neq i, j}}^{N} (x_{i} - x_{m})}{\prod_{\frac{m = 1}{m \neq i}}^{N} (x_{j} - x_{m})} (i, j = 1, 2, . . ., N; i \neq j) \\ \sum_{\frac{m = 1}{m \neq i}}^{N} \frac{1}{(x_{i} - x_{m})} (i = j = 1, 2, . . ., N) \end{matrix}$ $A^{(r)} = A^{(r - 1}$

Numerical result and discussion

In this article, the bending and buckling analysis of cylindrical micro shell with variable thickness made of FGMs based on modified couple stress theory (MCST) are investigated that the length of cylindrical shell was stated as follows: $\begin{matrix} l = 17.6 μ m, \frac{h_{0}}{l} = 10, \\ \frac{R}{h_{0}} = 50, \frac{L}{R} = 2, \end{matrix}$ where l and L are the material length scale parameter (defined in Eqs. (9c)) and length of cylindrical shell, respectively. h₀ and R are the initial thickness in x=0 according to Eq. (1), and mean radius of cylindrical shell,

Conclusion

Declaration of Competing Interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Acknowledgment

The authors would like to thank the referees for their valuable comments. Also, they are thankful to the Iranian Nanotechnology Development Committee for their financial support and the University of Kashan for supporting this work by Grant No. 682561/23.

References (19)

R.K. Bhangale et al.
Linear thermoelastic buckling and free vibration behavior of functionally graded truncated conical shells
J. Sound Vib.
(2006)
F. Mehralian et al.
Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell
Compos. Struct.
(2016)
F. Yang et al.
Couple stress based strain gradient theory for elasticity
Int. J. Solids Struct.
(2002)
F. Mehralian et al.
On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure
Int. J. Mech. Sci.
(2016)
Y. Tadi Beni et al.
Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory
Compos. Struct.
(2015)
R. Bellman et al.
Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations
J. Comput. Phys.
(1972)
U. Yücel et al.
Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems
Appl. Math. Modelling
(2012)
R. Javaheri et al.
Thermal Buckling of Functionally Graded Plates
AIAA J
(2002)
R. Gholami et al.
Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory
Meccanica
(2014)

There are more references available in the full text version of this article.

Cited by (12)

Free vibration analysis of bi-directional functionally graded cylindrical shells with varying thickness
2023, Aerospace Science and Technology
The article presents a semi-analytical solution for the free vibration problem in bi-directional functionally graded (FG) cylindrical shells with varying thickness, utilizing a modified shear deformation shell theory (MSDT). In this model, the mechanical properties of the cylindrical shells change along both the axial and thickness directions. Classical analytical methods for shell structures often face limitations in terms of boundary conditions, transverse shear deformation, and rotary inertia, making the use of MSDT and Galerkin procedure necessary for solving the problem of free vibration analysis in this study. The boundary conditions of the bi-directional FG cylindrical shells are chosen to be simply supported and clamped, and the presented method is compared to relevant results in the literature to validate its effectiveness. Moreover, a comprehensive parametric investigation is conducted to evaluate the influence of material and geometrical parameters on the natural frequencies. The results show that the natural frequency of the FG cylindrical shells is significantly affected by the material and geometrical parameters, including the power-law index, the thickness variation, the aspect ratio, and the boundary conditions.
Geometrically nonlinear analysis of sandwich panels with auxetic honeycomb core and nanocomposite enriched face-sheets under periodic and impulsive loads
2023, Aerospace Science and Technology
In this paper the shallow sandwich panels are considered to be made of the nanocomposite enriched face-sheets and the auxetic honeycomb core so that its Poisson's ratio is negative. In order to evaluate the geometrical nonlinear dynamic behavior of such sandwich panels, various periodic and impulsive loadings are considered. For the sake of solving the governing nonlinear differential equations system, an analytical approach on the basis of the Galerkin method is conducted which yields a closed form differential equation of motion which is numerically solved by the fourth-order Runge–Kutta method concept. The precision of the proposed analytical method is approved using comparison of the results related to some special examples and the relevant ones achieved by the other analytical and numerical methods in the references. The results of verification examples showed that the proposed method yields to the results with errors of less than 2% comparing to the results of previously published papers. The influences of different geometrical and mechanical parameters on the dynamic nonlinear responses of the proposed sandwich panels undergoing the various impulsive and periodic pressures are also evaluated in the framework of wide range of numerical study. The numerical studies revealed that by using the proposed method, performing the nonlinear dynamic analysis of the proposed panels without time-consuming computations is possible.
Dynamical responses of variable generatrix profile and thickness ceramic-matrix composite shells under electro-thermo-mechanical effects
2023, Thin-Walled Structures
In this paper, new structures of shells have been proposed with complex profiles and thicknesses varying through the length direction subjected to a thermal environment. The inner and outer surfaces of the shells are bonded to the piezoelectric actuators with changes of various voltages. The nonlinear motion equations of variable generatrix profile and thickness (VGPT) shells are solved to investigate the nonlinear vibration and chaotic responses using the theory of elasticity and Von Karman-Donnell’s geometrical nonlinearity assumption. Four kinds of VGPT shells are combined by two types of complex profiles and two patterns of thickness distribution. The used materials are made of Silicon carbide (SiC) and Borosilicate. Galerkin’s method is applied to solve the systems of differential equations and then the fundamental frequencies and nonlinear dynamic behaviors can be found. The effects of material (coefficient k), geometrical parameters, thickness, piezoelectric layers, and thermal environment on the dynamic responses of VGPT shells are examined and presented in tables and graphs.
On nonlinear forced vibration of micro scaled panels
2023, International Journal of Engineering Science
Citation Excerpt :
The Galerkin method is the other applicable basic method in order to extend the analytical solutions which was employed in Emadi et al. (2022), Tang & Dai (2021), Viet Hoang et al. (2021) for analysis of shell panels at macro scale and in Huang et al. (2021), Mirfatah et al. (2022) for analysis of micro-shell panels. In several studies such as (Abedini Baghbadorani & Kiani, 2021; Ghareghani et al., 2021), the Navier's method was employed to develop a closed-form solution for analysis of plates and shells constrained by simply-supported edges. Also in some other works such as (He et al., 2021; Matsunaga, 2007; Jalei et al. 2022), the expansion of power series was used to achieve the mentioned target.
The main target of this paper is to evaluate the geometrically nonlinear and size-dependent response of shallow sandwich curved micro-panels (SCMPs). In this regard, the governing equations are developed by using modified couple stress theory (MCST) based on the first-order shear deformation theory (FSDT). The proposed sandwich micro panels are assumed to be made of three layers so that the middle layer is a porous core and the other ones are the nanocomposite enriched face-sheets. The extended nonlinear differential equations are analytically solved by using the main concepts of the Galerkin method which results a closed-form formulation for the nonlinear equation of motion of the proposed micro-panels. Then, the obtained equation of motion is solved numerically using the forth-order Runge-Kutta method. Accuracy of the present method is validated by comparing the results of some examples with those are available in the other studies. Using an extensive parametric study, the effects of various mechanical and geometrical properties on the size-dependent forced vibration of proposed micro-panels are evaluated.
Size-dependent and piezoelectric effects on SH wave propagation in functionally graded plates
2022, Mechanics Research Communications
Citation Excerpt :
Then, FG small-scaled plates [14,15] and beam [16] models have been proposed. Size-dependent effects on bending and buckling [17], free vibration [18,19], dynamic stability [20] and wave propagation [21,22] in FG structures have been discussed. Considering piezoelectric structures, Li and Pan [23] investigated the static bending and free vibration of functionally graded piezoelectric (FGP) couple-stressed plates.
This paper is concerned with the propagation of shear horizontal waves in a functionally graded piezoelectric small-scaled plate. In the context of the modified couple stress theory, the Legendre orthogonal polynomial method is utilized to derive the solutions based on the explicit functionally graded model. Four different electric boundary conditions are considered. The obtained results are compared with those by the global matrix method and the advantages of the polynomial method are indicated. Furthermore, the couple stress effect and piezoelectric effect on the SH waves are investigated. Numerical results indicate that the couple stress and piezoelectric coupling effects on SH propagation are mutually inhibited.
Vibration, Buckling and Stability Analyses of Spinning Bi-Directional Functionally Graded Conical Shells
2024, International Journal of Structural Stability and Dynamics

View all citing articles on Scopus

View full text

Analysis of FGM micro cylindrical shell with variable thickness using Cooper Naghdi model: Bending and buckling solutions

Abstract

Introduction

Section snippets

Geometry and Simulation

Navier's method

DQ method

Numerical result and discussion

Conclusion

Declaration of Competing Interest

Acknowledgment

J. Sound Vib.

Compos. Struct.

Int. J. Solids Struct.

Int. J. Mech. Sci.

Compos. Struct.

J. Comput. Phys.

Appl. Math. Modelling

Thermal Buckling of Functionally Graded Plates

AIAA J

Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory

Meccanica