Analysis of FGM micro cylindrical shell with variable thickness using Cooper Naghdi model: Bending and buckling solutions
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 () (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 (ux(x,θ), vθ(x,θ), w(x, θ), βx(x,θ) and βθ(x,θ)) are considered as follows:
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:where and N are the weighting coefficient of nth derivative and the number of grid point, respectively [17].
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: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
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). In the present study, based on modified couple stress theory (MCST), the bending and buckling analysis of cylindrical micro shell with variable thickness made of FGMs are presented. In this theory, Power law distribution
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)
- et al.
Linear thermoelastic buckling and free vibration behavior of functionally graded truncated conical shells
J. Sound Vib.
(2006) - et al.
Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell
Compos. Struct.
(2016) - et al.
Couple stress based strain gradient theory for elasticity
Int. J. Solids Struct.
(2002) - et al.
On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure
Int. J. Mech. Sci.
(2016) - 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) - et al.
Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations
J. Comput. Phys.
(1972) - 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) - et al.
Thermal Buckling of Functionally Graded Plates
AIAA J
(2002) - et al.
Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory
Meccanica
(2014)
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