Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.     

Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads

This paper presents a simple analytical approach to investigate the stability of functionally graded plates under in-plane compressive, thermal and combined loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for functionally graded plates are derived by using the classical plate theory taking into account both geometrical nonlinearity in von Karman sense and initial geometrical imperfection. The resulting equations are solved by Galerkin procedure to obtain explicit expressions of postbuckling load–deflection curves. Stability analysis of a simply supported rectangular functionally graded plate shows the effects of the volume fraction index, plate geometry, in-plane boundary conditions, and imperfection on postbuckling behavior of the plate.     

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Summary. In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Extended three-dimensional generalized differential quadrature method: The basic equations and thermal vibration analysis of functionally graded fiber orientation rectangular plates

The present article deals with free vibration of functionally graded fiber orientation rectangular plates considering temperature effect. Three different types of fiber orientation distributions through the thickness of the plate are proposed. The properties of the plate are assumed to be temperature-dependent. Equations of motions are derived based on a three-dimensional theory of elasticity. General differential quadrature method is used to discretize these equations. Effects of temperature, fiber orientation, and boundary conditions besides some geometric parameters are presented. Also, some interesting conclusions are obtained since temperature and functionality of a functionally graded plate have a significant effect on the natural frequency of the plate.     

A closed-form solution for thermal buckling of piezoelectric FGM rectangular plates with temperature-dependent properties

A thermal buckling analysis is presented for functionally graded rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of thermal load and constant applied actuator voltage. The temperature-dependent material properties of the functionally graded plate are assumed to vary as a power form of the thickness coordinate. Derivation of the equations is based on the third-order shear deformation plate theory. Results for the critical buckling temperatures are obtained in closed-form solution, which are convenient to be used in engineering design applications. The effects of the applied actuator voltage, plate geometry, and volume fraction exponent of the functionally graded material on the buckling temperature are investigated.     

Wave propagation and transient response of a FGM plate under a point impact load based on higher-order shear deformation theory

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load are presented. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Based on the higher-order shear deformation theory and considering the effect of the rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Then, using the dispersion relation and integral transforms, exact integral solutions for the functionally graded plate under a point impact load are obtained. The transient response curves of the functionally graded plates are plotted and the influence of volume fraction distributions on transient response of functionally graded plates is analyzed. Finally, the solutions of the higher-order shear deformation theory and the first-order shear deformation theory are studied.     

Dynamic response of initially stressed functionally graded rectangular thin plates

This paper deals with the dynamic response of initially stressed functionally graded rectangular thin plates subjected to partially distributed impulsive lateral loads and without or resting on an elastic foundation. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on classical small deflection plate theory, and account for the plate–foundation interaction effects by a two-parameter model (Pasternak-type). A one-dimensional differential quadrature approximation and the Galerkin procedure are employed in the free vibration analysis, and the Modal Superposition Method is used to determine the transient response of the plate structure. A parametric study is carried out. Effects of constituent volume fraction index, foundation stiffness, plate aspect ratio, the shape and duration of impulsive load, as well as the initial membrane stresses on the dynamic response of FGM plates are studied. Comprehensive numerical results for silicon nitride/stainless steel rectangular plates are presented in dimensionless tabular and graphical forms.     

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates

Thermal buckling and postbuckling behavior is presented for functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) subjected to in-plane temperature variation. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, geometrically mid-plane symmetric FG-CNTRC plates and uniformly distributed CNTRC plates under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the plate can be increased as a result of a functionally graded reinforcement. It is found that in some cases the CNTRC plate with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces. Therefore, there is no need to use shear correction factor. Material properties of functionally graded plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as Pasternak foundation. Equations of motion are derived using Hamilton’s principle. Closed-form solution of rectangular plates is derived, and the obtained results are compared well with three-dimensional elasticity solutions and third-order shear deformation theory solutions. Finally, the influences of power law index, thickness ratio, foundation parameter, and boundary condition on the natural frequency of plates have been investigated.     

Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory

Free vibration analysis of simply supported functionally graded plates (FGP) resting on a Winkler–Pasternak elastic foundation are examined by a new higher shear deformation theory in this paper. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The equation of motion for FG rectangular plates resting on elastic foundation is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration of functionally graded plates on elastic foundation are presented, and compared with the ones available in the literature.     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

压电元件驱动的功能梯度弹性薄板的屈曲

考虑功能梯度薄板,其上下表面嵌有压电执行元件.根据逆压电效应将电压转换成作用于板上的等效电载荷.假设梯度材料的弹性参数为板厚度方向坐标的幂函数,基于经典板理论,导出了功能梯度弹性薄板小挠度屈曲平衡微分方程.利用双三角级数展开法,得到了四边简支具有压电元件的功能梯度矩形板的临界屈曲载荷,在此基础上通过数值例子讨论了弹性板的几何尺寸、材料梯度指数的变化对临界电压(载荷)的影响.研究结果表明,材料的梯度指数对临界电压有重要影响,并且通过调整作用于执行元件上的电压的大小和方向,可实现对结构稳定性的有效控制.     

四边简支功能梯度矩形板的热屈曲分析

基于经典板理论,假设材料性质为板厚度方向坐标的幂函数,推导了功能梯度材料矩形板在热荷载作用下的平衡方程和稳定方程。给出了四边简支的功能梯度板在均匀受热时临界屈曲温度变化的封闭解,讨论了板的几何外形尺寸、相对厚度、梯度指数以及中面变形等因素对临界屈曲温度变化的影响。     

Buckling and Postbuckling Behavior of Functionally Graded Nanotube-Reinforced Composite Plates in Thermal Environments

This paper investigates the buckling and postbuckling of simply supported, nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. The nanocomposite plates are assumed to be functionally graded in the thickness direction using single-walled carbon nanotubes (SWCNTs) serving as reinforcements and the plates' effective material properties are estimated through a micromechanical model. The higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity is used to model the composite plates and a two-step perturbation technique is performed to determine the buckling loads and postbuckling equilibrium paths. Numerical results for perfect and imperfect, geometrically mid-plane symmetric functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC plate, which is a special case in the present study, are compared with those of the FG-CNTRC plate. The results show that the buckling loads as well as postbuckling strength of the plate can be significantly increased as a result of a functionally graded nanotube reinforcement. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC plates.     

Bending analysis of functionally graded sectorial plates using Levinson plate theory

Based on the Levinson plate theory (LPT) and the first-order shear deformation plate theory (FST), the bending analysis of functionally graded (FG) thick circular sector plates is presented. The LPT solutions of FG sectorial plates are first expressed in terms of the solutions of the classical plate theory (CPT) for homogeneous sectorial plates and then presented using a direct method. It is assumed that the non-homogeneous mechanical properties of plate, graded through the thickness, are described by a power function of the thickness coordinate. The results are given in closed-form solutions and verified with the known data in the literature.     

Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations

Exact solutions for functionally graded thick plates are presented based on the three-dimensional theory of elasticity. The plate is assumed isotropic at any point, while material properties to vary exponentially through the thickness. The system of governing partial differential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into infinite dual series of trigonometric functions. Interactions between the Winkler–Pasternak elastic foundation and the plate are treated as boundary conditions. The problem is finally solved using the state space method. Effects of stiffness of the foundation, loading cases, and gradient index on mechanical responses of the plates are discussed. It is established that elastic foundations affects significantly the mechanical behavior of functionally graded thick plates. Numerical results presented in the paper can serve as benchmarks for future analyses of functionally graded thick plates on elastic foundations.     

Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory

Natural frequencies and buckling stresses of plates made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations and rotatory inertia. The modulus of elasticity of the plates is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional (2-D) higher-order theory for rectangular functionally graded (FG) plates is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG plates with simply supported edges. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Critical buckling stresses of FG plates subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG plates without in-plane stresses is presented. The distributions of modal displacements and modal stresses in the thickness direction are obtained accurately by satisfying the surface boundary conditions of a plate. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the top or bottom surface of a plate. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2-D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG plates.     

Nonlinear thermo-elastic bending of functionally graded carbon nanotube-reinforced composite plates resting on elastic foundations by dynamic relaxation method

In this study, the nonlinear thermo-elastic bending analysis of a functionally graded carbon nanotube-reinforced composite plate resting on two parameter elastic foundations is investigated. The material properties of the carbon nanotube-reinforced composite plates are assumed to be temperature dependent and graded in the thickness direction. The nonlinear formulations are based on a first-order shear deformation plate theory and large deflection von Karman equations. A dynamic relaxation method is employed to solve the plate nonlinear partial differential equations. The effects of volume fraction of carbon nanotubes, thermal gradient, temperature dependency, elastic foundation, boundary conditions, plate width-to-thickness ratio, aspect ratio, and carbon nanotubes distribution are studied in detail.     

A refined plate theory for functionally graded plates resting on elastic foundation

A refined plate theory for functionally graded plates resting on elastic foundation is developed in this paper. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns of present theory is four, as against five in other shear deformation theories. The material properties of plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as two-parameter Pasternak foundation. Equations of motion are derived using Hamilton’s principle. The closed-form solutions of rectangular plates are obtained. Numerical results are presented to verify the accuracy of present theory.     

Bending response of functionally graded plates by using a new higher order shear deformation theory

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.