Two-Variable Refined Plate Theory for Thermoelastic Bending Analysis of Functionally Graded Sandwich Plates

The thermoelastic bending analysis of functionally graded sandwich plates using the two-variable refined plate theory is presented in this paper. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson's ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. The influences played by the transverse shear deformation, thermal load, plate aspect ratio, and volume fraction distribution are studied. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated. It can be concluded that the proposed theory is accurate and simple in solving the thermoelastic bending behavior of functionally graded plates.     

Thermal Buckling Analysis of Perforated Functionally Graded Plates

In this research, the buckling behavior of functionally graded (FG) plates under thermal loading is investigated based on finite element analysis. It is assumed the plate is subjected to a uniform temperature rise across plate thickness. First-order shear deformation theory (FSDT) is utilized for developing the solution method. By using an appropriately designed mesh structure for a perforated plate, the critical thermal buckling temperature is obtained by numerical solution of the problem based on finite element method (FEM). The FG plate is perforated by multiple cutouts. The number of cutouts is assumed one, two, four, or six. Also different geometrical shapes of cutouts including triangle, square, rhombus, pentagon, hexagon, and circle are considered. The influence of the number of cutouts and their geometrical shapes on thermal buckling response is investigated. The effects of the number of sides of cutouts from three (triangle) to infinity (circle) are discussed. Two different boundary conditions are taken into account. Also the influences of the distance between the cutouts and the orientation of cutouts on critical buckling temperature are studied. In addition, the effects of the orientation of ellipse cutouts are studied. Some remarkable conclusions are gained that can be useful in practical applications.     

Thermal Buckling Analysis of Functionally Graded Thin Circular Plate Made of Saturated Porous Materials

This study presents the buckling analysis of thermal loaded solid circular plate made of porous material. It is assumed that the material properties of the porous plate vary across the thickness. The edge of the plate is clamped and the plate is assumed to be geometrically perfect. The geometrical nonlinearities are considered in the Love–Kirchhoff hypothesis sense. Equilibrium and stability equations, derived through the variational formulation, are used to determine the prebuckling temperatures and critical buckling temperatures. The equations are based on the Sanders non-linear strain-displacement relation.The porous plate is assumed of the form where pores are saturated with fluid. Also, the effect of pores distribution and thermal distribution on the critical buckling temperature is investigated.     

Thermal Buckling of Piezoelectric Functionally Graded Material Beams

Buckling analysis of functionally graded material (FGM) beams with surface-bonded piezoelectric layers which are subjected to both thermal loading and constant voltage is studied. The material nonhomogeneous properties are assumed to vary smoothly by distribution of power law through the beam thickness. The Euler-Bernoulli beam theory and nonlinear strain-displacement relation are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of thermal loading and boundary conditions, closed-form solutions are obtained. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of functionally graded material on the buckling temperature are investigated.     

Exact Solution for Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations with Various Boundary Conditions

In this article, we investigate the buckling analysis of plates that are made of functionally graded materials (FGMs) resting on two-parameter Pasternak's foundations under thermal loads. Three different thermal loads were considered, i.e., uniform temperature rise (UTR), linear and non-linear temperature distributions (LTD and NTD) through the thickness. The mechanical and thermal properties of functionally graded material (FGM) vary continuously along the plate thickness according to a simple power law distribution. Employing an analytical approach, the five coupled governing stability equations, which are derived based on first-order shear deformation plate theory, are converted into two uncoupled partial differential equations (PDEs). Considering the Levy-type solution, these two PDEs are reduced to two ordinary differential equations (ODEs) with variable coefficients. Then, the ODEs are solved using an exact analytical solution, which is called the power series Frobenius method. The appropriate convergence study and comparison with previously published related articles was employed to verify the accuracy of the proposed method. After such verifications, the effects of parameters such as the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the critical buckling temperature difference are illustrated and explained. The critical buckling temperatures of functionally graded rectangular plates with six various boundary conditions are reported for the first time and can serve as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Thermal Buckling Response of Functionally Graded Plates with Clamped Boundary Conditions

In this research work, an exact analytical solution for thermal buckling analysis of functionally graded material (FGM) plates with clamped boundary condition subjected to uniform, linear, and non-linear temperature rises across the thickness direction is developed. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory accounts for parabolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. The material properties of FGM plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations are solved analytically for a plate with simply supported boundary conditions. Resulting equations are employed to obtain the closed-form solution for the thermal force resultant for each loading case. Numerical examples covering the effects of the plate aspect ratio, side-to-thickness ratio and gradient index on thermal force resultant are discussed.     

Thermoelastic Buckling Analysis of Functionally Graded Circular Plates Integrated with Piezoelectric Layers

Thermal buckling of circular plates made of functionally graded materials with surface-bounded piezoelectric layers are studied. The material properties of the FG plates are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituent materials. The general thermoelastic nonlinear equilibrium and linear stability equations for the piezoelectric FG plate are derived using the variational formulations. Buckling temperatures are derived for solid circular plates under uniform temperature rise, nonlinear and linear temperature variation through the thickness for immovable clamped edge of boundary conditions. The effects of piezo-to-host thickness ratio, applied actuator voltage, boundary condition, and power law index of functionally graded plates on the buckling temperature of plate are investigated. The results are verified with the data in literature.     

Thermal Buckling of Functionally Graded Plates Resting On Elastic Foundations Using the Trigonometric Theory

In this article, thermal buckling analysis of functionally graded material (FGM) plates resting on two-parameter Pasternak's foundations is investigated. Equilibrium and stability equations of FGM plates are derived based on the trigonometric shear deformation plate theory and includes the plate foundation interaction and thermal effects. The material properties vary according to a power law form through the thickness coordinate. The governing equations are solved analytically for a plate with simply supported boundary conditions and subjected to uniform temperature rise and gradient through the thickness. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. The influences of the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the buckling temperature difference are discussed.     

Effect of Initial Imperfections on Thermal Buckling of Functionally Graded Plates

ABSTRACT

Thermal buckling analysis of rectangular functionally graded plates with initial geometrical imperfections is presented in this article. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the first-order shear deformation plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through the thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading, namely: uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect functionally graded plate. The influence of transverse shear on thermal buckling load is discussed.     

Size-Dependent Thermal Buckling and Postbuckling of Functionally Graded Annular Microplates Based on the Modified Strain Gradient Theory

This article reports on the thermal instability of functionally graded (FG) annular microplates with different boundary conditions. The modified strain gradient elasticity theory is employed to capture size effects. The non-linear governing equations and boundary conditions are derived based on the first-order shear deformation theory (FSDT) and virtual displacements principle. The generalized differential quadrature technique is implemented so as to discretize. To obtain the critical buckling temperature, the set of linear discretized governing equations is solved as an eigenvalue problem. Also, the non-linear problem of thermal postbuckling is solved by the pseudo arc-length continuation method. The effects of boundary conditions, length scale parameter, and the variation of material through the thickness and geometrical properties on both critical buckling temperature and thermal postbuckling behavior are studied.     

Thermo-Mechanical Bending of Functionally Graded Plates

In this work the deformations of a simply supported, functionally graded, rectangular plate subjected to thermo-mechanical loadings are analysed, extending Unified Formulation by Carrera. The governing equations are derived from the Principle of Virtual Displacements accounting for the temperature as an external load only. The required temperature field is not assumed a priori, but determined separately by solving Fourier's equation. Numerical results for temperature, displacement and stress distributions are provided for different volume fractions of the metallic and ceramic constituent as well as for different plate thickness ratios. They correlate very well with three-dimensional solutions given in the literature.     

Plane Thermal Stresses in a Functionally Graded Plate Subjected to a Partial Heating

This paper presents plane thermal stresses in a functionally graded plate (FGP) subjected to a partial heating. The heat conductivity, Young's modulus and the coefficient of the linear thermal expansion are expressed by exponential functions of the position. The analytical solution for the FGP with two-dimensional temperature distribution is obtained by use of the stress function method. General solution of the governing equation of the stress function is derived in the functionally graded materials (FGMs). The numerical calculations are carried out for ZrO₂/Ti-6 Al-4 V and ZrO₂{/} stainless (SUS304) functionally graded plates, when the ceramic surface is partially heated. The numerical results are shown in figures for two cases. Even though the FGM, suitable selection of the compositional materials does not produce thermal stresses in the FGP.     

Thermal Buckling of Axially Functionally Graded Thin Cylindrical Shell

In the present research, thermal buckling of shell made of functionally graded material (FGM) under thermal loads is investigated. The material properties of functionally graded materials (FGMs) are assumed to be graded in the axial direction according to a simple power law distribution in terms of the volume fractions of the constituents. In the previous articles that published, these properties are assumed to be graded in the thickness direction. Nonlinear kinematic (strain-displacement) relations are considered based on the first order shear deformation shell theory. By substituting kinematic and stress-strain relations of functionally graded shell in the total potential energy equation and employing Euler equations, the equilibrium equations are obtained. Applying Euler equations to the second variation of total potential energy equation leads to the stability equations. Then, buckling analysis of functionally graded shell under three types of thermal loads is carried out resulting into closed-form solutions.     

Thermal Instability of Functionally Graded Shallow Spherical Shell

In this paper, thermal instability of shallow spherical shells made of functionally graded material (FGM) is considered. The governing equations for a thin spherical shell based on the Donnell–Mushtari–Vlasov theory are obtained. The equations are derived using the Sanders simplified kinematic relations and variational method. It is assumed that the mechanical properties vary linearly through the shell thickness. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. Analytical solutions are obtained for three types of thermal loading including Uniform Temperature Rise (UTR), Linear Radial Temperature (LRT), and Nonlinear Radial Temperature (NRT). The results are validated with the known data in the literature.

     

Mechanical Behavior of Rectangular Plates with Functionally Graded Coefficient of Thermal Expansion Subjected to Thermal Loading

This study analyzed an elastic, rectangular, and simply supported functionally graded material (FGM) plate with medium thickness subjected to linear temperature change in the z direction. Young's modulus and Poisson ratio of the FGM plates are assumed to remain constant throughout the entire plate. However, the coefficient of thermal expansion of the FGM plate varies continuously throughout the thickness direction in relation to the volume fraction of constituents defined by power-law, sigmoid, or exponential functions. The series solutions for the power-law FGM (P-FGM), sigmoid FGM (S-FGM), or exponential FGM (E-FGM) plates subjected to thermal loading are obtained based on the classical plate theory and Fourier series expansion. The analytical solutions for P-, S-, and E-FGM plates are verified by numerical results obtained with the finite element technique.     

Thermal Effects on Stress Analysis in Large Amplitude Vibration of a Thick Annular Functionally Graded Plate

This paper deals with the nonlinear free and forced vibration of thick annular functionally graded material plates. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction. The formulations are based on the first-order shear deformation plate theory and von Kármán-type equation. The numerical illustrations concern with nonlinear vibration characteristics of functional graded plates with two constituent materials in thermal environments. Effects of material compositions and thermal loads on the vibration characteristics and stresses are examined.     

Effects of Thermal Loading on the Buckling and Vibration of Ring-Stiffened Functionally Graded Shell

A theoretical method is developed to investigate the effects of thermal load and ring stiffeners on buckling and vibration characteristics of the functionally graded cylindrical shells, based on the first-order shear deformation theory (FSDT) considering rotary inertia. Heat conduction equation across the shell thickness is used to determine the temperature distribution. Material properties are assumed to be graded across the shell wall thickness of according to a power-law, in terms of the volume fractions of the constituents. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. The effects of stiffener's number and size on natural frequency of functionally graded cylindrical shells are investigated. Moreover, the influences of material composition, thermal loading and shell geometry parameters on buckling and vibration are studied. The obtained results have been compared with the analytical results of other researchers, which showed good agreement. The new features of thermal vibration and buckling of ring-stiffened functionally graded cylindrical shells and some meaningful and interesting results obtained in this article are helpful for the application and the design of functionally graded structures under thermal and mechanical loads.     

Analysis of Thermal Stress in a Functionally Graded Coating on a Parabolic Substrate

The thermal stress field in a functionally graded coating on a parabolic substrate, where the material properties vary along the thickness direction, is considered. The closed-form solutions of thermal stresses related to compositional gradient, coating thickness and substrate curvature were obtained based on force and moment balances, and then numerical results are presented for several special examples. It is found that the magnitude and distribution of thermal stress in the functionally graded coating system with general geometrical shape can be designed properly by controlling the compositional gradient, coating thickness and substrate curvature.     

Thermal Buckling of Imperfect Deep Functionally Graded Spherical Shell

Thermal buckling analysis of deep imperfect functionally graded (FGM) spherical shell is considered in this paper. A mixture of ceramic and metal is considered for the FGM shell and the material properties, such as the modulus of elasticity and coefficient of thermal expansion, vary by a power law function through the thickness. Employing the Sanders non-linear kinematic relations, total potential energy function is derived and the equilibrium and stability equations are obtained for the imperfect shell. Approximate solutions satisfying the simply supported boundary condition are assumed and using the Galerkin method the error due to the approximation is minimized. The geometrically imperfect shell is considered and three types of thermal loadings, such as the uniform temperature rise (UTR), linear temperature rise through the thickness (LTR), and non-linear temperature rise through the thickness (NLTR) are considered and their associated buckling temperatures are obtained. The effects of different temperature functions and the magnitude of initial geometric imperfection are examined on the thermal buckling loads of the shell.     

Thermal Postbuckling Analysis of Functionally Graded Beams

Thermal buckling and postbuckling analysis of functionally graded (FG) beams is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. For discretizing the governing equations and the related boundary conditions differential quadrature method (DQM) as a simple and computationally efficient numerical tool is used. Based on displacement control method, a direct iterative method is employed to obtain thermal postbuckling behavior of FG beams with different boundary conditions and geometrical parameters.