THERMAL BUCKLING OF FUNCTIONALLY GRADED PLATES BASED ON HIGHER ORDER THEORY

Equilibrium and stability equations of a rectangular plate made of functionally graded material (FGM) under thermal loads are derived, based on the higher order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. The derived equilibrium and stability equations for functionally graded plates (FGPs) are identical to the equations for laminated composite plates. A buckling analysis of a functionally graded plate under four types of thermal loads is carried out and results in closed-form solutions. The critical buckling temperature relations are reduced to the respective relations for functionally graded plates with a linear composition of constituent materials and homogeneous plates. The results are compared with the critical buckling temperatures obtained for functionally graded plates based on classical plate theory given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of functionally graded plates, whereas the classical plate theory overestimates buckling temperatures.     

REFINED THEORY FOR THERMOELASTIC STABILITY OF FUNCTIONALLY GRADED CIRCULAR PLATES

Axisymmetric thermal and mechanical buckling of functionally graded circular plates is considered. Equilibrium and stability equations under thermal and mechanical loads are derived based on first-order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental ordinary differential equations is established. Buckling analysis of a functionally graded plate under uniform temperature rise, linear and nonlinear gradient through the thickness, and uniform radial compression are considered, and the critical buckling loads are derived for clamped edge plates. The results are compared with the buckling loads obtained for a functionally graded plate based on the classical plate theory given in the literature.     

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.     

Thermal Buckling of Functionally Graded Plates According to a Four-Variable Refined Plate Theory

In this article, a four-variable refined plate theory is presented for buckling analysis of functionally graded plates. The theory, which has strong similarity with classical plate theory in many aspects, 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. A power law distribution is used to describe the variation of volume fraction of material compositions. Equilibrium and stability equations are derived based on the present theory. The non-linear governing equations are solved for plates subjected to simply supported boundary conditions. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. The influences of many plate parameters on buckling temperature difference will be investigated. It is noticed that the present refined plate theory can predict accurately the critical temperatures of simply supported functionally graded plates.     

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.     

Thermal buckling analysis of functionally graded sandwich plates

Thermal buckling of functionally graded sandwich plates are presented in this article. Two common types of FGM sandwich plates, namely, homogeneous face layers with FGM core and FGM face layers with homogeneous core are considered. Material properties and thermal expansion coe?cient of FGM layers are assumed to vary continuously through-the-thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. Equilibrium and stability equations of FGM sandwich plate with simply supported boundary conditions are derived using the higher-order shear deformation plate theory. The influence of the plate aspect ratio, the relative thickness, the gradient index, and the thermal loading conditions on the critical buckling temperature of FGM sandwich plates are investigated. The thermal loads are assumed to be uniform, linear, and nonlinear distribution through-the-thickness. A new simple solution for thermal buckling of FGM sandwich plates under nonlinear temperature rise is presented.     

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 post-buckling of temperature dependent sandwich plates with FG-CNTRC face sheets

Present research investigates the thermal postbuckling of sandwich plates containing a stiff core and two thin carbon nanotube reinforced composite (CNTRC) face sheets. Properties of the core, carbon nanotubes (CNTs) and polymeric matrix of the faces are assumed to be temperature-dependent. It is assumed that CNTs as reinforcements may be distributed according to a functionally graded pattern. Plate is formulated based on the first-order shear deformation theory and von Kármán type of geometrical nonlinearity. The governing equations are obtained by the energy method with the aid of the Conventional Ritz method. Shape functions of the Ritz method are estimated according to the Chebyshev polynomials. A set of nonlinear eigenvalue equations is achieved. The obtained equations are homogeneous, coupled, and nonlinear in terms of both displacements and temperature. A successive displacement control strategy is implemented to trace the thermal postbuckling equilibrium path of the plate. It is shown that, with increasing the volume fraction of CNT, critical buckling temperature of sandwich plate increases and postbuckling deflection decreases. Furthermore, through a functionally graded distribution of volume fraction of CNTs across the thickness, critical buckling temperature of the sandwich plate may be enhanced and thermal postbuckling deflection may be alleviated.     

Thermal buckling and postbuckling behavior of functionally graded carbon-nanotube-reinforced composite plates resting on elastic foundations with tangential-edge restraints

This article presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes, resting on elastic foundations and subjected to thermal load due to uniform temperature rise or linear temperature change across the plate thickness. The material properties of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) are assumed to be temperature independent, graded in the thickness direction, and estimated by extended rule of mixture through a micromechanical model. Formulations are based on classical plate theory taking von Kármán nonlinearity, initial geometrical imperfection, Pasternak-type foundation interaction, and tangential-edge constraints into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to obtain closed-form expressions of buckling temperatures and temperature-deflection relations. The influences of carbon-nanotube volume fraction and distribution pattern, aspect ratios, stiffness of foundations, degree of tangential-edge constraints, and imperfection on the thermal buckling and postbuckling behavior of FG-CNTRC plates are analyzed and discussed.     

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

This article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.     

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 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.     

Size dependent thermal buckling and postbuckling of functionally graded circular microplates based on modified couple stress theory

In this article, size-dependent thermal buckling and postbuckling behavior of a functionally graded circular microplate under uniform temperature rise field and clamped boundary conditions is investigated. Material properties are assumed to gradually vary through the thickness according to a simple power law. Equilibrium equations and associated boundary conditions are derived using variational method and based on modified couple stress theory, classical plate theory and von Kármán geometric nonlinearity. The differential quadrature method is used to discretize the governing equations. This technique is accompanied by an iterative method to determine the thermal postbuckling behavior of microplate. Finally, effects of length scale parameter, power law index and ratio of thickness to radius on the thermal buckling and postbuckling behavior of FG circular microplate are investigated.     

Thermal Stability of FGM Sandwich Plates Under Various Through-the-Thickness Temperature Distributions

A thermal stability analysis of functionally graded material (FGM) isotropic and sandwich plates is carried out by virtue of a refined quasi-3D Equivalent Single Layer (ESL) and Zig-Zag (ZZ) plate models developed within the framework of the Carrera Unified Formulation (CUF) and implemented within the Hierarchical Trigonometric Ritz Formulation (HTRF). The Principle of Virtual Displacements (PVD) is used both to derive the thermal stability differential equations with natural boundary conditions and to develop the HTRF. Uniform, linear, and non-linear temperature rises through-the-thickness direction are taken into account. The non-linear temperature distribution is given in different forms: 1) functionally graded; 2) solution of the one-dimensional Fourier heat conduction equation; and 3) sinusoidal. Several FGM sandwich plate configurations are investigated. Parametric studies are carried out in order to evaluate the effects of significant parameters, such as volume fraction index, length-to-thickness ratio, boundary conditions, aspect ratio, sandwich plate type, and temperature distribution through-the-thickness direction, on the critical buckling temperatures.     

Thermal buckling analysis of moderately thick FGM plates based on the von Kármán nonlinearity and improved third order shear deformation theory

Abstract

In this study, thermal buckling of moderately thick functionally graded rectangular plates with all edges simply supported is analyzed by means of an improved third order shear deformation theory (improved TSDT). The plate is assumed to be under two types of thermal loadings, namely; uniform temperature rise and nonlinear temperature change across the thickness. The equilibrium and stability equations are derived based on the von Kármán type of geometrical nonlinearity and the improved third-order theory. By solving the stability equations, the value of buckling temperature difference is obtained. To calculate the critical buckling temperature difference, this value is minimized with respect to the half-wave parameters. The results are compared with the known data in literatures. The results indicate that, the values of critical buckling temperature difference which are obtained based on the improved TSDT, are lower in comparison with those obtained based on TSDT. Also, the results show that incorporation of the von Kármán type of geometrical nonlinearity with the improved third-order theory, gives the lower values of the critical buckling temperature difference.     

Thermal buckling of temperature-dependent composite super elliptical plates reinforced with carbon nanotubes

In this work, the problem of thermal buckling of composite plates reinforced with carbon nanotubes (CNTs) is investigated. Distribution of CNTs as reinforcements through the thickness direction of the plate is assumed to be either uniform or functionally graded (FG). Properties of the reinforcement and matrix are both temperature dependent. Properties of the composite media are obtained according to a refined rule of mixture approach where the e?ciency parameters are introduced. The plate is in a super elliptical shape where the simple elliptical shape and rectangular shapes are obtained as especial cases. In these types of plates due to the round corners, stress concentration phenomenon is eliminated. Based on the Ritz method where the shape functions are of the polynomial type, the governing equations are obtained. These equations are solved using an iterative eigenvalue problem since the properties are temperature dependent. Numerical results are validated for the simple case of an isotropic plate. Novel numerical results are provided for plates reinforced with CNTs in different shapes, various volume fractions and different patterns of CNT distribution. It is shown that FG-X pattern of CNTs in matrix results in the maximum critical buckling temperature.     

Mechanical and thermal buckling of exponentially graded sandwich plates

This article presents an analytical solution for the mechanical and thermal buckling of exponentially graded material (EGM) sandwich plates. The solution is obtained using a four-variable refined plate theory. Two types of sandwich plates are investigated: one with EGM face sheets and homogeneous core; the other with EGM core and homogeneous face sheets. The governing equations are derived based on the principle of virtual work and then solved through Navier method. The results on critical buckling load and temperature change of simply supported EGM sandwich plates are obtained. The influences of several parameters on buckling behaviors are discussed.     

Refined theories based on non-polynomial kinematics for the thermoelastic analysis of functionally graded plates

This article presents an analytical solution for the thermoelastic analysis of simply supported functionally graded sandwich plates using the Carrera unified formulation, which allows the automatic implementation of various structural theories. The governing equations for plates under thermal loads are obtained using the principal of virtual displacement and solved using the Navier method. Linear and nonlinear temperature fields through the thickness are taken into account. Particular attention is focused on plate theories with nonpolynomial refined kinematics. The results of the present displacement fields are compared with the classical polynomial ones, proposed by Carrera, for several orders of expansion.     

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.     

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.