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 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.     

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.     

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.     

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 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 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.     

Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

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.     

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.     

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 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 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.     

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

This paper investigates the nonlinear dynamic response of thick functionally graded materials (FGM) plates using the third-order shear deformation plate theory and stress function. The FGM plate is assumed to rest on elastic foundations and subjected to thermal and damping loads. Numerical results for dynamic response of the FGM plate are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the material properties, the elastic foundations, and thermal loads on the nonlinear dynamic response of FGM plates.     

Elasticity Solution for Bending Response of Functionally Graded Sandwich Plates Under Thermomechanical Loading

The thermomechanical bending response of functionally graded sandwich plates has been investigated by the use of the new four variable refined plate theories. The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, 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 no symmetric sandwich plate faces are made of isotropic, two-constituent (ceramic–metal) material distribution through the thickness. The core layer is still homogeneous and made of an isotropic metal material. Several kinds of no symmetric sandwich plates are presented. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order, and the other higher-order theories. Field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory are derived. Displacement and stress functions of the plate for different values of the power-law exponent and thickness to-side ratios are presented. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated.     

Thermal post-buckling behavior of simply supported sandwich panels with truss cores

This article presents a thermal post-buckling solution for sandwich panels with truss cores under simply supported conditions, when subjected to uniform temperature rise. The Reissner assumptions are adopted and truss cores are assumed to be continuous and homogeneous. Differential governing equations are developed based on the variational principle. The perturbation technique is employed to determine the thermal post-buckling path of sandwich panels with truss cores. Based on the present method, influences of truss core configuration, relative density, aspect ratio, and initial imperfection on the thermal post buckling behavior are discussed.     

Thermal Buckling of Piezolaminated Plates Subjected to Different Loading Conditions

A thermal buckling analysis is presented for simply supported rectangular laminated composite plates that are covered with top and bottom piezoelectric actuators, and subjected to the combined action of thermal load and constant applied actuator voltage. The thermomechanical properties of composite and piezoelectric materials are assumed to be linear functions of the temperature. The formulations of the equations are based on the higher-order laminated plate theory of Reddy and using the Sanders nonlinear kinematic relations. The closed-form solutions for the buckling temperature are obtained through the Galerkin procedure and solving the resultant eigenvalue problem, which are convenient to be used in engineering design applications. Numerical examples are presented to verify the proposed method. The effects of the plate geometry, fiber orientation in composite layers, lay-up configuration, different utilized piezoelectric materials, temperature dependency of material properties, thermal conductivity, and energy generation on the buckling load are investigated.     

THERMOELASTIC STABILITY OF ORTHOTROPIC CIRCULAR PLATES

In this article the thermoelastic buckling of a circular orthotropic composite plate is discussed. The plate is assumed to be geometrically perfect. The equilibrium and stability equations, derived via variational formulations, are used to determine the prebuckling forces and the buckling temperatures. The equations are based on the Love-Kirchhoff hypothesis and Sanders' nonlinear strain-displacement relation. Critical buckling temperatures associated with the uniform temperature rise, gradient through-the-thickness temperature, and linear temperature variation along the radius are obtained. The results are validated for the first type of loading with the known data in literature.     

Thermal Post-Buckling of Functionally Graded Material Circular Plates Subjected to Transverse Point-Space Constraints

Axisymmetrical thermal post-buckling of functionally graded material (FGM) circular plates with immovably clamped boundary and a transversely central point-space constraint was studied. The material properties of the plate were assumed to vary as power law functions in the thickness direction and the temperature rise field to change only in the thickness direction. Based on von Karman's non-linear plate theory, governing equations in terms of the displacements of the middle plane were established. Temperature rise field was obtained by solving the one-dimensional heat conduction equation associated with specified boundary conditions at the top and bottom surface of the plate. By using the shooting method, thermal post-buckling deformation of the FGM circular plate was obtained before and after the plate contacting the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates were discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of FGM circular plates were also examined.     

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.