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.     

Transient Thermoelastic and Piezothermoelastic Problems of Functionally Graded Materials

This paper is concerned with the theoretical treatment of transient thermoelastic problems involving functionally graded thick plate, laminated composite strip with an interlayer of functionally graded material, and functionally graded hollow cylinder, and transient piezothermoelastic problems involving functionally graded piezoelectric cylindrical panel. The thermal, thermoelastic and piezoelectric constants of the functionally graded materials are expressed as power functions of the radial coordinate variable or exponential functions of the thickness coordinate variable. The exact solutions for the two-dimensional temperature change in a transient state, and thermoelastic or piezothermoelastic response under the state of plane strain are presented herein. Some numerical results are shown in figures.     

Transient Thermoelastic Analysis for a Laminated Composite Strip with an Interlayer of Functionally Graded Material

This paper is concerned with the theoretical treatment of transient thermal stress problem involving a laminated composite thick strip with an interlayer of functionally graded material due to nonuniform heat supply in the width direction. The thermal and thermoelastic constants of the interlayer of functionally graded material are assumed to vary exponentially in the thickness direction. We obtain the exact solution for the two-dimensional temperature change in a transient state, and thermal stresses of a simple supported strip under the state of plane strain. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Furthermore, the influence of the thickness and position of the interlayer is investigated.     

Transient Thermal Stress Problem of a Functionally Graded Magneto-Electro-Thermoelastic Hollow Cylinder Due to a Uniform Surface Heating

This article is concerned with the theoretical analysis of the functionally graded magneto-electro-thermoelastic hollow cylinder due to uniform surface heating. We analyze the transient thermal stress problem for a functionally graded hollow cylinder constructed of the anisotropic and linear magneto-electro-thermoelastic materials using a laminated composite model as one of theoretical approximation under a plane strain state. As an illustration, we carry out numerical calculations for a functionally graded hollow cylinder constructed of piezoelectric and magnetostrictive materials and examined the behaviors in the transient state. We investigate the effects of the nonhomogeneity of material on the stresses, electric potential, and magnetic potential, and the effect of the applied electric potential on the thermal stress σ_θθ.     

THERMAL STRESSES INTENSITY FACTOR FOR FUNCTIONALLY GRADIENT PLATE WITH AN EDGE CRACK

Thermal stress problems of a functionally gradient plate as one of the advanced high-temperature materials capable of withstanding the extreme temperature environments, with and without an edge crack, are discussed One of the most important problems of the thermal stress in the functionally gradient plate with the crack are how to decrease the thermal stress intensity factor and how to determine the optimally continuous profile of the composition of the plate. The functionally gradient plate is subjected to a cycle of heating and cooling on the ceramics surface of the plate. The material properties of the functionally gradient plate are dependent on the temperature and the position. The optimally continuous profile of the composition of the plate is discussed. The numerical results for thermal stresses and the thermal stress intensity factor are shown for many temperature conditions and for many continuous profiles of the composition of the plate.     

TRANSIENT THERMAL STRESS ANALYSIS AND BENDING BEHAVIOR OF AN ANGLE-PLY LAMINATED SLAB

Abstract

This paper is concerned with a theoretical treatment of thermal stress and bending behavior in a transient state of a multilayered, nonisotropic, laminated slab. As an analytical model, we consider an infinitely long, laminated slab, which consists of obliquely directed layers with orthotropic material properties; the model corresponds to the so-called angle-ply laminate. We solve the thermoelastic problem for the slab under the condition of uniformly distributed heat supply from its one surface. Introducing a method of Laplace transforms to the temperature field, we obtain the temperature solution using the residue theorem, and we evaluate the thermal stresses in a transient state by using the elementary plate theory. As an example, we carry out numerical calculations for the five-layered angle-ply laminate, evaluate the thermal stress distributions and the bending behavior, and examine the influence of the ply angle on the thermal stress distribution.     

Thermal stress analysis of functionally graded annular fin

Thermal stress distributions in an annular fin with rectangular profile made of functionally graded material (FGM) are considered. The material properties of annular fin are assumed to be graded along the fin radius as a power-law function while the Poisson’s ratio is taken to be constant. The governing equations are solved analytically for specific value of inhomogeneity parameter of thermal conductivity and all numerical values of inhomogeneity parameters of modulus of elasticity and linear thermal expansion coefficient. The effect of the inhomogeneity parameters on temperature distribution and thermal stresses are presented in graphical form. The formulation is validated with benchmark results in the literature. It is also shown that functionally graded annular fin is subject to lower stresses, although it has higher tip temperature than the homogeneous one.     

Research on shakedown analysis for functionally graded material plate under thermomechanical loading

Shakedown is an important problem in the design and analysis of functionally graded structures subjected to cyclic, thermal, and mechanical loadings. Subjected to constant mechanical load and cyclic temperature change, static shakedown of a functionally graded material plate and its homogenous counterpart were analyzed in this article with the approach proposed by the authors previously. The functionally graded material plate is composed of an elastoplastic matrix Al and elastic particles SiC, and the particle volume fraction varies through the thickness. The distributions of the effective mechanical and thermal properties of the composites through the thickness are described graded continuously with an exponential law. The results show that a proper and continuously graded distribution of material properties can efficiently improve the shakedown capability of the functionally graded material plate and also show the significance of shakedown analysis and application for functionally graded material plates.     

Non-Axisymmetric Thermal Stress Around a Circular Hole in a Functionally Graded Infinite Plate

This paper presents the non-axisymmetric two-dimensional problem of thermal stresses in a functionally graded plate with a circular hole based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the plate having radial arbitrary elastic properties is derived when it is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the stress around the circular hole in the functionally graded material plate can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

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 Residual Stresses in One-Directional Functionally Graded Plates Subjected to In-Plane Heat Flux

This study carries out the transient thermal residual stress analyses of functionally graded clamped plates for different in-plane material compositions and in-plane heat fluxes. The heat conduction and Navier equations representing the two-dimensional thermoelastic problem were discretized using the finite-difference method, and the set of linear equations were solved using the pseudo singular value method. Both in-plane temperature distributions and the heat transfer period were affected considerably by the compositional gradient. The type of in-plane heat flux had a minor effect on the temperature profile, but on the heat transfer period. The high stress levels appeared in the ceramic-rich regions. The normal and equivalent stresses exhibited a sharp change in the plates with ceramic-rich as well as metal-rich compositions, and the concentrated on a narrow ceramic layer. A smooth stress variation was achieved through the graded region with a balanced composition of ceramic and metal-phases, and the stress discontinuities disappeared. The in-plane shear stress was negligible. The equivalent stress exhibited a linear temporal variation for both constant and sinusoidal heat fluxes, but a nonlinear variation for the exponential heat flux. In case the heat flux is applied along the metal edge (metal-to-ceramic plate) instead of the ceramic edge, the displacement and stress components exhibited similar distributions to those of a ceramic-to-metal plate but in the opposite direction. As a result, the distribution of in-plane material composition affects only normal stress distributions, whereas the peak stress levels occur in the ceramic-rich regions. Since the normal stresses concentrate along a narrow ceramic layer for ceramic-rich or metal-rich compositions, a balanced in-plane material composition distribution of ceramic and metal would be useful to avoid probable local ceramic fracture or damage.     

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 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.     

TRANSIENT THERMAL SINGULAR STRESSES OF MULTIPLE CRACKING IN A W-Cu FUNCTIONALLY GRADED DIVERTOR PLATE

We consider the transient thermal singular stresses of multiple cracking in a functionally graded divertor plate due to a thermal shock. The plate is made of a graded layer bonded between a homogeneous substrate and a homogeneous coating, and it is subjected to a cycle of heating and cooling on the coating surface of the plate. The surface layer contains a parallel array of embedded or edge cracks perpendicular to the boundaries. The thermal and elastic properties of the material are dependent on the temperature and the position. Finite element calculations are carried out, and the transient thermal stress intensity factors are shown graphically.     

Thermal and mechanical stresses in a functionally graded thick sphere

In this paper, a general solution for the one-dimensional steady-state thermal and mechanical stresses in a hollow thick sphere made of functionally graded material is presented. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. The material properties, except Poisson's ratio, are assumed to vary along the radius r according to a power law function. The analytical solution of the heat conduction equation and the Navier equation lead to the temperature profile, radial displacement, radial stress, and hoop stress as a function of radial direction.     

Mechanical and thermal stresses of a functionally graded circular hollow cylinder with finite length

By using a multi-layered approach based on the theory of laminated composites, the solutions of temperature, displacements, and thermal/mechanical stresses in a functionally graded circular hollow cylinder are presented in this paper. The cylinder has finite length and is subjected to axisymmetric thermal and mechanical loads. The material properties are assumed to be temperature-independent and radially dependent, but are assumed to be homogeneous in each layer. As an illustration, the numerical results for a mullite/molybdenum functionally graded circular hollow cylinder are also presented.     

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.     

TWO CRACK GROWTHS IN A FUNCTIONALLY GRADED PLATE UNDER THERMAL SHOCK

This article examines the problem of two thermal cracks under a transient temperature field in a ceramic/metal functionally graded plate. When the functionally graded plate is subjected to thermal shock, multiple cracks often occur on the ceramic surface. It is shown that the crack paths are influenced by interaction between multiple cracks and a compositional profile of the functionally graded plate. Transient thermal stresses are treated as a linear quasi-static thermoelastic problem for a plane strain state. The crack paths of two cracks are obtained using the finite element method with mode I and mode II stress intensity factors.     

COMBINED MACROSCOPIC AND MICROSCOPIC ANALYSIS OF THERMOELASTOPLASTIC STRESSES OF A FUNCTIONALLY GRADED MATERIAL PLATE

This article discusses the elastoplastic thermal stresses induced in a ceramic-metal functionally graded material plate (FGP) subjected to a thermal load taking the fabrication process into consideration. The FGP is divided into three regions. The first region near the cooling, metal, surface of the FGP is produced by ceramic particle-reinforced metal; while the second region near the heat-resistant, ceramic, surface is the opposite; and the third middle region is perfectly mixed by the metal and the ceramic. The first and second regions are governed by the particle-reinforced thermoelastoplastic constitutive equation, while the third region is expressed by the macroscopic analysis. Three cases of the temperature condition are studied: cooling from the fabricated temperature to room temperature, heating from the room temperature, and heating after cooling from the fabricated temperature. The temperature-dependent material properties are considered, and the particle volume fraction is assumed to vary according to a power function along the thickness direction of the FGP. The effect of the distribution parameter of the composition on the macroscopic stress, the stress in the matrix, and the stress in the particle in the FGP are discussed and illustrated in figures. Also, the effect of the fabricated temperature on the maximum tensile matrix stress is discussed.     

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.