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.     

Nonaxisymmetric Thermomechanical Analysis of Functionally Graded Hollow Cylinders

Analytical solutions for nonaxisymmetric, thermomechanical response of functionally graded hollow cylinders are obtained in this article. The hollow cylinders are assumed to be subjected to nonaxisymmetric mechanical and transient thermal loads. Properties of functionally graded material are considered as temperature-independent and continuously varying in radial direction. Employing complex Fourier series and Laplace transform techniques, analytical solutions of time-dependent temperature and thermomechanical stresses are obtained. Numerical values of temperature and stresses of a FGM hollow cylinder under assumed thermomechanical loads are presented in graphical form.     

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.     

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.     

Investigation of Aerothermoelastic Behaviors of Functionally Graded Panels in Supersonic Flows

Considering the potentials of Functionally Graded Panels (FGPs) in aerospace field, it is necessary to study the aerothermoelastic behaviors of FGPs in supersonic flows. In this study, Piston Theory Aerodynamics (PTA) and Eckert reference enthalpy method are used to model aerodynamic force and heating, respectively. The 2-D heat conduction equation is solved and the impact of elevated temperature on the mechanical properties of FGPs is considered to build an aerothermoelastic two-way coupling model of FGPs, and Finite Element Method (FEM) is used to approach the solution. As the results, it is found that there exist three different regions in the bifurcation diagram, namely, thermal buckling region, critical region and flutter region. Due to the inhomogeneous distribution of thermal expansion coefficient, the panel buckles up first and then buckles down via vibration, as thermal buckling happens. Also, irregular vibrations are observed in the critical region of bifurcation diagram. In the flutter region, the dynamic behavior of FGPs is discontinuous and very sensitive to initial conditions. With the impact of aerothermoelastic two-way coupling, different FGPs behaviors lead to the differences in temperature distribution. In particular, the final buckling position and vibration center move to lower positions, and lower temperature region near leading edge is left in the FGPs, because of thermal moment. Also, regular vibrations, rather than irregular vibrations, are easy to extract more principal and regular POD (Proper Orthogonal Decomposition) modes. The results presented could be applied to the analysis and design of Functionally Graded Panels in supersonic flows.     

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.     

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.     

DQM-Based Thermal Stresses Analysis of a Functionally Graded Cylindrical Shell Under Thermal Shock

The transient thermal stresses of a functionally graded (FG) cylindrical shell subjected to a thermal shock are investigated. The dynamic temperature fields of FG shells are obtained by using the Laplace transform and power series method. The differential quadrature method is developed to obtain the transient thermal stresses by solving dynamic governing equations in terms of displacements. The effects of the material constitutions on the transient temperature and the thermal stresses are analyzed in the cases of obverse thermal shock and reverse thermal shock. It turns out that the thermal stresses could be alleviated by means of changing the volume fractions of the constituents.     

Functionally Graded Materials Optimization Using Particle Swarm-Based Algorithms

This article presents an application of particle swarm-based algorithms for volume fraction optimization of functionally graded materials. A simple and efficient method for constraint handling is included too. To optimize the material composition, two numerical examples are done. In the first example, the peak residual stresses are minimized when the FGM is cooled from a reference temperature. In the second example, the factor of safety against yielding are maximized for a temperature-dependent FGM sphere subjected to thermal gradients. The algorithms PSO, PSOPC, PSACO and PSPCACO are used too and the optimum volume fraction are obtained. According comparison of the results, particle swarm-based algorithms can be used as a powerful tool for optimizing the FG materials.     

Non-Linear Response of Functionally Graded Cylindrical Shells under Mechanical and Thermal Loads

Considering rotary in-plane inertias, the geometrically non-linear vibrations of the functionally graded cylindrical shells under the combined effect of thermal fields and mechanical excitations are analysed by using the von Kármán non-linear theory. The coupled non-linear partial differential equations are discretized based on a series expansion of linear modes and a multiterm Galerkin's method. The non-linear equation of motion is then solved by the fourth-order Runge-Kutta numerical method. Parametric studies are carried out in order to study the influence of temperature change, volume fraction exponent of functionally graded materials and the geometry parameters on the non-linear dynamic response of the functionally graded cylindrical shells.     

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 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.     

Three-Dimensional Pyroelectric Analysis of a Functionally Graded Piezoelectric Hollow Sphere

A three-dimensional analytical piezothermoelastic solution is presented for a functionally graded piezoelectric spherical shell subjected to various thermal boundary conditions applied on the inner and outer surfaces. Material properties are assumed to vary along the radius, r, obeying a power law. Both the thermal field and the pyroelectric responses are resolved using the state space method. On introducing three displacement and two stress functions, two independent kinds of state equations are derived from the basic equations of piezoelectricity. It is interesting that the first kind is a homogeneous equation only related to the purely elastic behavior of the sphere, yet the second has an inhomogeneous term associated with the thermal effect to determine the pyroelectric responses. The state equations are solved by expanding the field variables into series of spherical harmonic functions. Numerical examples are performed to investigate the influence of material inhomogeneity on the pyroelectric responses of the spherical shell.     

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.     

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.     

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.     

Computation of Thermal Fracture Parameters for Inclined Cracks in Functionally Graded Materials Using J k -Integral

This article describes the formulation and implementation of the J _k -integral for the analysis of inclined cracks located in functionally graded materials (FGMs) that are subjected to thermal stresses. The generalized definition of the J _k -integral over a vanishingly small curve at the tip of an inclined crack is converted to a domain independent form that consists of area and line integrals defined over finite domains. A numerical procedure based on the finite element method is then developed, which allows the evaluation of the components of the J _k -integral, the modes I and II stress intensity factors and the T-stresses at the crack tips. The developed procedure is validated and the domain independence is demonstrated by providing comparisons to the results obtained by means of the displacement correlation technique (DCT). Detailed parametric analyses are conducted by considering an inclined crack in an FGM layer that is subjected to steady-state thermal stresses. Numerical results show the influences of the thermal conductivity and thermal expansion coefficient variation profiles and the crack inclination angle on the mixed-mode fracture parameters.     

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.