Nonlinear thermal bending response of FGM plates due to heat conduction

Nonlinear thermal bending analysis is presented for a simply supported, shear deformable functionally graded plate without or with piezoelectric actuators subjected to the combined action of thermal and electrical loads. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) 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, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations of an FGM plate are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. A two step perturbation technique is employed to determine the thermal load–deflection and thermal load–bending moment curves. The numerical illustrations concern nonlinear bending response of FGM plates without or with surface bonded piezoelectric actuators due to heat conduction and under different sets of electric loading conditions. The results reveal that for the case of heat conduction the nonlinear thermal bending responses are quite different to those of FGM plates subjected to transverse mechanical loads, and the temperature-dependency of FGMs could not be neglected in the thermal bending analysis.     

Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment

In the present study, finite element formulation based on higher order shear deformation plate theory is developed to analyze nonlinear natural frequencies, time and frequency responses of functionally graded plate with surface-bonded piezoelectric layers under thermal, electrical and mechanical loads. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The material properties of functionally graded material (FGM) are assumed temperature-dependent. The temperature field has uniform distribution over the plate surface and varies in the thickness direction. The considered electric field only has non-zero-valued component E_z. Numerical results are presented to study effects of FGM volume fraction exponent, applied voltage in piezoelectric layers, thermal load and vibration amplitude on nonlinear natural frequencies and time response of FGM plate with integrated piezoelectric layers. In addition, nonlinear frequency response diagrams of the plate are presented and effects of different parameters such as FGM volume fraction exponent, temperature gradient, and piezoelectric voltage are investigated.     

Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations

Nonlinear vibration, nonlinear bending and postbuckling analyses are presented for a sandwich plate with FGM face sheets resting on an elastic foundation in thermal environments. The material properties of FGM face sheets 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 equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both FGM face sheets and homogeneous core layer are assumed to be temperature-dependent. The numerical results reveal that the foundation stiffness and temperature rise have a significant effect on the natural frequency, buckling load, postbuckling and nonlinear bending behaviors of sandwich plates. The results also reveal that the core-to-face sheet thickness ratio and the volume fraction distribution of FGM face sheets have a significant effect on the natural frequency, buckling load and postbuckling behavior of the sandwich plate, whereas this effect is less pronounced for the nonlinear bending, and is marginal for the nonlinear to linear frequency ratios of the same sandwich plate.     

Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads

Based on Reddy's higher-order shear deformation plate theory, this article presents an analysis of the nonlinear dynamic response and vibration of imperfect functionally graded material (FGM) thick plates subjected to blast and thermal loads resting on elastic foundations. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Numerical results for the dynamic response and vibration of the FGM plates with two cases of boundary conditions are obtained by the Galerkin method and fourth-order Runge–Kutta method. The results show the effects of geometrical parameters, material properties, imperfections, temperature increment, elastic foundations, and boundary conditions on the nonlinear dynamic response and vibration of FGM plates.     

Optimization of material composition to minimize the thermal stresses induced in FGM plates with temperature-dependent material properties

A hybrid genetic algorithm with the complex method is developed for the optimization of the material composition of a multi-layered functionally graded material plate with temperature-dependent material properties in order to minimize the thermal stresses induced in the plate when it is subjected to steady-state thermal loads. In the formulation, the plate is artificially divided into an n _l -layered plate, and a weak-form-based finite layer method is developed to obtain the displacement and stress components induced in the n _l -layered plate using the Reissner mixed variational theorem. Two thermal conditions, namely the specified temperature and heat convection conditions, imposed on the top and bottom surfaces of the plate are considered. The through-thickness distributions of the volume fractions of the constituents are assumed as certain specific/non-specific function distributions, such as power-law, sigmoid, layerwise step and layerwise linear function distributions, and the effective material properties of the plate are estimated using the Mori–Tanaka scheme. Comparisons with regard to the minimization for the peak values of the stress ratios induced in the FGM plates with various optimal material compositions are conducted.     

Guided thermoelastic waves in functionally graded plates with two relaxation times

Functionally graded material (FGM) is a promising heat insulation material. Wave propagation in FGM structures has received much attention for the purpose of non-destructive testing and evaluation. Few literatures dealt with the thermoelastic wave in FGM structures although the thermal effect would cause attenuations of elastic waves. In this paper, guided thermoelastic waves in FGM plates subjected to stress-free, isothermal boundary conditions are investigated in the context of the Green–Lindsay (GL) generalized thermoelastic theories (with two relaxation times). Coupled wave equations and heat conduction equation are solved by the Legendre polynomial approach. Dispersion curves for a pure elastic graded plate are calculated to make a comparison with the published data. For the thermoelastic graded plate, dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Attenuation curves for graded plates with different relaxation times are compared. The influences of different material gradient shapes are discussed. Two homogeneous thermoelastic plates with different volume fractions are obtained to show their differences from graded plates. Finally, thermoelastic wave dispersion curves for a homogeneous plate and a graded plate are calculated in the context of the classical coupled thermoelastic theory (CT) to show its differences and similarities to the generalized theory.     

Generalized thermoelastic waves in functionally graded plates without energy dissipation

In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics.     

A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading

Due to the variation in material properties through the thickness, bifurcation buckling cannot generally occur for plates or beams made of functionally graded materials (FGM) with simply supported edges. Further investigation in this paper indicates that FGM beams subjected to an in-plane thermal loading do exhibit some unique and interesting characteristics in both static and dynamic behaviors, particularly when effects of transverse shear deformation and the temperature-dependent material properties are simultaneously taken into account. In the analysis, based on the nonlinear first-order shear deformation beam theory (FBT) and the physical neutral surface concept, governing equations for both the static behavior and the dynamic response of FGM beams subjected to uniform in-plane thermal loading are derived. Then, a shooting method is employed to numerically solve the resulting equations. The material properties of the beams 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, and to be temperature-dependent. The effects of material constants, transverse shear deformation, temperature-dependent material properties, in-plane loading and boundary conditions on the nonlinear behavior of FGM beams are discussed in detail.     

Thermal shock analysis for a functionally graded plate with a surface crack

Summary The fracture behavior of a functionally graded material (FGM) plate subjected to a thermal shock is studied. A surface crack is considered. The thermomechanical properties of the FGM plate are assumed to vary along the thickness direction. By using a perturbation method, the transient temperature field is solved. Then the transient thermal stresses and the corresponding thermal stress intensity factor (TSIF) are obtained. The transient thermal stresses and TSIF in an FGM ceramic/metal (ZrO₂/Ti-6Al-4V) plate are shown in figures. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

Thermal buckling of various types of FGM sandwich plates

The sinusoidal shear deformation plate theory is used to study the thermal buckling of functionally graded material (FGM) sandwich plates. This theory includes the shear deformation and contains the higher- and first-order shear deformation theories and classical plate theory as special cases. Material properties and thermal expansion coefficient of the sandwich plate faces 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 core layer is still homogeneous and made of an isotropic material. Several kinds of symmetric sandwich plates are presented. Stability equations of FGM sandwich plates include the thermal effects. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio, loading type and sandwich plate type on the critical buckling for sandwich plates.     

Vibrations of cracked rectangular FGM thick plates

Accurate first-of-its-kind solutions of the free vibration characteristics of side-cracked rectangular functionally graded material (FGM) thick plates are reported. From a brief review summary of available shear deformable plate theories, the well-established Reddy third-order plate theory apropos to cracked FGM thick plates is utilized. A novel Ritz procedure is developed incorporating special admissible functions – appropriately named in this study as crack functions – that properly account for the stress singularity behaviors in the neighborhood of a crack tip, and that properly account for the discontinuities of displacements and slops across a crack. Material properties of the FGM plates are assumed to vary continuously in the thickness direction according to the Mori–Tanaka scheme or a simple power law. The proposed special admissible functions accelerate the convergence of the extensive non-dimensional frequency solutions summarized. The first known non-dimensional frequencies of simply-supported and cantilevered cracked aluminum (Al) and ceramic (zirconia (ZrO₂)) or alumina (Al₂O₃) FGM thick plates of moderate thickness ratio (side-length to plate thickness, b/h = 10) are accurately determined. The effects of the volume fraction in the modeling of material distribution in the thickness direction and of cracks with different lengths, locations and orientations on the non-dimensional frequencies are investigated.     

Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM

The effects of three-parameter elastic foundations and thermo-mechanical loading on axisymmetric large deflection response of a simply supported annular FGM plate are investigated. An annular FGM plate, resting on a three-parameter elastic foundation under a transverse uniform loading and a transverse non-uniform temperature, is considered. The mechanical and thermal properties of the 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 mathematical modeling of the plate and the resulting nonlinear governing equations of equilibrium are derived based on the first-order shear deformation theory (FSDT) in conjunction with nonlinear von Karman assumptions. A polynomial-based differential quadrature method is used as a simple but powerful numerical technique to discretize the nonlinear governing equations and to implement the boundary conditions. Finally, the effects of certain parameters, such as nonlinear foundations stiffness, volume fraction index, and temperature, on the axisymmetric large deflection response of the FGM plate are obtained and discussed in detail.     

Investigation on a Two-dimensional Generalized Thermal Shock Problem with Temperature-dependent Properties

The dynamic response of a two-dimensional generalized thermoelastic problem with temperature-dependent properties is investigated in the context of generalized thermoelasticity proposed by Lord and Shulman. The governing equations are formulated, and due to the nonlinearity and complexity of the governing equations resulted from the temperature-dependent properties, a numerical method, i.e., finite element method is adopted to solve such problem. By means of virtual displacement principle, the nonlinear finite element equations are derived. To demonstrate the solution process, a thermoelastic half-space subjected to a thermal shock on its bounding surface is considered in detail. The nonlinear finite element equations for this problem are solved directly in time domain. The variations of the considered variables are obtained and illustrated graphically. The results show that the effect of the temperature-dependent properties on the considered variables is to reduce their magnitudes, and taking the temperature-dependence of material properties into consideration in the investigation of generalized thermoelastic problem has practical meaning in predicting the thermoelastic behaviors accurately. It can also be deduced that directly solving the nonlinear finite element equations in time domain is a powerful method to deal with the thermoelastic problems with temperature-dependent properties.     

Stochastic nonlinear bending response of functionally graded material plate with random system properties in thermal environment

This study deals with the stochastic nonlinear bending response of functionally graded materials (FGMs) plate with uncertain system properties subjected to transverse uniformly distributed load in thermal environments. The system properties such as material properties of each constituent’s material, volume fraction index and transverse load are taken as independent random input variables. The material properties are assumed to be temperature independent (TID) and temperature dependent (TD). The basic formulation is based on higher order shear deformation theory with von-Karman nonlinear strain kinematics using modified C ⁰ continuity. A direct iterative based nonlinear finite element method in conjunction with first-order perturbation technique developed by last two authors for the composite plate is extended for the FGM plate to compute the second order statistics (mean and standard deviation) of the nonlinear bending response of the FGM plates. Effects of TD, TID material properties, aspect ratios, volume fraction index and boundary conditions, uniform temperature and non-uniform temperature distribution on the nonlinear bending are presented in detail through parametric studies. The present outlined approach has been validated with the results available in the literature and independent Monte Carlo simulation.     

Inverse problem of material distribution for desired fracture characteristics in a thick-walled functionally graded material cylinder with two diametrically-opposed edge cracks

This study concerns the inverse problem of evaluating the optimum material distribution for desired fracture characteristics in a thick-walled functionally graded material (FGM) cylinder containing two diametrically-opposed edge cracks emanating from the inner surface of the cylinder. The thermal eigenstrain developed in the cylinder material due to nonuniform coefficient of thermal expansion as a result of cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress intensity factors developed in a previous study, an inverse method is developed to optimize material distribution intending to realize prescribed apparent fracture toughness in the FGM cylinder. To present some numerical results, a TiC/Al₂O₃ FGM cylinder is considered and the inverse problems are solved to evaluate material distributions for two examples of prescribed apparent fracture toughness. The effect of cylinder wall thickness on the material distribution and comparison of material distributions corresponding to a single and two cracks are also discussed. The numerical results reveal that the apparent fracture toughness of FGM cylinders can be controlled by choosing the material distributions properly.     

Minimizing Thermal Residual Stress in Ni/Al₂O₃ Functionally Graded Material Plate by Volume Fraction Optimization

The thermal residual stress in the fabrication of functionally graded material (FGM) systems can give rise to various mechanical failures. For a FGM system under a given fabrication environment, the thermal residual stresses are determined by the spatial distribution of its constituent components. In this study, we optimize a Ni/Al₂O₃ FGM plate aiming at minimizing the thermal residual stresses through controlling its compositional distribution. Material properties are graded in the thickness direction following a power law distribution in terms of the volume fractions of constituents (P-FGM). An analytical model and a hybrid genetic algorithm with the pattern search are employed to predict and to minimize the thermal residual stresses, respectively. Simulation results show that an optimal design of the FGM plate could help fulfill its potential in reducing the thermal residual stresses.     

Nonlinear Dynamics of a Functionally Graded Thin Simply-Supported Plate Under a Hypersonic Flow

An analysis on the nonlinear dynamics of an FGM plate in hypersonic flow subjected to an external excitation and uniform temperature change is presented in this article. Material properties of the constituents are assumed to be temperature-dependent and the effective properties of FGM plate are graded in thickness direction according to a simple power law function in terms of the volume fractions. Based on von Karman type nonlinear strain–displacement relationship, the nonlinear governing equations of motion are derived by using Hamilton's principle. Galerkin's method is then utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system, including the quadratic and cubic nonlinear terms. A third-order piston theory is applied for the hypersonic aerodynamic load. The Mach number and in-plane loading are used as bifurcation parameters. Numerical results including the bifurcations, waveform, phase plots, and Poincare maps are presented, for an FGM plate to show the influences of material gradient index, Mach number, and in-plane load on the nonlinear dynamics.     

非保守力和热载荷作用下FGM梁的稳定性

研究了在热载荷和切向均布随从力作用下FGM梁的稳定性问题。假设材料常数(即弹性模量和密度)随温度及沿截面高度连续变化,且材料常数按各材料的体积分数以幂率变化,温度分布满足一维热传导方程,计算了不同梯度指标和不同温度下FGM梁的弹性模量随截面高度变化情况。基于Euler-Bernoulli梁理论,建立梁的控制微分方程,用小波微分求积法(WDQ法)求解,分析了梯度指标、温度、随从力等参数对简支FGM梁振动特性与稳定性的影响。     

Thermal Buckling Analysis of Piezoelectric Functionally Graded Plates with Temperature-Dependent Properties

In this study, the thermal buckling analysis of hybrid laminated plates made of two-layered functionally graded materials (FGMs) that are integrated with surface-bonded piezoelectric actuators referred to as (P/FGM)_s are investigated. Material properties for both substrate FGM layers and piezoelectric layers are temperature-dependent. Uniform temperature rise as a thermal load and constant applied actuator voltage are considered for this analysis. By definition of four new analytic functions, the five coupled governing stability equations, which are derived based on the first-order shear deformation plate theory, are converted into fourth-order and second-order decoupled partial differential equations (PDEs). Considering a Levy-type solution, these two PDEs are reduced to two ordinary differential equations. One of these equations is solved using an accurate analytical solution, which is named as power series Frobenius method. The effects of parameters, such as the plate aspect ratio, ratio of piezoelectric layer thickness to thickness of FGM layer, gradient index, actuator voltage, and the temperature dependency on the critical buckling temperature difference, are illustrated and explained. The critical buckling temperatures of (P/FGM)_s with six various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.