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.     

A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates

A meshfree model is presented for the static and dynamic analyses of functionally graded material (FGM) plates based on the radial point interpolation method (PIM). In the present method, the mid-plane of an FGM plate is represented by a set of distributed nodes while the material properties in its thickness direction are computed analytically to take into account their continuous variations from one surface to another. Several examples are successfully analyzed for static deflections, natural frequencies and dynamic responses of FGM plates with different volume fraction exponents and boundary conditions. The convergence rate and accuracy are studied and compared with the finite element method (FEM). The effects of the constituent fraction exponent on static deflection as well as natural frequency are also investigated in detail using different FGM models. Based on the current material gradient, it is found that as the volume fraction exponent increases, the mechanical characteristics of the FGM plate approach those of the pure metal plate blended in the FGM.     

热流温度场下功能梯度板的热问题研究

功能梯度材料因其内部组分沿着空间位置连续变化,能有效缓解热应力集中等现象,在高超音速飞行器的热防护系统设计中具有良好的应用前景.以金属-陶瓷功能梯度板为研究对象,探讨在不同热环境下功能梯度板热传导、热变形和热应力的变化规律.首先,基于功能梯度材料的幂律分布模型,分析了线性温度场、正弦温度场、热流温度场和非线性温度场四种...     

Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments. 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 shell 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 are based on a higher order shear deformation theory with a von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of FGM hybrid laminated cylindrical shells of finite length. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of pressure-loaded, perfect and imperfect, FGM cylindrical shells with fully covered piezoelectric actuators under different sets of thermal and electric loading conditions. The results reveal that temperature dependency, temperature change and volume fraction distribution have a significant effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells. In contrast, the control voltage only has a very small effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells.     

A closed-form solution for thermal buckling of piezoelectric FGM rectangular plates with temperature-dependent properties

A thermal buckling analysis is presented for functionally graded rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of thermal load and constant applied actuator voltage. The temperature-dependent material properties of the functionally graded plate are assumed to vary as a power form of the thickness coordinate. Derivation of the equations is based on the third-order shear deformation plate theory. Results for the critical buckling temperatures are obtained in closed-form solution, which are convenient to be used in engineering design applications. The effects of the applied actuator voltage, plate geometry, and volume fraction exponent of the functionally graded material on the buckling temperature are investigated.     

Optimal shape control of functionally graded smart plates using genetic algorithms

This paper deals with optimal shape control of functionally graded smart plate containing patches of piezoelectric sensors and actuators. The genetic algorithm (GA) is designed to search for optimal actuator voltage and displacement control gains for the shape control of the functionally graded material (FGM) plates. The work extends the earlier finite element formulations of the two leading authors, so that it can be readily treated using genetic algorithms. Numerical results have been obtained to study the effect of the shape control of the FGM plates under a temperature gradient by optimising (i) the voltage distribution for the open loop control, and (ii) the displacement control gain values for the closed loop feedback control. The effect of the constituent volume fractions of zirconia, through varying the volume fraction exponent n, on the optimal voltages and gain values has also been examined.     

A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators

Compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise are presented for a simply supported, shear deformable functionally graded plate with piezoelectric fiber reinforced composite (PFRC) actuators. 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 PFRC layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. A two step perturbation technique is employed to determine buckling loads (temperature) and postbuckling equilibrium paths. The numerical illustrations concern the compressive and thermal postbuckling behaviors of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded PFRC actuators under different sets of thermal and electric loading conditions. The results for monolithic piezoelectric actuator, which is a special case in the present study, are compared with those of PFRC actuators. The results reveal that, in the compressive buckling case, the applied voltage usually has a small effect on the postbuckling load–deflection curves of the plate with PFRC actuators, whereas in the thermal buckling case, the effect of applied voltage is more pronounced for the plate with PFRC actuators, compared to the results of the same plate with monolithic piezoelectric actuators.     

Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams

In this article, buckling analysis of functionally graded material (FGM) beams with or without surface-bonded piezoelectric layers subjected to both thermal loading and constant voltage is studied. Thermal and mechanical properties of FGM layer is assumed to follow the power law distribution in thickness direction, except Poisson’s ratio which is considered constant. The Timoshenko beam theory and nonlinear strain-displacement relations 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 boundary conditions, existence of bifurcation-type buckling is examined and for each case of thermal loading and boundary conditions, closed-form solutions are obtained which are easily usable for engineers and designers. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of FGM beam on critical buckling temperature difference are examined.     

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.     

Static and free vibration analysis of multilayered functionally graded shells and plates using an efficient zigzag theory

Accurate zigzag theory is presented for static and free vibration analysis of multilayered functionally graded material (FGM) cylindrical shells and rectangular plates by approximating inplane displacements as a combination of linear layerwise and cubic global terms. Governing equations of motion are derived using Hamilton’s principle. The theory yields accurate results for displacements, stresses and natural frequencies in simply-supported functionally graded multilayered cylindrical shell panels and rectangular plates. Effect of changing the volume fraction ratio, aspect ratio and thickness of FGM layer between two homogeneous layers are investigated for a number of multilayered shell and plate laminates.     

Three-dimensional thermomechanical buckling analysis for functionally graded composite plates

Three-dimensional thermomechanical buckling analysis is investigated for functionally graded composite structures that composed of ceramic, functionally graded material (FGM), and metal layers. Material properties are assumed to be temperature dependent, and in FGM layer, they are varied continuously in the thickness direction according to a simple power law distribution in terms of the ceramic and metal volume fractions. The finite element model is adopted by using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. Temperature at each node is obtained by solving the thermomechanical equations. For a time discretization, Crank–Nicolson method is used. In numerical results, the thermal buckling behavior of FGM composite structures due to FGM thickness ratios, volume fraction distributions, and system geometric parameters are analyzed.     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

Nonlinear vibration of nanotube-reinforced composite plates in thermal environments

This paper deals with the large amplitude vibration of nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation in thermal environments. The SWCNTs are assumed aligned, straight and a uniform layout. Two kinds of carbon nanotube-reinforced composite (CNTRC) plates, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation plate theory that includes plate-foundation interaction. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies of CNTRC plates. Numerical results reveal that the natural frequencies as well as the nonlinear to linear frequency ratios are increased by increasing the CNT volume fraction. The results also show that the natural frequencies are reduced but the nonlinear to linear frequency ratios are increased by increasing the temperature rise or by decreasing the foundation stiffness. The results confirm that a functionally graded reinforcement has a significant effect on the nonlinear vibration characteristics of CNTRC plates.     

A 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers

This paper addresses three-dimensional (3-D) free vibration characteristic of thick circular/annular functionally graded (FG) plates with surface-bonded piezoelectric layers on the basis of 3-D Ritz solution. Three displacement components along with electrical potential field of the plate are expressed by a set of Chebyshev polynomials multiplied by geometry boundary functions. Both open-circuit and closed-circuit surface conditions are taken into account. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey either exponent or power law distribution of the volume fraction of the constituents. The effect of thickness-to-radius ratio, inner-to-outer radius ratio, piezo-to-host thickness ratio and gradient index on the natural frequencies of coupled piezoelectric FG circular/annular plates is investigated for different electrical and mechanical boundary conditions. It is observed that, unlike isotropic homogeneous circular/annular plates, frequency parameters of their piezoelectric coupled FG counterparts significantly increase with an enhancement in the host plate thickness to radius ratio. Results also show that the frequency parameters for open-circuit condition are higher than those for closed-circuit condition.     

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.     

A new analysis of nonlinear free vibration behavior of bi-functionally graded sandwich plates using the p-version of the finite element method

Geometrically nonlinear vibration of bi-functionally graded material (FGM) sandwich plates has been carried out by the p-version of the finite element method (FEM). The bi-FGM sandwich plate is made up of two face-sheet layers of two different FGM and one layer of homogeneous core. The nonlinear equations of motion of bi-FGM sandwich plates are establish using the harmonic balance method and solved iteratively by the linearized updated mode method. The effects of amplitude vibration, mechanical properties, geometrical parameters, thickness ratio of bi-FGM layers, and volume fraction exponent on the nonlinear vibration behavior of bi-FGM sandwich plates are plotted and investigated.     

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.     

Dynamic response of functionally graded circular cylindrical shells subjected to radial impulse load

Dynamic response of simply supported circular cylindrical shell made of functionally graded material (FGM) subjected to lateral impulse load is investigated. The effective material properties are assumed to vary continuously along the thickness direction according to a volume fraction power law distribution. First order shear deformation theory and Love’s first approximation theory are utilized in the equilibrium equations. Finally time response of displacement components is derived using mode superposition method. The influence of material composition, FGM configuration and geometrical parameters (length-to-radius and thickness-to-radius ratios) on the dynamic response is investigated. The results show that even though the shell is thin, the value of power law exponent has a considerable effect on the natural frequencies as well as the dynamic response of the functionally graded shell. Verification of the results of natural frequencies and time response of the FGM shell is achieved by making comparison with those available in the literature and those obtained using commercial software.     

Finite element modeling of active control of functionally graded shells in frequency domain via piezoelectric sensors and actuators

 A flat-shell element is presented for the active control of functionally graded material (FGM) shells through integrated piezoelectric sensor/actuator layers. The finite element formulation based on first-order shear deformation theory (FSDT) can be applied to shells ranging from relatively thin to moderately thick dimensions. A constant gain displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied to provide active control of the integrated FGM shell in a self-monitoring and self-controlling system. Frequency response characteristics of the FGM shell containing the piezoelectric sensors/actuators are analyzed in the frequency domain. The effects of constituent volume fraction and the influence of feedback control gain values on the dynamic responses of the FGM shell system are examined in detail. Received 13 November 2000     

Dynamic characteristics of functionally graded material sandwich plates in thermal environments

The dynamic instability of functionally graded material (FGM) sandwich plates under an arbitrary periodic load in a thermal environment is studied. The sandwich plate is made up of two layers of FGM face sheets and one layer of homogeneous metal core. The properties of a FGM layer vary continuously across the thickness according to a simple power law. A set of differential equations of Mathieu type is formed to determine the dynamic instability regions based on Bolotin's method. The dynamic stability of the FGM sandwich plates is sensitive to the temperature rise, volume fraction index, thickness ratio, and static and dynamic load factor.