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.     

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.     

Buckling and Postbuckling Behavior of Functionally Graded Nanotube-Reinforced Composite Plates in Thermal Environments

This paper investigates the buckling and postbuckling of simply supported, nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. The nanocomposite plates are assumed to be functionally graded in the thickness direction using single-walled carbon nanotubes (SWCNTs) serving as reinforcements and the plates' effective material properties are estimated through a micromechanical model. The higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity is used to model the composite plates and a two-step perturbation technique is performed to determine the buckling loads and postbuckling equilibrium paths. Numerical results for perfect and imperfect, geometrically mid-plane symmetric functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC plate, which is a special case in the present study, are compared with those of the FG-CNTRC plate. The results show that the buckling loads as well as postbuckling strength of the plate can be significantly increased as a result of a functionally graded nanotube reinforcement. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC plates.     

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.     

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 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.     

Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.     

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.     

压电元件驱动的功能梯度弹性薄板的屈曲

考虑功能梯度薄板,其上下表面嵌有压电执行元件.根据逆压电效应将电压转换成作用于板上的等效电载荷.假设梯度材料的弹性参数为板厚度方向坐标的幂函数,基于经典板理论,导出了功能梯度弹性薄板小挠度屈曲平衡微分方程.利用双三角级数展开法,得到了四边简支具有压电元件的功能梯度矩形板的临界屈曲载荷,在此基础上通过数值例子讨论了弹性板的几何尺寸、材料梯度指数的变化对临界电压(载荷)的影响.研究结果表明,材料的梯度指数对临界电压有重要影响,并且通过调整作用于执行元件上的电压的大小和方向,可实现对结构稳定性的有效控制.     

Nonlinear analysis of smart functionally graded annular sector plates using cylindrically orthotropic piezoelectric fiber reinforced composite

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.     

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates

Thermal buckling and postbuckling behavior is presented for functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) subjected to in-plane temperature variation. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, geometrically mid-plane symmetric FG-CNTRC plates and uniformly distributed CNTRC plates under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the plate can be increased as a result of a functionally graded reinforcement. It is found that in some cases the CNTRC plate with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

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 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 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.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.     

On the shear stress distribution between a functionally graded piezoelectric actuator and an elastic substrate and the reduction of its concentration

Recent advances in material processing technologies allow the production of piezoelectric materials with functionally graded material properties. We investigate the implications of functionally graded piezoelectric materials when used as actuators for structural control by examining the distribution of the actuating shear stress under a piezoelectric actuator of a functionally graded material (FGM) on an isotropic elastic half-space. It is shown that FGM materials can be used to adjust the shear stress distribution. In particular, the concentration near the edges of a conventional homogeneous piezoelectric actuator can be significantly reduced in an FGM actuator.     

Nonlinear buckling and postbuckling of imperfect piezoelectric S-FGM circular cylindrical shells with metal–ceramic–metal layers in thermal environment using Reddy's third-order shear deformation shell theory

Based on Reddy's third-order shear deformation shell theory, this paper studied the nonlinear buckling and postbuckling response of imperfect Sigmoid functionally graded circular cylindrical shells in a thermal environment with an outer surface-bonded piezoelectric actuator. Material properties are temperature dependent and graded in the thickness direction with two shell's outer surfaces rich of metal and ceramic in the middle (S-FGM). The shell is subjected to uniform external pressure, axial compressive, electrical loads and resting on elastic foundations. The obtained numerical results are validated by comparing with other results reported in the open literature.     

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.     

Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical thin shell of finite length subjected to compressive axial loads and in thermal environments. 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. The governing equations are based on the classical shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of axially-loaded, perfect and imperfect, cylindrical thin shells with two constituent materials and under different sets of thermal environments. The effects played by temperature rise, volume fraction distribution, shell geometric parameter, and initial geometric imperfections are studied.