Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

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.     

Postbuckling of functionally graded cylindrical shells with tangential edge restraints and temperature-dependent properties

This paper presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded cylindrical shells subjected to thermal and axial compressive loads. 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 constituents. The governing equations are established within the framework of classical thin shallow shell theory taking both geometrical nonlinearity in von Kármán–Donnell sense and initial imperfection into consideration. Thermal stability analysis also incorporates the effects of tangential edge constraints. A Galerkin procedure is applied to derive expressions of load-deflection relations from which the thermal buckling loads and postbuckling curves of the shells are obtained by an iteration. Effects played by material and geometrical properties, tangential stiffness, imperfection and buckling modes are discussed.     

Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to combined axial and radial mechanical loads in thermal environment. Two types 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 shell 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 boundary layer theory and associated singular perturbation technique are employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, FG-CNTRC cylindrical shells under combined action of external pressure and axial compression for different values of load-proportional parameters. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell.     

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.     

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.     

Out-of-plane free vibration of functionally graded circular curved beams in thermal environment

A first known formulation for the out-of-plane free vibration analysis of functionally graded (FG) circular curved beams in thermal environment is presented. The formulation is based on the first order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be temperature dependent and graded in the direction normal to the plane of the beam curvature. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle. Differential quadrature method (DQM), as an efficient and accurate numerical method, is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results, in the limit cases, with the available solutions in the literature for isotropic circular curved beams. In addition, for FG circular curved beams with soft simply supported edges, the results are compared with the obtained exact solutions. Then, the effects of temperature rise, boundary conditions, material and geometrical parameters on the natural frequencies are investigated.     

Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads

This paper presents a simple analytical approach to investigate the stability of functionally graded plates under in-plane compressive, thermal and combined 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. Equilibrium and compatibility equations for functionally graded plates are derived by using the classical plate theory taking into account both geometrical nonlinearity in von Karman sense and initial geometrical imperfection. The resulting equations are solved by Galerkin procedure to obtain explicit expressions of postbuckling load–deflection curves. Stability analysis of a simply supported rectangular functionally graded plate shows the effects of the volume fraction index, plate geometry, in-plane boundary conditions, and imperfection on postbuckling behavior of the plate.     

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.     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

Nonlinear analysis of an imperfect radially graded annular plate with a heated edge

The present work deals with a geometrically nonlinear finite element analysis of an imperfect radially graded annular plate with a heated edge. The geometrical imperfection of the graded annular plate is assumed in aspect of its little intrinsic transverse deflection. The analysis is mainly for investigating the effects of intrinsic geometrical imperfection and temperature in the graded annular plate on its nonlinear flexural behaviour under a transverse mechanical load. The temperature is uniformly distributed across the thickness of the plate and it varies along the radial direction only. The temperature-dependent material properties are radially graded according to a simple power-law that is formed by power-law exponent and material properties of constituent materials (ceramic and metal). Based on the Von Karman nonlinear strain–displacement relations for imperfect annular plates, the nonlinear finite element equations of equilibrium are derived employing the principle of minimum potential energy. A single nodal displacement-control solution strategy is described for numerical solutions of nonlinear finite element equations of equilibrium. The numerical illustrations show a significant role of geometrical imperfection of the annular plate for its unstable equilibrium and alteration of structural behaviour under thermo-mechanical load. The analysis reveals the usefulness of radially graded annular plate in order to mitigate the unstable equilibrium of imperfect monolithic annular plate under thermo-mechanical load. It is found that the radial location for maximum value of a stress component insignificantly depends on the magnitudes of power-law exponent and applied temperature. The effects of material properties and applied temperature on the critical mechanical load corresponding to the unstable equilibrium of the imperfect radially graded annular plate are also presented.     

Finite Element Multi-mode Approach to Thermal Postbuckling of Functionally Graded Plates

Postbuckling analysis of functionally graded ceramic-metal plates under temperature field is presented using finite element multi-mode method. The three-node triangular element based on the Mindlin plate theory is employed to account for the transverse shear strains, and the von-Karman nonlinear strain-displacement relation is utilized considering the geometric nonlinearity. The effective material properties are assumed to vary through the thickness direction according to the power law distribution of the volume fraction of constituents. The temperature distribution along the thickness is determined by one dimensional Fourier equations of heat conduction. The buckling mode shape solved from eigen-buckling analysis is adopted as the assumed mode function to reduce the degrees of freedom of nonlinear postbuckling equilibrium equations. The postbuckling response is obtained by solving the nonlinear equilibrium equations, and compared with the Newton- Raphson numerical results. The effects of boundary conditions, material gradient index and temperature distribution on postbuckling behavior are examined.     

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.     

Hygrothermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model

Hygrothermal and mechanical buckling responses of functionally graded (FG) plates resting on Winkler–Pasternak’s foundations are presented in this paper using a refined quasi-3D model. The effects due to transverse normal strain and shear deformation are both included. The present model exactly satisfies stress boundary conditions on the upper and lower surfaces of the FG plate without using shear correction factors. It is assumed that the material properties vary according to a power law of the thickness coordinate variable. The hygrothermal buckling equilibrium equations are derived from the principle of virtual work for FG plates resting on Winkler–Pasternak’s foundations with simply-supported boundary conditions. Two types of thermal and hygrothermal loading, uniform thermal and hygrothermal rise, linear thermal and hygrothermal distribution through the thickness are considered. Numerical results are presented to verify the accuracy of the present study. The effects played by Winkler–Pasternak’s parameters, plate aspect ratio, side-to-thickness ratio, gradient index, and loading type on the critical buckling of the FG plates are all investigated.     

Analytical solutions of thermal buckling and postbuckling of symmetric laminated composite beams with various boundary conditions

Based on the first-order shear deformation beam theory, considering geometric nonlinearity, the governing equations for symmetric laminated composite beams subjected to uniform temperature rise are derived by using Hamilton’s principle, and then three solving methods are presented to deal with it. By introducing an auxiliary function, which is shown in method one, the governing equations are reduced to be a single fourth-order integral-differential equation, and the exact solutions for the thermal buckling and postbuckling of symmetric laminated composite beams with combination of in-plane immovable simply supported and clamped boundary conditions are presented for the first time. On the basis of the results given in the method one, the explicit solutions for the thermal buckling and postbuckling of the beams are presented by giving accurate displacement functions (method two) and Ritz method (method three), respectively. Then, the effects of the transverse shear effects and boundary conditions on the thermal buckling and postbuckling of the beams are qualitatively discussed. What is more, a preliminary discussion on the probability and difference of extending the giving methods to the higher-order shear deformation beam theory with various boundary conditions is conducted. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and efficiency of the present analysis and numerical results. And then the symmetric cross-ply laminated composite beam (0/90/0) is taken as an example to numerically evaluate the effects of the length-to-thickness ratio, beam theories, and boundary conditions on the thermal buckling and postbuckling of symmetric laminated composite beams. Some meaningful conclusions have been drawn.     

Semi-analytical mechanical and thermal buckling analyses of 2D-FGM circular plates based on the FSDT

In the present paper, mechanical and thermal buckling analyses of two-directional functionally graded material (2D-FGM) circular plate are investigated. The motion equations have been derived based on the first-order shear deformation theory (FSDT) and power series method has been employed to solve the motion equations. Two different kinds of boundary condition including simply supported and fixed are considered. The material properties are assumed to vary in both transverse and radial directions according to power and exponential laws, respectively. Comparisons with available studies in the literature confirm the high accuracy of the current approach. The effects of geometrical parameters and 2D-FG power indices on the critical buckling load have been studied. It is shown that increase of modulus of elasticity of outer layers of plate due to higher presence of hard phase of FGM, in radius and thickness directions of the plate makes it possible to attain a more solid structure against mechanical buckling loads, while increase of coefficient of thermal expansion and coefficient of thermal conduction of outer layers of plate results in less stability against thermal buckling loads.     

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.     

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

Thermal postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to a uniform temperature rise. The SWCNTs are assumed to be aligned and straight with a uniform layout. 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. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, FG- and UD-CNTRC shells under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the shell can be increased as a result of a functionally graded reinforcement. It is found that in most cases the CNTRC shell with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

Nonlinear behavior of functionally graded circular plates with various boundary supports under asymmetric thermo-mechanical loading

The equilibrium equations of the first-order nonlinear von Karman theory for FG circular plates under asymmetric transverse loading and heat conduction through the plate thickness are reformulated into those describing the interior and edge-zone problems of the plate. A two parameter perturbation technique, in conjunction with Fourier series method is used to obtain analytical solutions for nonlinear behavior of functionally graded circular plates with various clamped and simply-supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified with known results in the literature. The load–deflection curves for different loadings, boundary conditions, and material constant in a solid circular plate are studied and discussed. It is shown that the behavior of FG plates with clamped or simply-supported boundary conditions are completely different. Under thermo-mechanical loading, snap-through buckling behavior is observed in simply-supported FG plates which are immovable in radial direction. Moreover, it is found that linear theory is inadequate for analyzing FG and also homogenous plates with immovable boundary supports in radial direction and subjected to thermal loading, even for deflections that are normally considered small.     

Nonlinear stability analysis of a perforated FGM plate under thermal load

The nonlinear thermal stability analysis of a perforated Ni/Al₂O₃ FGM plate with temperature-dependent thermoelastic material properties is carried out through a finite element approach based on the first-order shear deformation theory and the von-Karman's nonlinear kinematics. The thermoelastic material properties of the FGM plate are varied in the thickness direction, and the Mori-Tanaka homogenization scheme is applied to evaluate these properties. After validating the results of the current formulation, parametric studies are conducted to examine the effects of temperature-dependent material properties, material in-homogeneity, cutout shape and size, edge boundary conditions, aspect and slenderness ratios on the thermal buckling and postbuckling response of perforated FGM plate.