Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Abstract

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.     

Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM

Thermal buckling and vibration of functionally graded (FG) sinusoidal microbeams with temperature-dependent properties and three kinds of temperature distributions are investigated in this article. As one material length scale is introduced, the modified couple stress theory is capable of predicting the small-scale effects. Material properties of FG microbeams are calculated using the Mori–Tanaka method. Furthermore, temperature-dependent properties are taken into account to investigate the mechanical characteristics of FG microbeams in high–thermal-gradient environment. Motion equations and the associated boundary conditions are obtained simultaneously through variational principle. Then Navier procedure and the differential quadrature method incorporating an iterative procedure are used to solve the governing differential equations with temperature-dependent properties and general boundary conditions. Numerical examples are performed for demonstrating the influences of temperature distribution, beam thickness, material length scale, slenderness ratio, shear deformation, functionally graded index, boundary conditions, and temperature-dependent/independent properties on thermal buckling and free vibration behaviors of FG microbeams.     

Geometrical nonlinear free vibration analysis of FGM spherical panel under nonlinear thermal loading with TD and TID properties

In this article, the nonlinear free vibration behavior of functionally graded (FG) spherical shell panel is examined under nonlinear temperature field. The functionally graded material (FGM) constituents are assumed to be the function of temperature and the thermal conductivity. The effective \hboxmaterial properties of the FGM are obtained using the Voigt micromechanical model through power-law distribution. The mathematical model of the shell panel is developed using Green–Lagrange nonlinear kinematics in the framework of the higher order shear deformation theory. The desired governing \hboxequation of the FG shell panel under thermal environment is obtained using the classical Hamilton's principle. The domain is discretized with the help of the \hboxisoparametric finite element steps and the responses are computed using the direct \hboxiterative method. The convergence behavior of the present nonlinear numerical model has been checked and compared with the previous reported results. Numerous examples have been demonstrated for the FG spherical panel to show the influence of different geometrical and material parameters and support conditions on the linear and nonlinear frequency parameters.     

Effect of Initial Imperfections on Thermal Buckling of Functionally Graded Plates

ABSTRACT

Thermal buckling analysis of rectangular functionally graded plates with initial geometrical imperfections is presented in this article. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the first-order shear deformation plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through the thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading, namely: uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect functionally graded plate. The influence of transverse shear on thermal buckling load is discussed.     

Thermal buckling analysis of moderately thick FGM plates based on the von Kármán nonlinearity and improved third order shear deformation theory

Abstract

In this study, thermal buckling of moderately thick functionally graded rectangular plates with all edges simply supported is analyzed by means of an improved third order shear deformation theory (improved TSDT). The plate is assumed to be under two types of thermal loadings, namely; uniform temperature rise and nonlinear temperature change across the thickness. The equilibrium and stability equations are derived based on the von Kármán type of geometrical nonlinearity and the improved third-order theory. By solving the stability equations, the value of buckling temperature difference is obtained. To calculate the critical buckling temperature difference, this value is minimized with respect to the half-wave parameters. The results are compared with the known data in literatures. The results indicate that, the values of critical buckling temperature difference which are obtained based on the improved TSDT, are lower in comparison with those obtained based on TSDT. Also, the results show that incorporation of the von Kármán type of geometrical nonlinearity with the improved third-order theory, gives the lower values of the critical buckling temperature difference.     

Thermal post-buckling of temperature dependent sandwich plates with FG-CNTRC face sheets

Present research investigates the thermal postbuckling of sandwich plates containing a stiff core and two thin carbon nanotube reinforced composite (CNTRC) face sheets. Properties of the core, carbon nanotubes (CNTs) and polymeric matrix of the faces are assumed to be temperature-dependent. It is assumed that CNTs as reinforcements may be distributed according to a functionally graded pattern. Plate is formulated based on the first-order shear deformation theory and von Kármán type of geometrical nonlinearity. The governing equations are obtained by the energy method with the aid of the Conventional Ritz method. Shape functions of the Ritz method are estimated according to the Chebyshev polynomials. A set of nonlinear eigenvalue equations is achieved. The obtained equations are homogeneous, coupled, and nonlinear in terms of both displacements and temperature. A successive displacement control strategy is implemented to trace the thermal postbuckling equilibrium path of the plate. It is shown that, with increasing the volume fraction of CNT, critical buckling temperature of sandwich plate increases and postbuckling deflection decreases. Furthermore, through a functionally graded distribution of volume fraction of CNTs across the thickness, critical buckling temperature of the sandwich plate may be enhanced and thermal postbuckling deflection may be alleviated.     

Thermoelastic Buckling Analysis of Functionally Graded Circular Plates Integrated with Piezoelectric Layers

Thermal buckling of circular plates made of functionally graded materials with surface-bounded piezoelectric layers are studied. The material properties of the FG plates are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituent materials. The general thermoelastic nonlinear equilibrium and linear stability equations for the piezoelectric FG plate are derived using the variational formulations. Buckling temperatures are derived for solid circular plates under uniform temperature rise, nonlinear and linear temperature variation through the thickness for immovable clamped edge of boundary conditions. The effects of piezo-to-host thickness ratio, applied actuator voltage, boundary condition, and power law index of functionally graded plates on the buckling temperature of plate are investigated. The results are verified with the data in literature.     

Thermal Buckling of Piezoelectric Functionally Graded Material Beams

Buckling analysis of functionally graded material (FGM) beams with surface-bonded piezoelectric layers which are subjected to both thermal loading and constant voltage is studied. The material nonhomogeneous properties are assumed to vary smoothly by distribution of power law through the beam thickness. The Euler-Bernoulli beam theory and nonlinear strain-displacement relation are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of thermal loading and boundary conditions, closed-form solutions are obtained. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of functionally graded material on the buckling temperature are investigated.     

Size dependent thermal buckling and postbuckling of functionally graded circular microplates based on modified couple stress theory

In this article, size-dependent thermal buckling and postbuckling behavior of a functionally graded circular microplate under uniform temperature rise field and clamped boundary conditions is investigated. Material properties are assumed to gradually vary through the thickness according to a simple power law. Equilibrium equations and associated boundary conditions are derived using variational method and based on modified couple stress theory, classical plate theory and von Kármán geometric nonlinearity. The differential quadrature method is used to discretize the governing equations. This technique is accompanied by an iterative method to determine the thermal postbuckling behavior of microplate. Finally, effects of length scale parameter, power law index and ratio of thickness to radius on the thermal buckling and postbuckling behavior of FG circular microplate are investigated.     

Hygrothermal buckling analysis of magnetically actuated embedded higher order functionally graded nanoscale beams considering the neutral surface position

In this article, the effects of humidity and thermal loads on buckling behavior of functionally graded (FG) nanobeams resting on elastic foundation and subjected to a unidirectional magnetic field is investigated. The nanobeam is modeled using different higher order refined beam theories which capture shear deformation influences needless of shear correction factors. The neutral axis position for all proposed beam models is determined. The material properties of FG nanobeam are temperature dependent and change gradually in spatial coordinate through the sigmoid and power-law models. Small-scale behavior of the nanobeam is described applying nonlocal elasticity theory of Eringen. Nonlocal governing equations for an embedded nanosize functionally graded material beam under hygrothermal loads obtained from Hamilton's principle are solved by an analytic method which satisfies various boundary conditions including S–S, C–S, and C–C. The validation of developed refined beam model has been proved with comparison to a previously published work on FG nanobeams. Numerical results are calculated for various beam theories to reveal the influences of moisture and temperature rise, elastic medium, nonlocality, volume fraction index, boundary conditions, and longitudinal magnetic field on the hygrothermal buckling responses of nanoscale P-FGM and S-FGM beams. The present study would be useful in the design of the nanoscale systems as one of the most demanded technologies in the near future.     

Thermal buckling and postbuckling analysis of piezoelectric FGM beams based on high-order shear deformation theory

This article deals with the thermal buckling and postbuckling of functionally graded material (FGM) beams with surface-bonded piezoelectric actuators based on physical neutral surface concept and high-order shear deformation theory including von Kármán strain–displacement relationships. The beams are exposed to a uniform temperature field and electric field, the material properties of FGM layers are temperature-dependent and vary in the thickness direction. The approximate solutions of piezoelectric FGM beams for thermal buckling and postbuckling are obtained by a two-step perturbation method, meanwhile, the analytical solutions of Timoshenko beam model and Euler beam model are also presented. The validity of the present work can be confirmed by comparisons with previous results. The effects of the applied actuator voltage, beam geometry as well as volume fraction index of FGM beam on the critical buckling temperature, and postbuckling load–deflection relationships are investigated.     

Thermal Postbuckling Analysis of Functionally Graded Beams

Thermal buckling and postbuckling analysis of functionally graded (FG) beams is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. For discretizing the governing equations and the related boundary conditions differential quadrature method (DQM) as a simple and computationally efficient numerical tool is used. Based on displacement control method, a direct iterative method is employed to obtain thermal postbuckling behavior of FG beams with different boundary conditions and geometrical parameters.     

Thermal environment effects on wave dispersion behavior of inhomogeneous strain gradient nanobeams based on higher order refined beam theory

In this article, an analytical model for the wave propagation analysis of inhomogeneous functionally graded (FG) nanobeam in thermal environment is developed based on nonlocal strain gradient theory, in which the stress accounts for not only the nonlocal elastic stress field but also the strain gradients stress field. The nanobeam is modeled through a higher order shear deformable refined beam theory which has a trigonometric shear stress function. The temperature field supposed to have a nonlinear distribution across the nanobeam thickness. Temperature-dependent material properties of nanobeams are spatially graded based on Mori–Tanaka model. The governing equations of the temperature-dependent functionally graded (FG) nanobeam are derived using the Hamilton’s principle. Numerical examples show that the characteristics of the wave propagation of FG nanobeam are influenced by various parameters such as nonlocality parameter, length scale parameter, gradient index, and temperature changes.     

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.     

Thermal Buckling of Piezolaminated Plates Subjected to Different Loading Conditions

A thermal buckling analysis is presented for simply supported rectangular laminated composite plates that are covered with top and bottom piezoelectric actuators, and subjected to the combined action of thermal load and constant applied actuator voltage. The thermomechanical properties of composite and piezoelectric materials are assumed to be linear functions of the temperature. The formulations of the equations are based on the higher-order laminated plate theory of Reddy and using the Sanders nonlinear kinematic relations. The closed-form solutions for the buckling temperature are obtained through the Galerkin procedure and solving the resultant eigenvalue problem, which are convenient to be used in engineering design applications. Numerical examples are presented to verify the proposed method. The effects of the plate geometry, fiber orientation in composite layers, lay-up configuration, different utilized piezoelectric materials, temperature dependency of material properties, thermal conductivity, and energy generation on the buckling load are investigated.     

Thermal postbuckling of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Thermal postbuckling response of a sandwich beam made of a stiff host core and carbon nanotube (CNT)-reinforced face sheets is analyzed in this research. Distribution of CNTs across the thickness of face sheets may be uniform or functionally graded. Material properties of the constituents are considered as temperature dependent. Properties of the face sheets are obtained by means of a modified rule of mixture approach. First-order shear deformation theory and von Kármán type of geometrical nonlinearity are incorporated with the virtual displacement principle. Ritz method with polynomial basis functions is applied to the virtual displacement principle to obtain the matrix representation of the governing equations. An iterative displacement control algorithm is applied to solve the nonlinear eigenvalue problem and trace the postbuckling equilibrium path. It is shown that, graded profile of CNTs, length to thickness ratio, host thickness to face thickness ratio, volume fraction of CNTs, boundary conditions, and temperature dependency, all are important factors on critical buckling temperature and postbuckling equilibrium path of sandwich beams with CNT-reinforced face sheets. However, influence of host thickness to face thickness ratio is ignorable.     

THERMAL AND MECHANICAL INSTABILITY OF AN IMPERFECT SHALLOW SPHERICAL CAP

In this article, the thermal and mechanical buckling loads of a cap of a shallow spherical shell of isotropic material and geometrically imperfect shell are considered. The equilibrium and stability equations are based on Donnell-Mushtari-Velasov (DMV) theory and are derived using the variational method. The Sander's nonlinear strain-displacement relations are used. The shell is under external pressure for mechanical loading and uniform temperature rise and radial temperature difference for thermal loadings. A simply supported boundary condition is assumed. The solutions for thermal and mechanical buckling loads are obtained using the stability equations and the Galerkin method. One-term approximation for the middle-plane shell displacement is considered. The expressions for the thermal and mechanical buckling loads are obtained analytically and are given by closed-form solutions.     

THERMAL BUCKLING OF FUNCTIONALLY GRADED PLATES BASED ON HIGHER ORDER THEORY

Equilibrium and stability equations of a rectangular plate made of functionally graded material (FGM) under thermal loads are derived, based on the higher order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. The derived equilibrium and stability equations for functionally graded plates (FGPs) are identical to the equations for laminated composite plates. A buckling analysis of a functionally graded plate under four types of thermal loads is carried out and results in closed-form solutions. The critical buckling temperature relations are reduced to the respective relations for functionally graded plates with a linear composition of constituent materials and homogeneous plates. The results are compared with the critical buckling temperatures obtained for functionally graded plates based on classical plate theory given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of functionally graded plates, whereas the classical plate theory overestimates buckling temperatures.     

Thermoelastic vibration analysis of functionally graded skew plate using nonlinear finite element method

Numerical investigation of nonlinear free vibration of functionally graded skew (FGS) plate in the thermal environment is presented. The mathematical model is proposed for the first time based on higher order shear deformation theory in conjunction with Green–Lagrange-type geometric nonlinearity for the FGS plate subjected to a thermal load. The material properties are considered to be temperature dependent and are graded along the thickness direction as per simple power law of distribution in terms of volume fraction of the constituent phase. The governing algebraic equations are derived using Hamilton’s principle, and the solutions are obtained using the direct iterative method. The proposed finite element model has discretized into an eight-noded quadratic serendipity elements. To validate the model, the obtained results are compared with the available literature. The influence of volume fraction index, skew angle, temperature change, aspect ratio, side–thickness ratio, and boundary conditions on the linear and nonlinear frequency of skew functionally graded material plate is examined and discussed in detail.     

Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells

This article presents analytical solutions for the nonlinear static and dynamic stability of imperfect eccentrically stiffened functionally graded material (FGM) higher order shear deformable double curved shallow shell on elastic foundations in thermal environments. It is assumed that the shell’s properties depend on temperature and change according to the power functions of the shell thickness. The shell is reinforced by the eccentrically longitudinal and transversal stiffeners made of full metal. Equilibrium, motion, and compatibility equations are derived using Reddy’s higher order shear deformation shell theory and taking into account the effects of initial geometric imperfection and the thermal stress in both the shells and stiffeners. The Galerkin method is applied to determine load–deflection and deflection–time curves. For the dynamical response, motion equations are numerically solved using Runge–Kutta method. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. The influences of inhomogeneous parameters, dimensional parameters, stiffeners, elastic foundations, initial imperfection, and temperature increment on the nonlinear static and dynamic stability of thick FGM double curved shallow shells are discussed in detail. Results for various problems are included to verify the accuracy and e?ciency of the approach.