Influence of three-parameter viscoelastic medium on vibration behavior of a cylindrical nonhomogeneous microshell in thermal environment: An exact solution

In this article, the free vibration behavior of a functionally graded (FG) size-dependent microshell surrounded by viscoelastic foundation subjected to various thermal loading conditions is analytically studied. The material properties of the cylindrical FG microshell are supposed to be temperature dependent and vary continuously along the thickness direction according to the modified rule of mixtures. The size-dependent FG microshell is analyzed based on the modified couple stress theory. The analytical modeling is developed using the first-order shear deformation theory and the equations of motion are derived by the principle of minimum total potential energy. Then the governing equations for the free vibration behavior of a simply supported FG cylindrical microshell subjected to thermal loading are solved using the Navier procedure. The effects of some important parameters, such as material length scale parameter, stiffness and damping of the visco-Pasternak foundation, temperature changes, axial and circumferential wave number, and length of the microshell on the natural frequency are investigated and discussed.     

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.     

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels

Based on the three-dimensional elasticity theory, free vibration analysis of functionally graded (FG) curved thick panels under various boundary conditions is studied. Panel with two opposite edges simply supported and arbitrary boundary conditions at the other edges are considered. Two different models of material properties variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential distribution of the material properties through the thickness are considered. Differential quadrature method in conjunction with the trigonometric functions is used to discretize the governing equations. With a continuous material properties variation assumption through the thickness of the curved panel, differential quadrature method is efficiently used to discretize the governing equations and to implement the related boundary conditions at the top and bottom surfaces of the curved panel and in strong form. The convergence of the method is demonstrated and to validate the results, comparisons are made with the solutions for isotropic and FG curved panels. By examining the results of thick FG curved panels for various geometrical and material parameters and subjected to different boundary conditions, the influence of these parameters and in particular, those due to functionally graded material parameters are studied.     

Thermomechanical Vibration Behavior of FG Nanobeams Subjected to Linear and Non-Linear Temperature Distributions

In this study, thermomechanical vibration analysis of functionally graded (FG) nanobeams subjected to in-plane thermal loads are carried out by presenting a Navier-type solution and employing a semi-analytical differential transform method (DTM) for the first time. Two types of thermal loading, namely, linear and non-linear temperature rises through the thickness direction are considered. Thermomechanical properties of FG nanobeam are supposed to vary smoothly and continuously throughout the thickness based on power-law model and material properties are assumed to be temperature-dependent. Eringen non-local elasticity theory is exploited to describe the size dependency of FG nanobeam. Using Hamilton's principle, the non-local equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FG nanobeams including size effect and they are solved applying DTM. According to numerical results, it was revealed that the proposed modeling and semi-analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. A parametric study is included to examine the effects of several parameters, such as temperature rise, gradient index, small-scale parameter and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour of a FG nanobeams is significantly influenced by these effects. The new results can be used as benchmark solutions for analyses of FG nanobeams.     

Stochastic Assessment of Thermo-Elastic Wave Propagation in Functionally Graded Materials (FGMs) with Gaussian Uncertainty in Constitutive Mechanical Properties

The main objective of this article is focused on stochastic analysis of wave propagation and effects of uncertainty in mechanical properties on transient behaviors of displacement and temperature fields in functionally graded materials under thermo-mechanical shock loading. The problem is studied in a cylindrical domain and the governing equations of a functionally graded thick hollow cylinder are solved. To assess the wave propagation, the generalized coupled thermoelasticty equations based on Green-Naghdi theory (without energy dissipation) are analyzed in a FG thick hollow cylinder. The FG cylinder is considered to have infinite length and axisymmetry conditions. The constitutive mechanical properties of FGM are assumed as random variables with Gaussian distribution and also the mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The FG cylinder is divided into many elements across thickness of cylinder and hybrid numerical method (Galerkin finite element and Newmark finite difference methods) along with the Monte Carlo simulation are employed to solve the statistical coupled equations. The effects of uncertainty in functionally graded materials on thermal and elastic waves, transient behaviors of radial displacement and temperature fields and variance and maximum values of displacement and temperature are discussed in details for various grading patterns in FGMs and various points on thickness at several times.     

Effects of Thermal Loading on the Buckling and Vibration of Ring-Stiffened Functionally Graded Shell

A theoretical method is developed to investigate the effects of thermal load and ring stiffeners on buckling and vibration characteristics of the functionally graded cylindrical shells, based on the first-order shear deformation theory (FSDT) considering rotary inertia. Heat conduction equation across the shell thickness is used to determine the temperature distribution. Material properties are assumed to be graded across the shell wall thickness of according to a power-law, in terms of the volume fractions of the constituents. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. The effects of stiffener's number and size on natural frequency of functionally graded cylindrical shells are investigated. Moreover, the influences of material composition, thermal loading and shell geometry parameters on buckling and vibration are studied. The obtained results have been compared with the analytical results of other researchers, which showed good agreement. The new features of thermal vibration and buckling of ring-stiffened functionally graded cylindrical shells and some meaningful and interesting results obtained in this article are helpful for the application and the design of functionally graded structures under thermal and mechanical loads.     

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.     

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.     

Exact solution for thermal damping of functionally graded Timoshenko microbeams

This article deals with the thermoelastic damping problem in a functionally graded (FG) Timoshenko microbeam. Thermal and mechanical properties of the microbeam vary in the thickness direction according to the power law relation. Employing Timoshenko beam theory, the governing dynamic equation coupled with thermal effects of the FG microbeam is developed. Afterwards, Using the Taylor series expansion for material properties, the heat conduction equation is solved analytically for temperature in the form of a power series. The free vibration of the FG microbeam is analyzed to achieve the natural frequencies and thermal damping ratio of the FG microbeam. The effect of FG index on the thermoelastic damping ratio is investigated in different aspect ratios. Also comparison studies are made between the results obtained from the models based on the Euler–Bernoulli and Timoshenko beam theories.     

DQM-Based Thermal Stresses Analysis of a Functionally Graded Cylindrical Shell Under Thermal Shock

The transient thermal stresses of a functionally graded (FG) cylindrical shell subjected to a thermal shock are investigated. The dynamic temperature fields of FG shells are obtained by using the Laplace transform and power series method. The differential quadrature method is developed to obtain the transient thermal stresses by solving dynamic governing equations in terms of displacements. The effects of the material constitutions on the transient temperature and the thermal stresses are analyzed in the cases of obverse thermal shock and reverse thermal shock. It turns out that the thermal stresses could be alleviated by means of changing the volume fractions of the constituents.     

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.     

Size-Dependent Thermal Buckling and Postbuckling of Functionally Graded Annular Microplates Based on the Modified Strain Gradient Theory

This article reports on the thermal instability of functionally graded (FG) annular microplates with different boundary conditions. The modified strain gradient elasticity theory is employed to capture size effects. The non-linear governing equations and boundary conditions are derived based on the first-order shear deformation theory (FSDT) and virtual displacements principle. The generalized differential quadrature technique is implemented so as to discretize. To obtain the critical buckling temperature, the set of linear discretized governing equations is solved as an eigenvalue problem. Also, the non-linear problem of thermal postbuckling is solved by the pseudo arc-length continuation method. The effects of boundary conditions, length scale parameter, and the variation of material through the thickness and geometrical properties on both critical buckling temperature and thermal postbuckling behavior are studied.     

Vibration and Critical Speed of Orthogonally Stiffened Rotating FG Cylindrical Shell Under Thermo-Mechanical Loads Using Differential Quadrature Method

In this article, an approximate solution using differential quadrature method is presented to investigate the effects of thermo-mechanical loads and stiffeners on the natural frequency and critical speed of stiffened rotating functionally graded cylindrical shells. Transverse shear deformation and rotary inertia, based on first-order shear deformation shell theory (FSDT), are taken into consideration. The equations of motion are derived by the Hamilton's principle while the stiffeners are treated as discrete elements. Material properties are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents. The temperature field is assumed to be varied in the thickness direction. The equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations applying the DQM. The results obtained include the relationship between frequency characteristics of different power-law index, rotating velocities, thermal loading and amplitude of axial load. To validate the present analysis, the comparison is made with a number of particular cases in literature. Excellent agreement is observed and a new range of results are presented for stiffened rotating FG cylindrical shell under thermo-mechanical loads which can be used as a benchmark to approximate solutions.     

Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity

In this article, thermal buckling and natural frequency of a curved functionally graded (FG) nanobeam in a thermal environment based on Eringen’s theory is investigated. Dimension of structure is in small scale, its geometric is curved, and properties of material vary in radial direction. In order to develop differential equation and boundary condition, Hamilton’s principle is adopted. Properties of material are a function of two variables of radial thickness and temperature. After developing equation of motion in thermal environment, analytical solution has been employed in order to obtain the amount of frequency and thermal buckling. Free vibration of a curved FG nanobeam subjected to in-plane thermal load may show zero frequency magnitude at a certain temperature, which specifies the existence of bifurcation type of instability. In numerical section, frequency responses have been studied one time based on temperature-dependent material property and another time based on temperature-independent material property and influences for parameters such as nonlocal parameter, power-law, mode number, temperature changes, and arc angle on natural frequency and critical temperature have been investigated. Results have shown that if properties of material are dependent on temperature, then expected frequency will be less than the case in which properties are independent of temperature. Performed validation certifies correctness of obtained results. Results indicate that critical temperature increasing the arc angle leads to a decrease in amount of dimensionless frequency, and this matter represents the importance of specification of critical temperature in curved structures.     

Transient Thermoelastic and Piezothermoelastic Problems of Functionally Graded Materials

This paper is concerned with the theoretical treatment of transient thermoelastic problems involving functionally graded thick plate, laminated composite strip with an interlayer of functionally graded material, and functionally graded hollow cylinder, and transient piezothermoelastic problems involving functionally graded piezoelectric cylindrical panel. The thermal, thermoelastic and piezoelectric constants of the functionally graded materials are expressed as power functions of the radial coordinate variable or exponential functions of the thickness coordinate variable. The exact solutions for the two-dimensional temperature change in a transient state, and thermoelastic or piezothermoelastic response under the state of plane strain are presented herein. Some numerical results are shown in figures.     

Temperature-dependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field

This study is focused on the wave propagation analysis of nanoplate made of temperature-dependent porous functionally graded (FG) materials rested on Winkler–Pasternak foundation under in-plane magnetic field. The material properties of FG nanoplate are supposed to vary through the thickness direction and described by power-law rule, in which the porosity distribution is considered as an even pattern. Hamilton’s principle is utilized to derive the governing equations on basis of second-order shear deformation theory in conjunction with nonlocal strain gradient theory. The influence of small-length parameters, thermal distribution, magnetic field, material composition, porosity, and Winkler–Pasternak foundation on wave dispersion is explored.     

Mechanical and thermal stresses in a functionally graded rotating disk with variable thickness due to radially symmetry loads

Rotating disks have many applications in the aerospace industry such as gas turbines and gears. These disks normally work under thermo mechanical loads. Minimizing the weight of such components can help reduce the overall payload in aerospace industry. For this purpose, a rotating functionally graded (FG) disk with variable thickness under a steady temperature field is considered in this paper. Thermo elastic solutions and the weight of the disk are related to the material grading index and the geometry of the disk. It is found that a disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared to a uniform thickness disk. Maximum radial stress due to centrifugal load in the solid disk with parabolic thickness profile may not be at the center unlike uniform thickness disk. Functionally graded disk with variable thickness has smaller stresses due to thermal load compared to those with uniform thickness. It is seen that for a given value of grading index, the FG disk having concave thickness profile is the lightest in weight whereas the FG disk with uniform thickness profile is the heaviest. Also for any given thickness profile, the weight of the FG disk lies in between the weights of the all-metal and the all-ceramic disks.     

Dynamic Thermal Postbuckling Analysis of Shear Deformable Piezoelectric-FGM Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is studied. The shell material is graded across the thickness according to a power law. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the Sanders nonlinear kinematic relations, which account for the transverse shear strains, and the third-order shear deformation shell theory is employed. Hamilton's principle is used to derive the equations of motion governing piezoelectric FGM cylindrical shells. A finite difference approximation combined with the Runge-Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent in the power-law variation of the FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known results in the literature.     

Dynamic Thermal Postbuckling Analysis of Piezoelectric Functionally Graded Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is analyzed using an incremental numerical technique. The shell is graded across the thickness according to a power law form function. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the classical shell theory with Sanders' nonlinear kinematic relations. Then, using Hamilton's principle, equations of motion are derived for the piezoelectric FGM cylindrical shell. A finite difference based method combined with the Runge–Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent of FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known data in the literature.