Thermal effects on wave propagation characteristics of rotating strain gradient temperature-dependent functionally graded nanoscale beams

In the present article, wave dispersion behavior of a temperature-dependent functionally graded (FG) nanobeam undergoing rotation subjected to thermal loading is investigated according to nonlocal strain gradient theory, in which the stress enumerates for both nonlocal stress field and the strain gradient stress field. Mori–Tanaka distribution model is considered to express the gradual variation of material properties across the thickness. The governing equations are derived as a function of axial force due to centrifugal stiffening and displacements by applying Hamilton’s principle according to Euler–Bernoulli beam theory. By applying an analytical solution, the dispersion relations of rotating FG nanobeam are obtained by solving an eigenvalue problem. Obviously, numerical results indicate that various parameters such as angular velocity, gradient index, temperature change, wave number, and nonlocality parameter have significant influences on the wave characteristics of rotating FG nanobeams. Hence, the results of this research can provide useful information for the next generation studies and accurate design of nanomachines including nanoscale molecular bearings and nanogears, etc.     

Through-the-length temperature distribution effects on thermal vibration analysis of nonlocal strain-gradient axially graded nanobeams subjected to nonuniform magnetic field

Thermomechanical vibration analysis of axially functionally graded (AFG) nanobeams under nonuniform longitudinal magnetic field is investigated based on nonlocal strain-gradient theory. This theory contains two scale parameters for modeling of size-dependent behavior of AFG nanobeam accurately. This theory takes into account both nonlocal stress field and strain-gradient effects on the response of nanostructures. The nanobeam is subjected to uniform and linear through-the-length temperature distributions. A power-law model is used to describe the distribution of temperature-dependent material properties along the axial direction. A Galerkin-based solution technique is implemented to solve the governing equation obtained from Hamilton’s principle. Natural frequencies of functionally graded nanobeam are verified with those of previous articles. It is shown that vibration frequencies of AFG nanobeams are significantly influenced by temperature rise, power-law index, nonlocal parameter, length-scale parameter, magnetic field intensity, and boundary conditions.     

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.     

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.     

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.     

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.     

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 dynamic buckling of temperature-dependent sandwich nanocomposite quadrilateral microplates using visco-higher order nonlocal strain gradient theory

This research deals with the nonlocal temperature-dependent dynamic buckling analysis of embedded laminated quadrilateral micro plates reinforced by functionally graded carbon nanotubes (FG-CNTs). The material properties of structure are assumed viscoelastic based on Kelvin–Voigt model. The effective material properties of structure are considered based on mixture rule. The elastic medium is simulated by orthotropic visco-Pasternak medium. The motion equations are derived applying Sinusoidal shear deformation theory (SSDT) in which the size effects are considered using higher order nonlocal strain gradient theory. The transformed weighing (TW) and differential quadrature (DQ) method in conjunction with the Bolotin’s method are applied for calculating resonance frequency and dynamic instability region (DIR) of structure. The effects of different parameters such as volume percent of CNTs, distribution type of CNTs, temperature, nonlocal parameter and structural damping on the dynamic instability of visco-system are shown. The results are compared with other published works in the literature. Results indicate that the CNTs have an important role in dynamic stability of structure and FGX distribution type is the better choice.     

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.     

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.     

Free vibration analysis of size-dependent functionally graded porous cylindrical microshells in thermal environment

In this article, the free vibration analysis of a functionally graded (FG) porous cylindrical microshell subjected to a thermal environment is investigated on the basis of the first-order shear deformation shells and the modified couple stress theories. The material properties are assumed to be temperature dependent and are graded in the thickness direction. The equations of motion and the related boundary conditions are derived using the principle of minimum potential energy and they are solved analytically. The model is validated by comparing the benchmark results with the obtained ones. The effects of material length scale parameter, temperature changes, volume fraction of the porosity, FG power index, axial and circumferential wave number, and length on the vibration behavior of the FG porous cylindrical microshell are studied. The results can have many applications such as in modeling of microrobots and biomedical microsystems.     

Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams

In this work, the thermal effect on the buckling response of the axially functionally graded (AFG) nanobeams is studied based on the nonlocal thermoelasticity theory. Size effects of elastic deformation and heat conduction are considered simultaneously. Non-uniform distribution of temperature along the longitudinal direction of the AFG nanobeams is taken into account and determined by the nonlocal heat conductive law. Equations of motion and the corresponding boundary conditions are derived with the aid of the variational principle within the sinusoidal shear deformation theory and the nonlocal thermoelasticity theory. Ritz method is used to obtain the solutions for the thermal buckling response of the AFG nanobeams with various boundary conditions. Numerical results addressing the significance of the AFG index, the nonlocal parameters of elasticity and heat conduction, and the transverse shear deformation on the buckling behavior are displayed. It is found that, in addition to the nonlocal effect of elasticity, the nonlocal heat conduction plays an important role in analyzing the thermal–mechanical behaviors of the FG nanostructures.     

Hygrothermal effects on static stability of embedded single-layer graphene sheets based on nonlocal strain gradient elasticity theory

Abstract

This article develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygrothermal environments. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, elastic foundation and geometrical parameters on buckling characteristics a graphene sheets are examined.     

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.     

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.     

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.     

Thermoelastic interaction in functionally graded thick hollow cylinder with temperature-dependent properties

The prediction of thermoelastic behavior induced by transient thermal shock is important to evaluate the durability of functionally graded materials. The purpose of this article is to study the axisymmetric thermoelastic interaction in a functionally graded thick hollow cylinder by an asymptotic approach. The governing equations with variable material properties, which are spatially graded and temperature dependent, are proposed based on the generalized theory of thermoelasticity with one relaxation time (L–S theory). The Laplace transform technique is used to derive the general solutions with the cylinder divided into thin cylinders and material properties assumed constant in each thin cylinder. The inverse Laplace transform is then conducted analytically by some approximations in the time domain, and the short-time solution of the problem with its interior boundary subjected to a sudden temperature rise and the outer surface maintained at constant temperature are obtained. Utilizing these asymptotic solutions, the propagation of thermal and thermoelastic waves are studied, which display dependence of each wave’s propagation upon the relaxation time, volume fraction parameter and temperature. The distributions of the radial displacement, temperature and stresses are also plotted and discussed. These results reveal effects of these variable material properties with spatial position and temperature on thermoelastic behavior.     

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 Buckling Analysis of a Mindlin Rectangular FGM Microplate Based on the Strain Gradient Theory

This article is aimed at developing a nonclassical Mindlin rectangular functionally graded material (FGM) microplate based on the strain gradient theory (SGT) to study the thermal buckling behavior of microplates with different boundary conditions. This theory comprises material length scale parameters to interpret size effects. The developed model encompasses classical and modified couple stress Mindlin microplate models, if all the material length scale parameters or two of them are taken to be zero, respectively. The Mindlin rectangular FGM microplate is considered to be made of a mixture of metal and ceramic of which the volume fraction is described through a power low function. According to Hamilton's principle and the generalized differential quadrature (GDQ) method, the stability equations and associated boundary conditions are obtained and discretized, respectively. Current formulations provide a possibility to have all types of boundary conditions which herein, FGM microplates with three commonly used boundary conditions are considered. Three different types of thermal loads including uniform, linear and nonlinear temperature rises along the thickness of FGM microplates are considered. The dimensionless critical buckling temperature difference (DCBTD) predicted by SGT is compared with that of modified couple stress theory (CST) and classical theory (CT) which it is found that CST and CT underestimate the DCBTD. Also, effects of the boundary conditions, length scale parameter and material gradient index of FGM microplates on the DCBTD are judiciously investigated.     

Nonlocal Thermoelasticity Theory for Thermal-Shock Nanobeams with Temperature-Dependent Thermal Conductivity

In this work, a model of nonlocal generalized thermoelasticity with one thermal relaxation time is used to consider the vibration behavior of an Euler-Bernoulli (E-B) nanobeam. The thermal conductivity of the nanobeam is assumed to be temperature-dependent. The nonlocality brings in an internal length scale in the formulation and, thus, allows for the interpretation of size effects. The governing partial differential equations are solved in the Laplace transform domain by adopting the state-space approach of modern control theory. The inverse of Laplace transforms are numerically computed using Fourier expansion techniques. The distributions of the lateral vibration, the temperature, the axial displacement and the bending moment of the nanobeam are determined. The effect of thickness and variability of thermal conductivity, as well as the influence of the nonlocal parameter are investigated.