A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory

In the present research, free vibration study of functionally graded (FG) nanobeams with graded nonlocality in thermal environments is performed according to the third-order shear deformation beam theory. The present nanobeam is subjected to uniform and nonlinear temperature distributions. Thermo-elastic coefficients and nonlocal parameter of the FG nanobeam are graded in the thickness direction according to power-law form. The scale coefficient is taken into consideration implementing nonlocal elasticity of Eringen. The governing equations are derived through Hamilton's principle and are solved analytically. The frequency response is compared with those of nonlocal Euler–Bernoulli and Timoshenko beam models, and it is revealed that the proposed modeling can accurately predict the vibration frequencies of the FG nanobeams. The obtained results are presented for the thermo-mechanical vibrations of the FG nanobeams to investigate the effects of material graduation, nonlocal parameter, mode number, slenderness ratio, and thermal loading in detail. The present study is associated to aerospace, mechanical, and nuclear engineering structures that are under thermal loads.     

Transient response of FG higher-order nanobeams integrated with magnetostrictive layers using modified couple stress theory

The present paper studies the transient response of a functionally graded nanobeam integrated with magnetostrictive layers. The material properties of sandwich nanobeam are temperature dependent and assumed to vary in the thickness direction. In order to consider small-scale effects, the modified couple stress theory is also taken into consideration. Using a unified beam theory that contains various beam models and energy method as well as Hamilton's principle, the governing motion equations and related boundary conditions are obtained. The obtained results in this paper can be used as sensors and actuators in sensitive applications.     

Forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments

Size-dependent forced vibration behavior of functionally graded (FG) nanobeams subjected to an in-plane hygro-thermal loading and lateral concentrated and uniform dynamic loads is investigated via a higher-order refined beam theory, which captures shear deformation influences needless of any shear correction factor. The nanobeam is in contact with a three-parameter Kerr foundation consisting of upper and lower spring layers as well as a shear layer. Hygro-thermo-elastic material properties of the nanobeam are described via power-law distribution considering exact position of the neutral axis. Through nonlocal elasticity theory of Eringen and Hamilton's principle, the governing equations of higher-order FG nanobeams on Kerr foundation under dynamic loading are derived. These equations are solved for simply-supported and clamped-clamped boundary conditions. A detailed parametric study is performed to show the importance of moisture concentration rise, temperature rise, material composition, nonlocality, Kerr foundation parameters, and boundary conditions on forced vibration characteristics and resonance frequencies of FG nanobeams. As a consequence, Kerr foundation parameters lead to a significant delay in the occurrence of resonance frequencies.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.     

A unified nonlocal formulation for bending,buckling and free vibration analysis of nanobeams

Abstract

A unified nonlocal formulation is developed for the bending, buckling, and vibration analysis of nanobeams. Theoretical formulations of eighteen nonlocal beam theories are presented by using unified formulation. Small scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The governing equations of motion and associated boundary conditions of the nanobeam are derived using Hamilton's principle. Closed form solutions are presented for a simply supported boundary condition using Navier's solution technique. Numerical results for axial and transverse shear stress are first time presented in this study which will serve as a benchmark for the future research.     

A higher-order micro-beam model with application to free vibration

The free vibration of micro-beams is analyzed employing three different beam models. The previously obtained equations of motion for Bernoulli–Euler and Timoshenko models are solved analytically. A higher-order model is devised, which satisfies the lateral boundary conditions of micro-beams. The equations of motion with associated boundary conditions are derived by means of Hamilton's principle. The generalized differential quadrature method is used for the solution of the equations. The first five natural frequencies are obtained for micro-beams with three length-scale parameter/height ratios and five different boundary conditions.     

Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment

In this paper, the size-dependent free vibration of magneto-electro-elastic (MEE) nanobeams in thermal environment is investigated. Size effects are taken into account using the modified couple stress theory, which is capable of accounting for higher-order electromechanical coupling, and the equations are developed on the basis of Euler–Bernoulli beam model and using von Karman nonlinear strain. The vibration of hinged–hinged nanobeams is investigated by way of example. Effects of various parameters such as temperature, thickness, and length on natural frequencies are demonstrated, and it is indicated that increased length and decreased thickness lead to decreased nanobeam natural frequencies.     

Longitudinal varying elastic foundation effects on vibration behavior of axially graded nanobeams via nonlocal strain gradient elasticity theory

In this research, vibration characteristics of axially functionally graded nanobeams resting on variable elastic foundation are investigated based on nonlocal strain gradient theory. This nonclassical nanobeam model contains a length scale parameter to explore the influence of strain gradients and also a nonlocal parameter to study the long-range interactions between the particles. The present model can degenerate into the classical models if the material length scale parameter and the nonlocal stress field parameter are both taken to be zero. Elastic foundation consists of two layers: a Winkler layer with variable stiffness and a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. Material properties are graded axially via a power-law distribution scheme. Hamilton's principle is employed to derive the governing equations that are solved applying a Galerkin-based solution for different boundary edges. Comparison study is also performed to verify the present formulation with those of previous papers. Results are presented to investigate the influences of the nonlocal and length scale parameters, various material compositions, elastic foundation parameters, type of foundation and various boundary conditions on the vibration frequencies of AFG nanobeams in detail.     

A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

In the present work, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through-thickness stretching. The formulation is general in the sense that it can be used to compare the influence of different structural theories, through static and dynamic analyses of curved nanobeams. The governing equations derived here are solved introducing a 3-nodes beam element. The formulation is validated considering problems for which solutions are available. A comparative study is done here by different theories obtained through the formulation. The effects of various structural parameters such as thickness ratio, beam length, rise of the curved beam, loadings, boundary conditions, and nonlocal scale parameter are brought out on the static bending behaviors of curved nanobeams.     

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.     

The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations

ABSTRACT

The modified couple stress theory (MCST) is utilized to investigate the bending of viscoelastic nanobeams laying on visco-Pasternak elastic foundations based on a new shear and normal deformations beam theory. This model consists of the material length scale coefficient that captures the size impact on small-scale beams. The simply supported beam is made of viscoelastic material, subjected to time harmonic transverse load. The nanobeam is presumed to be laying on double layers of foundations. The first layer is modeled as Kelvin–Voigt viscoelastic model and the second is taken as a shear layer. Based on the proposed beam theory and MCST, the differential motion equations are deduced using Hamilton’s principle. To check the validity of the obtained formulations, the predicted results are compared with those available in the open literature. In addition, the influences of various parameters such as the material length scale parameter, length-to-depth ratio, viscoelastic damping structure, the stiffness and damping coefficients of the viscoelastic substrate, and shear and normal strains on the deflection and stresses are illustrated.     

Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory

In this article, nonlocal free vibration analysis of curved functionally graded piezoelectric (FGP) nanobeams is conducted using a Navier-type solution method. The model contains a nonlocal stress field parameter and also a nonlocal strain-electric field gradient parameter to capture the size effects. Inclusion of these nonlocal parameters introduces both stiffness-softening and stiffness-hardening effects in the analysis of curved nanobeams. Nonlocal governing equations of curved FGP nanobeam are obtained from Hamilton's principle based on the Euler–Bernoulli beam model. The results are validated with those of curved FG nanobeams available in the literature. Finally, the influences of electric voltage, length scale parameter, nonlocal parameter, opening angle, material composition, and slenderness ratio on vibrational characteristics of nanosize curved FG piezoelectric beams are explored. These results may be useful in accurate analysis and design of smart nanostructures constructed from piezoelectric materials.     

Effect of three-parameter viscoelastic medium on vibration behavior of temperature-dependent non-homogeneous viscoelastic nanobeams in a hygro-thermal environment

In this article, free vibration of functionally graded (FG) viscoelastic nanobeams resting on viscoelastic foundation subjected to hygrothermal loading is investigated employing a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. The three-parameter viscoelastic medium consists of parallel springs and dashpots as well as a shear layer. Temperature-dependent material properties of FGM beam are graded across the thickness via the power-law model. Employing non-local elasticity theory of Eringen and Hamilton's principle, non-local governing equations of a size-dependent viscoelastic nanobeam are obtained and solved analytically for various boundary conditions. To verify the reliability of the developed model, the results of the current work are compared with those available in literature. The effects of viscoelastic foundation parameters, internal damping coefficient, hygrothermal loading, non-local parameter, gradient index, mode number, and slenderness ratio on the vibrational characteristics of nanoscale viscoelastic FG beams are explored.     

Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is 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. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of an FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

Small scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.     

Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory

This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The nonlinear governing equations and boundary conditions are derived by using the Hamilton principle and discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. A detailed parametric study is conducted to study the influences of the nonlocal parameter, temperature change and external electric voltage on the size-dependent nonlinear vibration characteristics of the piezoelectric nanobeams.     

A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation

Based on a modified couple stress theory, a model for composite laminated beam with first order shear deformation is developed. The characteristics of the theory are the use of rotation–displacement as dependent variable and the use of only one constant to describe the material’s micro-structural characteristics. The present model of beam can be viewed as a simplified couple stress theory in engineering mechanics. An example as a cross-ply simply supported beam subjected to cylindrical bending loads of f_w = q₀ sin (πx/L) is adopted and explicit expression of analysis solution is obtained. Numerical results show that the present beam model can capture the scale effects of microstructure, and the deflections and stresses of the present model of couple stress beam are smaller than that by the classical beam mode. Additionally, the present model can be reduced to the classical composite laminated Timoshenko beam model, Isotropic Timoshenko beam model of couple stress theory, classical isotropic Timoshenko beam, composite laminated Bernoulli–Euler beam model of couple stress theory and isotropic Bernoulli–Euler beam of couple stress theory.     

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory

This article proposes a higher-order shear deformation beam theory for free vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in a thermal environment. The temperature-dependent material properties of functionally graded carbon nanotube-reinforced composite beams are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. The governing equations and boundary conditions are derived by using Hamilton's principle, and the Navier solution procedure is used to achieve the natural frequencies of the sandwich beam in a thermal environment. A parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio, and core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Numerical results are also presented in order to compare the behavior of sandwich beams including uniformly distributed carbon nanotube-reinforced composite face sheets to those including functionally graded carbon nanotube-reinforced composite face sheets.     

The modified couple stress functionally graded cylindrical thin shell formulation

In this article, the functionally graded (FG) cylindrical thin shell formulation is developed by using modified couple stress theory. The equations of motion and classical and nonclassical boundary conditions are extracted based on Hamilton's principle. As a special case, the equations of motion in conjunction with the boundary conditions for simply supported FG cylindrical shell are obtained, and then Navier solution procedure is used for analysis free vibration of nano shell. Afterwards, the influences of different parameters like length scale parameter, distribution of FG properties, and length to radius ratio on dimensionless natural frequency are investigated and compared with classical theory.     

Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory

ABSTRACT

This article investigates the nonlinear vibration of piezoelectric nanoplate with combined thermo-electric loads under various boundary conditions. The piezoelectric nanoplate model is developed by using the Mindlin plate theory and nonlocal theory. The von Karman type nonlinearity and nonlocal constitutive relationships are employed to derive governing equations through Hamilton's principle. The differential quadrature method is used to discretize the governing equations, which are then solved through a direct iterative method. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, external electric voltage, and temperature rise on the nonlinear vibration characteristics of piezoelectric nanoplates.