Electrical performance analysis of functionally graded flexoelectric nanobeams considering surface effects and an elastic foundation

In this article, shear vibration and buckling of double-layer orthotropic nanoplates resting on elastic foundations are analyzed subjected to in-plane loadings including surface and nonlocal effects. These effects are considered by Gurtin---Murdoch's theory. Using the principle of virtual work, the governing equations for bulk and surface of double-layer orthotropic nanoplate are derived using refined plate theory. Differential quadrature method (DQM) is implemented. DQM solutions are validated by Navier's method and journal references. The influences of nonlocal parameter, van der Waals, Winkler, shear modulus, orthotropic material properties, boundary conditions, and in-plane biaxial, uniaxial, and shear loadings, are investigated on the surface effects of buckling and vibration modes of out-of-phase and in-phase. Results demonstrate that by augmenting nonlocal parameter, the surface effects on the vibration and buckling modes of both out-of-phase and in-phase increase. This result is in contrast with the works of other researchers in the field. Moreover, by enhancing in-plane loadings, the degree of surface effects on the vibration increase. On the other hand, the effects of nonlocal parameter on the vibration and buckling under in-plane shear load are more influential than those of biaxial and uniaxial, while the surface effects on the biaxial vibration and buckling are more remarkable than those of shear vibration and buckling.

Free vibration analysis of a functionally graded porous nanoplate in a hygrothermal environment resting on an elastic foundation

For micro/nano structures, surface elasticity, surface stress and surface mass strongly affect mechanical behaviors of 1D beam-columns. This article studies dynamic stability of microcantilevers on an elastic foundation or embedded in an elastic matrix when subjected to a subtangential follower force, where the surface effects are taken into account. An exact characteristic equation is derived for clamped–free end supports. For differential tangency coefficients, the force–frequency interaction diagram is displayed and the influences of surface elasticity modulus, residual surface tension, surface mass and the elastic foundation are analyzed for conservative and non-conservative compressive forces. When the tangency coefficient vanishes, a cantilever column subjected to a conservative tip force is reduced, and conventional Euler buckling for a compressive axial load is recovered. When the tangency coefficient does not vanish, a generalized Beck’s column with the surface effects is tackled. When the tangency coefficient exceeds certain critical value, flutter instability take places. For a fixed frequency, the critical divergency and flutter loads as a function of the tangency coefficient are given for various surface influences from residual surface tension, surface elasticity, surface mass and the stiffness of the elastic foundation. The boundary map of stability, divergence and flutter domain is shown.

Nonlocal and surface effects on nonlinear vibration characteristics of a flexoelectric nanobeams under magnetic field are examined. Eringen’s nonlocal elasticity as well as surface elasticity theories are employed to describe the size-dependency of the flexoelectric nanobeam. Also, flexoelectricity is an important size-dependent phenomena for piezoelectric structures at nanoscale, related to the strain gradient-electric polarization coupling. After the derivation of governing equation via Hamilton’s principle, Galerkin method is employed to satisfy boundary conditions. Also, analytical procedures are implemented to obtain the closed-form nonlinear frequency of flexoelectric nanobeam. It is showed that magnetic field intensity, flexoelectric parameter, nonlocal parameter, elastic foundation and applied voltage on the top surface of the nanobeam have great influences on nonlinear vibration frequency.

In this research, vibration characteristics of a flexoelectric nanobeam in contact with Winkler-Pasternak foundation is investigated based on the nonlocal elasticity theory considering surface effects. This nonclassical nanobeam model contains flexoelectric effect to capture coupling of strain gradients and electrical polarizations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, flexoelectric and surface effects are omitted. Hamilton's principle is employed to derive the governing equations and the related boundary conditions which are solved applying a Galerkin-based solution. Natural frequencies are verified with those of previous papers on nanobeams. It is illustrated that flexoelectricity, nonlocality, surface stresses, elastic foundation and boundary conditions affects considerably the vibration frequencies of piezoelectric nanobeams.

This paper develops a more comprehensive theoretical model for functionally graded material (FGM) piezoelectric nanobeams. The model incorporates a Winkler–Pasternak linear elastic foundation and fully accounts for the effects of dynamic flexoelectric, surface effects, and higher-order electric fields. The purpose of this study is to investigate the bending behavior and free vibration characteristics of Euler–Bernoulli beam models considering functionally graded materials. The governing equations and boundary conditions are produced using Hamilton’s variational principle. The Fourier series expansion approach is used to create the analytical solution for the bending problem. Then the analytical equation for the natural frequencies is obtained using the Navier method. The bending performance, electromechanical coupling characteristics, and normalized natural frequencies of FGM piezoelectric nanobeams are all significantly impacted by higher-order electric fields, gradient index, dynamic flexoelectric effects, surface effects, and the Winkler–Pasternak elastic foundation, according to numerical analysis. For the design and optimization of micro/nano energy harvesters and resonators, this paper offers theoretical insights and references.

In this research, vibration characteristics of a flexoelectric nanoplate in contact with Winkler-Pasternak foundation are investigated based on nonlocal elasticity theory considering surface effects. This non-classical nanoplate model contains flexoelectric effect to capture coupling of strain gradients and electrical polarizations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, flexoelectric and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations and the related boundary conditions which are solved applying a Galerkin-based solution. Natural frequencies are verified with those of previous papers on flexoelectric nanoplates. It is illustrated that flexoelectricity, nonlocality, surface stresses, elastic foundation and boundary conditions affects considerably the vibration frequencies of piezoelectric nanoplates.

In this research, vibration analysis of axially preloaded nonlocal flexoelectric nanobeams considering surface and magnetic field effects is performed. The main propose of present paper is to fill the gap in previous analyzes of flexoelectric nanobeams in which nonlocal and surface effects are ignored. Flexoelectricity in a nanobeam results in larger vibration frequencies than a typical piezoelectric nanobeam. Via Galerkin’s technique, the governing equations of the flexoelectric nanobeam are numerically solved. The vibration frequency of the nanobeam is validated with previous one in the literature. The illustrative results demonstrate the influences of the flexoelectricity, nonlocal stress field, surface energy, magnetic field, axial load and beam thickness on vibrational characteristics of a nanosize preloaded flexoelectric beam.

The present paper deals with the free vibration and buckling problem with consideration of surface properties of circular nanobeams and nanoarches. The Gurtin-Murdach theory is used for investigating the surface effects parameters including surface tension, surface density and surface elasticity. Both linear and nonlinear elastic foundation effect are considered on the circular curved nanobeam. The analytically Navier solution is employed to solve the governing equations. It is obviously detected that the natural frequencies of a curved nanobeams is substantially influenced by the elastic foundations. Besides, it is revealed that by increasing the thickness of curved nanobeam, the influence of surface properties and elastic foundations reduce to vanished, and the natural frequency and critical buckling load turns into to the corresponding classical values.

This paper scrutinizes the simultaneous impacts of surface elasticity, initial stress, residual surface stress and nonlocality on the nonlinear vibration of carbon nanotube conveying fluid resting while resting on linear and nonlinear elastic foundations and operating in a thermo-magnetic environment. The derived partial differential equation is decomposed into spatial and temporal equations using Galerkin method of decomposition. Thereafter, the temporal differential equation is solved with the aid of method of homotopy perturbation. Studies of the significance of the model parameters reveal that the negative value of the surface stress enhances the frequency ratio while the positive value of the surface stress abates the ratio. At any given value of nonlocal parameters, the surface effect is lessened for enhancing value of the length of the nanotube. The frequency ratio is abated as strength of the magnetic field, nonlocal parameter and the length of the nanotube are increased. The nonlocality lessens the surface effects and ratio of the frequencies. At high values of nonlocal parameter and nanotube length, the natural frequency of the structure gradually approaches nonlinear Euler–Bernoulli beam limit. The ratio of the frequencies is heightened when the temperature change is reduced at high temperature while at room/low temperature, such ratio is enhanced as the temperature change is augmented. Also, the frequency ratio at low temperatures is lower than at high temperatures. The present work will be very useful in the design and control of carbon nanotubes in thermo-magnetic environment while resting on elastic foundations.

In this article, simultaneous impacts of surface elasticity, initial stress, residual surface tension and nonlocality on the nonlinear vibration of single-walled carbon conveying nanotube resting on linear and nonlinear elastic foundation and operating in a thermo-magnetic environment are studied. The developed equation of motion is solved using Galerkin’s decomposition and Temini and Ansari method. The studies of the impacts of various parameters on the vibration problems revealed that the ratio of the nonlinear to linear frequencies increases with the negative value of the surface stress while it decreases with the positive value of the surface stress. The surface effect reduces for increasing in the length of the nanotube. Ratio of the frequencies decreases with increase in the strength of the magnetic field, nonlocal parameter and the length of the nanotube. Increase in temperature change at high temperature causes decrease in the frequency ratio. However, at room or low temperature, the frequency ratio of the hybrid nanostructure increases as the temperature change increases. The natural frequency of the nanotube gradually approaches the nonlinear Euler–Bernoulli beam limit at high values of nonlocal parameter and nanotube length. Nonlocal parameter reduces the surface effects on the ratio of the frequencies. Also, the ratio of the frequencies at low temperatures is lower than at high temperatures. It is hoped that the present work will enhance the control and design of carbon nanotubes operating in thermo-magnetic environment and resting on elastic foundations.

This paper deals with the free vibration analysis of circular alumina (Al2O3) nanobeams in the presence of surface and thermal effects resting on a Pasternak foundation. The system of motion equations is derived using Hamilton’s principle under the assumptions of the classical Timoshenko beam theory. The effects of the transverse shear deformation and rotary inertia are also considered within the framework of the mentioned theory. The separation of variables approach is employed to discretize the governing equations which are then solved by an analytical method to obtain the natural frequencies of the alumina nanobeams. The results show that the surface effects lead to an increase in the natural frequency of nanobeams as compared with the classical Timoshenko beam model. In addition, for nanobeams with large diameters, the surface effects may increase the natural frequencies by increasing the thermal effects. Moreover, with regard to the Pasternak elastic foundation, the natural frequencies are increased slightly. The results of the present model are compared with the literature, showing that the present model can capture correctly the surface effects in thermal vibration of nanobeams.

This article investigates vibrational behavior of a multi-phase nanocrystalline nanobeam resting on Winkler–Pasternak foundation and subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations and the related boundary conditions which are solved applying differential transform method. The frequencies are compared with those of nonlocal and couple stress based beams. It is showed that vibration frequencies of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, elastic foundation, magnetic field, surface effect, nonlocality and boundary conditions.

Nonlocal dynamic stability analysis of a Timoshenko nanobeam subjected to a sequence of moving nanoparticles considering surface effects

The nonlinear buckling and postbuckling characteristics of cylindrical nanoshells embedded in elastic foundations are investigated based on a strain-consistent elastic shell model including the both surface tension and the induced residual stress. For this purpose, in contrast to the previous models on the basis of the Gurtin-Murdoch elasticity theory, a new non-classical shell model based on Ru's surface elasticity theory is developed in which the non-strain displacement gradient terms are eliminated from the surface stress-strain relations. Using the virtual work's principle, the governing differential equations incorporating surface effects are derived. After that, the size-dependent governing equations are deduced to a boundary layer type problem which is subsequently solved through employing a two-stepped singular perturbation technique. It is revealed that because the edge supports of nanoshells are movable, before applying the axial compression, surface effects lead to an initial shortening due to induced residual strains, but the terms related to the residual strain and initial surface tension vanish in the size-dependent nonlinear governing equations. As a result, it is observed that before applying the axial compressive load, the surface effects cause an initial end-shortening for very thin nanoshells and these effects quickly diminish by increasing the shell thickness.Communicated by Krzysztof Kamil Żur