Effects Of Small Scale Parameter Research Articles

The Nonlinear Bending of Sector Nanoplate via Higher-Order Shear Deformation Theory and Nonlocal Strain Gradient Theory

In this context, the nonlinear bending investigation of a sector nanoplate on the elastic foundation is carried out with the aid of the nonlocal strain gradient theory. The governing relations of the graphene plate are derived based on the higher-order shear deformation theory (HSDT) and considering von Karman nonlinear strains. Contrary to the first shear deformation theory (FSDT), HSDT offers an acceptable distribution for shear stress along the thickness and removes the defects of FSDT by presenting acceptable precision without a shear correction parameter. Since the governing equations are two-dimensional and partial differential, the extended Kantorovich method (EKM) and differential quadrature (DQM) have been used to solve the equations. Furthermore, the numeric outcomes were compared with a reference, which shows good harmony between them. Eventually, the effects of small-scale parameters, load, boundary conditions, geometric dimensions, and elastic foundations are studied on maximum nondimensional deflection. It can be concluded that small-scale parameters influence the deflection of the sector nanoplate significantly.

Effects of strain gradient on Moore-Gibson-Thompson generalized coupled non-fickian diffusion-thermoelasticity analysis in a Love-Bishop nanorod resonator: A size dependent meshless implementation

An effective meshless method based on the generalized finite difference (GFD) technique is developed to find the effects of strain gradient on the size-dependent Moore-Gibson-Thompson (MGT) generalized coupled non-Fickian diffusion-thermoelasticity analysis in a Love-Bishop nanorod resonator. A new MGT-based model with considering the strain gradient is proposed to obtain the shock-induced wave propagations in displacement, temperature and molar concentration fields in a Love-Bishop nanorod resonator for the first time. The transient governing equations of the size-dependent MGT generalized coupled non-Fickian diffusion-thermoelasticity are derived using the strain gradient equations of motion based on Love-Bishop theory, the size-dependent MGT-based energy balance equation and the size-dependent non-Fickian mass diffusion equation. The governing equations are transferred to Laplace domain and an effective GFD-based meshless method is employed to solve them. To obtain the temporal variation of fields’ variables, a proper Laplace inversion technique is employed in the problem. The small-scale effects in the MGT coupled non-Fickian diffusion-thermoelasticity analysis for the nano-sized Love-Bishop rods are taken into account using six small-scale parameters including three higher-order materials length parameters, the micro-length inertia, the micro-length thermal and the micro-length molar concentration parameters. The effects of small-scale parameters on wave propagations in all fields are discussed in detail.

Modified couple stress and thickness-stretching included formulation of a sandwich micro shell subjected to electro-magnetic load resting on elastic foundation

This paper studies electromagnetoelastic static investigation of a sandwich doubly curved microshell subjected to multi-field loading based on a new thickness stretching included refined higher order shear/normal deformable model. Modified-couple-stress-theory (MCST) is used for accounting small-scale-dependency. The numerical results are derived using an analytical method. The effect of small scale parameter in micro scale, initial electric and magnetic potentials and foundation parameters is studied on the electromagnetoelastic bending results. It is confirmed an enhancing in stiffness of small scale shell with an increase in micro length scale parameter.

Accurate solution for size-dependent free vibration analysis of functionally graded micro-plates with Levy boundary conditions

This paper has developed the Levy solution of a size-dependent model for free vibration analysis functionally graded thin rectangular plate according to the theories of first-order shear deformation as well as modified couple stress. The couple stress theory is used to calculate the effects of small-scale parameter, while the first-order shear deformation theory is employed to calculate the effects of shear deformation. By taking the displacement field according to the Mindlin plate theory, application of the Hamilton principle aims at obtaining five governing equations of motion and desired six boundary conditions. The governing equations of motion have been solved based on Levy boundary conditions by using four auxiliary functions. Numerical findings have been used to determine the impacts of various parameters including small-scale parameter changes, power-law index changes, as well as aspect ratios changes with different boundary conditions. As the results show, an increase in the size effects makes that plate tougher and consequently the natural frequency is increased. Also, it can be seen that in little thicknesses of plate, the effects of small-scale parameter are noticeable, but as the plate becomes thicker, the effects will be insignificant.

A size-dependent differential quadrature element model for vibration analysis of FG CNT reinforced composite microrods based on the higher order Love-Bishop rod model and the nonlocal strain gradient theory

In the current paper, a size-dependent differential quadrature element model using the higher-order Love-Bishop rod theory in conjunction with the nonlocal strain gradient theory for axial vibration analysis in the embedded functionally graded (FG) carbon nanotube (CNT) reinforced composite microrods with varying cross-section is presented for the first time. The CNT agglomeration effect is considered applying the two-parameter Eshelby-Mori-Tanaka homogenization scheme. The differential equation form of nonlocal strain gradient theory is employed to take into account the size-dependent effects. Based on the Hamilton's principle, the governing differential equation of the model is derived and solved using differential quadrature element method (DQEM). The higher-order variationally consistent boundary conditions are obtained using the weighted residual approach. The numerical results indicate that the present model is simple and it can be successfully employed for analysis of the small-sized structures. Moreover, the effects of small-scale parameters, the CNT distribution's patterns, the CNT agglomeration parameters, the non-uniformity of the cross section and the stiffness of the surrounding medium on the frequency are studied and discussed in details.

Nonlocal coupled photo-thermoelasticity analysis in a semiconducting micro/nano beam resonator subjected to plasma shock loading: A Green-Naghdi-based analytical solution

In this article, coupled photo-thermoelasticity analysis is carried out using an analytical method in a semiconducting micro/nano beam resonator, considering Green – Naghdi theory (with energy dissipation) and small scale effects. The governing equations for temperature and displacement fields are derived using Eringen nonlocal theory combined with Rayleigh beam theory. One end of the assumed semiconducting MEMS/NEMS is excited by three types of suddenly increasing carrier density and temperature as the plasma and thermal shock loading. The transient behaviours of carrier density field are studied and the effects of disturbances in plasma field on other fields including temperature and deflection are obtained using the proposed analytical solution. The presented analytical solution is based on Laplace transform. To find the dynamic and transient behaviours of fields’ variables in time domain, an inversion Laplace technique is utilized, which is called Talbot method. The effects of small scale parameter and dimensions of the semiconducting micro/nano beam on the dynamic behaviours of fields’ variables are discussed in detail. The axial wave propagation and the distribution of fields’ variables along axial direction are studied at various times.

Non-local orthotropic elastic shell model for vibration analysis of protein microtubules

Vibrational analysis in microtubules is examined based on the nonlocal theory of elasticity. The complete analytical formulas for wave velocity are obtained and the results reveal that the small scale effects can reduce the frequency, especially for large longitudinal wave-vector and large circumferential wave number. It is seen that the small scale effects are more significant for smaller wave length. The methods and results may also support the design and application of nano devices such as micro sound generator etc. The effects of small scale parameters can increase vibrational frequencies of the protein microtubules and cannot be overlooked in the analysis of vibrating phenomena. The results for different modes with nonlocal effect are checked.

A GN-based modified model for size-dependent coupled thermoelasticity analysis in nano scale, considering nonlocality in heat conduction and elasticity: An analytical solution for a nano beam with energy dissipation

This investigation deals with a size-dependent coupled thermoelasticity analysis based on Green-Naghdi (GN) theory in nano scale using a new modified nonlocal model of heat conduction, which is based on the GN theory and nonlocal Eringen theory of elasticity. In the analysis based on the proposed model, the nonlocality is taken into account in both heat conduction and elasticity. The governing equations including the equations of motion and the energy balance equation are derived using the proposed model in a nano beam resonator. An analytical solution is proposed for the problem using the Laplace transform technique and Talbot technique for inversion to time domain. It is assumed that the nano beam is subjected to sinusoidal thermal shock loading, which is applied on the one of beam ends. The transient behaviors of fields\' quantities such as lateral deflection and temperature are studied in detail. Also, the effects of small scale parameter on the dynamic behaviors of lateral deflection and temperature are obtained and assessed for the problem. The proposed GN-based model, analytical solution and data are verified and also compared with reported data obtained from GN coupled thermoelasticity analysis without considering the nonlocality in heat conduction in a nano beam.

Application of hetero junction CNTs as mass nanosensor using nonlocal strain gradient theory: An analytical solution

This paper aims to propose an analytical solution for dynamic analysis of the hetero junction carbon nanotubes (HJCNTs)-based mass nanosensors using a nonlocal strain gradient Timoshenko beam model. To have a more precise nanosensor, it is necessary to have deep information about the vibration characteristics of the nanostructure. So, two main goals are followed in this paper. At first, the vibration of HJCNTs with general (elastic) boundary conditions and without attached mass are studied using a proposed analytical solution. Afterward, the HJCNT is applied as a cantilever mass sensor for sensing light as well as heavy masses attached to its tip. For the large and heavy masses, the rotary inertia of the attached mass is also considered in the analysis. The governing differential equations are derived based on the Hamilton's principle and solved by an analytical method, which is based on the modified Fourier series. The weighted residual method is employed for obtaining the variationally consistent boundary conditions using the known equations of motion of the structure. The field quantities are obtained in the closed forms. The convergence and accuracy of the proposed solution are validated through some special cases available in the literature. The effects of small scale parameters and the elastic boundaries on the frequency and mode shapes of HJCNTs are studied. Moreover, the factors that affect the frequency shift of HJCNT-mass sensor are discussed. The obtained results introduce HJCNTs as new mass nanosensors that can operate more efficiently than uniform CNTs. This paper can be greatly useful in designing HJCNT-mass sensors and may serve as a benchmark for the future research in this field.

Nonlocal FEM Formulation for Vibration Analysis of Nanowires on Elastic Matrix with Different Materials

In this study, free vibration behaviors of various embedded nanowires made of different materials are investigated by using Eringen’s nonlocal elasticity theory. Silicon carbide nanowire (SiCNW), silver nanowire (AgNW), and gold nanowire (AuNW) are modeled as Euler–Bernoulli nanobeams with various boundary conditions such as simply supported (S-S), clamped simply supported (C-S), clamped–clamped (C-C), and clamped-free (C-F). The interactions between nanowires and medium are simulated by the Winkler elastic foundation model. The Galerkin weighted residual method is applied to the governing equations to gain stiffness and mass matrices. The results are given by tables and graphs. The effects of small-scale parameters, boundary conditions, and foundation parameters on frequencies are examined in detail. In addition, the influence of temperature change on the vibrational responses of the nanowires are also pursued as a case study.

Free vibration analysis Silicon nanowires surrounded by elastic matrix by nonlocal finite element method

Higher-order theories are very important to investigate the mechanical properties and behaviors of nanoscale structures. In this study, a free vibration behavior of SiNW resting on elastic foundation is investigated via Eringen\'s nonlocal elasticity theory. Silicon Nanowire (SiNW) is modeled as simply supported both ends and clamped-free Euler-Bernoulli beam. Pasternak two-parameter elastic foundation model is used as foundation. Finite element formulation is obtained nonlocal Euler-Bernoulli beam theory. First, shape function of the Euler-Bernoulli beam is gained and then Galerkin weighted residual method is applied to the governing equations to obtain the stiffness and mass matrices including the foundation parameters and small scale parameter. Frequency values of SiNW is examined according to foundation and small scale parameters and the results are given by tables and graphs. The effects of small scale parameter, boundary conditions, foundation parameters on frequencies are investigated.

A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects

• A unified size-dependent plate model for buckling analysis of nanoplates is developed. • The governing equations are derived based on nonlocal strain gradient theory incorporating surface effects. • Various higher-order plate theories as well as Kirchhoff and Mindlin plate theories are included. • Analytical solutions for critical buckling load of rectangular nanoplates under various boundary conditions are obtained. • The effects of small scale parameters, geometric parameters , boundary condition, shear deformation and surface energy are studied. Based on the nonlocal strain gradient theory and surface elasticity theory, a unified size-dependent plate model is developed for buckling analysis of rectangular nanoplates. The developed model is capable of capturing nonlocal effect, strain gradient effect as well as surface energy effects simultaneously. Moreover, by selecting appropriate shape function, the present model can be reduced to not only Kirchhoff and Mindlin plate models but also various higher-order shear deformation plate models. The non-classical governing equations and associated boundary conditions are established by using the principle of minimum potential energy. Analytical solutions for critical buckling load of rectangular nanoplates under various boundary conditions are obtained. Verification of the proposed model is carried out by comparing the degenerated results with those reported in open literature. The effects of nonlocal parameter, material length scale parameter, geometric parameters, shear deformation and surface energy on the buckling behavior of rectangular nanoplates under different boundary conditions are discussed in detail. The numerical results show that the critical buckling load evaluated by nonlocal strain gradient theory is lower than that predicted by classical continuum theory when the nonlocal parameter is larger than the material length scale parameter, and is higher than that evaluated by classical continuum theory when the nonlocal parameter is smaller than the material length scale parameter. However, when taking surface effects into account, the critical buckling load is mainly affected by surface effects at large length-to-thickness ratio, and depends on the combined effects of nonlocality, strain gradient and surface energy at small length-to-thickness ratio.

Bending Response of Nanobeams Resting on Elastic Foundation

In the present study, the finite element method is developed for the static analysis of nano-beams under the Winkler foundation and the uniform load. The small scale effect along with Eringen's nonlocal elasticity theory is taken into account. The governing equations are derived based on the minimum potential energy principle. Galerkin weighted residual method is used to obtain the finite element equations. The validity and novelty of the results for bending are tested and comparative results are presented. Deflections according to different Winkler foundation parameters and small scale parameters are tabulated and plotted. As it can be seen clearly from figures and tables, for simply-supported boundary conditions, the effect of small scale parameter is very high when the Winkler foundation parameter is smaller. On the other hand, for clamped-clamped boundary conditions, the effect of small scale parameter is higher when the Winkler foundation parameter is high. Although the effect of the small scale parameter is adverse on deflection for simply-supported and clamped-clamped boundary conditions.

Wave Propagation Analysis of Piezoelectric Nanoplates Based on the Nonlocal Theory

Based on the nonlocal theory, this paper develops the Kirchhoff nanoplate and Mindlin nanoplate models for the wave propagation analysis of piezoelectric nanoplates. The effects of small scale parameter and thermo-electro-mechanical loads are incorporated in the nanoplate models. The Hamilton’s principle is employed to derive the governing equations of the nanoplate, which are solved analytically to obtain the dispersion relation for piezoelectric nanoplates. The results show that the nonlocal parameter, temperature change, mechanical load and external electric potential have significant influence on the wave propagation characteristics of the piezoelectric nanoplates. The cut-off wave number is observed to exist for piezoelectric nanoplates subjected to positive electric potential, axial tensile force and temperature rise.

Vibration and Stability Analysis of DWCNT-Based Spinning Nanobearings

This paper aims at investigating free vibrations and stability of double-walled carbon nanotube (DWCNT)-based spinning nanobearings. The so-called nanobearing consists of two coaxial carbon nanotubes (CNTs) where either of the two CNTs can be a rotor while the other takes the role of stator. Euler–Bernoulli beam model along with the Eringen’s nonlocal theory of elasticity are employed to obtain governing equations of transverse vibrations for the CNTs. The coupling of the two CNTs originates from the van-der-Waals (vdW) forcing present in the interface of the two CNTs. The coupling is taken into account as distributed spring foundation with an equivalent elastic stiffness. Based on the obtained model, effects of small-scale parameter, vdW interaction between CNTs and the diameter ratio of CNTs on the critical spinning speed are investigated. Finally, the stability margins of the nanobearings are determined and some general conclusions are drawn.

On buckling and postbuckling behavior of nanotubes

For the first time, the size-dependent thermal buckling and post-buckling behavior of nanotubes made of functionally graded materials (FGMs) with porosities is investigated by using a refined beam theory. This non-classical nanotube model is based on Eringen nonlocal elasticity model which incorporates the small scale effect. Two types of porosity distribution, including even and uneven distribution, are taken into account. The material properties of the nanotubes are temperature-dependent and vary in the radial direction. The size-dependent governing differential equations are derived by employing the generalized variation principle and solved by using a two-step perturbation method. The effects of small scale parameter, porosity volume fraction, the volume fraction index and boundary conditions on thermal buckling and post-buckling of FGM nanotubes are studied by several numerical examples. It can be concluded that the porosity volume fraction and small scale parameter change the buckling and post-buckling behavior of the nanotubes.

Nonlocal vibration and pull-in instability analysis of electrostatic carbon-nanotube based NEMS devices

The objective of this paper is to investigate dynamical pull-in behavior of an electrostatic actuated nano-device based on Eringen's nonlocal elasticity theory. The Euler–Bernoulli beam model is used to establish the dynamic equation of motion of the nano-device subjected to both electrostatic and intermolecular forces. The nanobeam is considered with axially immovable ends and the geometrically nonlinearity due to mid-plane stretching is incorporated to the model as well. A new intermolecular attractive force model based on the macroscopic interactions of a circular cross section nanobeam and a flat surface is presented for the carbon nanotube based nano-device. The nonlinear static equation and the linear dynamic equation are treated by the Differential Quadrature Method (DQM) and through a comprehensive survey, the effect of small scale parameter, DC voltage, intermolecular force, residual stress and geometrical nonlinearity on the fundamental natural frequency and pull-in phenomenon are studied. The results are verified with the literature and a good agreement is achieved.

On the nonlinear mechanics of layered microcantilevers

The nonlinear mechanics of three-layered microcantilevers under base excitation is investigated numerically. Employing the modified version of the couple stress-based theory, together with the Bernoulli-Euler beam theory, the potential energy of the three-layered microsystem is derived, while accounting for size effects. Obtaining the kinetic energy, the equations of motion in the longitudinal and transverse directions are derived via Hamilton's principle. Application of the inextensibility condition reduces the two equations of motion to one nonlinear integro-partial differential equation for the transverse oscillation, consisting of geometrical and inertial nonlinearities. The nonlinear equation of partial-differential type is reduced to set of equations of ordinary-differential type through use of a weighted-residual method. Solving the resultant set of discretised equations via a continuation technique gives the frequency-amplitude and force-amplitude responses of the microsystem. The nonlinear response is investigated for different layer composition and different layer thicknesses. The effect of small-scale parameter, as well as base excitation amplitude, is also examined.

Modeling and analysis of out-of-plane behavior of curved nanobeams based on nonlocal elasticity

Abstract The purpose of this study is to analytically investigate the out-of-plane static behavior of curved nanobeams with non-uniform distributed loads. Nonlocal constitutive equations are implemented into the governing equilibrium equations in cylindrical coordinates. Use of these equations enables revealing the scale effect on static response of curved nanobeams. Shear deformation is considered in the formulation of out-of-plane static behavior of nanobeams. The exact analytical solution to the theoretical model is derived by using initial value method. The fundamental matrix required for the initial value method is obtained analytically. Then, the displacement, slopes and stress resultants are found analytically using the fundamental matrix. Illustrative examples for such nano structures with non-uniform distributed loads are given in subsequent sections. Results showed that the scale effect plays crucial role at nano scale which agree with the results given in the literature. The effects of small scale parameter, slenderness ratio and opening angle on the displacements ratio, rotation angles ratio and stress resultants ratio are also investigated in details. Considering scale effect and shear deformation in the exact solutions of the governing equilibrium equations presented here, the results can be considered as a reference for nonlocal theories of curved nanobeams.

Size-dependent in vibration analysis of magnetostrictive sandwich composite micro-plate in magnetic field using modified couple stress theory

In the present study, free vibration of magnetostrictive sandwich composite micro plate with magnetostrictive core and composite face sheets are investigated. The modified couple stress theory is taken into account so as to consider the small scale effects. The surrounding elastic medium is simulated as visco-Pasternak foundation to study the effects of both damping and shear effects. Using energy method, Hamilton’s principle and first-order shear deformation theory, the governing equations of motion and related boundary conditions are obtained. Finally, the differential quadrature method is employed to analysis the vibration of magnetostrictive sandwich composite micro plate. In this regard, the dimensionless frequency are plotted to study the effects of small scale parameter, surrounding elastic medium, magnetic field, composite fiber angle, aspect ratio, thickness ratio, and boundary conditions. The results indicate that the magnetic field and composite fiber angle play a key role in the dimensionless frequency of magnetostrictive sandwich composite micro plate. The obtained results in this article can be used to design sensors and actuators, aerospace industry, and control of vibration response of systems.