Effects Of Small Scale Parameter Research Articles

Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method

This study investigates the small scale effect on the flapwise bending vibrations of a rotating nanoplate. The nanoplate is modeled with a classical plate theory and considering cantilever and propped cantilever boundary conditions. Due to the rotation, the axial forces are included in the model as true spatial variation. Hamilton's principle is used to derive the governing equation and boundary conditions of the classical plate theory based on Eringen's nonlocal elasticity theory. The generalized differential quadrature method is employed to solve the governing equation. The effect of small-scale parameter, non-dimensional angular velocity, non-dimensional hub radius, aspect ratio, and different boundary conditions in the first four non-dimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nano-machines such as nano-motors and nano-turbines and other nanostructures.

Nonlocal nonlinear first-order shear deformable beam model for postbuckling analysis of magneto-electro-thermo elastic nanobeams

In this study, the size-dependent postbuckling behavior of magneto-electro-thermo-elastic (METE) nanobeams with different edge supports is investigated. Based on the nonlocal first-order shear deformation beam theory and considering the von Karman hypothesis, a size-dependent nonlinear METE nanobeam model is developed, in which the effects of small scale parameter and thermo-electro-magnetic-mechanical loadings are incorporated. A numerical solution procedure based on the generalized differential quadrature (GDQ) and pseudo arc-length continuation methods is utilized to describe the size-dependent postbuckling behavior of METE nanobeams under various boundary conditions. The effects of different parameters such as nonlocal parameter, external electric voltage, external magnetic potential and temperature rise on the postbuckling path of METE nanobeams are explored. The results indicate that increasing the non-dimensional nonlocal parameter, imposed positive voltage, negative magnetic potential and temperature rise decrease the critical buckling load and post-buckling load-carrying capacity of METE nanobeams, whilean increase in the negative voltage, positive magnetic potential lead to a considerable increase of critical buckling load as well as postbuckling strength of the METE nanobeams.

Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method

ABSTRACTIn this study, thermal and small-scale effects on the flapwise bending vibrations of a rotating nanoplate, which can be the basis of nano-turbine design, have been analyzed. The nano-turbine is made of an orthotropic nanoplate with a setting angle that is modeled based on the classical plate theory (CPT) with cantilever boundary conditions. The axial forces are also included in the model as the true spatial variation due to the rotation and temperature change. The governing equations and boundary conditions are derived according to Hamilton's principle and the governing equations are solved with the aid of the generalized differential quadrature method. The effects of small-scale parameter, nondimensional angular velocity, temperature change, and setting angles in the first four nondimensional frequencies are discussed. Due to the consideration of the rotating effects, results of this study are applicable in nano-machines, such as nano-motors, nano-rotor, and other rotating nano-structures. Also, by considering the effect of thermal loading on rotation of a nanoplate, the results are useful in the design of nano-turbines.

Non-linear bending analysis of multi-layer orthotropic annular/circular graphene sheets embedded in elastic matrix in thermal environment based on non-local elasticity theory

In this paper, the large deflection of multi-layer orthotropic annular/circular graphene sheets is investigated based on the non-local elasticity theory. The plate is considered to be in thermal environment. The equilibrium equations are derived in terms of generalized displacements and rotations considering the FSDT non-local elasticity theory and the van der Waals interaction between the layers. In order to solve the governing equations, the differential quadrature method (DQM) which is an accurate numerical method and a new semi-analytical polynomial method (SAPM) are applied. By applying DQM or SAPM, the ODE's would be converted to non-linear algebraic equations. In continue, the Newton–Raphson iterative scheme is applied to solve the obtained non-linear algebraic equations. The results of DQM and SAPM are compared. Although, the SAPM's formulation is considerably simpler than DQM, however, the results of two methods are so close to each other. The results are validated with the other available researches. The effect of small scale parameter, temperature effects on non-local results, the value of van der Waals interaction between the layers for bi-layer and triple layers graphene sheet, different values of elastic foundation matrix and load for various small scale parameters, the comparison between local and non-local deflections and linear to non-linear analysis are investigated.

Analytical modeling of wave propagation in viscoelastic functionally graded carbon nanotubes reinforced piezoelectric microplate under electro-magnetic field

Wave propagation analysis of a functionally graded carbon nanotubes reinforced piezoelectric composite (FG-CNTRPC) microplate is the major main of the present research. In order to present a realistic model, the material properties of the system are assumed viscoelastic and the Kelvin–Voigt model is applied. The viscoelastic FG-CNTRPC microplate is subjected to longitudinal magnetic and three-dimensional electric fields. The distribution of carbon nanotubes in FG-CNTRPC microplate is supposed as uniform distribution and surrounding circumference is simulated as Visco-Pasternak foundation. The original formulation of the quasi-three-dimensional sinusoidal shear deformation plate theory is here extended to the wave propagation analysis and the size effects are considered based on Eringen’s nonlocal theory. In order to calculate the dimensionless frequency, cut-off and escape frequencies analytical solution is applied. In this article, the influences of the volume fraction of carbon nanotubes, electro-magnetic fields and elastic medium on the dimensionless frequency of viscoelastic FG-CNTRPC microplate are investigated. Furthermore, the effect of small-scale parameter on the cut-off and escape frequencies of the system will be studied. Results demonstrate that the dimensionless cut-off and escape frequencies decrease with increasing the magnitude of small-scale parameter. In addition, the imposed magnetic field and external voltage are significant parameters for controlling wave propagation of the viscoelastic FG-CNTRPC microplate. Results of this investigation can be helpful for the study and design of composite systems based on smart control and sensor applications.

Steady state response of functionally graded nano-beams resting on viscous foundation to super-harmonic excitation

This article attempts to investigate the effects of small scale parameter on steady state response of functionally graded nano-beams resting on a viscous foundation to super-harmonic excitation. A simple power-law distribution is used to model the variation of material property graded in the thickness direction. The dimensionless partial differential equation of motion is derived by using Euler-Bernoulli beam theory, von-Karman geometric nonlinearity and Eringen’s nonlocal elasticity theory. Using multiple scale method, one can find the governing equations of steady state response of functionally graded nano-beams excited by distributed harmonic force. The small scale parameter (e0a) is changed between 0 and 2 to investigate the effects of small scale on steady state response of excited functionally graded nano-beams due to lack of information. The study of the effects of small scale parameter on backbone curves shows that an increase in the small scale parameter often decreases the dimensionless peak response although the type of loading can change the relationship between small scale parameter and the dimensionless peak response.

Nonlinear thermo-elastic bending behavior of graphene sheets embedded in an elastic medium based on nonlocal elasticity theory

This paper investigates the large deflection behavior of orthotropic single layered graphene sheet (SLGS) embedded in a Winkler–Pasternak elastic medium under a uniform transverse load in thermal environments. Using the nonlocal differential constitutive relations of Eringen, the SLGS is modeled as a nonlocal orthotropic plate. Using the principle of virtual work, the coupled nonlinear equilibrium equations are obtained based on first-order shear deformation theory (FSDT) and the von Kármán geometrical model. The differential quadrature (DQ) discretized form of the governing equations with clamped and simply supported boundary conditions is derived. The Newton–Raphson iterative scheme is used to solve the resulting system of nonlinear algebraic equations. Effects of small scale parameter, thermal environment, width-to-length elasticity ratio, aspect ratio, thickness, elastic foundation, load value and boundary conditions are considered in detail. The results show that, unlike the simply supported boundary conditions, increase of small scale parameter plays a decreasing role in effect of thermal environment on the deflections of nanoplates with clamped edges. It is also observed that increasing the plate thickness, thermal load effects decline noticeably and depending on the small scale value the thermal loads do not have any significant effect on the results beyond a specified thickness.

A new mixed method for nonlinear fuzzy free vibration analysis of nanobeams on nonlinear elastic foundation

A new mixed method for nonlinear fuzzy free vibration analysis of nanobeams on nonlinear elastic foundation is introduced. The governing equations are derived based on the first-order shear deformation theory (FSDT) in conjunction with the von-Kármán’s assumptions and the Eringen’s nonlocal elasticity theory. The differential quadrature method (DQM) is employed to discretize the governing equations and the related boundary conditions. The direct displacement control iterative method is used to solve the discretized system of equations. The fuzzy transformation method (FTM) is coupled with the solution to incorporate effects of different uncertainties such as the small scale effect parameter, nonlinear elastic foundation parameters and vibration amplitude of the nanobeam. Applicability, rapid rate of convergence and high accuracy of the presented method are shown and significant effects of the nonlinearity on the response of nanobeams are investigated via solving some examples.

Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler–Bernoulli microbeams

The target of this paper is to present an exhaustive study on the small scale effect on vibrational behavior of a rotary tapered axially functionally graded (AFG) microbeam on the basis of Timoshenko and Euler–Bernoulli beam and modified couple stress theories. The variation of the material properties and cross section along the longitudinal direction of the microbeam are taken into consideration as a linear function. Hamilton's principle is used to derive the equations for cantilever and propped cantilever boundary conditions and the generalized differential quadrature method (GDQM) is employed to solve the equations. By parametric study, the effects of small-scale parameter, rates of cross section change of the microbeam and angular velocity on the fundamental and second frequencies of the microbeam are studied. Also, comparison between the frequencies of Timoshenko and Euler–Bernoulli microbeams are presented. The results can be used in many applications such as micro-robots and biomedical microsystems.

Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of Eringen

Abstract Nonlinear bending analysis of orthotropic annular/circular graphene sheets has been studied based on the non-local elasticity theory. The first order shear deformation theory (FSDT) is applied in combination with the nonlinear Von-Karman strain field. The obtained differential equations are solved by using two methods, first the differential quadrature method (DQM) and a new semi-analytical polynomial method (SAPM) which is innovated by the authors. Applying the DQM or SAPM, the differential equations are transformed to nonlinear algebraic equations system. Then the Newton–Raphson iterative scheme is used. First, the obtained results from DQM and SAPM are compared and it is concluded that although the SAPM’s formulation is considerably simpler than DQM, however, the SAPM’s results are so close to DQM. The results are validated with available papers. Finally, the effects of small scale parameter on the results, the comparison between local and non-local theories, and linear to nonlinear analyses are investigated.

Linear free vibration in pre/post-buckled states and nonlinear dynamic stability of lipid tubules based on nonlocal beam model

In this paper, considering the small scale effect, the linear free vibration in pre/post-buckled states and nonlinear dynamic stability of lipid tubules with in-plane movable ends are studied. The small scale effect is characterized by nonlocal elasticity theory. The vibration in pre/post-buckled regions is solved by the differential quadrature method (DQM), and the nonlinear dynamic stability is solved by incremental harmonic balance method (IHBM). In numerical results, the effects of small scale parameter, types of lipid tubule on vibration in pre/post-buckled states and nonlinear dynamic stability are discussed.

Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics

In this paper, the nonlinear bending behavior of bilayer orthotropic rectangular graphene sheets resting on a two parameter elastic foundation is studied subjected to uniform transverse loads using the non-local elasticity theory. The non-local theory consider the small scale effects. Considering the non-local differential constitutive relations of Eringen theory based on first order shear deformation theory (FSDT) and using the von-Karman strain field, the nonlinear formulations are derived. Equilibrium partial differential equations are expressed in terms of generalized displacements and rotations. Because of nonlinear partial differential equations, if it is not impossible but it is too complicated to find an analytical solution, so, the differential quadrature method (DQM) that is a high accurate numerical method is investigated to solve the governing equations. The Newton–Raphson iterative scheme is applied to solve the obtained nonlinear algebraic equations system. Different boundary conditions including clamped, simply supports and free edges are considered. Since there is not any researches available for nonlinear bending of bilayer rectangular graphene sheets with FSD theory, so considering the monolayer, the results are compared with available papers. Finally, the small scale effect parameter due to various types of conditions such as thickness ratio, boundary conditions, stiffness of elastic foundation, the van der Waals interactions between the layers, nonlinear to linear FSDT analysis and the differences between non-local and local elasticity theories are investigated.

Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories

The active vibration control of nanotubes under action of a moving nanoscale particle is carried out using nonlocal elasticity theory of Eringen. The nanotube is simulated by an equivalent continuum structure under simply supported boundary conditions within the framework of nonlocal Rayleigh as well as Timoshenko beam theory. A Dirac-delta function is applied to model the location of the moving mass along the nanotube as well as its inertial effects. In the following a linear classical optimal control algorithm with a time varying gain matrix and displacement-velocity feedback is applied to vibration suppression in nanotube structure. The effects of the moving nanoparticle velocity, slenderness ratio of nanotube and small scale effect parameter on the dynamic deflection are studied in some detail for both the Rayleigh and Timoshenko theories. Finally the proficiency of the control algorithm in suppressing the response of the nanostructure under the effect of moving nanoparticle with different number of controlled modes and control forces is studied. The results of the present work can be used as a benchmark in future studies of nanomachines and nanosystems.

Nonlinear free vibration of skew nanoplates with surface and small scale effects

The nonlinear free flexural vibration of skew nanoplates is studied by considering the influences of free surface energy and size effect (small scale) simultaneously. The formulations are derived based on classical plate theory (CPT) in conjunction with nonlocal and surface elasticity theories using Hamilton's principle. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The free surfaces are modeled as two-dimensional membranes adhering to the underlying bulk material without slipping. The solution algorithm is based on the transformation of the governing differential equation from the physical domain to a rectangular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The effect of small scale parameter and surface effect together with the geometrical parameters and boundary conditions on the nonlinear frequency parameters of the skew nanoplates are studied.

Nonlinear bending analysis of orthotropic nanoscale plates in an elastic matrix based on nonlocal continuum mechanics

In this paper, nonlinear bending behavior of the orthotropic single layered graphene sheet (SLGS) subjected to a transverse uniform load and resting on an elastic matrix as Pasternak foundation model is investigated using the nonlocal elasticity theory. The nanoplate equilibrium equations are derived in terms of the generalized displacements based on first-order shear deformation theory (FSDT) using the nonlocal differential constitutive relations of Eringen and the von Karman nonlinear strains. The differential quadrature (DQ) discretized form of the governing equations with the various types of boundary conditions are derived. The Newton–Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Numerical results obtained by the present theory are compared with available solutions in the literature and those developed by finite difference method (FDM) and dynamic relaxation method (DRM) in this work. Excellent agreement between the results of different solution method and presented results is observed. Finally, effects of small scale parameter, width ratio, thickness of plate, elastic foundation properties, load value, boundary conditions and nonlinearity are studied for both nonlocal FSDT and classical plate theory (CPT) in detail.

CALIBRATION OF NONLOCAL SCALE EFFECT PARAMETER FOR BENDING SINGLE-LAYERED GRAPHENE SHEET UNDER MOLECULAR DYNAMICS

In this article, the small scale effect parameter e0 of single-layered graphene sheets (SLGSs) is calibrated for the bending problem. Taking the SLGSs as a rectangular plate, the normal displacement of the simply supported plate under concentrated force was analyzed by both nonlocal elasticity theory and molecular dynamics (MD) simulations, then the small scale effect parameter e0 of SLGSs with different size was obtained by matching the displacement of the nonlocal elasticity theory and that obtained from MD simulations. The results show that the value of e0 is not a constant but has a relationship with the size of SLGSs, and the relationship of armchair-graphene sheets and zigzag-graphene sheets is different.

Effect of small scale on the dynamic characteristic of carbon nanotubes under axially oscillating loading

This paper presents the report of an investigation into the effect of small scale parameter on the natural frequency and the dynamic stability of carbon nanotubes (single-wall and double-walls) under axially oscillating loading, based on a non-local shell model and a non-local effective model of multi-walled carbon nanotubes. It is seen from example calculations that effects of the small scale parameter on dynamic characteristics of SWNT and DWNTs under axial loadings are dependent on the loading form and the geometrical size of carbon nanotubes. The results show that utilizing a non-local effective model to calculate the dynamic characteristics of MWNTs considering small scale parameters, under axial tensional or compression loadings, is simple and feasible.

Free vibration of nanoplates based on a nonlocal two-variable refined plate theory

The application of a two-variable refined plate theory is extended to the free vibration of nanoplates. The theory takes into account the effects of small scale and the quadratic variation of the transverse shear strains through the thickness of the nanoplate and hence, it does not require the use of shear correction factors. The equations of motion and the related boundary conditions are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined plate theory via Hamilton’s principle. The obtained equations are only dynamically coupled. Exact solution for the simply supported nanoplates is obtained to verify the theory by comparing its results with other available solutions in the open literature. Then, using the differential quadrature method (DQM) as an accurate and computationally efficient numerical method, the effect of small scale parameter on the frequency parameters of the nanoplates with some different types of boundary conditions are studied.

Large amplitude vibration of a bilayer graphene embedded in a nonlinear polymer matrix

In the present article, nonlinear free and forced vibration of a bilayer graphene embedded in a polymer medium is studied based on the nonlocal elasticity theory. As a nonlinear function of deflection of graphene sheets, a refined pressure expression is established to describe the van der Waals (vdW) interactions between graphene layers and polymer medium. Assuming the large displacements and anisotropic model for graphene layers, the nonlinear couple partial differential equations of a double layered graphene sheet (DLGS) are obtained. The in-phase and out of phase nonlinear to linear natural frequencies are shown for both zigzag and armchair geometries. The effects of small scale parameter, nonlinear coefficients of vdW between two layers, nonlinear factor of polymer matrix and geometric properties on the nonlinear vibrational behavior of a DLGS are discussed in detail. It is found that the nonlinear vdW coefficient has a significant effect on the out of phase frequencies while the nonlinear polymer coefficient has considerable influence on in-phase frequencies.

Stability analysis of nanocones under external pressure and axial compression using a nonlocal shell model

A nonlocal continuum shell model is developed to study the stability of nanocones under combined loading: external pressure and compression force. The nonlinear governing equations of motion of nanocone are obtained using Hamilton's principle and the external loads are considered as prestress. Based on Eringen's nonlocal elasticity theory the small-scale effect is accounted in the governing equations of motion. To obtain the critical loads, the equations are solved using Galerkin technique and the effect of small-scale parameter and geometry on the stability of nanocone is studied.