Values Of Nonlocal Parameter Research Articles

Reflection of plane waves in a non-local viscothermoelastic half-space under variable thermal conductivity and microelongation effects

The present work is implicated with the investigation of wave propagation in a homogeneous, isotropic, non-local microelongated viscothermoelastic solid half space with variable thermal conductivity under the dual phase lag (DPL) model. It is found analytically that three coupled longitudinal waves and one independent transverse wave exist in the medium which travel with distinct speeds. Reflection coefficients and energy ratios of various reflected waves are obtained analytically and presented graphically for aluminum epoxy-like material. The variations of these reflection coefficients are analyzed for different values of non-local parameter, microelongational parameter, viscosity and variable thermal conductivity. The expressions of energy ratios acquired in explicit form are also represented graphically as functions of the angle of incidence in accordance with the defined rule of conservation of energy.

Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.

Thermodynamical interactions in a nonlocal initially stressed fiber-reinforced thermoelastic medium with microtemperatures under GN-II model

The aim of this paper is to study the thermoelastic interactions in a nonlocal initially stressed fiber-reinforced transversely isotropic half-space with microtemperatures. The formulation is established in the context of Green-Naghdi theory of thermoelasticity without energy dissipation. Normal mode technique is used to obtain the general solution for a set of boundary conditions. The expressions of different physical variables such as displacement distribution, stress distribution, temperature field and microtemperatures are obtained in the physical domain and numerical computations are carried out with the help of MATLAB software. Comparisons are made among the results for different values of nonlocal parameter and time along with the results obtained in the presence and absence of fiber-reinforcement and microtemperature. The outcomes point out a strong impact of the nonlocal parameter, fiber-reinforcement, microtemperatures and time on the field variables. Some particular cases of special interest have also been deduced to show the comparison between the present results and previous investigations.

Propagation of waves with nonlocal effects for vibration response of armchair double-walled CNTs

In this paper, vibration characteristics of double-walled carbon nanotubes (CNTs) are studied based upon nonlocal elastic shell theory. The significance of small scale is being perceived by developing nonlocal Love shell model. The wave propagation approach has been utilized to frame the governing equations as eigen value system. The influence of nonlocal parameter subjected to diverse end supports has been overtly analyzed. An appropriate selection of material properties and nonlocal parameter has been considered. The influence of changing mechanical parameter Poisson's ratio has been investigated in detail. It is found that the frequencies decrease as nonlocal parameter increases and for the certain values of nonlocal parameter against range of Poisson ratio rise slowly with length double-walled CNTs. The dominance of boundary conditions via nonlocal parameter is shown graphically. The results generated furnish the evidence regarding applicability of nonlocal shell model and also verified by earlier published literature.

Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions

AbstractIn the present work, axial vibrational behavior of nanorods with different boundary conditions is researched. Bishop's rod theory is implemented to simulate the axial deflection. Size‐dependency is captured by using Eringen's nonlocal elasticity theory. Based on nonlocal deformable boundary conditions and Stokes’ transformation, a system of linear equations is derived and then constructed an Eigen value problem. Several numerical examples are presented to investigate the significance of various parameters such as geometric parameters, vibrational modes, various values of nonlocal parameter and axial spring parameters on the axial frequencies of nanorods. The numerical examples indicate that the deformable boundary conditions and small scale parameter have considerable effects on the axial vibration response.

Exact analytical solutions to the problem of relative post-buckling stiffness of thin nonlocal graphene sheets

Abstract The main purpose of this research is to obtain analytical formulations for exact calculation of relative post-buckling stiffness of nonlocal graphene sheets. In addition to calculating the post-buckling stiffness reduction, the buckling and initial post-buckling responses of these structures, when they are subjected to end-shortening strain, have also been studied. To investigate these phenomena, a new technique called semi-Galerkin technique is used in which the out-of-plane deflection function is firstly postulated as the only displacement field and then, exact nonlocal stress function is calculated through a complete solution of the von-Karman compatibility equation. Finally, Galerkin's method is used to solve the unknown parameter considered in the proposed technique. The nano-sheets are modeled as an orthotropic layer with Kirchhoff assumptions and nonlocal differential elasticity theory is employed to achieve the buckling loads and exact relative stiffness values. For in-plane movements of the longitudinal edges of the nano-sheets, two essential and natural boundary conditions are adopted to be “straightly movable” or “freely movable”. The effects of aspect ratio and nonlocal parameter have been studied for each type of boundary conditions and for graphene sheets with different materials. The stress distribution along the length and the width of the nano-sheets is also investigated and discussed by the two local and nonlocal theories and for various values of nonlocal parameter.

Dynamic Stability of Temperature-Dependent Graphene Sheet Embedded in an Elastomeric Medium

This work applies the first-order shear deformation theory (FSDT) to study the dynamic stability of orthotropic temperature-dependent single-layered graphene sheet (SLGS) embedded in a temperature-dependent elastomeric medium and subjected to a biaxial oscillating loading in a thermal environment. Possible thermal effects are considered in the size-dependent governing equations of the problem. These last ones are derived by means of the Hamilton’s variational principle combined with the Eringen’s differential constitutive model. Navier’s solution as well as Bolotin’s approach are applied to obtain the dynamic instability region (DIR) of the graphene sheet. Thus, a parametric study is carried out to explore the sensitivity of the DIR of the graphene sheet to the temperature variation, the static load factor, the aspect ratio, the foundation type, and the nonlocal parameter (NP). Results indicate that the dimensionless pulsation frequency reduces for increasing values of temperature and NP, whereas the size effect becomes even more pronounced for increasing temperatures. In addition, the adoption of temperature-dependent mechanical properties, rather than independent ones, yields a global shift of the DIR to smaller pulsating frequencies. This proves the relevance of the temperature-dependent mechanical properties to obtain reliable results, in a physical sense.

Analysis of the buckling of carbon nanotube under self-weight: The power series and differential quadrature approaches

ABSTRACTThe buckling of carbon nanotube (CNT) under self-weight and resting on elastic foundations is examined in this article. The system is mathematically modeled as nonlocal Euler-Bernoulli beam. Two different numerical schemes are employed to calculate the values of buckling loads, namely power series method (PSM) and differential quadrature method (DQM). Both methods bring out buckling loads which are very close to each other for considered boundary conditions and for chosen values of nonlocal parameter. The spring constant plays an increasing role on the buckling loads for all boundary conditions while the shear constant decreases the buckling loads of cantilevered CNT.

Study of Large Deflection in Nano-Beams Using the Nonlocal Elasticity Theory

A nonlocal elastic beam model is developed by incorporating Eringen’s nonlocal constitutive equation into the large deflection beam theory for a nano-cantilever Euler–Bernoulli beam. The equilibrium equations are solved in an iterative manner using the shooting method. Deformed configurations of beam for different values of nonlocal parameter are plotted, and the final length of the beam after deflection is obtained. It is demonstrated that the beam scale has a significant impact on its flexural behavior under the large deflection, such that the nano-beam deflection can be either larger or smaller than the classical theory, depending on the loading level.

Free Vibrational Response of Single-Layered Graphene Sheets Embedded in an Elastic Matrix using Different Nonlocal Plate Models

In this paper, the small scale effects are incorporated into the free vibration analysis of single-layered graphene sheets (SLGSs) embedded in an elastic medium. To this end, Eringen’s nonlocal elasticity continuum are applied to the different types of plate theory namely as the classical plate theory (CLPT), first order shear deformation theory (FSDT), and higher order shear deformation theory (HSDT). Winkler and Pasternak foundation models used to simulate the surrounding elastic medium are compared with each other. Explicit expressions are derived to calculate the natural frequencies of square SLGSs corresponding to each type of nonlocal plate model. Selected numerical results are given to indicate the influence of the nonlocal parameter, Winkler and Pasternak elastic moduli, mode number, and the side length of SLGSs in detail. Also, comparison is made between the vibrational responses of SLGSs obtained through different nonlocal plate theories. It is found that the elastic foundation and value of nonlocal parameter have quite significant effects on the natural frequencies of SLGSs and these effects are influenced by mode number as well as side length.DOI: http://dx.doi.org/10.5755/j01.mech.23.5.14883

Effect of nonlocal elasticity on the performance of a flexoelectric layer as a distributed actuator of nanobeams

The paper is concerned with the development of finite element model for the static analysis of smart nanobeams integrated with a flexoelectric layer on its top surface, using nonlocal elastic theory. The flexoelectric layer acts as a distributed actuator of the nanobeam. A layerwise displacement theory has been used to derive the element stiffness matrices from variational principles incorporating nonlocal effects. The finite element model for nonlocal response of the beams has been validated with the exact solution for the case of a simply supported standalone flexoelectric layer. Also, the finite element model of the simply supported smart beam has been validated with exact solutions and numerical models for the local elastic case. The performance of the flexoelectric actuator has been compared for different values of nonlocal parameters and different combinations of nonlocal and local elastic substrate and flexoelectric layer. Further, the model developed has been utlized for investigating the performance of the active flexoelectric layer in case of cantilever beam, for which the exact solutions are not available.

Nonlocal and surface effects on the flutter instability of cantilevered nanotubes conveying fluid subjected to follower forces

On the basis of nonlocal elasticity theory, this paper studies the dynamic structural instability behavior of cantilever nanotubes conveying fluid incorporating end concentrated follower force and distributed tangential load, resting on the visco-Pasternak substrate. In order to improve the accuracy of the results, surface effects, i.e. surface elasticity and residual stresses are considered. Extended Hamilton’s principle is implemented to obtain the nonlocal governing partial differential equation and related boundary conditions. Then, the extended Galerkin technique is used to convert partial differential equations into a general set of ordinary differential equations. Numerical results are expressed to reveal the variations of the critical flow velocity for flutter phenomenon of cantilever nanotubes with the various values of nonlocal parameter, mass ratios, nanotubes thickness, surface effects, various parameters of the visco-Pasternak medium, constant follower force and distributed compressive tangential load. Some numerical results of this research illustrated that the values of critical flutter flow velocity and stable region increase by considering surface effects. Also, critical flutter flow velocity decreases towards zero by increasing the value of the distributed compressive tangential load and constant follower force.

NONLOCAL FLUGGE SHELL MODEL FOR THE AXIAL BUCKLING OF SINGLE-WALLED CARBON NANOTUBES: AN ANALYTICAL APPROACH

In this paper, the stability characteristics of single-walled carbon nanotubes (SWCNTs) under the action of axial load are investigated. To this end, a nonlocal Flugge shell model is developed to accommodate the small length scale effects. The analytical Rayleigh-Ritz method with beam functions is applied to the variational statement derived from the Flugge-type buckling equations. Molecular dynamics (MD) simulations are performed to obtain the critical axial buckling loads of (8,8) armchair SWCNTs with different types of end conditions. Through comparison of the results obtained from the present analytical solution and the ones from molecular dynamics simulations, the appropriate values of nonlocal parameter are proposed for SWCNTs with different kinds of boundary conditions. The effects of nonlocal parameter and boundary conditions on the critical buckling load are also examined. Moreover, in spite of the uncertainty that exists in defining the in-plane stiffness and bending rigidity of nanotube, by adjusting the nonlocal parameter, the present nonlocal shell model is shown to be capable of predicting the MD simulations results.

Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method

In this article, buckling analysis of double-orthotropic nanoplates (DONP) embedded in elastic media under biaxial, uniaxial and shear loading is numerically studied. The analysis is based on non-local theory. Both two-variable refined plate theory (TVRPT) and first-order shear deformation plate theory (FSDT) are used to derive the governing equations. Generalized differential quadrature method (GDQM) is utilized to solve the governing equations. In buckling analysis, both in-phase and out-of-phase modes are studied. A graphene sheet is selected as the case study to investigate the numerical results. GDQM results are validated by comparing with the Navier’s solutions. After validating the formulation and method of solution, the effect of non-local parameter, geometrical parameters and boundary conditions on the critical buckling load of the double-orthotropic nanoplate are investigated and discussed in detail. It is shown that the effects of non-local parameter for shear buckling are more noticeable than that of biaxial buckling. Moreover, for higher values of non-local parameter, the shear buckling is not dependent on the van der Waals and Winkler moduli.

Nonlocal plate model for dynamic pull-in instability analysis of circular higher-order shear deformable nanoplates including surface stress effect

In the present paper, dynamic pull-in instability and free vibration characteristics of circular higher-order shear deformable nanoplates subjected to hydrostatic and electrostatic forces are studied including surface stress effect. For this purpose, Eringen’s nonlocal elasticity continuum in conjunction with the Gurtin-Murdoch elasticity theory is incorporated into the classical higher-order shear deformation plate theory to develop size-dependent plate model able to consider both of small scale and surface stress effects. The non-classical governing differential equations are then discretized along with simply supported and clamped edge supports by employing generalized differential quadrature (GDQ) method. To evaluate the size-dependent pull-in voltage of nanoplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. It is demonstrated that the pull-in instability occurs at lower voltages for nanoplates with higher values of nonlocal parameters. Moreover, it is found that surface stress effect can increase or decrease the pull-in voltage of nanoplates which depends on the sign of surface elastic constants.

An analytical study on the buckling and free vibration of rectangular nanoplates using nonlocal third-order shear deformation plate theory

Abstract In this paper, analytical closed-form solutions in explicit forms are presented to investigate small scale effects on the buckling and the transverse vibration behavior of Levy-type rectangular nanoplates based on the Reddy's nonlocal third-order shear deformation plate theory. Two other edges of Levy-type rectangular nanoplates may be restrained by different combinations of free, simply supported, or clamped boundary conditions. Hamilton's principle is used to derive the nonlocal equations of motion and natural boundary conditions of the nanoplate. Two comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate the high accuracy of the present new formulation. Comprehensive benchmark results with considering the small scale effects on frequency ratios, buckling load ratios, non-dimensional fundamental natural frequencies and non-dimensional buckling loads of rectangular nanoplates with different combinations of boundary conditions are presented for various values of nonlocal parameters, aspect ratios and thickness to length ratios. It is observed that except for SFSF rectangular nanoplates, as the aspect ratio increases, buckling load and natural frequency decreases, while keeping all other parameters fixed. For SFSF rectangular nanoplates, by increasing the aspect ratio, the values of the buckling load and frequency ratio increase.

Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity

Using three-dimensional (3-D) nonlocal elasticity theory of Eringen, this paper presents closed-form solutions for in-plane and out-of-plane free vibration of simply supported functionally graded (FG) rectangular micro/nanoplates. Elasticity modulus and mass density of FG material are assumed to vary exponentially through the thickness of micro/nanoplate, whereas Poisson's ratio is considered to be constant. By employing appropriate displacement fields for the in-plane and out-of-plane modes that satisfy boundary conditions of the plate, ordinary differential equations of free vibration are obtained. Boundary conditions on the lateral surfaces are imposed on the analytical solutions of the equations to yield the natural frequencies of FG micro/nanoplate. The natural frequencies of FG micro/nanoplate are obtained for different values of nonlocal parameter and gradient index of material properties. The results of this investigation can be used as a benchmark for the future numerical, semi-analytical and analytical studies on the free vibration of FG micro/nanoplates.

Analytical Treatment of the Free Vibration of Single-Walled Carbon Nanotubes Based on the Nonlocal Flugge Shell Theory

In the current work, the vibration characteristics of single-walled carbon nanotubes (SWCNTs) under different boundary conditions are investigated. A nonlocal elastic shell model is utilized, which accounts for the small scale effects and encompasses its classical continuum counterpart as a particular case. The variational form of the Flugge type equations is constructed to which the analytical Rayleigh–Ritz method is applied. Comprehensive results are attained for the resonant frequencies of vibrating SWCNTs. The significance of the small size effects on the resonant frequencies of SWCNTs is shown to be dependent on the geometric parameters of nanotubes. The effectiveness of the present analytical solution is assessed by the molecular dynamics simulations as a benchmark of good accuracy. It is found that, in contrast to the chirality, the boundary conditions have a significant effect on the appropriate values of nonlocal parameter.

Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics

In the present study, the free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. Eringen's nonlocal elasticity equations are incorporated into the classical Donnell shell theory accounting for small scale effects. The Rayleigh–Ritz technique is applied to consider different sets of boundary conditions. The displacements are represented as functions of polynomial series to implement the Rayleigh–Ritz method to the governing differential equations of nonlocal shell model and obtain the natural frequencies of DWCNTs relevant to different values of nonlocal parameter and aspect ratio. To extract the proper values of nonlocal parameter, molecular dynamics (MD) simulations are employed for various armchair and zigzag DWCNTs, the results of which are matched with those of nonlocal continuum model through a nonlinear least square fitting procedure. It is found that the present nonlocal elastic shell model with its appropriate values of nonlocal parameter has the capability to predict the free vibration behavior of DWCNTs, which is comparable with the results of MD simulations.