Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen’s Nonlocal Elasticity Theory and the Perturbation Method

A three-dimensional(3D)asymptotic theory is reformulated for the static analysis of simply-supported,isotropic and orthotropic single-layered nanoplates and graphene sheets(GSs),in which Eringen’s nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these.The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional(2D)nonlocal plate problems,the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory(CST),although with different nonhomogeneous terms.Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions,we can obtain the Navier solutions of the leading-order problem,and the higher-order modifications can then be determined in a hierarchic and consistent manner.Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.

TRANSVERSE VIBRATION OF A HANGING NONUNIFORM NANOSCALE TUBE BASED ON NONLOCAL ELASTICITY THEORY WITH SURFACE EFFECTS

The aim of this paper is to study the free transverse vibration of a hanging nonuni- form nanoscale tube. The analysis procedure is based on nonlocal elasticity theory with surface effects. The nonlocal elasticity theory states that the stress at a point is a function of strains at all points in the continuum. This theory becomes significant for small-length scale objects such as micro- and nanostructures. The effects of nonlocality, surface energy and axial force on the natural frequencies of the nanotube are investigated. In this study, analytical solutions are formulated for a clamped-free Euler-Bernoulli beam to study the free vibration of nanoscale tubes.

VIBRATION OF FLUID-FILLED MULTI-WALLED CARBON NANOTUBES SEEN VIA NONLOCAL ELASTICITY THEORY

Vibration characteristics of fluid-filled multi-walled carbon nanotubes axe studied by using nonlocal elastic Fliigge shell model. Vibration governing equations of an N-layer carbon nanotube are formulated by considering the scale effect. In the numerical simulations, the effects of different theories, small-scale, variation of wavenumber, the innermost radius and length of double- walled and triple-walled carbon nanotubes are considered. Vibrational frequencies decrease with an increase of scale coefficient, the innermost radius, length of nanotube and effects of wall number are negligible. The results show that the cut-off frequencies can be influenced by the wall number of nanotubes.

NEW POINTSOF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS( Ⅲ) ——RE_DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

In this paper, it is proven that the balance equation of energy is the first integral of the balance equation of momentum in the linear theory of nonlocal elasticity. In other words, the balance equation of energy is not an independent one. It is also proven that the residual of nonlocal body force identically equals zero. This makes the transform formula of the nonlocal residual of energy much simpler. The linear nonlocal constitutive equations of elastic bodies are deduced in details, and a new formula to calculate the antisymmetric stress is given.

NONLOCAL THEORY STUDY ON FRACTURE TOUGHNESS OF CERAMIC MATERIALS

The theoretical calculation formulas for the plane strain fracture toughness of mode Ⅰand Ⅱcracks of ceramic materials are deduced in this paper by using the nonlocal elasticity theory and maximum tensile stress criterion The deduced formulas, which are independent of crack geometry,bear a relation to material parameters.It is shown through experiment that the theoretical value of fracture toughness is the lower limit of testing value. The theoretical calculation formulas for fracture toughness relate the macro-mechanical performance of materials with the micro-structural parameters and,therefore, are beneficial to fully understanding the physical mechanism of material rupture.

Memory response in a nonlocal micropolar double porous thermoelastic medium with variable conductivity under Moore-Gibson-Thompson thermoelasticity theory

The present study enlightens the two-dimensional analysis of the thermo-mechanical response for a mi-cropolar double porous thermoelastic material with voids(MDPTMWV)by virtue of Eringen’s theory of nonlocal elasticity.Moore-Gibson-Thompson(MGT)heat equation is introduced to the considered model in the context of memory-dependent derivative and variable conductivity.By employing the normal mode technique,the non-dimensional coupled governing equations of motion are solved to determine the an-alytical expressions of the displacements,temperature,void volume fractions,microrotation vector,force stress tensors,and equilibrated stress vectors.Several two-dimensional graphs are presented to demon-strate the influence of various parameters,such as kernel functions,thermal conductivity,and nonlocality.Furthermore,different generalized thermoelasticity theories with variable conductivity are compared to visualize the variations in the distributions associated with the prior mentioned variables.Some particu-lar cases are also discussed in the presence and absence of different parameters.

Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method

Free vibration analysis of quadrilateral multilayered graphene sheets（MLGS） embedded in polymer matrix is carried out employing nonlocal continuum mechanics.The principle of virtual work is employed to derive the equations of motion.The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis.The dependence of small scale effect on thickness,elastic modulus,polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated.The non-dimensional natural frequencies of skew,rhombic,trapezoidal and rectangular MLGS are obtained with various geometrical parameters and mode numbers taken into account,and for each case the effects of the small length scale are investigated.

Nonlocal isogeometric analysis for bidirectional functionally graded porous curved microbeams with arbitrary boundary conditions

This research aims to analyze the static bending and free vibration behaviors of bidirectional functionally graded porous(BFGP)curved microbeams with elastic boundary conditions resting on a Pasternak’s elastic foundation in a hygro-thermal environment.Isogeometric analysis based on non-uniform rational B-splines,first-order shear deformation theory,nonlocal elasticity theory combined with the modified strain gradient theories,modified Timoshenko beam theory are formulated to depict bending and shear deformations of BFGP curved microbeams.Especially,because using the modified Timoshenko beam theory,this study removes the requirement for a shear correction factor and describes shear stress by zero at the upper and lower cross-sectional sites of the BFGP curved microbeam.Different from traditional boundary conditions,where the beginning and end positions of a curved beam are connected by an elastic system of straight and torsion springs.This allows for greater flexibility in controlling the stiffness of the springs to obtain arbitrary boundaries.To assess the accuracy and convergence of the proposed approach,validation numerical examples were conducted in the various examples.

An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets

A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach. A Hamiltonian system is established by introducing a total unknown vector consisting of the displacement amplitude, rotation angle, shear force, and bending moment. The high-order governing differential equation of the vibration of SLGSs is transformed into a set of ordinary differential equations in symplectic space. Exact solutions for free vibration are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary conditions. Vibration modes are expressed in terms of the symplectic eigenfunctions. In the numerical examples, comparison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given. A parametric study of the natural frequency is also included.

Nonlinear free vibration of piezoelectric cylindrical nanoshells

The nonlinear vibration characteristics of the piezoelectric circular cylindrical nanoshells resting on an elastic foundation are analyzed. The small scale effect and thermo-electro-mechanical loading are taken into account. Based on the nonlocal elasticity theory and Donnell's nonlinear shell theory, the nonlinear governing equations and the corresponding boundary conditions are derived by employing Hamilton's principle. Then,the Galerkin method is used to transform the governing equations into a set of ordinary differential equations, and subsequently, the multiple-scale method is used to obtain an approximate analytical solution. Finally, an extensive parametric study is conducted to examine the effects of the nonlocal parameter, the external electric potential, the temperature rise, and the Winkler-Pasternak foundation parameters on the nonlinear vibration characteristics of circular cylindrical piezoelectric nanoshells.

Nanoscale mass sensing based on vibration of single-layered graphene sheet in thermal environments

Based on vibration analysis, single-layered graphene sheet （SLGS） with multiple attached nanoparticles is developed as nanoscale mass sensor in thermal environments. Graphene sensors are assumed to be in simplysupported configuration. Based on the nonlocal plate the- ory which incorporates size effects into the classical theory, closed-form expressions lot the frequencies and relative fre- quency shills of SLGS-based mass sensor are derived using the Galerkin method. The suggested model is justified by a good agreement between the results given by the present model and available data in literature. The effects of tem- perature difference, nonlocal parameter, the location of the nanoparticle and the number of nanoparticles on the relative frequency shift of the mass sensor are also elucidated. The obtained results show that the sensitivity of the SLGS- based mass sensor increases with increasing temperature difference.

Vibration and buckling analyses of nanobeams embedded in an elastic medium

Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient,and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.

Size-Dependent Geometrically Nonlinear Free Vibration of First-Order Shear Deformable Piezoelectric-Piezomagnetic Nanobeams Using the Nonlocal Theory

This article investigates the geometrically nonlinear free vibration of piezoelectric-piezomagnetic nanobeams subjected to magneto-electro-thermal loading taking into account size effect using the nonlocal elasticity theory.To this end,the sizedependent nonlinear governing equations of motion and corresponding boundary conditions are derived according to the nonlocal elasticity theory and the first-order shear deformation theory with von K´arm´an-type of kinematic nonlinearity.The effects of size-dependence,shear deformations,rotary inertia,piezoelectric-piezomagnetic coupling,thermal environment and geometrical nonlinearity are taken into account.The generalized differential quadrature(GDQ)method in conjunction with the numerical Galerkin method,periodic time differential operators and pseudo arclength continuation method is utilized to compute the nonlinear frequency response of piezoelectric-piezomagnetic nanobeams.The influences of various parameters such as non-dimensional nonlocal parameter,temperature change,initial applied electric voltage,initial applied magnetic potential,length-to-thickness ratio and different boundary conditions on the geometrically nonlinear free vibration characteristics of piezoelectric-piezomagnetic nanobeams are demonstrated by numerical examples.It is illustrated that the hardening spring effect increases with increasing the non-dimensional nonlocal parameter,positive initial applied voltage,negative initial applied magnetic potential,temperature rise and decreases with increasing the negative initial applied voltage,positive initial applied magnetic potential and length-tothickness ratio.

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs)are presented.The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness.To estimate the associated effective material properties,various homogenization schemes including the Reuss model,the Voigt model,the Mori-Tanaka model,and the Hashin-Shtrikman bound model are used.The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates.The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem.It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index.Also,by increasing the skew angle,the critical shear buckling load of an FGM skew nanoplate enhances.This pattern becomes a bit less significant for a higher value of the material property gradient index.Furthermore,among various homogenization models,the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads,and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.

Band structures of elastic SH waves in nanoscale multi-layered functionally graded phononic crystals with/without nonlocal interface imperfections by using a local RBF collocation method

A meshless radial basis function （RBF） collocation method based on the Eringen nonlocal elasticity theory is developed to calculate the band structures of ternary and quaternary nanoscale multi-layered phononic crystals （PNCs） with functionally graded （FG） interlayers. Detailed calculations are performed for anti-plane transverse waves propagating in such PNCs. The influences of FG and homogeneous interlayers, component number, nonlocal interface imperfections and nanoscale size on cut-off frequency and band structures are investigated in detail. Numerical results show that these factors have significant effects on band structures at the macroscopic and microscopic scales.

Electromechanical buckling behavior of smart piezoelectrically actuated higher-order size-dependent graded nanoscale beams in thermal environment

In the present work,thermo-electro-mechanical buckling behavior of functionally graded piezoelectric(FGP)nanobeams is investi-gated based on higher-order shear deformation beam theory.The FGP nanobeam is subjected to four types of thermal loading including uniform,linear,and sinusoidal temperature rise as well as heat conduction through the beam thickness.Thermo-electro-mechanical properties of FGP nanobeam are supposed to change continuously in the thickness direction based on power-law model.To consider the influences of small-scale sizes,Eringen’s nonlocal elasticity theory is adopted.Applying Hamilton’s princi-ple,the nonlocal governing equations of an FGP nanobeam in thermal environments are obtained and are solved using Navier-type analytical solution.The significance of various parameters,such as thermal loadings,external electric voltage,power-law index,nonlocal parameter,and slenderness ratio on thermal buck-ling response of size-dependent FGP nanobeams is investigated.

Free Vibration of Elastically Constrained Single-Layered MoS_(2)

Free vibration of elastically constrained rectangular single-layered M0S_(2)is investigated by using a nonlocal Kirchhoff plate model with an initial stress.The variationally consistent elastically constrained boundary conditions are obtained by using the weighted residual method,while the governing equations of the nonlocal Kirchhoff plate model are known.A modified Fourier series method is applied to study the vibrational behaviors of elastically constrained nonlocal Kirchhoff plate models.The convergence and reliability of the modified Fourier series method is verified via comparison with the finite element method.A comprehensive parametric study is performed to show the influences of the boundary elastic constant,nonlocal parameter and initial stress on the vibrational behaviors of single-layered M0S2.The results should be good for the design of nanoresonators.