Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method

This paper addresses the elastic buckling and vibration characteristics of isotropic and orthotropic nanoplates using finite strip method. In order to consider small scale effect, Eringen’s nonlocal continuum elasticity is employed. The governing nanoplate equations are derived using the principle of virtual work while B3-spline finite strip method is applied to the buckling and vibration analyses. The buckling load and vibration frequency of graphene sheets, which are subjected to biaxial compression and pure shear loading, are determined whilst the effects of different parameters such as sheet size, nonlocal parameter, aspect ratio and boundary conditions are investigated. The interaction curves of the critical biaxial compression loading as well as the interaction curves of the critical uniaxial compression and shear loading are also obtained. It is shown that small scale effect plays considerable role in the analysis of small sizes plates.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.     

A general nonlocal nonlinear model for buckling of nanobeams

This study presents a unified model for the nonlocal response of nanobeams in buckling and postbuckling states. The formulation is suitable for the classical Euler–Bernoulli, first-order Timoshenko, and higher-order shear deformation beam theories. The small-scale effect is modeled according to the nonlocal elasticity theory of Eringen. The equations of equilibrium are obtained using the principle of virtual work. The stress resultants are developed taking into account the nonlocal effect. Analytical solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state are obtained. It is found out that as the nonlocal parameter increases, the critical buckling load reduces and the amplitude of buckling increases. Numerical results showing variation of the critical buckling load and the amplitude of buckling with the nonlocal parameter and the length-to-height ratio for simply supported and clamped–clamped nanobeams are presented.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Vibration monitoring of cylindrical shells using piezoelectric sensors

From linear vibration theory for beams and plates, one can express the response as a linear combination of its natural modes. For beams, these eigenfunctions can be shown to be mutually orthogonal for any boundary conditions. For plates, orthogonality of the modes is not guaranteed, but can be proven for various boundary conditions. Modal analysis for beams and plates allows the system response to be broken down into simpler vibration models, due to the orthogonality of the modes. Here the modal analysis approach is extended to the vibration of thin cylindrical shells. The longitudinal, radial, and circumferential displacements are coupled with each other, due to Poisson's ratio and the curvature of the shell. As will be shown, the mode shapes can be solved analytically with numerically determined coefficients. The immediate application of this work will be for modal sensing of cylindrical shell vibrations using thin piezoelectric films.     

Electro-thermo-mechanical torsional buckling of a piezoelectric polymeric cylindrical shell reinforced by DWBNNTs with an elastic core

The effect of partially filled poly ethylene (PE) foam core on the behavior of torsional buckling of an isotropic, simply supported piezoelectric polymeric cylindrical shell made from polyvinylidene fluoride (PVDF), and subjected to combined electro-thermo-mechanical loads has been analyzed using energy method. The shell is reinforced by armchair double walled boron nitride nanotubes (DWBNNTs). The core is modeled as an elastic environment containing Winkler and Pasternak modules. Using representative volume element (RVE) based on micromechanical modeling, mechanical, electrical and thermal characteristics of the equivalent composite were determined. Critical buckling load is calculated using strains based on Donnell theory, the coupled electro-thermo-mechanical governing equations and principle of minimum potential energy. The results indicate that buckling strength increases substantially as harder foam cores are employed i.e. as E_c/E_s is increased. The most economic in-fill foam core is at η = 0.6, as cost increases without much significant improvement in torsional buckling at higher η’s.     

Surface and nonlocal effects on response of linear and nonlinear NEMS devices

Nonlocal and surface effects become important for nanoscale devices. To model these effects on frequency response of linear and nonlinear nanobeam subjected to electrostatic excitation, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. Subsequently, we apply Galerkin’s method with exact mode shape including nonlocal and surface effects to get static and dynamic modal equations. After validating the procedure with the available results, we analyze the variation of pull-in voltage and frequency resonance by varying surface and nonlocal parameters. To do frequency analysis of nonlinear system, we solve nonlinear dynamic equation using the method of multiple scale. We found that the frequency response of nonlinear system reduces for fixed excitation as the surface and nonlocal effects increase. Also, we found that the nature of nonlinearity can be tuned from hardening to softening by increasing the nonlocal effects.     

Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models

In this paper, the size-effects in the torsional and axial response of microtubules by using the nonlocal continuum rod model is investigated. To this end, continuous and discrete rod models are performed for modeling of microtubules. A simple finite element procedure is used for modeling and solution of nonlocal discrete system equation for microtubules. The influence of the small length scale on the vibration frequencies is examined both torsional and axial vibration cases. Some parametric results are also presented for examination of the accuracy and performances of discrete and continuous models.     

A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams

An analytical approach for static bending and buckling analyses of curved nanobeams using the differential constitutive law, consequent to Eringen’s strain-driven integral model coupled with a higher-order shear deformation accounting for through thickness stretching is presented. The formulation is general in the sense that it can be deduced to examine the influence of different structural theories, for static and dynamic analyses of curved nanobeams. The governing equations derived using Hamiltons principle are solved in conjunction with Naviers solutions. The formulation is validated considering problems for which solutions are available. A comparative study is made here by various theories obtained through the formulation. The effects various structural parameters such as thickness ratio, beam length, rise of the curved beam, and nonlocal scale parameter are brought out on bending and stability characteristics of curved nanobeams.     

Active control of dynamic behaviors of graded graphene reinforced cylindrical shells with piezoelectric actuator/sensor layers

This work investigates the active vibration control and vibration characteristics of a sandwich thin cylindrical shell whose intermediate layer is made of the graphene reinforced composite that is bonded with integrated piezoelectric actuator and sensor layers at its outer and inner surfaces. The volume fraction of graphene platelets in the intermediate layer varies continuously in the shell's thickness direction, which generates position-dependent effective material properties. The constitutive relations of the graphene reinforced composite and piezoelectric materials are given by taking one-dimensional steady thermal field into account. Considering Donnell's shell theory, a final equation of motion in terms of the generalized radial displacement is derived by using Hamilton's principle and Galerkin method. Shell's natural frequencies are derived considering influences of the thermo-electro-elastic field. Introducing a constant velocity feedback control algorithm, active vibration control of the sandwich cylindrical shell is presented by employing the Runge-Kutta method. The feedback control gain has a pronounced effect on the damping, as well as the inertia of the system. Comparisons between the present results and those in other papers are done to validate the present solutions. Influences of weight fractions, distribution patterns and geometrical sizes of graphene platelets, temperature variations, thicknesses of layers and the feedback control gain on the vibration characteristics and active vibration control behaviors of the novel sandwich cylindrical shell are discussed.     

Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory

Thermal buckling of nanocolumns considering nonlocal effect and shear deformation is investigated based on the nonlocal elasticity theory and the Timoshenko beam theory. By expressing the nonlocal stress as nonlinear strain gradients and based on the variational principle and von Kármán nonlinearity, new higher-order differential governing equations with corresponding higher-order nonlocal boundary conditions both in transverse and axial directions for instability of nanocolumns are derived. New analytical solutions for some practical examples on instability of nanocolumns are presented and analyzed in detail. The paper concluded that the critical buckling load is significantly increased in the presence of nonlocal stress and the results confirm that nanocolumn stiffness is enhanced by nanoscale size effect and reduced by shear deformation. The critical temperature change is increased with larger diameter to length ratio and higher nonlocal nanoscale. It is also concluded that at low and room temperatures the buckling load of nanocolumns increases with increasing temperature change, while at high temperature the buckling load decreases with increasing temperature change.     

Non-linear buckling for the surface rectangular delamination of laminated piezoelectric shells

An analytical method is presented to study the non-linear buckling characteristic of rectangular local delamination near the surface of fiber-reinforced piezoelectric lamination shells under coupled mechanical and electric loads. The stacking sequence of fiber reinforced lamination shells with piezoelectric layers is considered as symmetry, but the stacking sequence of rectangular local delamination sub-shells is arbitrary. Based on the nonlinear displacement mode of delaminated sub-shells, the effects of electric fields, the geometrical, physical parameters and stacking sequences of piezoelectric laminated shells on the non-linear local buckling for rectangular delamination near the surface of piezoelectric laminated base-shells are solved.     

Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory

This paper studies the electro-mechanical shear buckling analysis of piezoelectric nanoplate using modified couple stress theory with various boundary conditions.In order to be taken electric effects into account, an external electric voltage is applied on the piezoelectric nanoplate. The simplified first order shear deformation theory (S-FSDT) has been employed and the governing differential equations have been obtained using Hamilton's principle and nonlinear strains of Von-Karman. The modified couple stress theory has been applied to considering small scale effects. An analytical approach was developing to obtain exact results with various boundary conditions. After all, results have been presented by change in some parameters, such as; aspect ratio, effect of various boundary conditions, electric voltage and length scale parameter influences. At the end, results showed that the effect of external electric voltage on the critical shear load occurring on the piezoelectric nanoplate is insignificant.     

Instability of travelling wave solutions of a population model with nonlocal effects

The authors study a single-species population model in the formof a scalar reaction-diffusion equation incorporating a timedelay which, because of the assumption that the animals aremoving, leads to an integral term in both space and time. Ina previous paper, it was shown that small-amplitude periodictravelling wave solutions of the equation arise via bifurcationfrom a uniform steady state. In this paper, it is shown, usinga multiscale perturbation expansion, that these solutions areunstable. Numerical evidence suggesting in certain cases theexistence of large-amplitude steady solutions is also presented.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

Graphene-based mass sensors: Chaotic dynamics analysis using the nonlocal strain gradient model

Nonlinear oscillations of graphene resonators are unavoidable due to enhancing the mass sensitivity of graphene-based mass sensors and the nonlinear behavior of the systems provides the route to chaos. In this paper, the nonlinear and chaotic behavior of the graphene-based mass sensor is investigated. The nano-mechanical sensor includes an electrostatically actuated fully clamped single-graphene sheet as a nano-resonator with an attached concentrated mass. By neglecting the rotary inertia, the equation of motion of the nano- resonator and the attached mass is derived using the nonlocal strain gradient theory of elasticity. The nano-resonator is modeled as a Kirchhoff nano-plate with the von Kármán-type geometric nonlinearity. Applying the Galerkin decomposition method to the partial differential equation of motion leads to the ordinary differential equation. Based on the Melnikov's integral method two analytical criteria are derived which provide necessary conditions that determine the chaotic region of the system. The chaotic dynamics of the system are also scrutinized and verified through plotting the Lyapunov exponent diagram, phase plane trajectories and Poincaré maps.     

Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory

Graphene-polymer nano-composites are one of the most applicable engineering nanostructures with superior mechanical properties. In the present study, a finite element (FE) approach based on the size dependent nonlocal elasticity theory is developed for buckling analysis of nano-scaled multi-layered graphene sheets (MLGSs) embedded in polymer matrix. The van der Waals (vdW) interactions between the graphene layers and graphene-polymer are simulated as a set of linear springs using the Lennard-Jones potential model. The governing stability equations for nonlocal classical orthotropic plates together with the weighted residual formulation are employed to explicitly obtain stiffness and buckling matrices for a multi-layered super element of MLGS. The accuracy of the current finite element analysis (FEA) is approved through a comparison with molecular dynamics (MD) and analytical solutions available in the literature. Effects of nonlocal parameter, dimensions, vdW interactions, elastic foundation, mode numbers and boundary conditions on critical in-plane loads are investigated for different types of MLGS. It is found that buckling loads of MLGS are generally of two types namely In-Phase (INPH) and Out-of-Phase (OPH) loads. The INPH loads are independent of interlayer vdW interactions while the OPH loads depend on vdW interactions. It is seen that the decreasing effect of nonlocal parameter on the OPH buckling loads dwindles as the interlayer vdW interactions become stronger. Also, it is found that the small scale and polymer substrate have noticeable effects on the buckling loads of embedded MLGS.     

Data-driven estimation of atomistic support for continuum stress using the Gaussian mixture model

The notion of stress being an inherent continuum concept has been a matter of discussion at the atomistic level. The atomistic stress measure at a given spatial position contains a space averaging volume over nearby atoms to provide an averaged macroscopic stress measure. Previous work on atomistic stress measures introduce the characteristic length as an a priori given parameter. In this contribution we learn the characteristic length directly from the atomistic data itself. Central to our proposed approach is the grouping of atoms with highly similar values of position and stress into the same atomistic sub-population. We hypothesise that atoms with similar values for position and stress are those atoms which harbour the greatest influence over each other and therefore should be contained within the same space averaging volume. Consequently the characteristic length can be computed directly from the discovered sub-populations by averaging over the maximum extent of each sub-population. We motivate the Gaussian mixture model (GMM) as a principled probabilistic method of estimating the similarity between atoms within position-stress space. The GMM parameters are learnt from the atomistic data using the Expectation Maximization (EM) algorithm. To form a parsimonious representation of the dataset we regularise our model using the Bayesian Information Criterion (BIC) which maintains a balance between too few and too many atomistic sub-populations. We use the GMM to segment the atoms into homogeneous sub-populations based on the probability of each atom belonging to a particular sub-population. Thorough evaluation is conducted on a numerical example of an edge dislocation in a single crystal. We derive estimates of the space averaging volume which are in very close agreement to the corresponding analytical solution. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Barlow-wu continuum systems: A discussion on the model

Summary Barlow-Wu continuum structure functions have been introduced in the past as one particularly interesting family of continuum structure functions. In this paper we provide an alternative characterization for such continuum structure functions, showing other interesting properties. Research supported by Dirección General de Investigación Científica y Técnica (DGICYT), national grants number PB91-0389 and number BE91-225.