Nonlinear magneto-thermo-elastic vibration of mass sensor armchair carbon nanotube resting on an elastic substrate

Abstract The present paper aims at studying the nonlinear ultrasonic waves in a magneto-thermo-elastic armchair single-walled (SW) carbon nanotube (CNT) with mass sensors resting on a polymer substrate. The analytical formulation accounts for small scale effects based on the Eringen's nonlocal elasticity theory. The mathematical model and its differential equations are solved theoretically in terms of dimensionless frequencies while assuming a nonlinear Winkler-Pasternak-type foundation. The solution is obtained by means of ultrasonic wave dispersion relations. A parametric work is carried out to check for the effect of the nonlocal scaling parameter, together with the magneto-mechanical loadings, the foundation parameters, the attached mass, boundary conditions and geometries, on the dimensionless frequency of nanotubes. The sensitivity of the mechanical response of nanotubes investigated herein, could be of great interest for design purposes in nano-engineering systems and devices

Nonlinear scale-dependent deformation behaviour of beam and plate structures

Improving the knowledge of the mechanics of small-scale structures is important in many microelectromechanical and nanoelectromechanical systems. Classical continuum mechanics cannot be utilised to determine the mechanical response of small-scale structures, since size effects become significant at small-scale levels. Modified elasticity models have been introduced for the mechanics of ultra-small structures. It has recently been shown that higher-order models, such as nonlocal strain gradient and integral models, are more capable of incorporating scale influences on the mechanical characteristics of small-scale structures than the classical continuum models. In addition, some scaledependent models are restricted to a specific range of sizes. For instance, nonlocal effects on the mechanical behaviour vanish after a particular length. Scrutinising the available literature indicates that the large amplitude vibrations of small-scale beams and plates using two-parameter scaledependent models and nonlocal integral models have not been investigated yet. In addition, no twoparameter continuum model with geometrical nonlinearity has been introduced to analyse the influence of a geometrical imperfection on the vibration of small-scale beams. Analysing these systems would provide useful results for small-scale mass sensors, resonators, energy harvesters and actuators using small-scale beams and plates. In this thesis, scale-dependent nonlinear continuum models are developed for the time-dependent deformation behaviour of beam-shaped structures. The models contain two completely different size parameters, which make it able to describe both the reduction and increase in the total stiffness. The first size parameter accounts for the nonlocality of the stress, while the second one describes the strain gradient effect. Geometrical nonlinearity on the vibrations of small-scale beams is captured through the strain-displacement equations. The small-scale beam is assumed to possess geometrical imperfections. Hamilton’s approach is utilised for deriving the corresponding differential equations. The coupled nonlinear motion equations are solved numerically employing Galerkin’s method of discretisation and the continuation scheme of solution. It is concluded that geometrical imperfections would substantially alter the nonlinear vibrational response of small-scale beams. When there is a relatively small geometrical imperfection in the structure, the small-scale beam exhibits a hardeningtype nonlinearity while a combined hardening- and softening-type nonlinearity is found for beams with large geometrical imperfections. The strain gradient influence is associated with an enhancement in the beam stiffness, leading to higher nonlinear resonance frequencies. By contrast, the stress nonlocality is related to a remarkable reduction in the total stiffness, and consequently lower nonlinear resonance frequencies. In addition, a scale-dependent model of beams is proposed in this thesis to analyse the influence of viscoelasticity and geometrical nonlinearity on the vibration of small-scale beams. A nonlocal theory incorporating strain gradients is used for describing the problem in a mathematical form. Implementing the classical continuum model of beams causes a substantial overestimation in the beam vibrational amplitude. In addition, the nonlinear resonance frequency computed by the nonlocal model is less than that obtained via the classical model. When the forcing amplitude is comparatively low, the linear and nonlinear damping mechanisms predict almost the same results. However, when forcing amplitudes become larger, the role of nonlinear viscoelasticity in the vibrational response increases. The resonance frequency of the scale-dependent model with a nonlinear damping mechanism is lower than that of the linear one. To simulate scale effects on the mechanical behaviour of ultra-small plates, a novel scale-dependent model of plates is developed. The static deflection and oscillation of rectangular plates at small-scale levels are analysed via a two-dimensional stress-driven nonlocal integral model. A reasonable kernel function, which fulfil all necessary criteria, is introduced for rectangular small-scale plates for the first time. Hamilton and Leibniz integral rules are used for deriving the non-classical motion equations of the structure. Moreover, two types of edge conditions are obtained for the linear vibration. The first type is the well-known classical boundary condition while the second type is the nonclassical edge condition associated with the curvature nonlocality. The differential quadrature technique as a powerful numerical approach for implementing complex boundary conditions is used. It is found that while the Laplacian-based nonlocal model cannot predict size influences on the bending of small-scale plates subject to uniform lateral loading, the bending response is remarkably size-dependent based on the stress-driven plate model. When the size influence increases, the difference between the resonance frequency obtained via the stress-driven model and that of other theories substantially increases. Moreover, the resonance frequency is higher when the curvature nonlocality increases due to an enhancement in the plate stiffness. It is also concluded that more constraint on the small-scale plate causes the system to vibrate at a relatively high frequency. In addition to the linear vibration, the time-dependent large deformation of small-scale plates incorporating size influences is studied. The stress-driven theory is employed to formulate the problem at small-scale levels. Geometrical nonlinearity effects are taken into account via von Kármán’s theory. Three types of edge conditions including one conventional and two nonconventional conditions are presented for nonlinear vibrations. The first non-classical edge condition is associated with the curvature nonlocality while the second one is related to nonlocal in-plane strain components. A differential quadrature technique and an appropriate iteration method are used to compute the nonlinear natural frequencies and maximum in-plane displacements. Molecular dynamics simulations are also performed for verification purposes. Nonlinear frequency ratios are increased when vibration amplitudes increase. Furthermore, the curvature nonlocality would cause the small-scale pate to vibrate at a lower nonlinear frequency ratio. By contrast, the nonlocal in-plane strain has the opposite effect on the small-scale system. The outcomes from this thesis will be useful for engineers to design vibrating small-scale resonators and sensors using ultra-small plates.Thesis (Ph.D.) -- University of Adelaide, School of Mechanical Engineering, 202

Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1

Computational hygro-thermal vibration and buckling analysis of functionally graded sandwich microbeams

In this study, for the first time, hygro-thermal behaviour of functionally graded (FG) sandwich microbeams based on nonlocal elasticity theory is investigated. Temperature-dependent material properties are considered for the FG microbeam, which are assumed to change continuously through the thickness based on the power-law form. The equations of motion are obtained on the basis of first-order shear deformation beam theory via Hamilton's principle. The size effects are considered in the framework of the nonlocal elasticity theory of Eringen. The detailed variational and finite element procedure for FG sandwich microbeams are presented with a five-noded beam element and numerical examinations are performed. The influence of several parameters such as temperature and moisture gradients, material graduation, nonlocal parameter, face-core-face and span to depth ratios on the critical buckling temperature and the nondimensional fundamental frequencies of the FG sandwich microbeams are analysed. Based on the results of this study, temperature and moisture rise soften the FG sandwich microbeam and result in the reduction of the critical buckling load and vibration frequency. In addition, the FG sandwich microbeam with a thicker ceramic core can resist higher temperature and moisture gradients

Vibrating nonlocal multi-nanoplate system under inplane magnetic field

The recent development in nanotechnology resulted in growing of various nanoplate like structures. High attention was devoted to graphene sheet nanostructure, which enforced the scientist to start developing various theoretical models to investigate its physical properties. Magnetic field effects on nanoplates, especially graphene sheets, have also attracted a considerable attention of the scientific community. Here, by using the nonlocal theory, we examine the influence of in-plane magnetic field on the viscoelastic orthotropic multi-nanoplate system (VOMNPS) embedded in a viscoelastic medium. We derive the system of m partial differential equations describing the free transverse vibration of VOMNPS under the uniaxial in-plane magnetic field using the Eringen's nonlocal elasticity and Kirchhoff's plate theory considering the viscoelastic and orthotropic material properties of nanoplates. Closed form solutions for complex natural frequencies are derived by applying the Navier's and trigonometric method for the case of simply supported nanoplates. The results obtained with analytical method are validated with the results obtained by using the numerical method. In addition, numerical examples are given to show the effects of nonlocal parameter, internal damping, damping and stiffness of viscoelastic medium, rotary inertia and uniaxial in-plane magnetic force on the real and the imaginary parts of complex natural frequencies of VOMNPS. This study can be useful as a starting point for the research and design of nanoelectromechanical devices based on graphene sheets

Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions

none5siThis work aims at investigating the free torsional vibration of one-directional nanostructures with an elliptical shape, under different boundary conditions. The equation of motion is derived from Hamilton’s principle, where Eringen’s nonlocal theory is applied to analyze the small-scale effects. The analytical Galerkin method is employed to rewrite the equation of motion as an ordinary differential equation (ODE). After a preliminary validation check of the proposed formulation, a systematic study investigates the influence of the nonlocal parameters, boundary conditions, geometrical and mechanical parameters on the natural frequency of nanorods; the objective is to provide useful findings for design and optimization purposes of many nanotechnology applications, such as, nanodevices, actuators, sensors, rods, nanocables, and nanostructured aerospace systems.openFarshad Khosravi; Seyyed Amirhosein Hosseini; Babak Alizadeh Hamidi; Rossana Dimitri; Francesco TornabeneKhosravi, Farshad; Amirhosein Hosseini, Seyyed; Alizadeh Hamidi, Babak; Dimitri, Rossana; Tornabene, Francesc