Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis

This study aims to investigate the postbuckling response of functionally graded (FG) nanoplates by using the nonlocal elasticity theory of Eringen to capture the size effect. In addition, Reddy’s third-order shear deformation theory is adopted to describe the kinematic relations, while von Kaman’s assumptions are used to account for the geometrical nonlinearity. In order to calculate the effective material properties, the Mori-Tanaka scheme is adopted. Governing equations are derived based on the principle of virtual work. Isogeometric analysis (IGA) is employed as a discretization tool, which is able to satisfy the C1-continuity demand efficiently. The Newton-Raphson iterative technique with imperfection is employed to trace the postbuckling paths. Various numerical studies are carried out to examine the influences of gradient index, nonlocal effect, ratio of compressive loads, boundary condition, thickness ratio and aspect ratio on the postbuckling behaviour of FG nanoplates

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

Buckling Analysis of CNTRC Curved Sandwich Nanobeams in Thermal Environment

none6siThis paper presents a mathematical continuum model to investigate the static stability buckling of cross-ply single-walled (SW) carbon nanotube reinforced composite (CNTRC) curved sandwich nanobeams in thermal environment, based on a novel quasi-3D higher-order shear deformation theory. The study considers possible nano-scale size effects in agreement with a nonlocal strain gradient theory, including a higher-order nonlocal parameter (material scale) and gradient length scale (size scale), to account for size-dependent properties. Several types of reinforcement material distributions are assumed, namely a uniform distribution (UD) as well as X- and O- functionally graded (FG) distributions. The material properties are also assumed to be temperature-dependent in agreement with the Touloukian principle. The problem is solved in closed form by applying the Galerkin method, where a numerical study is performed systematically to validate the proposed model, and check for the effects of several factors on the buckling response of CNTRC curved sandwich nanobeams, including the reinforcement material distributions, boundary conditions, length scale and nonlocal parameters, together with some geometry properties, such as the opening angle and slenderness ratio. The proposed model is verified to be an effective theoretical tool to treat the thermal buckling response of curved CNTRC sandwich nanobeams, ranging from macroscale to nanoscale, whose examples could be of great interest for the design of many nanostructural components in different engineering applications.openAhmed Amine Daikh; Mohammed Sid Ahmed Houari; Behrouz Karami; Mohamed A. Eltaher; Rossana Dimitri; Francesco TornabeneAmine Daikh, Ahmed; Sid Ahmed Houari, Mohammed; Karami, Behrouz; Eltaher, Mohamed A.; Dimitri, Rossana; Tornabene, Francesc

Thermal buckling of functionally graded piezomagnetic micro- and nanobeams presenting the flexomagnetic effect

Galerkin weighted residual method (GWRM) is applied and implemented to address the axial stability and bifurcation point of a functionally graded piezomagnetic structure containing flexomagneticity in a thermal environment. The continuum specimen involves an exponential mass distributed in a heterogeneous media with a constant square cross section. The physical neutral plane is investigated to postulate functionally graded material (FGM) close to reality. Mathematical formulations concern the Timoshenko shear deformation theory. Small scale and atomic interactions are shaped as maintained by the nonlocal strain gradient elasticity approach. Since there is no bifurcation point for FGMs, whenever both boundary conditions are rotational and the neutral surface does not match the mid-plane, the clamp configuration is examined only. The fourth-order ordinary differential stability equations will be converted into the sets of algebraic ones utilizing the GWRM whose accuracy was proved before. After that, by simply solving the achieved polynomial constitutive relation, the parametric study can be started due to various predominant and overriding factors. It was found that the flexomagneticity is further visible if the ferric nanobeam is constructed by FGM technology. In addition to this, shear deformations are also efficacious to make the FM detectable

Nonlinear Dynamics of Silicon Nanowire Resonator Considering Nonlocal Effect

In this work, nonlinear dynamics of silicon nanowire resonator considering nonlocal effect has been investigated. For the first time, dynamical parameters (e.g., resonant frequency, Duffing coefficient, and the damping ratio) that directly influence the nonlinear dynamics of the nanostructure have been derived. Subsequently, by calculating their response with the varied nonlocal coefficient, it is unveiled that the nonlocal effect makes more obvious impacts at the starting range (from zero to a small value), while the impact of nonlocal effect becomes weaker when the nonlocal term reaches to a certain threshold value. Furthermore, to characterize the role played by nonlocal effect in exerting influence on nonlinear behaviors such as bifurcation and chaos (typical phenomena in nonlinear dynamics of nanoscale devices), we have calculated the Lyapunov exponents and bifurcation diagram with and without nonlocal effect, and results shows the nonlocal effect causes the most significant effect as the device is at resonance. This work advances the development of nanowire resonators that are working beyond linear regime

On a 3D material modelling of smart nanocomposite structures

Smart composites (SCs) are utilized in electro-mechanical systems such as actuators and energy harvesters. Typically, thin-walled components such as beams, plates, and shells are employed as structural elements to achieve the mechanical behavior desired in these composites. SCs exhibit various advanced properties, ranging from lower order phenomena like piezoelectricity and piezomagneticity, to higher order effects including flexoelectricity and flexomagneticity. The recently discovered flexomagneticity in smart composites has been investigated under limited conditions. A review of the existing literature indicates a lack of evaluation in three-dimensional (3D) elasticity analysis of SCs when the flexomagnetic effect (FM) exists. To address this issue, the governing equations will incorporate the term ∂/∂z, where z represents the thickness coordinate. The variational technique will guide us in further developing these governing equations. By using hypotheses and theories such as a 3D beam model, von Kármán's strain nonlinearity, Hamilton's principle, and well-established direct and converse FM models, we will derive the constitutive equations for a thick composite beam. Conducting a 3D analysis implies that the strain and strain gradient tensors must be expressed in 3D forms. The inclusion of the term ∂/∂z necessitates the construction of a different model. It should be noted that current commercial finite element codes are not equipped to accurately and adequately handle micro- and nano-sized solids, thus making it impractical to model a flexomagnetic composite structure using these programs. Therefore, we will transform the derived characteristic linear three-dimensional bending equations into a 3D semi-analytical Polynomial domain to obtain numerical results. This study demonstrates the importance of conducting 3D mechanical analyses to explore the coupling effects of multiple physical phenomena in smart structures

Nonlocal nonlinear mechanics of imperfect carbon nanotubes

In this article, for the first time, a coupled nonlinear model incorporating scale influences is presented to simultaneously investigate the influences of viscoelasticity and geometrical imperfections on the nonlocal coupled mechanics of carbon nanotubes; large deformations, stress nonlocality and strain gradients are captured in the model. The Kelvin-Voigt model is also applied in order to ascertain the viscoelasticity effects on the mechanics of the initially imperfect nanoscale system. The modified coupled equations of motion are then derived via the Hamilton principle. A solution approach for the derived coupled equations is finally developed applying a decomposition-based procedure in conjunction with a continuation-based scheme. The significance of many parameters such as size parameters, initial imperfections, excitation parameters and linear and nonlinear damping effects in the nonlinear mechanical response of the initially imperfect viscoelastic carbon nanotube is assessed. The present results can be useful for nanoscale devices using carbon nanotubes since the viscoelasticity and geometrical imperfection are simultaneously included in the proposed model

A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition

A drawback to the material composition of thick functionally graded materials (FGM) beams is checked out in this research in conjunction with a novel hyperbolic-polynomial higher-order elasticity beam theory (HPET). The proposed beam model consists of a novel shape function for the distribution of shear stress deformation in the transverse coordinate. The beam theory also incorporates the stretching effect to present an indirect effect of thickness variations. As a result of compounding the proposed beam model in linear Lagrangian strains and variational of energy, the system of equations is obtained. The Galerkin method is here expanded for several edge conditions to obtain elastic critical buckling values. First, the importance of the higher-order beam theory, as well as stretching effect, is assessed in assorted tabulated comparisons. Next, with validations based on the existing and open literature, the proposed shape function is evaluated to consider the desired accuracy. Some comparative graphs by means of well-known shape functions are plotted. These comparisons reveal a very good compliance. In the final section of the paper, based on an inappropriate mixture of the SUS304 and Si3C4 as the first type of FGM beam (Beam-I) and, Al and Al2O3 as the second type (Beam-II), the results are pictured while the beam is kept in four states, clamped–clamped (C–C), pinned–pinned (S–S), clamped-pinned (C–S) and in particular cantilever (C–F). We found that the defect impresses markedly an FGM beam with boundary conditions with lower degrees of freedom

VIBRATION AND STABILITY OF A NONLINEAR NONLOCAL STRAIN-GRADIENT FG BEAM ON A VISCO-PASTERNAK FOUNDATION

This study investigates the stability of periodic solutions of a nonlinear nonlocal strain gradient functionally graded Euler–Bernoulli beam model resting on a visco-Pasternak foundation and subjected to external harmonic excitation. The nonlinearity of the beam arises from the von Kármán strain-displacement relation. Nonlocal stress gradient theory combined with the strain gradient theory is used to describe the stress-strain relation. Variations of material properties across the thickness direction are defined by the power-law model. The governing differential equation of motion is derived by using Hamilton's principle and discretized by the Galerkin approximation. The methodology for obtaining the steady-state amplitude-frequency responses via the incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the numerical integration method and stability analysis is performed by utilizing the Floquet theory