In recent years, mechanical characteristics including vibration, bending, buckling and postbuckling, stability and instability, etc. of small-scaled structures (such as micro/nanowires, micro and nano plates, micro and nano tubes, micro and nano beams, carbon micro/nano-tubes, micro and nano shells, micro and nano probes, etc.) have been broadly investigated, because of recent advances in nanotechnologies and due to their astonishing characteristics. Also, these small-scaled structures are extensively applied in several areas. However, comprehending the mechanical characteristics of these small-scaled structures is of great prominence and yet a daunting task. To understand the mechanical behaviors and capture the effects (including small-scale effect) of these different advanced small-scaled structures (at micro and nano levels), it is very essential to implement them into classical/nonclassical continuum elasticity based on a suitable classical/nonclassical elasticity formulation or theory. Thus, this work searches the literature and hereby presents an ample literature review study with a special focus primarily on the novel progress of vibration analyses of microplates and nanoplates using nonlocal/nonclassical continuum theories of elasticity. Scopus and web of science databases were used as the primary scholarly-databases to acquire the documents reviewed in the current research. Furthermore, several perspectives of classifications are considered while reviewing and grouping these acquired documents. The main purpose was to illustrate existing research trends in the vibrational analysis of small-scaled plate-based structures utilizing nonlocal plate-based theories and nonclassical continuum elasticity theories, and to provide a foundation and guidance for future research on the mechanics of small-scaled plate-based structures. 

A comprehensive review on the vibration analyses of small-scaled plate-based structures by utilizing the nonclassical continuum elasticity theories

Parametric vibrations of the single-layered graphene sheet (SLGS) are studied in the presented work. The equations of motion govern geometrically nonlinear oscillations. The appearance of small effects is analysed due to the application of the nonlocal elasticity theory. The approach is developed for rectangular simply supported small-scale plate and it employs the Bubnov–Galerkin method with a double mode model, which reduces the problem to investigation of the system of the second-order ordinary differential equations (ODEs). The dynamic behaviour of the micro/nanoplate with varying excitation parameter is analysed to determine the chaotic regimes. As well the influence of small-scale effects to change the nature of vibrations is studied. The bifurcation diagrams, phase plots, Poincare sections and the largest Lyapunov exponent are constructed and analysed. It is established that the use of nonlocal equations in the dynamic analysis of graphene sheets leads to a significant alteration in the character of oscillations, including the appearance of chaotic attractors.

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Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory

In this study, analytical solution is presented to calculate the free vibration frequencies of nonuniform microbeams. Scale effects are modelled using modified couple stress theory and the microbeam is assumed to be thin while Poisson's ratio effects are also taken into account. Nonuniformity is presented by exponentially varying width among the microbeam while the thickness remains constant. Results are presented for simply-supported, cantilever and clamped boundary conditions. First five natural frequency parameter are calculated for different scale and nonuniformity parameters and effects of each parameter on the results are discussed. Also, to understand the effects of Poisson's ratio, small scale and nonuniformity on the first frequency of the nonuniform microbeam and resonance domain, a comprehensive parametric study is done. This research is important in understanding the dynamic behavior of microbeams and effective designs using variable cross section in this type of microstructures.

https://www.ije.ir/article_72890_626e793a5d1a730fb34d270dfe3fcf79.pdf

Free Vibration Analysis of Nonuniform Microbeams Based on Modified Couple Stress Theory: an Analytical Solution


 Nonlinear vibrations of the orthotropic nanoplates subjected to an influence of in-plane magnetic field are considered. The model is based on the nonlocal elasticity theory. The governing equations for geometrically nonlinear vibrations use the von Kármán plate theory. Both the stress formulation and the Airy stress function are employed. The influence of the magnetic field is taken into account due to the Lorentz force yielded by Maxwell's equations. The developed approach is based on applying the Bubnov–Galerkin method and reducing partial differential equations to an ordinary differential equation. The effect of the magnetic field, elastic foundation, nonlocal parameter, and plate aspect ratio on the linear frequencies and the nonlinear ratio is illustrated and discussed.

Nonlinear Vibrations of Embedded Nanoplates Under In-Plane Magnetic Field Based on Nonlocal Elasticity Theory

Porous materials are used extensively to manufacture beams, plates, and shells, and have been studies from different perspectives. Composite porous plates in various shapes and dimensions have nume...

Free vibration analysis of a composite elliptical plate made of a porous core and two piezoelectric layers

In this article, classical plate theory (CPT) is reformulated using the nonlocal differential constitutive relations of Eringen to develop an equivalent continuum model for orthotropic triangular nanoplates. The equations of motion are derived and the Galerkin’s approach in conjunction with the area coordinates is used as a basis for the solution. Nonlocal theories are employed to bring out the effect of the small scale on natural frequencies of nano scaled plates. Effect of nonlocal parameter, lengths of the nanoplate, aspect ratio, mode number, material properties, boundary condition and in-plane loads on the natural frequencies are investigated. It is shown that the natural frequencies depend highly on the non-locality of the nanoplate, especially at the very small dimensions, higher mode numbers and stiffer edge condition.

http://jsm.iau-arak.ac.ir/article_524396_7cecd41820c10e59eb055588ab0559a8.pdf

Vibration analysis of orthotropic triangular nanoplates using nonlocal elasticity theory and galerkin method

Mistuning in cyclic symmetric systems increases severely the forced response of system and splits the modes. This paper concerns with nonlinear behavior of mistuned cyclic systems. A nonlinear, mistuned model based on the method of multiple scales is proposed and formulated in which nonlinearity and mistuning parameter is assumed to be in of low order. Next, two mistuned systems were considered and solved by the multiple scale technique. Numerical results demonstrate that mistuning can lead to repeating and scattering of jump phenomena during the excitation frequency whereas in tuned cyclic system it occurs simultaneously (synchronously).

/pdf/effects-of-mistuning-on-dynamic-behavior-of-nonlinear-cyclic-1ob1dj7ki6.pdf

Effects of mistuning on dynamic behavior of nonlinear cyclic systems with lump masses and cubic nonlinearity

In the present study, small scale effect on critical buckling loads of triangular nano- composite plates under uniform in-plane compression is studied. Since at nano-scale the structure of the plate is discrete, the size dependent nonlocal elasticity theory is employed to develop an equivalent continuum plate model for this nanostructure incorporating the changes in its mechanical behavior. The two-parameter, Winkler-Pasternak, elastic medium is used for precisely modelling the behavior of matrix surrounding the nano-plate. The governing stability equations are then derived using the classical plate theory and the principle of virtual work for a perfect uniform triangular nano-plate composite. The well-known numerical Galerkin method in conjunction with the areal coordinates system is used as the basis for the solution. The solution procedure views the entire nano-composite plate as a single super-continuum element. Effects of nonlocal parameter, length, aspect ratio, mode number, anisotropy, edge supports and elastic medium on the buckling loads are rigorously investigated. All of these parameters are seen to have significant effect on the stability of nano-composite plate. It is shown that the results depend obviously on the non-locality of buckled nano-composite plate, especially at very small dimensions, small aspect ratios, higher mode numbers, higher anisotropy degree, stiffer edge supports and softer elastic mediums. Also, it is seen that the classical continuum mechanics overestimates the buckling results which can lead to deficient design and analysis of these widely used nanostructures.

https://www.ije.ir/article_72692_dd162a8f80206b0c8debeeaa05e90e42.pdf

E. Raeisi Estabragh

Papers

Vibration analysis of orthotropic triangular nanoplates using nonlocal elasticity theory and galerkin method

Effects of mistuning on dynamic behavior of nonlinear cyclic systems with lump masses and cubic nonlinearity

Nonlocal Effect on Buckling of Triangular Nano-composite Plates