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

2020 ◽  
Vol 15 (12) ◽  
Author(s):  
Olga Mazur ◽  
Jan Awrejcewicz
Keyword(s):  
Magnetic Field ◽  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Plate Aspect Ratio ◽  
Account Due ◽  
The Magnetic Field

Abstract 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.

scholarly journals Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Scale Effects ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Phase Portraits ◽  
Small Scale ◽  
Governing System ◽  
Account Due

AbstractIn this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

Wave propagation analysis of size-dependent rotating inhomogeneous nanobeams based on nonlocal elasticity theory

2017 ◽  
Vol 24 (17) ◽  
pp. 3809-3818 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Mohammad Reza Barati ◽  
Parisa Haghi
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Wave Dispersion ◽  
Functionally Graded ◽  
Wave Number ◽  
Graded Material ◽  
Fg Nanobeam

The present research deals with the wave dispersion behavior of a rotating functionally graded material (FGMs) nanobeam applying nonlocal elasticity theory of Eringen. Material properties of rotating FG nanobeam are spatially graded according to a power-law model. The governing equations as functions of axial force due to centrifugal stiffening and displacements are obtained by employing Hamilton’s principle based on the Euler–Bernoulli beam theory. By using an analytical model, the dispersion relations of the FG nanobeam are derived by solving an eigenvalue problem. Numerical results clearly show that various parameters, such as angular velocity, gradient index, wave number and nonlocal parameter, are significantly effective to characteristics of wave propagations of rotating FG nanobeams. The results can be useful for next generation study and design of nanomachines, such as nanoturbines, nanoscale molecular bearings and nanogears, etc.

On the application of the partition of unity method for nonlocal response of low-dimensional structures

2014 ◽  
Vol 23 (5-6) ◽  
pp. 153-168 ◽  
Author(s):  
Sundararajan Natarajan
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Gradient Elasticity ◽  
Partition Of Unity ◽  
Nonlocal Elasticity Theory ◽  
B Splines ◽  
Plate Aspect Ratio ◽  
Gradient Elasticity Theory ◽  
Low Dimensional

AbstractThe main objectives of the paper are to (1) present an overview of nonlocal integral elasticity and Aifantis gradient elasticity theory and (2) discuss the application of partition of unity methods to study the response of low-dimensional structures. We present different choices of approximation functions for gradient elasticity, namely Lagrange intepolants, moving least-squares approximants and non-uniform rational B-splines. Next, we employ these approximation functions to study the response of nanobeams based on Euler-Bernoulli and Timoshenko theories as well as to study nanoplates based on first-order shear deformation theory. The response of nanobeams and nanoplates is studied using Eringen’s nonlocal elasticity theory. The influence of the nonlocal parameter, the beam and the plate aspect ratio and the boundary conditions on the global response is numerically studied. The influence of a crack on the axial vibration and buckling characteristics of nanobeams is also numerically studied.

Nonlinear Vibration Analysis of Nano-Disks Based on Nonlocal Elasticity Theory Using Homotopy Perturbation Method

2019 ◽  
Vol 11 (02) ◽  
pp. 1950011 ◽  
Author(s):  
Mohammad Shishesaz ◽  
Mojtaba Shariati ◽  
Amin Yaghootian ◽  
Ali Alizadeh
Keyword(s):  
Perturbation Method ◽  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Scale Effects ◽  
Plate Theory ◽  
Free Field ◽  
Nonlocal Elasticity Theory ◽  
Polar Coordinates ◽  
Small Scale ◽  
Classical Plate Theory

This paper introduces a novel approach for small-scale effects on nonlinear free-field vibration of a nano-disk using nonlocal elasticity theory. The formulation of a nano-disk is based on the nonlinear model of von Kármán strain in polar coordinates and classical plate theory. To analyze the nonlinear geometric and small-scale effects, the differential equation based on nonlocal elasticity theory was extracted from Hamilton principle, while the inertial and shear-stress effects were neglected. The equation of motion was discretized using the Galerkin method on selecting an appropriate function based on the boundary condition used for the nano-disk. Due to presence of nonlinear terms, the homotopy method was used in conjunction with the perturbation method (HPM) to ease up the solution and completely solve the problem. For further comparison, the nonlinear equations were solved by the fourth-order Runge–Kutta method, the solution of which was compared with that of HPM. Excellent agreements in results were observed between the two methods, indicating that the latter method can simplify the solution, and hence, can be applied to nonlinear nano-disk problems to seek their solution with a high accuracy.

NONLOCAL THEORY FOR BUCKLING OF NANOPLATES

2011 ◽  
Vol 11 (03) ◽  
pp. 411-429 ◽  
Author(s):  
S. C. PRADHAN ◽  
J. K. PHADIKAR
Keyword(s):  
Elasticity Theory ◽  
Scale Effect ◽  
Nonlocal Elasticity ◽  
Plate Theory ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Theory ◽  
Theoretical Development ◽  
Small Scale ◽  
Classical Plate Theory ◽  
Small Scale Effect

Classical plate theory (CLPT) and first-order shear deformation plate theory (FSDT) of plates are reformulated using the nonlocal elasticity theory. Developed nonlocal plate theories have been applied to study buckling behavior of nanoplates. Nonlocal elasticity theory, unlike traditional elasticity theory introduces a length scale parameter into the formulation to take into account the discrete structure of the material to some extent. Both single-layered and multilayered nanoplates have been included in the analysis. Navier's approach has been used to obtain exact solutions for buckling loads for simply supported boundary conditions. Dependence of the small scale effect on various geometrical and material parameters has been investigated. Present study reveals the presence of significant small scale effect on the buckling response of nanoplates. The theoretical development and the numerical results presented in the present work are expected to promote the use of nonlocal theories for more accurate prediction of stability behavior of nanoplates and nanoshells.

Size-dependent vibration of functionally graded rotating nanobeams with different boundary conditions based on nonlocal elasticity theory

2021 ◽  
pp. 095440622110380
Author(s):  
Jianshi Fang ◽  
Bo Yin ◽  
Xiaopeng Zhang ◽  
Bin Yang
Keyword(s):  
Angular Velocity ◽  
Boundary Conditions ◽  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Slenderness Ratio ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Different Boundary Conditions ◽  
Gradient Variation

The free vibration of rotating functionally graded nanobeams under different boundary conditions is studied based on nonlocal elasticity theory within the framework of Euler-Bernoulli and Timoshenko beam theories. The thickness-wise material gradient variation of the nanobeam is considered. By introducing a second-order axial shortening term into the displacement field, the governing equations of motion of the present new nonlocal model of rotating nanobeams are derived by the Hamilton’s principle. The nonlocal differential equations are solved through the Galerkin method. The present nonlocal models are validated through the convergence and comparison studies. Numerical results are presented to investigate the influences of the nonlocal parameter, angular velocity, material gradient variation together with slenderness ratio on the vibration of rotating FG nanobeams with different boundary conditions. Totally different from stationary nanobeams, the rotating nanobeams with relatively high angular velocity could produce larger fundamental frequencies than local counterparts. Additionally, the axial stretching-transverse bending coupled vibration is perfectly shown through the frequency loci veering and modal conversion.

scholarly journals Static analysis of rectangular nanoplates using trigonometric shear deformation theory based on nonlocal elasticity theory

2013 ◽  
Vol 4 ◽  
pp. 968-973 ◽  
Author(s):  
Mohammad Rahim Nami ◽  
Maziar Janghorban
Keyword(s):  
Elasticity Theory ◽  
Shear Deformation ◽  
Static Analysis ◽  
Nonlocal Elasticity ◽  
Deformation Theory ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Nonlocal Elasticity Theory ◽  
Shear Stresses ◽  
Rectangular Plates

In this article, a new higher order shear deformation theory based on trigonometric shear deformation theory is developed. In order to consider the size effects, the nonlocal elasticity theory is used. An analytical method is adopted to solve the governing equations for static analysis of simply supported nanoplates. In the present theory, the transverse shear stresses satisfy the traction free boundary conditions of the rectangular plates and these stresses can be calculated from the constitutive equations. The effects of different parameters such as nonlocal parameter and aspect ratio are investigated on both nondimensional deflections and deflection ratios. It may be important to mention that the present formulations are general and can be used for isotropic, orthotropic and anisotropic nanoplates.

The effect of bi-axial in-plane loads on the natural frequency of nano-plates

2017 ◽  
Vol 24 (19) ◽  
pp. 4513-4528 ◽  
Author(s):  
Seyed Ghasem Enayati ◽  
Morteza Dardel ◽  
Mohammad Hadi Pashaei
Keyword(s):  
Natural Frequency ◽  
Nonlocal Elasticity ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Classical Plate Theory ◽  
First Order ◽  
Compressive Loads ◽  
Comprehensive Study

In this paper, natural frequencies of nano-plates subjected to two-sided in-plane tension or compressive loads, based on Eringen nonlocal elasticity theory and displacement field of first-order shear deformation plate theory (FSDT), are investigated. By considering total rotational variables as the two rotations due to bending and shear, another formulation form of FSDT nano-plate is achieved, that can simultaneously consider classical plate theory (CLPT) and FSDT. In a comprehensive study, the effects of different parameters such as a nonlocal parameter, aspect ratio, thickness to length ratio, mode number, boundary conditions and also length of nano-plate are examined on the dimensionless natural frequency. The results show that simultaneously applying two-sided tension and compressive in-plane loads changes frequency in a manner which is different to one-directional loading.

scholarly journals Nonlinear Vibrations of a SWCNT with Geometrical Imperfection Using Nonlocal Elasticity Theory

2017 ◽  
Vol 11 (10) ◽  
pp. 91 ◽  
Author(s):  
Mu’tasim S. Abdel-Jaber ◽  
Ahmad A Al-Qaisia ◽  
Nasim K Shatarat
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Multiple Scales ◽  
Nonlocal Parameter ◽  
Initial Imperfection ◽  
Nonlocal Elasticity Theory ◽  
Equation Of Motion ◽  
Assumed Mode Method ◽  
Geometrical Imperfection ◽  
Mode Method

The nonlinear free vibration and frequency veering of a single wall carbon nanotube (SWCNT) based on nonlocal elasticity theory is studied and investigated in this paper. The carbon nanotubes (CNT) is assumed to have an imperfection modeled as half sine and clamped at both ends. The Euler-Bernoulli Beam and Hamilton’s principle were used to derive the nonlinear equation of motion of the SWCNT. The effect of; nonlocal elasticity, geometric initial rise/imperfection, and the effect of the axial force induced by mid-plane stretching are accounted for in the derivation of the nonlinear mathematical model of the CNT. The governing partial differential equation includes quadratic and cubic nonlinearities due to the initial imperfection and the mid-plane stretching. The derived equation of motion is discretized using the assumed mode method by inserting the exact linear eigen mode shape. The resulting nonlinear temporal equation was solved using the method of multiple scales (MMS) to obtain results for the nonlinear natural frequencies of the first three modes of vibrations, for different values of rise/imperfection amplitude, and for different values of the nonlocal parameter. The results are presented in non-dimensional characteristic curves to show the effect of variation of rise/imperfection amplitude and nonlocal parameter on the vibrational behavior of the CNT.

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