Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions

2014 ◽  
Vol 30 (5) ◽  
pp. 443-453 ◽  
Author(s):  
M. Sobhy
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Graphene Sheets ◽  
Elastic Foundations ◽  
Various Boundary Conditions ◽  
Mechanics Of Composite Materials

AbstractIn this article, the analyses of the natural frequency and buckling of orthotopic nanoplates, such as single-layered graphene sheets, resting on Pasternak's elastic foundations with various boundary conditions are presented. New functions for midplane displacements are suggested to satisfy the different boundary conditions. These functions are examined by comparing their results with the results obtained by using the functions suggested by Reddy (Reddy JN. Mechanics of Composite Materials and Structures: Theory and Analysis. Boca Raton, FL: CRC Press; 1997). Moreover, these functions are very simple comparing with Reddy's functions, leading to ease of calculations. The equations of motion of the nonlocal model are derived using the sinusoidal shear deformation plate theory (SPT) in conjunction with the nonlocal elasticity theory. The present SPT are compared with other plate theories. Explicit solution for buckling loads and vibration are obtained for single-layered graphene sheets with isotropic and orthotropic properties; and under biaxial loads. The formulation and the method of the solution are firstly validated by executing the comparison studies for the isotropic nanoplates with the results being in literature. Then, the influences of nonlocal parameter and the other parameters on the buckling and vibration frequencies are investigated.

scholarly journals Modeling of smart magnetically affected flexoelectric/piezoelectric nanostructures incorporating surface effects

2017 ◽  
Vol 7 ◽  
pp. 184798041771310 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Mohammad Reza Barati
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Surface Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Surface Effects ◽  
Geometrical Parameters ◽  
Various Boundary Conditions ◽  
Buckling Behavior

In this article, electromechanical buckling behavior of size-dependent flexoelectric/piezoelectric nanobeams is investigated based on nonlocal and surface elasticity theories. Flexoelectricity represents the coupling between the strain gradients and electrical polarizations. Flexoelectric/piezoelectric nanostructures can tolerate higher buckling loads compared with conventional piezoelectric ones, especially at lower thicknesses. Nonlocal elasticity theory of Eringen is applied for analyzing flexoelectric/piezoelectric nanobeams for the first time. The flexoelectric/piezoelectric nanobeams are assumed to be in contact with a two-parameter elastic foundation which consists of infinite linear springs and a shear layer. The residual surface stresses which are usually neglected in modeling of flexoelectric nanobeams are incorporated into nonlocal elasticity to provide better understanding of the physics of the problem. Applying an analytical method which satisfies various boundary conditions, the governing equations obtained from Hamilton’s principle are solved. The reliability of the present approach is verified by comparing the obtained results with those provided in literature. Finally, the influences of nonlocal parameter, surface effects plate geometrical parameters, elastic foundation, and boundary conditions on the buckling characteristics of the flexoelectric/piezoelectric nanobeams are explored in detail.

scholarly journals A nonlocal sinusoidal plate model for micro/nanoscale plates

2014 ◽  
Vol 228 (14) ◽  
pp. 2652-2660 ◽  
Author(s):  
Huu-Tai Thai ◽  
Thuc P Vo ◽  
Trung-Kien Nguyen ◽  
Jaehong Lee
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Transverse Shear ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Plate Model ◽  
Proposed Model ◽  
Small Scale Effect

A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen’s nonlocal elasticity theory and sinusoidal shear deformation plate theory. The small-scale effect is considered in the former theory while the transverse shear deformation effect is included in the latter theory. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundary conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending, buckling, and vibration of simply supported plates are presented, and the obtained results are compared with the existing solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated.

Torsional Vibration of Cracked Nanobeam Based on Nonlocal Stress Theory with Various Boundary Conditions: An Analytical Study

2015 ◽  
Vol 07 (03) ◽  
pp. 1550036 ◽  
Author(s):  
Omid Rahmani ◽  
S. A. H. Hosseini ◽  
M. H. Noroozi Moghaddam ◽  
I. Fakhari Golpayegani
Keyword(s):  
Boundary Conditions ◽  
Torsional Vibration ◽  
Analytical Study ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Stress Theory ◽  
Various Boundary Conditions ◽  
Crack Location ◽  
Nonlocal Stress ◽  
Torsional Spring

In this paper, the torsional vibration of cracked nanobeam was studied based on a nonlocal elasticity theory. The location of the crack is simulated by a torsional spring which links segments of nanobeam together. Also, different boundary conditions, including clamped–free, clamped–clamped and clamped–torsional spring, were considered. Furthermore, a detailed parametric study was conducted to investigate the influence of crack location, nonlocal parameter, length of nanobeam, spring constant and end supports on the torsional vibration.

scholarly journals Size-Dependent Free Vibration of Axially Moving Nanobeams Using Eringen’s Two-Phase Integral Model

2018 ◽  
Vol 8 (12) ◽  
pp. 2552 ◽  
Author(s):  
Yuanbin Wang ◽  
Zhimei Lou ◽  
Kai Huang ◽  
Xiaowu Zhu
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Critical Velocity ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Integral Model ◽  
Two Phase ◽  
Size Dependent ◽  
Axially Moving

In this paper, vibration of axially moving nanobeams is studied using Eringen’s two-phase nonlocal integral model. Geometric nonlinearity is taken into account for the integral model for the first time. Equations of motion for the beam with simply supported and fixed–fixed boundary conditions are obtained by Hamilton’s Principle, which turns out to be nonlinear integro-differential equations. For the free vibration of the nanobeam, the critical velocity and the natural frequencies are obtained numerically. Furthermore, the effects of parameters on critical velocity and natural frequency are analyzed. We have found that, for the two-phase nonlocal integral model, regardless of the boundary conditions considered, both the critical velocity and the natural frequency increase with the nonlocal parameter and the geometric parameter.

Free vibration analysis of multistepped nonlocal Bernoulli–Euler beams using dynamic stiffness matrix method

2020 ◽  
pp. 107754632093347
Author(s):  
Moustafa S Taima ◽  
Tamer A El-Sayed ◽  
Said H Farghaly
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Stiffness Matrix ◽  
Matrix Method ◽  
Dynamic Stiffness ◽  
Nonlocal Parameter ◽  
Free Vibration Analysis ◽  
Nonlocal Elasticity Theory ◽  
Dynamic Stiffness Matrix

The free vibration of multistepped nanobeams is studied using the dynamic stiffness matrix method. The beam analysis is based on the Bernoulli–Euler theory, and the nanoscale analysis is based on the Eringen’s nonlocal elasticity theory. The nanobeam is attached to linear and rotational elastic supports at the start, end, and intermediate boundary conditions. The effect of the nonlocal parameter, boundary conditions, and step ratios on the nanobeam natural frequency is investigated. The results of the dynamic stiffness matrix methods are validated by comparing selected cases with the literature, which give excellent agreement with those literatures. The results show that the dimensionless natural frequency parameter is inversely proportional to the nonlocal parameters except in the first mode for clamped-free boundary conditions. Also, the gap between every two consecutive modes decreases with the increasing of the nonlocal parameter.

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 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.

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

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Nonlinear Oscillations ◽  
Scale Effects ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Largest Lyapunov Exponent ◽  
Graphene Sheets ◽  
Parametric Vibrations

AbstractParametric 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, Poincaré 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.

A nonlocal strain gradient theory for rotating thermo-mechanical characteristics on magnetically actuated viscoelastic functionally graded nanoshell

2020 ◽  
Vol 31 (12) ◽  
pp. 1511-1523
Author(s):  
Mohammad Mahinzare ◽  
Hossein Akhavan ◽  
Majid Ghadiri
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Smart Structure ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Nonlocal Strain Gradient

In this article, a first-order shear deformable model is expanded based on the nonlocal strain gradient theory to vibration analysis of smart nanostructures under different boundary conditions. The governing equations of motion of rotating magneto-viscoelastic functionally graded cylindrical nanoshell in the magnetic field and corresponding boundary conditions are obtained using Hamilton’s principle. To discretize the equations of motion, the generalized differential quadrature method is applied. The aim of this work is to investigate the effects of the temperature changes, nonlocal parameter, material length scale, viscoelastic coefficient, various boundary conditions, and the rotational speed of this smart structure on natural frequencies of rotating cylindrical nanoshell made of magneto-viscoelastic functionally graded material.

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