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.

Thermo-Electro-Mechanical Vibration Characteristics of Graphene/Piezoelectric/Graphene Sandwich Nanobeams

2017 ◽  
Vol 35 (1) ◽  
pp. 65-79 ◽  
Author(s):  
N. Kammoun ◽  
H. Jrad ◽  
S. Bouaziz ◽  
M. B. Amar ◽  
M. Soula ◽  
...  
Keyword(s):  
Nonlocal Elasticity ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Mechanical Vibration ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Mode Shapes ◽  
Governing Equations ◽  
Numerical Examples ◽  
Generalized Differential

AbstractThis paper reports an investigation on thermo-electro-mechanical vibration of graphene/piezoelectric graphene/piezoelectric/graphene sandwich nanobeams. Based on the nonlocal elasticity theory, Timoshenko beam theory and Hamilton's principles, the governing equations are developed and solved using generalized differential quadrature (GDQ) method. The effects of the nonlocal parameter, external electrical voltage, temperature change and axial force on vibration of graphene/piezoelectric/graphene sandwich nanobeams are examined. The performance and the accuracy of the presented model are highlighted through numerical examples with different boundary conditions. This study reports that the nonlocal parameter and thermo-electro-mechanical loadings have important effect on the natural frequencies and the deflection mode shapes of the graphene/piezoelectric/graphene sandwich nanobeam. The present work can serve as guideline for the design of a nanoscale graphene/piezoelectric/graphene beams based electromechanical resonator sensors.

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.

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.

Size-Dependent Nonlinear Vibrations of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

2016 ◽  
Vol 08 (04) ◽  
pp. 1650053 ◽  
Author(s):  
Reza Ansari ◽  
Raheb Gholami
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Nonlinear Free Vibration ◽  
First Order ◽  
Size Dependent

This paper deals with the size-dependent geometrically nonlinear free vibration of magneto-electro-thermo elastic (METE) nanoplates using the nonlocal elasticity theory. The mathematical formulation is developed based on the first-order shear deformation plate theory, von Kármán-type of kinematic nonlinearity and nonlocal elasticity theory. The influences of geometric nonlinearity, rotary inertia, transverse shear deformation, magneto-electro-thermal loading and nonlocal parameter are considered. First, the generalized differential quadrature (GDQ) method is utilized to reduce the nonlinear partial differential equations to a system of time-dependent nonlinear ordinary differential equations. Afterwards, the numerical Galerkin method, periodic time differential operators and pseudo-arc length continuation algorithm are employed to compute the nonlinear frequency versus the amplitude for the METE nanoplates. The presented methodology enables one to describe the large-amplitude vibration characteristics of METE nanoplates with various sets of boundary conditions. A detailed parametric study is carried out to analyze the important parameters such as the nondimensional nonlocal parameter, external electric potential, external magnetic potential, temperature change, length-to-thickness ratio, aspect ratio and various edge conditions on the nonlinear free vibration characteristics of METE nanoplates. The results demonstrate that considering the size effect on the vibration response of METE nanoplate results in decreasing the natural frequency, a remarkable increasing effect on the hardening behavior and subsequently increasing the nonlinear-to-linear frequency ratio.

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.

Size-Dependent Buckling and Postbuckling Analyses of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

2017 ◽  
Vol 17 (01) ◽  
pp. 1750014 ◽  
Author(s):  
R. Ansari ◽  
R. Gholami
Keyword(s):  
Boundary Conditions ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
First Order ◽  
Size Dependent ◽  
Shear Deformable

This paper presents a nonlocal nonlinear first-order shear deformable plate model for investigating the buckling and postbuckling of magneto-electro-thermo elastic (METE) nanoplates under magneto-electro-thermo-mechanical loadings. The nonlocal elasticity theory within the framework of the first-order shear deformation plate theory along with the von Kármán-type geometrical nonlinearity is used to derive the size-dependent nonlinear governing partial differential equations and associated boundary conditions, in which the effects of shear deformation, small scale parameter and thermo-electro-magneto-mechanical loadings are incorporated. The generalized differential quadrature (GDQ) method and pseudo arc-length continuation algorithm are used to determine the critical buckling loads and postbuckling equilibrium paths of the METE nanoplates with various boundary conditions. Finally, the influences of the nonlocal parameter, boundary conditions, temperature rise, external electric voltage and external magnetic potential on the critical buckling load and postbuckling response are studied.

FREE ASYMMETRIC TRANSVERSE VIBRATION OF CIRCULAR MONOLAYER GRAPHENE

NANO ◽  
2014 ◽  
Vol 09 (07) ◽  
pp. 1450072
Author(s):  
WIN-JIN CHANG ◽  
HAW-LONG LEE
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Mode Shape ◽  
Transverse Vibration ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Flexural Vibration ◽  
Nonlocal Elasticity Theory ◽  
Monolayer Graphene ◽  
Nodal Lines

The free vibration equations of circular monolayer graphene are analytically derived based on nonlocal elasticity theory. The natural frequency and mode shape for the axisymmetric and asymmetric circular monolayer graphenes are analyzed using the equations. The results show that the natural frequency of the asymmetric graphene is larger than that of the axisymmetric one. The natural frequency decreases with increasing nonlocal parameter for the axisymmetric and asymmetric graphenes. In addition, the diametrical nodal lines and nodal circles for the flexural vibration of the circular monolayer graphene are investigated. To avoid vibration failure of the sensors, they can be placed at the diametrical nodal lines and nodal circles.

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.

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