A Novel Refined Plate Theory for Free Vibration Analyses of Single-Layered Graphene Sheets Lying on Winkler-Pasternak Elastic Foundations

2019 ◽  
Vol 58 ◽  
pp. 151-164 ◽  
Author(s):  
Fatima Boukhatem ◽  
Aicha Bessaim ◽  
Abdelhakim Kaci ◽  
Abderrahmane Mouffoki ◽  
Mohammed Sid Ahmed Houari ◽  
...  
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Displacement Field ◽  
Scale Effects ◽  
Plate Theory ◽  
Small Scale ◽  
Graphene Sheets ◽  
Refined Plate Theory ◽  
Foundation Parameters

In this article, the analyses of free vibration of nanoplates, such as single-layered graphene sheets (SLGS), lying on an elastic medium is evaluated and analyzed via a novel refined plate theory mathematical model including small-scale effects. The noteworthy feature of theory is that the displacement field is modelled with only four unknowns, which is even less than the other shear deformation theories. The present one has a new displacement field which introduces undetermined integral variables, the shear stress free condition on the top and bottom surfaces of the plate is respected and consequently, it is unnecessary to use shear correction factors. The theory involves four unknown variables, as against five in case of other higher order theories and first-order shear deformation theory. By using Hamilton’s principle, the nonlocal governing equations are obtained and they are solved via Navier solution method. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all examined. From this work, it can be observed that the small-scale effects and elastic foundation parameters are significant for the natural frequency.

scholarly journals Free Vibration Analysis of Simply Supported P-FGM Nanoplate Using a Nonlocal Four Variables Shear Deformation Plate Theory

2019 ◽  
Vol 69 (4) ◽  
pp. 9-24 ◽  
Author(s):  
Chikh Abdelbaki
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Power Law ◽  
Material Properties ◽  
Scale Effects ◽  
Plate Theory ◽  
Functionally Graded ◽  
Small Scale ◽  
Simply Supported ◽  
Shear Deformation Plate Theory

AbstractThis paper shows an analysis of the free vibration of functionally graded simply supported nanoplate. The nonlocal four variables shear deformation plate theory is used to predict the free vibration frequencies of functionally graded nanoplate simply supported using non-local elasticity theory with the introduction of small-scale effects. The effect of the material properties, thickness-length ratio, aspect ratio, the exponent of the power law, the vibration mode is presented, the current solutions are compared to those obtained by other researchers. Equilibrium equations are obtained using the virtual displacements principle. P-FGM Power law is used to have a distribution of material properties that vary across the thickness. The results are in good agreement with those of the literature.

Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory

2011 ◽  
Vol 226 (7) ◽  
pp. 1896-1906 ◽  
Author(s):  
AR Setoodeh ◽  
P Malekzadeh ◽  
AR Vosoughi
Keyword(s):  
Free Vibration ◽  
Virtual Work ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Mindlin Plate ◽  
Small Scale ◽  
Graphene Sheets ◽  
Nonlinear Free Vibration ◽  
Mindlin Plate Theory

This article deals with the small-scale effect on the nonlinear free vibration of orthotropic single-layered graphene sheets using the nonlocal elasticity plate theory. The formulations are based on the Mindlin plate theory, and von Karman-type nonlinearity is considered in strain displacement relations. Virtual work principle is used to derive the nonlinear nonlocal plate equations in which the effects of rotary inertia and transverse shear are included. The differential quadrature method is employed to reduce the governing nonlinear partial differential equations to a system of nonlinear algebraic eigenvalue equations. The efficiency and accuracy of the method are demonstrated by comparing the developed result with those available in literature. The methodology is capable of studying large-amplitude vibration characteristics of nanoplates with different sets of boundary conditions. The effects of various parameters on the nonlinear vibrations of nanoplates are presented.

Free Vibration and Buckling Analysis of Functionally Graded Plates Resting on Elastic Foundation Using Higher Order Theory

2018 ◽  
Vol 18 (04) ◽  
pp. 1850049 ◽  
Author(s):  
Smita Parida ◽  
Sukesh Chandra Mohanty
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Volume Fraction ◽  
Plate Theory ◽  
Higher Order ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Pasternak Elastic Foundation

This paper deals with the free vibration and buckling analysis of functionally graded material (FGM) plates, resting on the Winkler–Pasternak elastic foundation. The higher order shear deformation plate theory (HSPT) is adopted for the realistic variation of transverse displacement through the thickness, using the power law distribution to describe the variation of the material properties. Both the effects of shear deformation and rotary inertia are considered. In the present model, the plate is discretised into [Formula: see text] eight noded serendipity quadratic elements with seven nodal degrees of freedom (DOFs). The validation study is carried out by comparing the calculated values with those given in the literature. The effects of various parameters like the Winkler and Pasternak modulus coefficients, volume fraction index, aspect ratio, thickness ratio and different boundary conditions on the behaviour of the FGM plates are studied.

Thermo-mechanical vibration, buckling, and bending of orthotropic graphene sheets based on nonlocal two-variable refined plate theory using finite difference method considering surface energy effects

2017 ◽  
Vol 231 (3) ◽  
pp. 111-130 ◽  
Author(s):  
Morteza Karimi ◽  
Ali Reza Shahidi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Surface Energy ◽  
Difference Method ◽  
Plate Theory ◽  
Small Scale ◽  
Graphene Sheets ◽  
Governing Equations ◽  
Refined Plate Theory ◽  
Temperature Changes

In this article, the influence of temperature change on the vibration, buckling, and bending of orthotropic graphene sheets embedded in elastic media including surface energy and small-scale effects is investigated. To take into account the small-scale and surface energy effects, the nonlocal constitutive relations of Eringen and surface elasticity theory of Gurtin and Murdoch are used, respectively. Using Hamilton’s principle, the governing equations for bulk and surface of orthotropic nanoplate are derived using two-variable refined plate theory. Finite difference method is used to solve governing equations. The obtained results are verified with Navier’s method and validated results reported in the literature. The results demonstrated that for both isotropic and orthotropic material properties, by increasing the temperature changes, the degree of surface effects on the buckling and vibration of nanoplates could enhance at higher temperatures, while it would diminish at lower temperatures. In addition, the effects of surface and temperature changes on the buckling and vibration for isotropic material property are more noticeable than those of orthotropic. On the contrary, these results are totally reverse for bending problem.

scholarly journals Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 515
Author(s):  
Olga Mazur ◽  
Jan Awrejcewicz
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Magnetic Parameter ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Ritz Method ◽  
Graphene Sheets ◽  
Classical Plate Theory ◽  
Foundation Parameters

Vibrations of single-layered graphene sheets subjected to a longitudinal magnetic field are considered. The Winkler-type and Pasternak-type foundation models are employed to reproduce the surrounding elastic medium. The governing equation is based on the modified couple stress theory and Kirchhoff–Love hypotheses. The effect of the magnetic field is taken into account due to the Lorentz force deriving from Maxwell’s equations. The developed approach is based on applying the Ritz method. The proposed method is tested by a comparison with results from the existing literature. The numerical calculations are performed for different boundary conditions, including the mixed ones. The influence of the material length scale parameter, the elastic foundation parameters, the magnetic parameter and the boundary conditions on vibration frequencies is studied. It is observed that an increase of the magnetic parameter, as well as the elastic foundation parameters, brings results closer to the classical plate theory results. Furthermore, the current study can be applied to the design of microplates and nanoplates and their optimal usage.

Buckling Analysis of Orthotropic Nanoscale Plates Resting on Elastic Foundations

2018 ◽  
Vol 55 ◽  
pp. 42-56 ◽  
Author(s):  
Belkacem Kadari ◽  
Aicha Bessaim ◽  
Abdelouahed Tounsi ◽  
Houari Heireche ◽  
Abdelmoumen Anis Bousahla ◽  
...  
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Scale Effects ◽  
Plate Theory ◽  
Constitutive Relations ◽  
Shear Deformation Theory ◽  
Nonlocal Elasticity Theory ◽  
Higher Order ◽  
Small Scale ◽  
Classical Plate Theory

This work presents the buckling investigation of embedded orthotropic nanoplates by using a new hyperbolic plate theory and nonlocal small-scale effects. The main advantage of this theory is that, in addition to including the shear deformation effect, the displacement field is modeled with only three unknowns and three governing equation as the case of the classical plate theory (CPT) and which is even less than the first order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). A shear correction factor is, therefore, not required. Nonlocal differential constitutive relations of Eringen is employed to investigate effects of small scale on buckling of the rectangular nanoplate. The elastic foundation is modeled as two-parameter Pasternak foundation. The equations of motion of the nonlocal theories are derived and solved via Navier's procedure for all edges simply supported boundary conditions. The proposed theory is compared with other plate theories. Analytical solutions for buckling loads are obtained for single-layered graphene sheets with isotropic and orthotropic properties. The results presented in this study may provide useful guidance for design of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates. Verification studies show that the proposed theory is not only accurate and simple in solving the buckling nanoplates, but also comparable with the other higher-order shear deformation theories which contain more number of unknowns. Keywords: Buckling; orthotropic nanoplates; a simple 3-unknown theory; nonlocal elasticity theory; Pasternak’s foundations. * Corresponding author; [email protected]

scholarly journals Static bending and free vibration analysis of functionally graded porous plates laid on elastic foundation using the meshless method

2021 ◽  
Vol 15 (2) ◽  
pp. 141-159
Author(s):  
Vu Tan Van ◽  
Nguyen Huynh Tan Tai ◽  
Nguyen Ngoc Hung
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Meshless Method ◽  
Vibration Analysis ◽  
Plate Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Kriging Interpolation ◽  
Moving Kriging Interpolation

This paper presents a numerical approach for static bending and free vibration analysis of the functionally graded porous plates (FGPP) resting on the elastic foundation using the refined quasi-3D sinusoidal shear deformation theory (RQSSDT) combined with the Moving Kriging–interpolation meshfree method. The plate theory considers both shear deformation and thickness-stretching effects by the sinusoidal distribution of the in-plane displacements, satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate without shear correction coefficient. The advantage of the plate theory is that the displacement field of plate is approximated by only four variables leading to reduce computational efforts. Comparison studies are performed for the square FGPP with simply supported all edges to verify the accuracy of the present approach. The effect of the aspect ratio, volume fraction exponent, and elastic foundation parameters on the static deflections and natural frequency of FGPP are also investigated and discussed. Keywords: meshless method; Moving Kriging interpolation; refined quasi-3D theory; porous functionally\break graded plate; Pasternak foundation.

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.

Sign in / Sign up

Export Citation Format

Share Document