scholarly journals Bending Analysis of Functionally Graded Nanoscale Plates by Using Nonlocal Mixed Variational Formula

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1162 ◽  
Author(s):  
Ashraf M. Zenkour ◽  
Zahra S. Hafed ◽  
Ahmed F. Radwan
Keyword(s):  
Virtual Work ◽  
Variational Formula ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Small Scale ◽  
Governing Equations ◽  
Bending Analysis ◽  
Nano Scale ◽  
Edge Conditions ◽  
The Impact

This work is devoted to the bending analysis of functionally graded (FG) nano-scale plate by using the nonlocal mixed variational formula under simply supported edge conditions. According to Eringen’s nonlocal elasticity theory, the mixed formula is utilized in order to obtain the governing equations. The system of equations is derived by using the principle of virtual work. The governing equations include both the small and the mechanical effects. The impact of the small-scale parameter, aspect and thickness nano-scale plate ratios, and gradient index on the displacement and stresses are explored, numerically presented, and discussed in detail. Different comparisons are made to check the precision and validity of the bending outcomes obtained from the present analysis of FG nano-scale plates. Parametric examinations are then performed to inspect the impacts of the thickness of the plate on the by and large mechanical reaction of the practically evaluated plates. The displayed outcomes are valuable for the configuration procedures of keen structures and examination from materials.

Thermal Post-Buckling Analysis of Functionally Graded Composite Plates with Nonlinear Aerodynamic Forces

2010 ◽  
Vol 123-125 ◽  
pp. 280-283
Author(s):  
Chang Yull Lee ◽  
Ji Hwan Kim
Keyword(s):  
Material Properties ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Composite Plates ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Functionally Graded Composite ◽  
Post Buckling

The post-buckling of the functionally graded composite plate under thermal environment with aerodynamic loading is studied. The structural model has three layers with ceramic, FGM and metal, respectively. The outer layers of the sandwich plate are different homogeneous and isotropic material properties for ceramic and metal. Whereas the core is FGM layer, material properties vary continuously from one interface to the other in the thickness direction according to a simple power law distribution in terms of the volume fractions. Governing equations are derived by using the principle of virtual work and numerical solutions are solved through a finite element method. The first-order shear deformation theory and von-Karman strain-displacement relations are based to derive governing equations of the plate. Aerodynamic effects are dealt by adopting nonlinear third-order piston theory for structural and aerodynamic nonlinearity. The Newton-Raphson iterative method applied for solving the nonlinear equations of the thermal post-buckling analysis

A NEW SIMPLE HYPERBOLIC SHEAR DEFORMATION THEORY FOR FUNCTIONALLY GRADED PLATES RESTING ON WINKLER–PASTERNAK ELASTIC FOUNDATIONS

2014 ◽  
Vol 11 (06) ◽  
pp. 1350098 ◽  
Author(s):  
ABDERRAHMANE SAID ◽  
MOHAMMED AMEUR ◽  
ABDELMOUMEN ANIS BOUSAHLA ◽  
ABDELOUAHED TOUNSI
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Functionally Graded ◽  
Functionally Graded Plates ◽  
Governing Equations ◽  
Pasternak Elastic Foundation ◽  
Shear Deformation Theories

An improved simple hyperbolic shear deformation theory involving only four unknown functions, as against five functions in case of first or other higher-order shear deformation theories, is introduced for the analysis of functionally graded plates resting on a Winkler–Pasternak elastic foundation. The governing equations are derived by employing the principle of virtual work and the physical neutral surface concept. The accuracy of the present analysis is demonstrated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories.

scholarly journals Elastic Wave Characteristics of Graphene Reinforced Polymer Nanocomposite Curved Beams Including Thickness Stretching Effect

Polymers ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 2194 ◽  
Author(s):  
Pouyan Talebizadehsardari ◽  
Arameh Eyvazian ◽  
Farayi Musharavati ◽  
Roohollah Babaei Mahani ◽  
Tamer A. Sebaey
Keyword(s):  
Elastic Wave ◽  
Strain Gradient ◽  
Effective Properties ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Opening Angle ◽  
Small Scale ◽  
Curved Beams ◽  
Wave Characteristics

This work aims at analyzing elastic wave characteristics in a polymeric nanocomposite curved beam reinforced by graphene nanoplatelets (GNPs). GNPs are adopted as a nanofiller inside the matrix to enhance the effective properties, which are approximated through Halpin-Tasi model and a modified rule of mixture. A higher-order shear deformation theory accounting for thickness stretching and the general strain gradient model to have both nonlocality and strain gradient size-dependency phenomena are adopted to model the nanobeam. A virtual work of Hamilton statement is utilized to get the governing motion equations and is solved in conjunction with the harmonic solution procedure. A comparative study shows the effects of small-scale coefficients, opening angle, weight fraction, the total number of layers in GNPs, and wave numbers on the propagation of waves in reinforced nanocomposite curved beams. This work is also developed for two different distribution of GNPs in a polymeric matrix, namely uniformly distribution and functionally graded one.

Nonlinear Vibration Analysis of Prestressed Double Layered Nanoscale Viscoelastic Plates

2019 ◽  
Vol 24 (3) ◽  
pp. 394-407
Author(s):  
Farzad Ebrahimi ◽  
S. Hamed S. Hamed S. Hossei
Keyword(s):  
Plate Theory ◽  
Differential Quadrature Method ◽  
Geometrical Nonlinearity ◽  
Flexural Vibration ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Nonlinear Frequency ◽  
Governing Equations ◽  
Viscoelastic Coefficient ◽  
Vdw Interaction

In the present study, the nonlinear flexural vibration behavior of a double layered prestressed viscoelastic nanoplate under shear in-plane load is investigated based on nonlocal elasticity theory. Using nonlinear strain-displacement relations, the geometrical nonlinearity is modeled. Both nonlocal plate theory and Hamilton’s principle are utilized for deriving the governing equations. The differential quadrature method (DQM) is employed for the computation of nonlinear frequency of the nanoplate. The detailed parametric study is conducted, focusing on the influences of small scale, aspect ratio of the plate, Winkler and Pasternak effects, van der Walls (vdW) interaction, temperature, the effect of pre-stress under shear in-plane load, and the viscidity of the plate. The influence of the viscoelastic coefficient is also discussed. The plots for the ratio of nonlinear to linear frequencies versus maximum transverse amplitude for double layered viscoelastic nanoplate are presented.

Size-dependent thermoelastic analysis of a functionally graded nanoshell

2018 ◽  
Vol 32 (03) ◽  
pp. 1850033 ◽  
Author(s):  
Mohammad Arefi ◽  
Ashraf M. Zenkour
Keyword(s):  
Virtual Work ◽  
Shear Deformation Theory ◽  
Nonlocal Elasticity Theory ◽  
Longitudinal Direction ◽  
Functionally Graded ◽  
Two Dimensional ◽  
Power Law Distribution ◽  
Thermoelastic Analysis ◽  
Radial And Axial Displacements ◽  
Eigenvalue And Eigenvector

In this paper, two-dimensional thermoelastic analysis of a functionally graded nanoshell is presented based on nonlocal elasticity theory. To formulate this problem, first-order shear deformation theory (FSDT) is used for axial and radial deformations simultaneously. Material properties are assumed to be mixture of ceramic and metal based on a power law distribution. Principle of virtual work is used for derivation of the governing equations. The analytical approach is presented based on eigenvalue and eigenvector method to derive four unknown functions including radial and axial displacements and rotations along the longitudinal direction. In addition, the influence of nonlocal and in-homogeneous index parameter is studied on the responses of the system. Two-dimensional results are presented along the radial and longitudinal directions.

Dynamic Behavior Study of Functionally Graded Porous Nanoplates

2021 ◽  
Vol 33 ◽  
pp. 83-92
Author(s):  
Hadj Mostefa Adda ◽  
Bouchafa Ali ◽  
Merdaci Slimane
Keyword(s):  
Natural Frequencies ◽  
Plate Theory ◽  
Functionally Graded ◽  
Small Scale ◽  
Dimensionless Frequency ◽  
Equilibrium Equations ◽  
Local Principle ◽  
Behavior Study ◽  
Small Scale Effect ◽  
The Impact

This paper introduces the analytical solutions of complex behavior analysis utilizing high-order shear deformation plate theory of functionally graded FGM nano-plate content consisting of a mixture of metal and ceramics with porosity. To incorporate the small-scale effect, the non-local principle of elasticity is used. The impact of variance of material properties such as thickness-length ratio, aspect ratio, power-law exponent and porosity factor on natural frequencies of FG nano-plate is examined. Compared to those achieved from other researchers, the latest solutions are. Using the simulated displacements theory, equilibrium equations are obtained. Current solutions of the dimensionless frequency are compared with those of the finite element method. The effect of geometry, material variations of nonlocal FG nano-plates and the porosity factor on their natural frequencies are investigated in this review. The results are in good agreement with those of the literature.

Magnetic field effect on free vibration of smart rotary functionally graded nano/microplates: A comparative study on modified couple stress theory and nonlocal elasticity theory

2018 ◽  
Vol 29 (11) ◽  
pp. 2492-2507 ◽  
Author(s):  
Mohammad Hassan Shojaeefard ◽  
Hamed Saeidi Googarchin ◽  
Mohammad Mahinzare ◽  
Seyed Ahmad Eftekhari
Keyword(s):  
Elastic Material ◽  
Nonlocal Elasticity ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Nonlocal Elasticity Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

In this article, free vibration behavior of a rotating nano/microcircular plate constructed from functionally graded magneto-elastic material is simulated with the first-order shear deformation theory. For the sake of comparison, the nonlocal elasticity theory and the modified couple stress theory are employed to implement the small size effect in the natural frequencies behavior of the nano/microcircular plate. The governing equations of motion for functionally graded magneto-elastic material nano/microcircular plates are derived based on Hamilton’s principle; comparing the obtained results with those in the literature, they are in a good agreement. Finally, the governing equations are solved using the differential quadrature method. It is shown that the vibrational characteristics of functionally graded magneto-elastic material nano/microcircular plates are significantly affected by non-dimensional angular velocity, size dependency of the Eringen’s and the modified couple stress theories, and power law index for clamped and hinged boundary conditions. Results show that a critical point occurs by increasing the angular velocity and the effect of several parameters are changed after this point.

Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams using the Most General form of Strain Gradient Elasticity

2013 ◽  
Vol 30 (2) ◽  
pp. 161-172 ◽  
Author(s):  
R. Ansari ◽  
M. Faghih Shojaei ◽  
V. Mohammadi ◽  
R. Gholami ◽  
H. Rouhi
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Vibration ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Small Scale ◽  
Length Scale ◽  
Beam Model ◽  
Strain Gradient Elasticity ◽  
Governing Equations

ABSTRACTBased on the Timoshenko beam model, the nonlinear vibration of microbeams made of functionally graded (FG) materials is investigated under different boundary conditions. To consider small scale effects, the model is developed based on the most general form of strain gradient elasticity. The nonlinear governing equations and boundary conditions are derived via Hamilton's principle and then discretized using the generalized differential quadrature technique. A pseudo-Galerkin approach is used to reduce the set of discretized governing equations into a time-varying set of ordinary differential equations of Duffing-type. The harmonic balance method in conjunction with the Newton-Raphson method is also applied so as to solve the problem in time domain. The effects of boundary conditions, length scale parameters, material gradient index and geometrical parameters are studied. It is found that the importance of the small length scale is affected by the type of boundary conditions and vibration mode. Also, it is revealed that the classical theory tends to underestimate the vibration amplitude and linear frequency of FG microbeams.

Flutter Analysis of Functionally Graded Panel Based on Neutral Surface

2015 ◽  
Vol 1125 ◽  
pp. 526-530
Author(s):  
Jung Hwan Kim ◽  
Ji Hwan Kim
Keyword(s):  
Virtual Work ◽  
Functionally Graded ◽  
Reference Plane ◽  
Neutral Surface ◽  
Governing Equations ◽  
Virtual Work Principle ◽  
Temperature Distributions ◽  
Linear Rule ◽  
Flutter Analysis ◽  
Graded Material

In this work, flutter behavior of Functionally Graded Material (FGM) panel is investigated based on the physical neutral surface. The panel is made with ceramic and metal according to linear rule of mixture. The virtual work principle is applied including pressure due to aero-dynamic load. Then governing equations are derived using von Karman's strain-displacement relations. Conventionally, mid-plane is used as a reference plane for laminate structures, while this concept is not appropriate for materially asymmetry of a panel such as FGMs. For this reason, physical neutral surface is defined as the origin of coordinate system in the structure. Numerical results are discussed and compared with previous studies. Finally, flutter behavior is investigated according to the volume fractions, temperature distributions and aero-dynamic pressures.

VIBRATION OF INITIALLY STRESSED MICRO- AND NANO-BEAMS

2007 ◽  
Vol 07 (04) ◽  
pp. 555-570 ◽  
Author(s):  
C. M. WANG ◽  
Y. Y. ZHANG ◽  
S. KITIPORNCHAI
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Initial Stress ◽  
Transverse Shear ◽  
Virtual Work ◽  
Nonlocal Elasticity Theory ◽  
Rotary Inertia ◽  
Small Scale ◽  
Transverse Shear Deformation ◽  
Vibration Problem

This paper is concerned with the vibration problem of initially stressed micro/nano-beams. The vibration problem is formulated on the basis of Eringen's nonlocal elasticity theory and the Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of initial stress, transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. These equations are solved analytically for the vibration frequencies of micro/nano-beams with different initial stress values and boundary conditions. The effect of the initial stress on the fundamental frequency and vibration mode shape of the beam is investigated. The solutions obtained provide a better representation of the vibration behavior of initially stressed micro/nano-beams which are stubby and short, since the effects of small scale, transverse shear deformation and rotary inertia are significant and cannot be neglected.

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