Strain Gradient Finite Element Analysis on the Vibration of Double-Layered Graphene Sheets

2016 ◽  
Vol 13 (03) ◽  
pp. 1650011 ◽  
Author(s):  
Wei Xu ◽  
Lifeng Wang ◽  
Jingnong Jiang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Scale Effect ◽  
Strain Gradient ◽  
Virtual Work ◽  
Gradient Elasticity ◽  
Kirchhoff Plate ◽  
Small Scale ◽  
Graphene Sheets ◽  
Simply Supported

A nonlocal Kirchhoff plate model with the van der Waals (vdW) interactions taken into consideration is developed to study the vibration of double-layered graphene sheets (DLGS). The dynamic equations of multi-layered Kirchhoff plate are derived based on strain gradient elasticity. An explicit formula is derived to predict the natural frequency of the DLGS with all edges simply supported. Then a 4-node 24-degree of freedom (DOF) Kirchhoff plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration by the theory of virtual work. It can be directly used to predict the scale effect on the vibrational DLGS with different boundary conditions. A good agreement between finite element method (FEM) results and theoretical natural frequencies of the vibration simply supported double-layered graphene sheet (DLGS) validates the reliability of the FEM. Finally, this new FEM is used to investigate the effect of vdW coefficients, sizes, nonlocal parameters, vibration mode and boundary conditions on the vibration behaviors of DLGS.

scholarly journals A Finite Element Formulation for Kirchhoff Plates in Strain-gradient Elasticity

2019 ◽  
Author(s):  
Alireza Beheshti
Keyword(s):  
Finite Element ◽  
Strain Gradient ◽  
Virtual Work ◽  
Hermite Polynomials ◽  
Gradient Elasticity ◽  
Weak Form ◽  
Element Formulation ◽  
Strain Gradient Elasticity ◽  
Rectangular Plates ◽  
Kirchhoff Plates

The current contribution is centered on bending of rectangular plates using the finite element method in the strain-gradient elasticity. To this aim, following introducing stresses and strains for a plate based on the Kirchhoff hypothesis, the principle of the virtual work is adopted to derive the weak form. Building upon Hermite polynomials and by deeming convergence requirements, four rectangular elements for the static analysis of strain-gradient plates are presented. To explore the performance of the proposed elements, particularly in small scales, some problems are solved and the results are compared with analytical solutions.

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.

scholarly journals Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity

2019 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Mustafa Özgür Yayli
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Scale Parameter ◽  
Gradient Elasticity ◽  
Buckling Load ◽  
Elastic Matrix ◽  
Small Scale ◽  
Strain Gradient Elasticity ◽  
Critical Buckling Load ◽  
Fourier Sine Series

AbstractThe buckling of rotationally restrained microbars embedded in an elastic matrix is studied within the framework of strain gradient elasticity theory. The elastic matrix is modeled in this study as Winkler’s one-parameter elastic matrix. Fourier sine series with a Fourier coefficient is used for describing the deflection of the microbar. An eigenvalue problem is obtained for buckling modes with the aid of implementing Stokes’ transformation to force boundary conditions. This mathematical model bridges the gap between rigid and the restrained boundary conditions. The influences of rotational restraints, small scale parameter and surrounding elastic matrix on the critical buckling load are discussed and compared with those available in the literature. It is concluded from analytical results that the critical buckling load of microbar is dependent upon rotational restraints, surrounding elastic matrix and the material scale parameter. Similarly, the dependencies of the critical buckling load on material scale parameter, surrounding elastic medium and rotational restraints are significant.

Finite Element Simulation of Laser Beam Welding Induced Residual Stresses and Distortions in a T-Joint Configuration for Aeronautic Structures

2009 ◽  
Author(s):  
Muhammad Zain-ul-abdein ◽  
Daniel Ne´lias ◽  
Jean-Franc¸ois Jullien ◽  
Dominique Deloison
Keyword(s):  
Finite Element ◽  
Residual Stresses ◽  
Laser Beam ◽  
Boundary Conditions ◽  
Laser Beam Welding ◽  
Temperature Fields ◽  
Heat Transfer Analysis ◽  
Small Scale ◽  
Joint Configuration ◽  
Simulation Results

Laser beam welding has found its application in the aircraft industry for the fabrication of fuselage panels in a T-joint configuration. However, the inconveniences like distortions and residual stresses are inevitable consequences of welding. The effort is made in this work to experimentally measure and numerically simulate the distortions induced by laser beam welding of a T-joint with industrially used thermal and mechanical boundary conditions on the thin sheets of aluminium 6056-T4. Several small scale experiments were carried out with various instrumentations to establish a database necessary to verify the simulation results. Finite element (FE) simulation is performed with Abaqus and the conical heat source is programmed in FORTRAN. Heat transfer analysis is performed to achieve the required weld pool geometry and temperature fields. Mechanical analysis is then performed with industrial loading and boundary conditions so as to predict the distortion and the residual stress pattern. A good agreement is found amongst the experimental and simulation results.

Dynamic Instability Analysis and Design Charts of Curved Panels with Linearly Varying Periodic In-Plane Load

2021 ◽  
pp. 2150130
Author(s):  
G. Patel ◽  
A. N. Nayak ◽  
A. K. L. Srivastava
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dynamic Instability ◽  
Excitation Frequency ◽  
Instability Region ◽  
Concentrated Load ◽  
Simply Supported ◽  
Finite Element Software ◽  
Design Charts ◽  
Curved Panels

The present paper reports an extensive study on dynamic instability characteristics of curved panels under linearly varying in-plane periodic loading employing finite element formulation with a quadratic isoparametric eight nodded element. At first, the influences of three types of linearly varying in-plane periodic edge loads (triangular, trapezoidal and uniform loads), three types of curved panels (cylindrical, spherical and hyperbolic) and six boundary conditions on excitation frequency and instability region are investigated. Further, the effects of varied parameters, such as shallowness parameter, span to thickness ratio, aspect ratio, and Poisson’s ratio, on the dynamic instability characteristics of curved panels with clamped–clamped–clamped–clamped (CCCC) and simply supported-free-simply supported-free (SFSF) boundary conditions under triangular load are studied. It is found that the above parameters influence significantly on the excitation frequency, at which the dynamic instability initiates, and the width of dynamic instability region (DIR). In addition, a comparative study is also made to find the influences of the various in-plane periodic loads, such as uniform, triangular, parabolic, patch and concentrated load, on the dynamic instability behavior of cylindrical, spherical and hyperbolic panels. Finally, typical design charts showing DIRs in non-dimensional forms are also developed to obtain the excitation frequency and instability region of various frequently used isotropic clamped spherical panels of any dimension, any type of linearly varying in-plane load and any isotropic material directly from these charts without the use of any commercially available finite element software or any developed complex model.

Vibration of Single- and Double-Layered Graphene Sheets

2011 ◽  
Vol 2 (1) ◽  
Author(s):  
Behrouz Arash ◽  
Quan Wang
Keyword(s):  
Molecular Dynamics ◽  
Boundary Conditions ◽  
Molecular Dynamics Simulations ◽  
Nonlocal Parameter ◽  
Small Scale ◽  
Graphene Sheets ◽  
Elastic Model ◽  
Resonant Frequencies ◽  
Nonlocal Continuum Theory ◽  
Dynamics Simulations

Free vibration of single- and double-layered graphene sheets is investigated by employing nonlocal continuum theory and molecular dynamics simulations. Results show that the classical elastic model overestimated the resonant frequencies of the sheets by a percentage as high as 62%. The dependence of small-scale effects, sizes of sheets, boundary conditions, and number of layers on vibrational characteristic of single- and double-layered graphene sheets is studied. The resonant frequencies predicted by the nonlocal elastic plate theory are verified by the molecular dynamics simulations, and the nonlocal parameter is calibrated through the verification process. The simulation results reveal that the calibrated nonlocal parameter depends on boundary conditions and vibrational modes. The nonlocal plate model is found to be indispensable in vibration analysis of grapheme sheets with a length less than 8 nm on their sides.

scholarly journals Study on the small-scale effect on wave propagation characteristics of fluid-filled carbon nanotubes based on nonlocal fluid theory

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401882332
Author(s):  
Yang Yang ◽  
Wuhuai Yan ◽  
Jinrui Wang
Keyword(s):  
Wave Propagation ◽  
Fluid Flow ◽  
Carbon Nanotube ◽  
Scale Effect ◽  
Strain Gradient ◽  
Small Scale ◽  
Beam Model ◽  
Governing Equations ◽  
Mode Wave ◽  
Fluid Theory

In this article, Timoshenko’s beam model is established to investigate the wave propagation behaviors for a fluid-conveying carbon nanotube when employing the nonlocal stress–strain gradient coupled theory and nonlocal fluid theory. The governing equations of motion for the carbon nanotube are derived. The small-scale influences induced by the nanotube are simulated by nonlocal and strain gradient effects, and the scale effect induced by fluid flow is first investigated applying nonlocal fluid theory. Numerical results obtained by solving the governing equations indicate that the nonlocal effect induced by the nanotube leads to wave damping and a decrease in stiffness, while the strain gradient effect contributes to wave promotion and an enhancement in stiffness. The scale effect caused by the inner fluid only leads to a decay for a high-mode wave since there is no influence from fluid flow on the low-mode wave. The numerical solution is validated by comparing with Monte Carlo simulation and interval analysis method.

Sign in / Sign up

Export Citation Format

Share Document