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.

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 Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Mass Density ◽  
Functionally Graded ◽  
Governing Equations ◽  
Mass System ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

Determination of the Steady-State Response of Viscoelastically Supported Rectangular Orthotropic Mass Loaded Plates by an Energy-Based Finite Difference Method

2005 ◽  
Vol 11 (12) ◽  
pp. 1535-1552 ◽  
Author(s):  
Gökhan Altintaş ◽  
Muhiddin Bağci
Keyword(s):  
Steady State ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Plate Theory ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Elastic Rectangular Plate ◽  
Spring Coefficient

A method based on a variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics and steady-state response to a sinusoidally varying force applied orthotropic elastic rectangular plate carrying masses. Using the energy-based finite difference method, the problem reduced to the solution of a system of algebraic equations. Due to the significance of the fundamental natural frequency of the plate, its variation is investigated with respect to the mechanical properties of the plate material, the translational spring coefficient of the supports, the mass distribution, the mass locations and the quantity of mass. The steady-state response of the viscoelastically supported plates was also investigated numerically for the damping coefficient of the supports and the force distribution in addition to the characteristics of the plate system. Many new results are presented and the validity of the present approach is demonstrated by comparing the results with other solutions based on the Kirchhoff-Love plate theory.

A Study of Photo-Thermoelastic Wave in Semiconductor Materials With Spherical Holes Using Analytical-numerical Methods

2021 ◽  
Author(s):  
Faris S. Alzahrani ◽  
Ibrahim Abbas
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Governing Equations ◽  
Thermoelastic Wave ◽  
Spherical Cavities ◽  
Implicit Finite Difference ◽  
Implicit Finite Difference Method

Abstract Analytical and numerical solutions are two basic tools in the study of photothermal interaction problems in semiconductor medium. In this paper, we compare the analytical solutions with the numerical solutions for thermal interaction in semiconductor mediums containing spherical cavities. The governing equations are given in the domain of Laplace transforms and the eigenvalues approaches are used to obtained the analytical solution. The numerical solutions are obtained by applying the implicit finite difference method (IFDM). A comparison between the numerical solutions and analytical solution are presented. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.

A Mathematical Model for the Analysis of Fluid Flow in a Scroll

1986 ◽  
Vol 108 (1) ◽  
pp. 6-11 ◽  
Author(s):  
Shou-Rue Chen ◽  
Samuel S. Lee ◽  
Yuan Mao Huang
Keyword(s):  
Mathematical Model ◽  
Fluid Flow ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Flow Condition ◽  
Three Dimensional ◽  
Coordinate Transformations ◽  
Governing Equations ◽  
The Finite Difference Method

A three-dimensional mathematical model has been developed to simulate the flow condition in a scroll. Coordinate transformations are used as an effective tool to make the model universal, and the final governing equations are solved by the finite difference method. Three cases of scroll geometry have been investigated and the results are compared with one another to show the effects of scroll geometry on the flow condition at the outlet of the scroll.

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Timoshenko Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Functionally Graded ◽  
Timoshenko Beams ◽  
Governing Equations ◽  
Mass System ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded non-prismatic Timoshenko beams (AFGNPTB) using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, shear moduli, and mass density) of axially functionally graded beams vary along the longitudinal axis. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFGNPTB was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFGNPTB, considering the damping. The results obtained in this study showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

The Study of the Difference Methods with Variable Grids Seismic Wave Numerical Simulation in Multi-Scale Complex Media

2014 ◽  
Vol 1055 ◽  
pp. 254-258
Author(s):  
Jie Zhang ◽  
Fan Shun Meng ◽  
Yang Sen Li
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Wave Field ◽  
Seismic Wave ◽  
Computational Efficiency ◽  
Difference Method ◽  
Staggered Grid ◽  
Small Scale ◽  
Simulation Accuracy

In the process of seismic wave field numerical simulation using finite difference method, the simulation accuracy and computational efficiency is one of the keys to the problem which is especially important to the numerical simulation of small scale geological body which velocity changes violently. In order to describe the local structure of medium subtly and guarantee the efficiency of the simulation, this article introduces the variable grid finite difference method to the staggered grid high-order finite difference numerical simulation on the basic of the traditional staggered grid finite difference algorithm to improve the staggered grid spatial algorithm and avoid the reduction of the simulation accuracy and computational efficiency caused by the interpolation factor. The results show that the variable staggered grid numerical simulation of finite difference algorithm can accurately depict the space variation of underground medium physical properties to further enhance the adaptability of numerical simulation of complex medium, it also can provide reliable basis for wave field imaging and the combined interpretation of p-wave and s-wave.

scholarly journals A REDUCED KINETIC MECHANISM FOR PROPANE FLAMES

2012 ◽  
Vol 11 (1-2) ◽  
pp. 37
Author(s):  
G. S. L. Andreis ◽  
R. S. Gomes ◽  
A. L. De Bortoli
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Flame ◽  
Difference Method ◽  
Kinetic Mechanism ◽  
Governing Equations ◽  
Propane Combustion ◽  
Eddy Simulation ◽  
Large Eddy ◽  
Reduced Kinetic Mechanism

Propane is one of the simplest hydrocarbons that can be a representative of higher hydrocarbons used in many applications. Therefore, this work develops a ten-step reduced kinetic mechanism among 14 reactive species for the propane combustion. The model is based on the solution of the flamelet equations. The equations are discretized using the second-order space finite difference method, using LES (Large-Eddy Simulation). Obtained results compare favorably with data in the literature for a propane jet diffusion flame. The main advantage of this strategy is the decrease of the work needed to solve the system of governing equations.

Sign in / Sign up

Export Citation Format

Share Document