scholarly journals On Bending of Bernoulli-Euler Nanobeams for Nonlocal Composite Materials

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Luciano Feo ◽  
Rosa Penna
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Scale Parameter ◽  
Elastic Equilibrium ◽  
Functionally Graded ◽  
Constitutive Law ◽  
Bending Moments ◽  
Innovative Methodology ◽  
Governing Differential Equation ◽  
Nonlocality Effect

Evaluation of size effects in functionally graded elastic nanobeams is carried out by making recourse to the nonlocal continuum mechanics. The Bernoulli-Euler kinematic assumption and the Eringen nonlocal constitutive law are assumed in the formulation of the elastic equilibrium problem. An innovative methodology, characterized by a lowering in the order of governing differential equation, is adopted in the present manuscript in order to solve the boundary value problem of a nanobeam under flexure. Unlike standard treatments, a second-order differential equation of nonlocal equilibrium elastic is integrated in terms of transverse displacements and equilibrated bending moments. Benchmark examples are developed, thus providing the nonlocality effect in nanocantilever and clampled-simply supported nanobeams for selected values of the Eringen scale parameter.

scholarly journals A Note on Torsion of Nonlocal Composite Nanobeams

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Luciano Feo ◽  
Rosa Penna
Keyword(s):  
Order Differential Equation ◽  
Scale Effects ◽  
Elastic Equilibrium ◽  
Functionally Graded ◽  
Small Scale ◽  
Cross Sectional ◽  
Graded Materials ◽  
First Order ◽  
Sectional Plane ◽  
Mixed Framework

The ERINGENelastic constitutive relation is used in this paper in order to assess small-scale effects in nanobeams. Structural behavior is studied for functionally graded materials in the cross-sectional plane and torsional loading conditions. The governing boundary value problem has been formulated in a mixed framework. Torsional rotations and equilibrated moments are evaluated by solving a first-order differential equation of elastic equilibrium with boundary conditions of kinematic-type. Benchmarks examples are briefly discussed, enlightening thus effectiveness of the proposed methodology.

scholarly journals Nonlinear Vibration Analysis of Axially Functionally Graded Microbeams Based on Nonlinear Elastic Foundation Using Modified Couple Stress Theory

2020 ◽  
Vol 64 (2) ◽  
pp. 97-108
Author(s):  
Mehdi Alimoradzadeh ◽  
Mehdi Salehi ◽  
Sattar Mohammadi Esfarjani
Keyword(s):  
Differential Equation ◽  
Forced Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, a non-classical approach was developed to analyze nonlinear free and forced vibration of an Axially Functionally Graded (AFG) microbeam by means of modified couple stress theory. The beam is considered as Euler-Bernoulli type supported on a three-layered elastic foundation with Von-Karman geometric nonlinearity. Small size effects included in the analysis by considering the length scale parameter. It is assumed that the mass density and elasticity modulus varies continuously in the axial direction according to the power law form. Hamilton's principle was implemented to derive the nonlinear governing partial differential equation concerning associated boundary conditions. The nonlinear partial differential equation was reduced to some nonlinear ordinary differential equations via Galerkin's discretization technique. He's Variational iteration methods were implemented to obtain approximate analytical expressions for the frequency response as well as the forced vibration response of the microbeam with doubly-clamped end conditions. In this study, some factors influencing the forced vibration response were investigated. Specifically, the influence of the length scale parameter, the length of the microbeam, the power index, the Winkler parameter, the Pasternak parameter, and the nonlinear parameter on the nonlinear natural frequency as well as the amplitude of forced response have been investigated.

Functionally Graded Varying Geometry Annular Fins

2018 ◽  
pp. 197-207
Author(s):  
Vivek Kumar Gaba ◽  
Anil Kumar Tiwari ◽  
Tarakant Bhanu ◽  
Shubhankar Bhowmick
Keyword(s):  
Differential Equation ◽  
Thermal Conductivity ◽  
Temperature Gradient ◽  
Power Function ◽  
Research Work ◽  
Functionally Graded ◽  
Annular Fin ◽  
Efficiency And Effectiveness ◽  
Governing Differential Equation ◽  
Annular Fins

The research work aims at investigating the performance of functionally graded annular fin. The work involves computation of efficiency and effectiveness of such fins and compares the fin performances for different geometry and grading parameters. It is also assumed that the temperature gradient only exists in one direction i.e. along the radius. A general second order governing differential equation has been derived for all the profiles and grading. The thickness of fin is considered to be a power function of radial co-ordinate. The functional grading of thermal conductivity is assumed to be a power function of radial co-ordinate and also consists of parameters, namely grading parameters, varying which different grading combinations can be carried out. The efficiency and effectiveness of the annular fin of different geometry and grading combinations have been calculated and plotted. The performance analysis reveals the dependence of thermal behaviour of annular fins on geometry and grading parameter.

SOME CONVENTIONAL AND UNCONVENTIONAL EDUCATIONAL COLUMN STABILITY PROBLEMS

2006 ◽  
Vol 06 (01) ◽  
pp. 139-151 ◽  
Author(s):  
I. ELISHAKOFF ◽  
C. GENTILINI ◽  
R. SANTORO
Keyword(s):  
Differential Equation ◽  
Inverse Method ◽  
Order Differential Equation ◽  
Flexural Rigidity ◽  
Functionally Graded ◽  
The Novel ◽  
Main Approach ◽  
Graded Materials ◽  
Closed Form Solutions ◽  
The Difference

Two interesting problems are considered for enriching the curriculum of the Strength of Materials course, in the light of recently developed functionally graded materials (FGMs), characterized with the smooth variation of the elastic modulus. These are problems associated with buckling of columns with variable flexural rigidity along the axis of the column. A simple semi-inverse method is proposed for determining closed-form solutions of axially inhomogeneous columns. In order for the presentation to be given in one package, the conventional problems are also recapitulated along with the novel ones. The main approach adopted here is the use of the second-order differential equation, instead of the fourth-order one, obtained by integrating twice and having a physical meaning, since it could be derived based on moment equilibrium. This is essentially the same idea as utilized in the textbook by Timoshenko and Gere;1 the difference is that this paper develops a semi-inverse formulation.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

On linear wave motions in magnetic-velocity shears

1977 ◽  
Vol 285 (1328) ◽  
pp. 607-636 ◽  
Keyword(s):  
Differential Equation ◽  
Critical Level ◽  
Velocity Shear ◽  
Linear Wave ◽  
Wave Amplification ◽  
Isothermal Fluid ◽  
Governing Differential Equation ◽  
Boussinesq Fluid ◽  
Propagation Properties ◽  
Background State

The propagation properties of linear wave motions in magnetic and/or velocity shears which vary in one coordinate z (say) are usually governed by a second order linear ordinary differential equation in the independent variable z. It is proved that associated with any such differential equation there always exists a quantity A which is independent of z. By employing A a measure of the intensity of the wave, this result is used to investigate the general propagation properties of hydromagnetic-gravity waves (e.g. critical level absorption, valve effects and wave amplification) in magnetic and/or velocity shears, using a full wave treatment. When variations in the basic state are included, the governing differential equation usually has more singularities than it has in the W.K.B.J. approximation, which neglects all variations in the background state. The study of a wide variety of models shows that critical level behaviour occurs only at the singularities predicted by the W.K.B.J. approximation. Although the solutions of the differential equation are necessarily singular at the irregularities whose presence is solely due to the inclusion of variations in the basic state, the intensity of the wave (as measured by A) is continuous there. Also the valve effect is found to persist whatever the relation between the wavelength of the wave and the scale of variations of the background state. In addition, it is shown that a hydromagnetic-gravity wave incident upon a finite magnetic and/or velocity shear can be amplified (or over-reflected) in the absence of any critical levels within the shear layer. In a Boussinesq fluid rotating uniformly about the vertical, wave amplification can occur if the horizontal vertically sheared flow and magnetic field are perpendicular. In a compressible isothermal fluid, on the other hand, wave amplification not only occurs in both magnetic-velocity and velocity shears but also in a magnetic shear acting alone.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

scholarly journals Even-order differential equation with continuous delay: nonexistence criteria of Kneser solutions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Muhib ◽  
M. Motawi Khashan ◽  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Delay Argument ◽  
Even Order ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.

scholarly journals A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 458
Author(s):  
Leobardo Hernandez-Gonzalez ◽  
Jazmin Ramirez-Hernandez ◽  
Oswaldo Ulises Juarez-Sandoval ◽  
Miguel Angel Olivares-Robles ◽  
Ramon Blanco Sanchez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytic Solution ◽  
Order Differential Equation ◽  
Electric Circuit ◽  
New Approach ◽  
In Doping ◽  
Electric Behavior ◽  
Spatio Temporal ◽  
Physical Phenomena

The electric behavior in semiconductor devices is the result of the electric carriers’ injection and evacuation in the low doping region, N-. The carrier’s dynamic is determined by the ambipolar diffusion equation (ADE), which involves the main physical phenomena in the low doping region. The ADE does not have a direct analytic solution since it is a spatio-temporal second-order differential equation. The numerical solution is the most used, but is inadequate to be integrated into commercial electric circuit simulators. In this paper, an empiric approximation is proposed as the solution of the ADE. The proposed solution was validated using the final equations that were implemented in a simulator; the results were compared with the experimental results in each phase, obtaining a similarity in the current waveforms. Finally, an advantage of the proposed methodology is that the final expressions obtained can be easily implemented in commercial simulators.

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