Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

2017 ◽  
Vol 34 (5) ◽  
pp. 1754-1770 ◽  
Author(s):  
Zaher Rahimi ◽  
Wojciech Sumelka ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Calculus ◽  
Free Vibration ◽  
Governing Equation ◽  
Stress Gradient ◽  
Local Theory ◽  
Small Scale ◽  
Content Type ◽  
Non Linear ◽  
Non Local ◽  
Fractional Parameter

Purpose Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems. Design/methodology/approach In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition. Findings The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio. Originality/value A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

Fundamental solutions of two 3D rectangular semi-permeable cracks in transversely isotropic piezoelectric media based on the non-local theory

2020 ◽  
Vol 16 (6) ◽  
pp. 1497-1520
Author(s):  
Haitao Liu ◽  
Liang Wang
Keyword(s):  
Electric Displacement ◽  
Transversely Isotropic ◽  
Elastic Behavior ◽  
Classical Solutions ◽  
Local Theory ◽  
Dual Integral Equations ◽  
Piezoelectric Media ◽  
Content Type ◽  
Present Solution ◽  
Non Local

PurposeThe paper aims to present the non-local theory solution of two three-dimensional (3D) rectangular semi-permeable cracks in transversely isotropic piezoelectric media under a normal stress loading.Design/methodology/approachThe fracture problem is solved by using the non-local theory, the generalized Almansi's theorem and the Schmidt method. By Fourier transform, this problem is formulated as three pairs of dual integral equations, in which the elastic and electric displacements jump across the crack surfaces. Finally, the non-local stress and the non-local electric displacement fields near the crack edges in piezoelectric media are derived.FindingsDifferent from the classical solutions, the present solution exhibits no stress and electric displacement singularities at the crack edges in piezoelectric media.Originality/valueAccording to the literature survey, the electro-elastic behavior of two 3D rectangular cracks in piezoelectric media under the semi-permeable boundary conditions has not been reported by means of the non-local theory so far.

Basic solution of two collinear mode-I cracks in the orthotropic medium by use of the non-local theory

2017 ◽  
Vol 13 (1) ◽  
pp. 100-115 ◽  
Author(s):  
Haitao Liu
Keyword(s):  
Stress Singularity ◽  
Local Theory ◽  
Mode I ◽  
Basic Solution ◽  
Orthotropic Medium ◽  
Dual Integral Equations ◽  
Content Type ◽  
Present Solution ◽  
Crack Tips ◽  
Non Local

Purpose The purpose of this paper is to present the basic solution of two collinear mode-I cracks in the orthotropic medium by the use of the non-local theory. Design/methodology/approach Meanwhile, the generalized Almansi’s theorem and the Schmidt method are used. By the Fourier transform, it is converted to a pair of dual integral equations. Findings Numerical examples are provided to show the effects of the crack length, the distance between the two collinear cracks and the lattice parameter on the stress field near the crack tips in the orthotropic medium. Originality/value The present solution exhibits no stress singularity at the crack tips in the orthotropic medium.

Chaotic Flexural Oscillations of Embedded Non-Local Nanotubes Subjected to Axial Harmonic Force

2017 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Chaotic Behavior ◽  
Scale Effects ◽  
Nonlinear Differential Equations ◽  
Mathematical Formulation ◽  
Time History ◽  
Critical Parameters ◽  
Local Theory ◽  
Small Scale ◽  
External Excitation ◽  
Non Local

Chaotic behavior of an embedded carbon nanotube subjected to an external excitation and the combinational static-dynamic axial loads is investigated. Mathematical formulation has been developed based on the non-local theory in order to reflect the small-scale effects. The tube is supported by the Kelvin-Voigt viscoelastic foundation and the Galerkin method is utilized to solve the governing nonlinear differential equations. The vibration behavior of the system for the parameters of a real model is studied and different vibration responses of the nanotube such as the periodic, quasi-periodic and the chaotic behaviors are detected. The bifurcation diagrams for several critical parameters, including the amplitude of external excitation and the axial applied load are presented. The time history diagram, phase-plane trajectories, and the Poincaré map are presented as the three appropriate techniques for diagnosing the system behavior under various conditions.

Nonlocal Dynamic Stability of DWCNTs Containing Pulsating Viscous Fluid Including Surface Stress and Thermal Effects Based on Sinusoidal Higher Order Shear Deformation Shell Theory

2017 ◽  
Vol 34 (3) ◽  
pp. 387-398 ◽  
Author(s):  
A. Mirafzal ◽  
A. Fereidoon ◽  
E. Andalib
Keyword(s):  
Viscous Fluid ◽  
Shear Deformation ◽  
Dynamic Stability ◽  
Surface Stress ◽  
Shell Theory ◽  
Fluid Velocity ◽  
Higher Order ◽  
Local Theory ◽  
Small Scale ◽  
Non Local

AbstractIn this study, the dynamic stability of double-walled carbon nanotubes (DWCNTs) conveying pulsating viscous fluid located on visco-Pasternak medium is investigated. The effects of small scale are considered using Eringen's non-local theory. The surrounding medium of DWCNT is anticipated as a visco-Pasternak foundation including the normal, shear, and damping forces. The van der Waals (vdWs) force is considered among the inner and outer layers of DWCNT. Three types of temperature changes, i.e. uniform, linear and sinusoidal temperature rise, through the thickness are studied. The equations of motion are obtained using higher order sinusoidal shear deformation shell theory (SSDT), in which the surface effects are included. Dynamical equations of DWCNT are extracted using the energy method and Hamilton's principle. Due to orthogonal conditions, the governing equations are solved based on Bolotin method. The effects of different parameters such as surface stress, non-local parameter, fluid velocity, Knudsen's number, fluid density, dimensions ratio and various temperature loadings on the dynamic stability regions of DWCNTs are discussed.

Non-local elastic plate theories

2007 ◽  
Vol 463 (2088) ◽  
pp. 3225-3240 ◽  
Author(s):  
Pin Lu ◽  
P.Q Zhang ◽  
H.P Lee ◽  
C.M Wang ◽  
J.N Reddy
Keyword(s):  
Free Vibration ◽  
Mindlin Plate ◽  
Small Scale ◽  
Plate Model ◽  
Simply Supported ◽  
Small Scale Effect ◽  
Non Local ◽  
Vibration Problems ◽  
Local Solutions ◽  
Basic Equations

A non-local plate model is proposed based on Eringen's theory of non-local continuum mechanics. The basic equations for the non-local Kirchhoff and the Mindlin plate theories are derived. These non-local plate theories allow for the small-scale effect which becomes significant when dealing with micro-/nanoscale plate-like structures. As illustrative examples, the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non-local solutions are discussed in relation to their corresponding local solutions.

scholarly journals Computational analysis of an infinite magneto-thermoelastic solid periodically dispersed with varying heat flow based on non-local Moore–Gibson–Thompson approach

2021 ◽  
Author(s):  
Ahmed E. Abouelregal ◽  
Hamid Mohammad-Sedighi ◽  
Ali H. Shirazi ◽  
Mohammad Malikan ◽  
Victor A. Eremeyev
Keyword(s):  
Magnetic Field ◽  
Heat Flow ◽  
Computational Analysis ◽  
Local Theory ◽  
Small Scale ◽  
Special Cases ◽  
The Laplace Transform ◽  
Non Local ◽  
Thermoelastic Solid ◽  
Periodic Heat

AbstractIn this investigation, a computational analysis is conducted to study a magneto-thermoelastic problem for an isotropic perfectly conducting half-space medium. The medium is subjected to a periodic heat flow in the presence of a continuous longitude magnetic field. Based on Moore–Gibson–Thompson equation, a new generalized model has been investigated to address the considered problem. The introduced model can be formulated by combining the Green–Naghdi Type III and Lord–Shulman models. Eringen’s non-local theory has also been applied to demonstrate the effect of thermoelastic materials which depends on small scale. Some special cases as well as previous thermoelasticity models are deduced from the presented approach. In the domain of the Laplace transform, the system of equations is expressed and the problem is solved using state space method. The converted physical expressions are numerically reversed by Zakian’s computational algorithm. The analysis indicates the significant influence on field variables of non-local modulus and magnetic field with larger values. Moreover, with the established literature, the numerical results are satisfactorily examined.

Differential quadrature solutions for the nonconservative instability of a class of single-walled carbon nanotubes

2018 ◽  
Vol 35 (1) ◽  
pp. 251-267 ◽  
Author(s):  
Maria Anna De Rosa ◽  
Maria Lippiello ◽  
Stefania Tomasiello
Keyword(s):  
Free Vibration ◽  
Scale Effects ◽  
Nonlocal Parameter ◽  
Added Mass ◽  
Differential Quadrature ◽  
Small Scale ◽  
Numerical Schemes ◽  
Content Type ◽  
Single Walled Carbon ◽  
Integration Schemes

Purpose The purpose of the present paper is to investigate the nonconservative instability of a single-walled carbon nanotube (SWCNT) with an added mass through nonlocal theories. The governing equations are discretized by means of the differential quadrature (DQ) rules, as introduced by Bellman and Casti. DQ rules have been largely used in engineering and applied sciences. Recently, they were applied to enhance some numerical schemes, such as step-by-step integration schemes and Picard-like numerical schemes. Design/methodology/approach In the present paper, the DQ rules are used to investigate the nonconservative instability of a SWCNT through nonlocal theories. Findings To show the sensitivity of the SWCNT to the values of added mass and the influence of nonlocal parameter on the fundamental frequencies values, some numerical examples have been performed and discussed. Yet, the effect of the different boundary conditions on the instability behaviour has been investigated. The validity of the present model has been confirmed by comparing some results against the ones available in literature. Originality/value Applying the nonlocal elasticity theory, this paper presents a re-formulation of Hamilton’s principle for the free vibration analysis of a uniform Euler–Bernoulli nanobeam. The main purpose of this paper is to investigate the free vibration response of an SWCNT with attached mass and for various values of small scale effects.

scholarly journals Free vibration of rectangular nanoplate strips

2019 ◽  
Author(s):  
Jaan Lellep ◽  
Mainul Hossain
Keyword(s):  
Free Vibration ◽  
Cross Sections ◽  
Vibration Analysis ◽  
Theory Of Elasticity ◽  
Local Theory ◽  
Natural Vibrations ◽  
Size Dependent ◽  
Nano Structures ◽  
Non Local ◽  
Local Compliance

Natural vibrations of nanobeams and nanosheets are investigated with the help of nonlocal theories of elasticity. The vibration analysis is based on the size-dependent non-local theory of elasticity developed by A. C. Eringen. It is assumed that the nano-structures under consideration have rectangular cross sections with piece wise constant dimensions and that the nanoplates are weakened with defects. The influence of the crack on the vibration of the nanoplate is assessed with the aid of additional local compliance developed in previous papers. Numerical results are presented for one- and two-stepped nanoplates.

On Free Vibration of Functionally Graded Euler-Bernoulli Beam Models Based on the Non-Local Theory

2012 ◽  
Author(s):  
Reza Moheimani ◽  
M. T. Ahmadian
Keyword(s):  
Free Vibration ◽  
Axial Load ◽  
Power Index ◽  
Theory Of Elasticity ◽  
Functionally Graded ◽  
Local Theory ◽  
Local Parameter ◽  
Bernoulli Beam ◽  
Non Local ◽  
Euler Bernoulli Beam

In this paper, the governing equations and boundary conditions of a functionally graded Euler-Bernoulli beam are developed based on the non-local theory of elasticity. Afterward, the free vibration is investigated and the effects of the axial load, the non-local parameter and the power index on the natural frequency of a hinged-hinged beam is assessed. The results indicate that the non-local parameter has a decreasing effect on the frequency while the power index has an increasing effect. It is also noted that the effect of the axial load is increasing too.

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