scholarly journals Fractional Gradient Elasticity from Spatial Dispersion Law

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Power Law ◽  
Lattice Model ◽  
Nonlocal Elasticity ◽  
Spatial Dispersion ◽  
Gradient Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Nonlocal Models ◽  
Special Cases ◽  
Derivatives Of ◽  
The Continuum

Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

scholarly journals Lattice model with power-law spatial dispersion for fractional elasticity

Open Physics ◽  
2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Vasily Tarasov
Keyword(s):  
Power Law ◽  
Lattice Model ◽  
Fractional Integral ◽  
Spatial Dispersion ◽  
Gradient Elasticity ◽  
Continuous Limit ◽  
Elastic Continuum ◽  
Laplacian Operators ◽  
Non Locality ◽  
The Continuum

AbstractA lattice model with a spatial dispersion corresponding to a power-law type is suggested. This model serves as a microscopic model for elastic continuum with power-law non-locality. We prove that the continuous limit maps of the equations for the lattice with the power-law spatial dispersion into the continuum equations with fractional generalizations of the Laplacian operators. The suggested continuum equations, which are obtained from the lattice model, are fractional generalizations of the integral and gradient elasticity models. These equations of fractional elasticity are solved for two special static cases: fractional integral elasticity and fractional gradient elasticity.

scholarly journals Analysis of Sigmoid Functionally Graded Material (S-FGM) Nanoscale Plates Using the Nonlocal Elasticity Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Woo-Young Jung ◽  
Sung-Cheon Han
Keyword(s):  
Elasticity Theory ◽  
Shear Deformation ◽  
Power Law ◽  
Functionally Graded Material ◽  
Nonlocal Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Vibration Response ◽  
First Order ◽  
Graded Material

Based on a nonlocal elasticity theory, a model for sigmoid functionally graded material (S-FGM) nanoscale plate with first-order shear deformation is studied. The material properties of S-FGM nanoscale plate are assumed to vary according to sigmoid function (two power law distribution) of the volume fraction of the constituents. Elastic theory of the sigmoid FGM (S-FGM) nanoscale plate is reformulated using the nonlocal differential constitutive relations of Eringen and first-order shear deformation theory. The equations of motion of the nonlocal theories are derived using Hamilton’s principle. The nonlocal elasticity of Eringen has the ability to capture the small scale effect. The solutions of S-FGM nanoscale plate are presented to illustrate the effect of nonlocal theory on bending and vibration response of the S-FGM nanoscale plates. The effects of nonlocal parameters, power law index, aspect ratio, elastic modulus ratio, side-to-thickness ratio, and loading type on bending and vibration response are investigated. Results of the present theory show a good agreement with the reference solutions. These results can be used for evaluating the reliability of size-dependent S-FGM nanoscale plate models developed in the future.

Pull-In Instability of Electrostatically Actuated Nano-Beams Using the Nonlocal-Gradient Elasticity Theory

2014 ◽  
Vol 488-489 ◽  
pp. 1256-1259
Author(s):  
Jian She Peng ◽  
Liu Yang
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Gradient Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Strain Gradient Theory ◽  
Small Scale ◽  
Gradient Theory ◽  
Beam Model ◽  
Electrostatically Actuated ◽  
Gradient Elasticity Theory

A nonlocal-gradient elasticity beam model with two independent gradient coefficients based on the classical nonlocal elasticity theory and strain gradient theory is used to study the pull-in instability of electrostatically actuated nanobeams. The numerical results show that the pull-in voltage of nanobeams are affected by the small scale. And the two independent gradient coefficients play the different roles in it. This paper broadens the way of studying scale effect.

scholarly journals Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Scale Effects ◽  
Plate Theory ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Phase Portraits ◽  
Small Scale ◽  
Governing System ◽  
Account Due

AbstractIn this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

scholarly journals Rate of Convergence for Ibragimov-Gadjiev-Durrmeyer Operators

2017 ◽  
Vol 50 (1) ◽  
pp. 119-129 ◽  
Author(s):  
Tuncer Acar
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
General Class ◽  
Durrmeyer Operators ◽  
Special Cases ◽  
Derivatives Of

Abstract The present paper deals with the rate of convergence of the general class of Durrmeyer operators, which are generalization of Ibragimov-Gadjiev operators. The special cases of the operators include somewell known operators as particular cases viz. Szász-Mirakyan-Durrmeyer operators, Baskakov-Durrmeyer operators. Herewe estimate the rate of convergence of Ibragimov-Gadjiev-Durrmeyer operators for functions having derivatives of bounded variation.

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