Dynamic Stability of Axially Loaded Nonlocal Rod on Generalized Pasternak Foundation

Zorica, Dušan; Atanacković, Teodor M.; Vrcelj, Zora; Novakovic, Branislava N.

doi:10.1061/(asce)em.1943-7889.0001090

Cited by 12 publications

(4 citation statements)

References 38 publications

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“…α 0 (|x − x | , e 1 a) and α 1 (|x − x | , e 1 a) are the nonlocal functions for the classical stress tensor and the strain gradient stress tensor, respectively. The nonlocal strain gradient theory states the total stress tensor accounts for not only nonlocal stress tensor but also strain gradient stress tensor [38] t i j = σ i j − ∇σ (1) i jm (2) Here ∇ is the Laplacian operator.…”

Section: Nonlocal Strain Gradient Theorymentioning

confidence: 99%

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Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

2016

International Journal of Mechanical Sciences

225

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Section: Nonlocal Strain Gradient Theorymentioning

confidence: 99%

“…It has been shown by many studies that the nonlocal effect potentially plays a important role in studying the sizedependent effect on the vibration of rods [2][3][4][5], beams [6][7][8][9][10][11][12][13] and plates [14][15][16][17]. Most works studied these size-dependent effects based on the nonlocal model in differential form.…”

Section: Introductionmentioning

confidence: 98%

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

2016

International Journal of Mechanical Sciences

225

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“…The choice of

and

, which is a generalization of the standard Zener model, was proposed in [ 235 ] and used in [ 245 ] to study the lateral vibrations of viscoelastic rod. DO models were also used to analyze the influence of viscoelastic foundations on the dynamic stability of local and nonlocal rods [ 246 ]. Similarly, Varghaei et al [ 247 ] investigated the nonlinear vibration of viscoelastic beams using a generalized Kelvin–Voigt model implemented via DO derivatives.…”

Section: Applications To Viscoelasticitymentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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“…Moreover, Zhang et al [13] applied Eringen's elasticity theory to present the scale parameter coefficients for the natural vibrations and buckling behavior of simply supported rectangular plates. Further studies about this topic were examined by other authors [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

2020

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The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

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Dynamic Stability of Axially Loaded Nonlocal Rod on Generalized Pasternak Foundation

Cited by 12 publications

References 38 publications

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

Applications of Distributed-Order Fractional Operators: A Review

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

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