A new method for solving dynamic problems of fractional derivative viscoelasticity

Rossikhin, Yury A.; Shitikova, Marina V.

doi:10.1016/s0020-7225(00)00025-2

Cited by 121 publications

(57 citation statements)

References 9 publications

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“…(3) is not so simple (Suarez and Shokooh (1995), Rossikhin and Shitnikova (2001)). To illustrate difficulties emerging in the evaluation of the inverse relationship (3) the simpler case of m = 2 and β 1 = 0 (C(β 1 ) = E) is taken into consideration.…”

Section: −β1mentioning

confidence: 99%

Fractional differential equations and related exact mechanical models

Paola

Pinnola

Zingales

2013

Computers & Mathematics with Applications

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Creep and relaxation tests, performed on various materials like polymers, rubbers and so on are well-fitted by power-laws with exponent β ∈ [0, 1] (Nutting (1921), Di Paola et al. (2011)). The consequence of this observation is that the stress-strain relation of hereditary materials is ruled by fractional operators (Scott Blair (1947), Slonimsky (1961). A large amount of researches have been performed in the second part of the last century with the aim to connect constitutive fractional relations with some mechanical models by means of fractance trees and ladders (see Podlubny (1999)). Recently, Di Paola and Zingales (2012) proposed a mechanical model that corresponds to fractional stress-strain relation with any real exponent and they have proposed a description of above model ). In this study the authors aim to extend the study to cases with more fractional phases and to fractional Kelvin-Voigt model of hereditariness.

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Section: −β1mentioning

confidence: 99%

Fractional differential equations and related exact mechanical models

Paola

Pinnola

Zingales

2013

Computers & Mathematics with Applications

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“…A large variety of such fluids and their industrial applications has represented a major motivation for research in non-Newtonian flows. In order to describe the rheological properties of a large class of materials, the rheological constitutive equations with fractional derivatives have been introduced and discussed for a long time by Bagley [1], Friedrich [2], Glöckle and Nonnenmacher [3], Mainardi [4] and Rossikhin and Shitikova [5,6]. Their results seem to be in good agreement with the experimental data.…”

Section: Introductionmentioning

confidence: 84%

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

Vieru

Athar

Fetecău

2008

Z. angew. Math. Phys.

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The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag-Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate. (2000). 76A05. Mathematics Subject Classification

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“…where 0 D β t is the Riemann-Liouville fractional derivative defined by Podlubny [7] (6) in which (·) is the Gamma function.…”

Section: The Governing Equationmentioning

confidence: 99%

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

Kang

Ming-yu

2008

Acta Mech Sin

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The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.

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A new method for solving dynamic problems of fractional derivative viscoelasticity

Cited by 121 publications

References 9 publications

Fractional differential equations and related exact mechanical models

Fractional differential equations and related exact mechanical models

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

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