Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models

Palade, Liviu Iulian; Attané, P.; Huilgol, R. R.; Mena, B.

doi:10.1016/s0020-7225(98)00080-9

Cited by 74 publications

(44 citation statements)

References 29 publications

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“…Using the notation, we observe from [12] that the composition rule for integration and differentiation obeys the simple form d p d q dp+q dtP dtq --dtP +q (4) for all numbers p and q, whether they are positive or negative.…”

Section: The Fractional Maxwell Model and Basic Equationsmentioning

confidence: 99%

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

2002

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The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. The flow near a wall suddenly set in motion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model. Exact solutions of velocity and stress are obtained by using the discrete inverse Laplace transform of the sequential fractional derivatives. It is found that the effect of the fractionM orders in the constitutive relationship on the flow field is significant. The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate, for large times the viscoelastic effects become weak.

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Section: The Fractional Maxwell Model and Basic Equationsmentioning

confidence: 99%

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

2002

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“…Anomalous diffusion has an important role in the literature to describe many physical phenomena, crowded systems or diffusion through porous media. On the other hand this phenomena is observed in heat baths [2], diffusion through also porous material [3,4], nuclear magnetic resonance diffusometry in disordered materials [4], behaviour of polymers in a glass transition [5,6]and also particle dynamics inside polymer network [7].Fractional order linear and nonlinear differential equations were examined by some researchers by using different methods [8][9][10][11][12]. In this study we consider the fractional subdiffusion equation in the following form [13]: Recently, because of their practical applications, anomalous subdiffusion equation has received much attention.…”

Section: Introductionmentioning

confidence: 99%

Numerical approach for solving time fractional diffusion equation

Koç

Gülsu

2017

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In this article one of the fractional partial differential equations was solved by finite difference scheme based on five point and three point central space method with discretization in time. We use between the Caputo and the Riemann-Liouville derivative definition and the Grünwald-Letnikov operator for the fractional calculus. The stability analysis of this scheme is examined by using von-Neumann method. A comparison between exact solutions and numerical solutions is made. Some figures and tables are included.

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“…The question of stability of constitutive equations containing fractional derivatives was raised by Palade et al [17], who showed that relaxation of an initial perturbation cannot be expressed in terms of a combination of exponential relaxations, questioning the usefulness of fractional calculus constitutive equations. However, this arises because of the self-similarity underlying fractional derivative descriptions, which breaks down on the scale of whole polymer chains.…”

Section: Introductionmentioning

confidence: 99%

Fractional Calculus Description of Non-Linear Viscoelastic Behaviour of Polymers

Heymans

2004

Nonlinear Dyn

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In recent decades, constitutive equations for polymers involving fractional calculus have been the object of ever increasing interest, due to their special suitability for describing self-similarity and memory effects, which are typical of viscoelastic behaviour in polymers. Thermodynamic validity of these equations can be ensured by obtaining them from analog models containing spring-pots with positive front factors. Failure of self-similarity in real polymers at short (local) and long (whole chain) scales has been addressed previously. In the past, interest in fractional differential descriptions of polymer viscoelasticity has been mainly concerned with linear viscoelasticity, despite the fact that in processing and end use conditions are largely in the non-linear range. In this paper, extension of fractional calculus models to the non-linear range of viscoelasticity is attempted, by accounting for stress activation of deformation and strain acceleration of annealing. Calculated stress-strain curves are compared with experimental results on an amorphous polymer (polycarbonate). The model adequately describes the general trends of yield and post-yield behaviour, but does not properly describe the gentle approach to yield observed experimentally.

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Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models

Cited by 74 publications

References 29 publications

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model

Numerical approach for solving time fractional diffusion equation

Fractional Calculus Description of Non-Linear Viscoelastic Behaviour of Polymers

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