A generalised fractional derivative approach to viscoelastic material properties measurement

Espíndola, José João de; Neto, João Morais da Silva; Lopes, Eduardo Márcio de Oliveira

doi:10.1016/j.amc.2004.06.099

Cited by 58 publications

(28 citation statements)

References 8 publications

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“…In the last 30 years, the use of the fractional derivatives to mimic the effects of the memory is largely used in electromagnetism (Jacquelin 1984), biology (Cesarone 2002), chaos (e.g., Mainardi 1996), economy (e.g., Caputo and Kolari 2001), and the rheology properties of solids (Le Mehaute and Crepy 1983;Bagley and Torvik 1986;De Espindola et al 2005;Adolfsson et al 2005). Concerning this last point, lately the memory, i.e., the history of the deformation and fractures of a solid under stress, is largely used.…”

Section: Introductionmentioning

confidence: 99%

Flux in Porous Media with Memory: Models and Experiments

2009

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The classic constitutive equation relating fluid flux to a gradient in potential (pressure head plus gravitational energy) through a porous medium was discovered by Darcy in the mid 1800s. This law states that the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix may cause a local variation of the permeability. For example, the flow may perturb the porous formation by causing particle migration resulting in pore clogging or chemically reacting with the medium to enlarge the pores or diminish the size of the pores. In order to adequately represent these phenomena, we modify the constitutive equations by introducing a memory formalism operating on both the pressure gradient-flux and the pressure-density variations. The memory formalism is then represented with fractional order derivatives. We perform a number of laboratory experiments in uniformly packed columns where a constant pressure is applied on the lower boundary. Both homogeneous and heterogeneous media of different characteristic particle size dimension were employed. The low value assumed by the memory parameters, and in particular by the fractional order, demonstrates that memory is largely influencing the experiments. The data and theory show how mechanical compaction can decrease permeability, and consequently flux

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Section: Introductionmentioning

confidence: 99%

Flux in Porous Media with Memory: Models and Experiments

2009

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“…It has been shown that the fractional order models of real systems are regularly more adequate than usually used integer order models. There are many systems where the use of fractional differential equations have turned out to be useful, for example systems involved with phenomena such as: viscoelasticity [7], dielectric polarization, electrode-electrolite polarization, stabilization using fractionalorder controllers [19] and electromagnetic waves [8].…”

Section: Introductionmentioning

confidence: 99%

Synchronization of nonlinear fractional order systems by means of PIrα reduced order observer

Cruz-Victoria

Martínez-Guerra

Pérez-Pinacho

et al. 2015

Applied Mathematics and Computation

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“…In order to numerically differentiate these levels, the loss factor peak will be used as damping index. As known [53,54,55,56,57], the loss factor of a frequency-domain constitutive equation is a real-valued and frequency-dependent function defined as the quotient between the imaginary and the real part of the complex stiffness. Here, the complex stiffness associated with the viscoelastic links of the structure shown in Fig.…”

Section: Example 2: Multiple Degree-of-freedom Systemsmentioning

confidence: 99%

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Lázaro

Pérez-Aparicio

Epstein

2012

Applied Mathematics and Computation

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Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly nonproportional viscoelastic structures is developed. The central idea is the construction of two complex-valued functions of a complex variable, whose fixed points are precisely the eigenvalues. This important property allows the use of these functions in a fixed-point iterative scheme. With help of some results in Fixed Point Theory, necessary conditions for global and local convergence are provided. It is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping. In addition, under certain conditions the recursive method can also be used for the calculation of non-viscous real eigenvalues. In order to validate the mathematical results, two numerical examples are analyzed, one for single degree-of-freedom systems and another for multiple ones.

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A generalised fractional derivative approach to viscoelastic material properties measurement

Cited by 58 publications

References 8 publications

Flux in Porous Media with Memory: Models and Experiments

Flux in Porous Media with Memory: Models and Experiments

Synchronization of nonlinear fractional order systems by means of PIrα reduced order observer

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

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