Rheological models containing fractional derivatives

Smit, W. M.; Vries, H. de

doi:10.1007/bf01985463

Cited by 126 publications

(51 citation statements)

References 5 publications

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“…On the other hand Nutting (1921) observed that the creep or the relaxation test performed on any real material follow a power law kernel instead of an exponential one. Based on this observation the constitutive law of any real material, including rubber, glass, asphalt mixture, is ruled by a fractional operator (Koeller 1984;Bagley, Torvik 1983, 1986Slonimsky 1967;Smit, de Vrie 1970;Soczkiewicz 2002;Di Paola et al 2011). The characterization of the real material by means of fractional derivative and integrals of real order produces strong variations on the response with respect to the characterization involving derivatives (or integrals) of integer order.…”

Section: Introductionmentioning

confidence: 99%

Linear and Nonlinear Fractional Hereditary Constitutive Laws of Asphalt Mixtures

Mino

Airey

Paola

et al. 2016

Journal of Civil Engineering and Management

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Abstract. The aim of this paper is to propose a fractional viscoelastic and viscoplastic model of asphalt mixtures using experimental data of several tests such as creep and creep recovery performed at different temperatures and at different stress levels. From a best fitting procedure it is shown that both the creep one and recovery curve follow a power law model. It is shown that the suitable model for asphalt mixtures is a dashpot and a fractional element arranged in series. The proposed model is also available outside of the linear domain but in this case the parameters of the model depend on the stress level.

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Section: Introductionmentioning

confidence: 99%

Linear and Nonlinear Fractional Hereditary Constitutive Laws of Asphalt Mixtures

Mino

Airey

Paola

et al. 2016

Journal of Civil Engineering and Management

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“…Many authors have suggested other dynamic shear modulus models, For instance, following the pioneering work of Smit and deVires [22] and Bagley and Torvik [23], fractional Maxwell models have gained popularity. Those models are often used in conjunction with analytical solutions for simple flows (mostly in the form of Mittag-Leffler functions), see for instance Enelund and Olsson [24], Mainardia and Goren [25], Herna'ndez-Jime'nez et al [26] and Wenchanga et al [27].…”

Section: Fractional-power Rheological Modelmentioning

confidence: 99%

Numerical Laplace inversion in rheological characterization

Abate

2004

Journal of Non-Newtonian Fluid Mechanics

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“…Using the phenomenological description of linear viscoelastic behavior, either the conventional dashpot is replaced with a fractional-order derivative dependent dashpot ("springpot") [8,9] or the spring and dashpot are replaced with a single composite element [6]. An example of a simple fractional derivative rheological equation is the "intermediate model" suggested by Smit and De Vries [10] …”

Section: Introductionmentioning

confidence: 99%

A variable order constitutive relation for viscoelasticity

Ramirez

Coimbra²

2007

Ann. Phys.

127

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A constitutive relation for linear viscoelasticity of composite materials is formulated using the novel concept of Variable Order (VO) differintegrals. In the model proposed in this work, the order of the derivative is allowed to be a function of the independent variable (time), rather than a constant of arbitrary order. We generalize previous works that used fractional derivatives for the stress and strain relationship by allowing a continuous spectrum of non-integer dynamics to describe the physical problem. Starting with the assumption that the order of the derivative is a measure of the rate of change of disorder within the material, we develop a statistical mechanical model that is in agreement with experimental results for strain rates varying more than eight orders of magnitude in value. We use experimental data for an epoxy resin and a carbon/epoxy composite undergoing constant compression rates in order to derive a VO constitutive equation that accurately models the linear viscoelastic deformation in time. The resulting dimensionless constitutive equation agrees well with all the normalized data while using a much smaller number of empirical coefficients when compared to available models in the literature.

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Rheological models containing fractional derivatives

Cited by 126 publications

References 5 publications

Linear and Nonlinear Fractional Hereditary Constitutive Laws of Asphalt Mixtures

Linear and Nonlinear Fractional Hereditary Constitutive Laws of Asphalt Mixtures

Numerical Laplace inversion in rheological characterization

A variable order constitutive relation for viscoelasticity

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