Formulation of thermodynamically consistent fractional Burgers models

Okuka, Aleksandar; Zorica, Dušan

doi:10.1007/s00707-018-2198-z

Cited by 14 publications

(15 citation statements)

References 40 publications

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“…where in (33) the highest order of fractional differentiation of strain is µ + η ∈ [1, 2] with η ∈ {α, β} , while the highest order of fractional differentiation of stress is either γ ∈ [0, 1] in the case of Model I (4), with (10), and (12), while in (34) one has 0 ≤ α ≤ β ≤ 1 and β + η ∈ [1,2] , with η = α in the case of Model VI (14) and η = β in the case of Model VII (16), while Model VIII (18) is obtained for η = β = α,ā 1 = a 1 + a 2 , andā 2 = a 3 .…”

Section: Fractional Burgers Models: Creep and Stress Relaxation Testsmentioning

confidence: 99%

“…Model VII, given by (16) and subject to thermodynamical restrictions (17), is obtained from the unified model (34) for η = β.…”

Section: Model VIImentioning

confidence: 99%

“…t −(ξ−n) Γ (1 − (ξ − n)) * y (t) , t > 0, see [11], with * denoting the convolution in time: f (t) * g (t) = t 0 f (u) g (t − u) du, t > 0. Thermodynamical consistency analysis of the fractional Burgers model (3), conducted in [16] by the use of storage and loss modulus non-negativity requirement, implied that the orders of fractional derivatives from interval [1,2] cannot be independent of the orders of fractional derivatives from interval [0, 1] , and this led to formulation of eight thermodynamically consistent fractional Burgers models.…”

Section: Introductionmentioning

confidence: 99%

“…were extensively used in modeling various fluid flow problems, see references in [16]. The thermodynamical consistency of the generalized fractional Burgers model for viscoelastic fluid…”

Section: Introductionmentioning

confidence: 99%

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Fractional Burgers models in creep and stress relaxation tests

Okuka

Zorica

2020

Applied Mathematical Modelling

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Classical and thermodynamically consistent fractional Burgers models are examined in creep and stress relaxation tests. Using the Laplace transform method, the creep compliance and relaxation modulus are obtained in integral form, that yielded, when compared to the thermodynamical requirements, the narrower range of model parameters in which the creep compliance is a Bernstein function while the relaxation modulus is completely monotonic. Moreover, the relaxation modulus may even be oscillatory function with decreasing amplitude. The asymptotic analysis of the creep compliance and relaxation modulus is performed near the initial time-instant and for large time as well.

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Section: Fractional Burgers Models: Creep and Stress Relaxation Testsmentioning

confidence: 99%

“…Model VII, given by (16) and subject to thermodynamical restrictions (17), is obtained from the unified model (34) for η = β.…”

Section: Model VIImentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…were extensively used in modeling various fluid flow problems, see references in [16]. The thermodynamical consistency of the generalized fractional Burgers model for viscoelastic fluid…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractional Burgers models in creep and stress relaxation tests

Okuka

Zorica

2020

Applied Mathematical Modelling

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“…Considering the rheological scheme of the classical Burgers model, with the dash-pot element replaced by the Scott-Blair (fractional) element, the fractional Burgers model (3) is derived in [27]. Moreover, using the requirement of storage and loss modulus non-negativity, the analysis of thermodynamical consistency for fractional Burgers model (3), conducted in [27], yielded that the orders of fractional derivatives γ, ν ∈ [1,2] cannot be independent of the orders of fractional derivatives α, β, µ ∈ [0, 1] , and this led to formulation of eight thermodynamically consistent fractional Burgers models, divided into two classes.…”

Section: Introductionmentioning

confidence: 99%

Fractional Burgers wave equation

2019

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Thermodynamically consistent fractional Burgers constitutive models for viscoelastic media, divided into two classes according to model behavior in stress relaxation and creep tests near the initial time instant, are coupled with the equation of motion and strain forming the fractional Burgers wave equations. Cauchy problem is solved for both classes of Burgers models using integral transform method and analytical solution is obtained as a convolution of the solution kernels and initial data. The form of solution kernel is found to be dependent on model parameters, while its support properties implied infinite wave propagation speed for the first class and finite for the second class. Spatial profiles corresponding to the initial Dirac delta displacement with zero initial velocity display features which are not expected in wave propagation behavior.

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Stress and power as a response to harmonic excitation of a fractional anti‐Zener and Zener type viscoelastic body

Jelić,

Zorica

2024

Z Angew Math Mech

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The stress as a response to strain prescribed as a harmonic excitation is examined in both transient and steady state regime for the viscoelastic body modeled by thermodynamically consistent fractional anti‐Zener and Zener models by the use of the Laplace transform method. Assuming strain as a sine function, the time evolution of power per unit volume, previously derived as a sum of time derivative of a conserved term, which represents the rate of change of stored energy, and a dissipative term, which represents dissipated power, is investigated when expressed through the relaxation modulus and creep compliance. Further, two forms of energy and two forms of dissipated power per unit volume are examined in order to see whether they coincide.

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Formulation of thermodynamically consistent fractional Burgers models

Cited by 14 publications

References 40 publications

Fractional Burgers models in creep and stress relaxation tests

Fractional Burgers models in creep and stress relaxation tests

Fractional Burgers wave equation

Stress and power as a response to harmonic excitation of a fractional anti‐Zener and Zener type viscoelastic body

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