Solvability and microlocal analysis of the fractional Eringen wave equation

Hörmann, Günther; Oparnica, Ljubica; Zorica, Dušan

doi:10.1177/1081286517726371

Cited by 3 publications

(5 citation statements)

References 13 publications

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Section: Introductionsupporting

confidence: 67%

“…In particular, it is found that both solid-like and fluid-like materials can have either infinite or finite wave speed. Singularity propagation properties of the memory and non-local type fractional wave equations are investigated in [17,18] using the tools of microlocal analysis, supporting the results obtained in [20]. Wave propagation phenomena in viscoelastic bodies, modeled by integer and fractional order models, including the question of wave speed and energy dissipation properties are analyzed in [8,9].…”

Section: Introductionsupporting

confidence: 56%

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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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Section: Introductionsupporting

confidence: 67%

Section: Introductionsupporting

confidence: 56%

Fractional Burgers wave equation

2019

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“…Generalizing the integer-order Eringen stress gradient non-local constitutive law, the fractional Eringen model (1.7) is postulated in [12], where the optimal values of the non-locality parameter and order of fractional differentiation are obtained with respect to the Born-Kármán model of lattice dynamics. Furthermore, wave propagation, as well as propagation of singularities, in nonlocal material described by the fractional Eringen model (1.7) is analysed in [13].…”

Section: Introductionmentioning

confidence: 99%

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

Zorica

Oparnica

2020

Phil. Trans. R. Soc. A.

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Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of motion, strain and fractional order constitutive models, that include the distributed-order constitutive law in which the integration is performed from zero to one generalizing all linear constitutive models of fractional and integer orders, as well as for the thermodynamically consistent fractional Burgers models, where the orders of fractional differentiation are up to the second order. In the case of non-local fractional wave equations, obtained using non-local constitutive models of Hooke- and Eringen-type in addition to the equation of motion and strain, a priori energy estimates yield the energy conservation, with the reinterpreted notion of the potential energy. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

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“…obtained by the Laplace transform inversion in (22) 1 and three expressed in terms of creep compliance…”

Section: Modelmentioning

confidence: 99%

“…Generalizing the integer-order Eringen stress gradient non-local constitutive law, the fractional Eringen model ( 7) is postulated in [11], where the optimal values of non-locality parameter and order of fractional differentiation are obtained with respect to the Born-Kármán model of lattice dynamics. Further, wave propagation, as well as propagation of singularities, in non-local material described by the fractional Eringen model ( 7) is analyzed in [22].…”

Section: Introductionmentioning

confidence: 99%

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

Zorica,

Oparnica

2019

Preprint

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Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of motion, strain, and fractional order constitutive models, that include the distributed-order constitutive law in which the integration is performed from zero to one generalizing all linear constitutive models of fractional and integer orders, as well as for the thermodynamically consistent fractional Burgers models, where the orders of fractional differentiation are up to the second order. In the case of non-local fractional wave equations, obtained using non-local constitutive models of Hooke-and Eringen-type in addition to the equation of motion and strain, a priori energy estimates yield the energy conservation, with the reinterpreted notion of the potential energy.

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Solvability and microlocal analysis of the fractional Eringen wave equation

Cited by 3 publications

References 13 publications

Fractional Burgers wave equation

Fractional Burgers wave equation

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations

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