Modelling of AAA in the framework of time-fractional damage hyperelasticity

Sumelka, Wojciech; Łuczak, Bartosz; Gajewski, Tomasz; Voyiadjis, George Z.

doi:10.1016/j.ijsolstr.2020.08.015

Cited by 43 publications

(14 citation statements)

References 57 publications

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“…In many applications, only a small modification of the medium is considered [ 16 ]. In this study, we focus on the frequency response of the modeled medium to monochromatic waves; the above-mentioned hereditary properties are more pronounced in the time domain studies, especially in the context of mechanical stress–strain relations and viscoelasticity [ 5 , 11 , 13 , 28 ]. In the field of electromagnetism, Gomez [ 14 ] has studied the step response of the fractional-derivative Drude model in the time domain, discussing the memory effects.…”

Section: Resultsmentioning

confidence: 99%

Fractional Derivative Modification of Drude Model

Karpiński

Zielińska-Raczyńska

Ziemkiewicz

2021

Sensors

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A novel, two-parameter modification of a Drude model, based on fractional time derivatives, is presented. The dielectric susceptibility is calculated analytically and simulated numerically, showing good agreement between theoretical description and numerical results. The absorption coefficient and wave vector are shown to follow a power law in the frequency domain, which is a common phenomenon in electromagnetic and acoustic wave propagation in complex media such as biological tissues. The main novelty of the proposal is the introduction of two separate parameters that provide a more flexible model than most other approaches found in the literature. Moreover, an efficient numerical implementation of the model is presented and its accuracy and stability are examined. Finally, the model is applied to an exemplary soft tissue, confirming its flexibility and usefulness in the context of medical biosensors.

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Section: Resultsmentioning

confidence: 99%

Fractional Derivative Modification of Drude Model

Karpiński

Zielińska-Raczyńska

Ziemkiewicz

2021

Sensors

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“…Thus, we obtain: By (13), the implicit nonlinear VOFBVP ( 12) is divided into two expressions as follows:…”

Section: Numerical Examplementioning

confidence: 99%

“…Fractional calculus has recently been discussed in various research works in multidisciplinary sciences due to its powerful applicability in modeling various scientific phenomena due to the property of the nonlocality and memory effect that some physical systems exhibit. Therefore, some interesting research works concerning the mathematical analysis and applications of fractional calculus have been discussed in [1][2][3][4][5][6][7][8][9][10][11][12][13]. The fractional calculus of variable order extends the theory of the constant order one.…”

Section: Introductionmentioning

confidence: 99%

Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings

et al. 2021

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In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.

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“…We mention the possibility of short-memory application of (1), referring the readers to the works [18] (model of abdominal aortic aneurysm phenomena), [19] (Noether's theorems of fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives), and the comprehensive book [20]. Therefore, as potential real-life applications of Equation (1), we bring (new) epidemiological systems via Caputo fractional order operator which is both a nonlocal operator and possesses features related to memory of the epidemic (see, for example [21]).…”

Section: Introductionmentioning

confidence: 99%

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

2021

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Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

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Modelling of AAA in the framework of time-fractional damage hyperelasticity

Cited by 43 publications

References 57 publications

Fractional Derivative Modification of Drude Model

Fractional Derivative Modification of Drude Model

Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

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