Storage and Dissipation of Energy in Prabhakar Viscoelasticity

Colombaro, Ivano; Giusti, Andrea; Vitali, Silvia

doi:10.3390/math6020015

Cited by 25 publications

(22 citation statements)

References 36 publications

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“…(2) Mittag-Leffler approximation of the material response resulting in constitutive equations expressed through the Atangana-Baleanu derivatives. (3) Prabhakar kernel as commented by Guisti [36,63,64]. (4) Bessel function based linear viscoelasticity [37][38][39].…”

Section: Some Final Commentsmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

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This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

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Section: Some Final Commentsmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

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“…Although anomalous dielectric relaxation is probably the most apparent exemplification of the need for a fractional theory of calculus based on the Prabhakar function, there exist also several other different physical systems in which this formalism naturally emerges. For instance, this scenario comes up when dealing with certain kind of anomalous diffusion processes [131-136, 139, 141], and when employing the formal duality between dielectric materials and viscoelastic systems [23,41,43]. More on these physical cases can be found in Section 6.…”

Section: Anomalous Physics: a Cry For Helpmentioning

confidence: 99%

“…For details and constraints on the parameter space see [43]. Furthermore, a thorough study of the processes of storage and dissipation of energy in materials described by (6.13) has been carried out by I. Colombaro, A. Giusti, and S. Vitali in [23].…”

Section: Linear Viscoelasticitymentioning

confidence: 99%

A Practical Guide to Prabhakar Fractional Calculus

Giusti

Colombaro

Garra

et al. 2020

Fract Calc Appl Anal

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158

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The Mittag-Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

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“…Therefore,(Equation 73) defines a Caputo-Fabrizio operators with respect to σ (t) and the fractional order β i is related to the scaled retardation timeλ i ( from the spectrum) as In the last two years (2016-2017) an interesting approach was developed by the group around Professor Mainardi [114][115][116][117][118][119] which could be considered as attempts to generalize the relaxation functions in the linear viscoelastic models, an approach also investigated in Colombaro et al [120] and Guisti and Colombaro [121]. The main idea comes from the possibility to represent the relaxation function in a viscoelastic Maxwelltype body by infinite discrete spectrum with times related to the zeros of Bessel functions of the first kind [118,119].…”

Section: Creep Compliance In Terms Of Caputo-fabrizio Operatormentioning

confidence: 99%

Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

Hristov

2018

Front. Phys.

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The study addresses the physical background and modeling of linear viscoelastic response functions and their reasonable relationships to the Caputo-Fabrizio fractional operator via the Prony (Dirichlet series) series decomposition. The problem of interconversion with power-law and exponential (single and multi-term functions) has been discussed. Special attentions have been paid on the Prony series decomposition approach, the related interconversion problems and the expression of the viscoelastic constitutive equations in terms of Caputo-Fabrizio fractional operator.

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Storage and Dissipation of Energy in Prabhakar Viscoelasticity

Cited by 25 publications

References 36 publications

Response functions in linear viscoelastic constitutive equations and related fractional operators

Response functions in linear viscoelastic constitutive equations and related fractional operators

A Practical Guide to Prabhakar Fractional Calculus

Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

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