One-dimensional dispersion phenomena in terms of fractional media

Sumelka, Wojciech; Zaera, R.; Fernández-Sáez, J.

doi:10.1140/epjp/i2016-16320-3

Cited by 7 publications

(6 citation statements)

References 36 publications

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“…The foundations for dynamic equilibrium for 1D sFCM body were formulated for a constant length scale in [34,35]. Subsequently, after introducing variable length scale concept to sFCM in [36], an extension of [34,35] was proposed in [32].…”

Section: Dynamic Equilibrium For 1d Sfcm Body Including Spatially Var...mentioning

confidence: 99%

Designing of Dynamic Spectrum Shifting in Terms of Non-Local Space-Fractional Mechanics

et al. 2021

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In this paper, the applicability of the space-fractional non-local formulation (sFCM) to design 1D material bodies with a specific dynamic eigenvalue spectrum is discussed. Such a formulated problem is based on the proper spatial distribution of material length scale, which maps the information about the underlying microstructure (it is important that the material length scale is one of two additional material parameters of sFCM compared to the classical local continuum mechanics—the second one, the order of fractional continua—is treated herein as a scaling parameter only). Technically, the design process for finding adequate length scale distribution is not trivial and requires the use of an inverse optimization procedure. In the analysis, the objective function considers a subset of eigenvalues reduced to a single value based on Kreisselmeier–Steinhauser formula. It is crucial that the total number of eigenvalues considered must be smaller than the limit which comes from the ratio of the sFCM length scale to the length of the material body.

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Section: Dynamic Equilibrium For 1d Sfcm Body Including Spatially Var...mentioning

confidence: 99%

Designing of Dynamic Spectrum Shifting in Terms of Non-Local Space-Fractional Mechanics

et al. 2021

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“…It is important that this concept was validated with experimental result in Sumelka et al (2015a), where the s-FCM concept was used to mimic the behaviour of micro-beams made of the polymer SU-8 (where strong scale effect was observed), and furthermore, in Sumelka et al (2016) it was presented that s-FCM can correctly mimic the Born-Von Karman (BK) lattice (discrete system). This result are crucial for correctness of physical interpretation of the results obtained below.…”

Section: D Inhomogeneous Fractional Elasticitymentioning

confidence: 99%

Identification of mechanical properties of 1D deteriorated non-local bodies

Szajek

Sumelka

2018

Struct Multidisc Optim

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Deterioration, understood as a change of mechanical properties of devices exhibiting a strong scale effect, is a phenomenon of great importance in modern industry. This effect is simultaneously an open question in theoretical mechanics, due to the fact that none of existing non-local theories (which are necessary to describe the scale effect) seem to be universal -thus constant development of new mathematical models is necessary. This paper presents the study of the above emphasised problem in the framework of fractional continuum mechanics, as an optimization (identification) task. In the identification routine, the steering parameters are: density of material, order of material, and material length scale.

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“…Eringen showed that the dispersion cure based on the non-local theory is in good agreement with Born–von-Kármán model of the lattice dynamics. More recently, Challamel et al (2013) by generalizing the Eringen non-local theory in terms of fractional calculus has provided a general approximation for the integral form of the constitutive relation which contains the integer and the non-integer orders of the stress gradient (Sumelka et al , 2016). They showed that the optimized fractional model shows perfect matching with Born–von-Kármánmodel of the lattice dynamics and is better than the Eringen model.…”

Section: Introductionmentioning

confidence: 99%

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

Rahimi

Sumelka

Yang

2017

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Purpose Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems. Design/methodology/approach In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition. Findings The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio. Originality/value A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

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One-dimensional dispersion phenomena in terms of fractional media

Cited by 7 publications

References 36 publications

Designing of Dynamic Spectrum Shifting in Terms of Non-Local Space-Fractional Mechanics

Designing of Dynamic Spectrum Shifting in Terms of Non-Local Space-Fractional Mechanics

Identification of mechanical properties of 1D deteriorated non-local bodies

Linear and non-linear free vibration of nano beams based on a new fractional non-local theory

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