A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation

Mashayekhi, Somayeh; Hussaini, M. Y.; Oates, William S.

doi:10.1016/j.jmps.2019.04.005

Cited by 57 publications

(22 citation statements)

References 24 publications

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“…This is believed to be a consequence of the non-local fractional order derivative. Its complexity has also been shown to be related to fractal geometry and power-law spectral dimensions of the underlying polymer dynamics (Mashayekhi et al, 2019) which offers additional opportunities to understand complex multiscale effects associated with nonlinear mechanics in soft materials.…”

Section: Discussionmentioning

confidence: 99%

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

Miles

Pash

Smith

et al. 2020

Journal of Intelligent Material Systems and Structures

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Dielectric elastomers are employed for a wide variety of adaptive structures. Many of these soft elastomers exhibit significant rate-dependencies in their response. Accurately quantifying this viscoelastic behavior is non-trivial and in many cases a nonlinear modeling framework is required. Fractional-order operators have been applied to modeling viscoelastic behavior for many years, and recent research has shown fractional-order methods to be effective for nonlinear frameworks. This implementation can become computationally expensive to achieve an accurate approximation of the fractional-order derivative. Accurate estimation of the elastomer’s viscoelastic behavior to quantify parameter uncertainty motivates the use of Markov Chain Monte Carlo (MCMC) methods. Since MCMC is a sampling based method, requiring many model evaluations, efficient estimation of the fractional derivative operator is crucial. In this paper, we demonstrate the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of estimating the uncertainty of a nonlinear viscoelastic model. We also demonstrate the use of parameter subset selection techniques to isolate parameters that are identifiable in the sense that they are uniquely determined by measured data. For those identifiable parameters, we employ Bayesian inference to compute posterior distributions for parameters. Finally, we propagate parameter uncertainties through the models to compute prediction intervals for quantities of interest.

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Section: Discussionmentioning

confidence: 99%

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

Miles

Pash

Smith

et al. 2020

Journal of Intelligent Material Systems and Structures

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show abstract

“…Experimental validation was provided on a dielectric elastomer, whose a distinctive feature is a significant rate-dependent deformation during uniaxial stress measurements; using Bayesian statistics for calibration, the authors have shown that the fractional-order models are more accurate than integer-order ones for deformation rates spanning several orders of magnitude [113]. In a further recent work, Mashayekhi et al [115] explored a physical connection between time-fractional derivative and fractal geometry of fractal media; on using thermodynamics law, the order of the fractional derivative in the linear fractional model of viscoelasticity was found to be a rate-dependent material property strongly correlated with fractal dimension and spectral dimension which characterizes diffusion in fractal media. The need for power-law functions to model phenomena in fractal media has been shown for the flux-time relations across fractal structures [116,117] as well as in the context of biomechanics of bone tissues [118].…”

Section: Fractal Mediamentioning

confidence: 99%

Advanced materials modelling via fractional calculus: challenges and perspectives

Failla

Zingales

2020

Phil. Trans. R. Soc. A.

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Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

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“…Since fractal media of interest to us exhibit spatial fractality only, a conventional time derivative can be used so that (∂/∂t) D = (∂/∂t). When the medium also displays the fractal characteristics of temporal response-as is sometimes the case in viscoelasticity-a fractional time derivative (such as Caputo) is to be used [22,23]. Clarification of a cause-effect relationship between the fractal spatial structure and a fractional-type calculus in viscoelastic and other constitutive (e.g.…”

Section: (B) Fractal Tensor Calculusmentioning

confidence: 99%

Thermo-poromechanics of fractal media

Ostoja–Starzewski

2020

Phil. Trans. R. Soc. A.

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This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick’s Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

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A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation

Cited by 57 publications

References 24 publications

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

Advanced materials modelling via fractional calculus: challenges and perspectives

Thermo-poromechanics of fractal media

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