A new visco–elasto-plastic model via time–space fractional derivative

Hei, X.; Chen, W.; Pang, Guofei; Zhang, C.

doi:10.1007/s11043-017-9356-x

Cited by 14 publications

(4 citation statements)

References 26 publications

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“…Numerical Scheme for Short-Time Integration. In the singularitycapturing scheme, given σ, in order to compute the error in (11) and the gradient (due to the error perturbation) (14), we need to compute the numerical solution u N n for n = 1, . .…”

Section: Stage-imentioning

confidence: 99%

“…Introduction. Fractional differential equations (FDEs) have been successfully applied in diverse problems that present the fingerprint of power-laws/heavytailed statistics, such as visco-elastic modeling of bio-tissues [5,30,31,32,33], cell rheology behavior [10], food preservatives [17], complex fluids [18], visco-elasto-plastic modeling for power-law-dependent stresses/strains [44,43,14], earth sciences [55], among others.…”

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confidence: 99%

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An Automated Singularity-Capturing Scheme for Fractional Differential Equations

Suzuki,

Zayernouri

2018

Preprint

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Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for accurate time-integration of single-and multiterm fractional differential equations. In the first stage, we formulate a self-singularity-capturing scheme, given available/observable data for diminutive time. In this approach, the fractional differential equation provides the necessary knowledge/insight on how the hidden singularity can bridge between the initial and the subsequent short-time solution data. We develop a new self-singularitycapturing finite-difference algorithm for automatic determination of the underlying power-law singularities nearby the initial data, employing gradient descent optimization. In the second stage, we can utilize the multi-singular behavior of solution in a variety of numerical methods, without resorting to making any ad-hoc/uneducated guesses for the solution singularities. Particularly, we employed an implicit finite-difference method, where the captured singularities, in the first stage, are taken into account through some Lubich-like correction terms, leading to an accuracy of order O(∆t 3−α ). Our computational results demonstrate that the developed framework can either fully capture or successfully control the solution error in the time-integration of fractional differential equations, especially in the presence of strong multi-singularities.

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Section: Stage-imentioning

confidence: 99%

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confidence: 99%

An Automated Singularity-Capturing Scheme for Fractional Differential Equations

Suzuki,

Zayernouri

2018

Preprint

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“…Hencky strain was used to formulate the theoretical framework and the model well predicted the rate-dependent mechanical response. In order to characterize visco-elasto-plastic behaviour, a time-space fractional derivative rheological model was proposed in (Hei et al 2018). The model was able to well-predict the rate effects, along with their viscoelastic and viscoplastic mechanical response.…”

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confidence: 99%

Mechanics of soft polymeric materials using a fractal viscoelastic model

Pramanik

Soni

Shanmuganathan

et al. 2021

Mech Time-Depend Mater

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Soft materials are known for their plethora of biomedical applications, intricate structure-property correlation and nonlinear mechanical response. Multiple length-time scale phenomena and hierarchical structure results in their nonlinearity. Phenomenological and continuum mechanical models have been developed to predict their mechanics, which have mostly been very material-specific with inability to predict the mechanics of different types of soft materials simultaneously. This shortcoming has been addressed in the present work, wherein a generic nonlinear viscoelastic model has been proposed to predict the mechanical response of hydrogels, sponges, and xerogels. A fractal derivative viscoelastic model is proposed considering a fractal Maxwell model in parallel with a nonlinear spring. In particular, this model is chosen to qualitatively mimic the material nonlinearity inherent in soft materials. The fractal dashpot in combination with the nonlinear spring accounts for the power law time-dependent rheology of generic soft materials. These two different aspects in the form of nonlinear stiffness and non-Newtonian rheology account for mechanics of most soft materials. The present model is shown to fit well the existing literature results for mechanical response of a multitude of soft material classes with different test conditions and loading rates, which is one of the salient features of the model, apart from its simplistic mathematical framework. Further, a parametric study is reported on the mechanics of nanocellulose loaded poly(vinyl alcohol) xerogel. The model predictions are observed to be in conjunction with the experimental observations.

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“…With particular arrangements of SB and standard elements, fractional models were applied, e.g., to describe the far from equilibrium power-law dynamics of multifractional visco-elastic [23,26,[37][38][39][40], distributed visco-elastic [17] and visco-elastoplastic [25,50,51,54,59] complex materials. Concurrently, significant advances in numerical methods allowed numerical solutions to time-and space-fractional partial differential equations (FPDEs) for smooth/non-smooth solutions, such as finitedifference (FD) schemes [32,34], fractional Adams methods [16,60], implicit-explicit (IMEX) schemes [11,63], spectral methods [44,45], fractional subgrid-scale modeling [43], fractional sensitivity equations [29], operator-based uncertainty quantification [28] and self-singularity-capturing approaches [53].…”

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confidence: 99%

A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

Suzuki¹,

Zhou²,

D’Elia³

et al. 2019

Preprint

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We develop a thermodynamically consistent, fractional visco-elasto-plastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco-elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix-vector operations for each time-step, allowing us to reduce the computational complexity from O(N 3 ) to O(N 2 ) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates.

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A new visco–elasto-plastic model via time–space fractional derivative

Cited by 14 publications

References 26 publications

An Automated Singularity-Capturing Scheme for Fractional Differential Equations

An Automated Singularity-Capturing Scheme for Fractional Differential Equations

Mechanics of soft polymeric materials using a fractal viscoelastic model

A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

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