A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures

AbstractThis study proposes a new variable-order fractional diffusion equation model to describe the coupled chloride diffusion-binding processes in reinforced concrete, in which the order of fractional derivative term is a variable function instead of a constant in the standard fractional model. The concentration influence coefficient k is introduced to capture the effect of concentration dependency on chloride transport due to the chloride binding behavior. The two parameters in the proposed model can be determined directly by a statistical analysis of measurement data. Four test cases illustrate that the proposed variable-order fractional derivative model agrees significantly better with experimental data than the most commonly used traditional model governed by the classical Fick’s second law, especially when a large concentration coefficient k is involved. That proposed model is also verified by accurately predicting chloride concentration profiles in a period of 200 days.

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Prediction of Beyond Design and Residual Performances of Viscoelastic Dampers by a Simplified Fractional Derivative Model

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455421500814 ◽

2021 ◽

pp. 2150081

Author(s):

Shiang-Jung Wang ◽

Qun-Ying Zhang ◽

Chung-Han Yu

Keyword(s):

Energy Dissipation ◽

Fractional Derivative ◽

Main Shock ◽

Full Scale ◽

The Other ◽

Ambient Temperatures ◽

Viscoelastic Dampers ◽

Fractional Derivative Model ◽

Proposed Model ◽

Residual Performance

When subjected to excessive shear deformation, viscoelastic (VE) dampers may inevitably suffer from damages, due to their VE material layers with limited thickness. Under the circumstance, their stiffness and energy dissipation capabilities may deteriorate but not totally vanish. To estimate the seismic performances of viscoelastically damped structures, the beyond design and residual performances of damaged VE dampers are crucial to protect structures from severe failure during the following main shock or aftershocks. On the other hand, for new viscoelastically damped structures under the normal design earthquakes, neglecting the residual performance of damaged VE dampers may result in nonconservative design. Thus, this study aims to provide approaches to analytically characterize the beyond design and residual performances of damaged full-scale VE dampers. Based on the simplified fractional derivative model, the analytical predictions have been compared with the experimental results. The proposed model works well for the design performance of the intact full-scale VE dampers. Particularly, it can also reproduce the beyond design and residual performances of damaged full-scale VE dampers, if due consideration is taken of the effects of excitation frequencies, ambient temperatures, temperature rises, softening, and hardening.

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Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete

European Journal of Environmental and Civil engineering ◽

10.1080/19648189.2015.1116467 ◽

2015 ◽

Vol 21 (3) ◽

pp. 319-331 ◽

Cited By ~ 15

Author(s):

Song Wei ◽

Wen Chen ◽

Jianjun Zhang

Keyword(s):

Reinforced Concrete ◽

Fractional Derivative ◽

Chloride Ions ◽

Fractional Derivative Model ◽

Time Fractional Derivative

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Thermomechanical Fractional Derivative Model of Viscoelastic Isolators

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.952.219 ◽

2014 ◽

Vol 952 ◽

pp. 219-222

Author(s):

Zhen Li Zhang ◽

Chao Shang ◽

Wei Hua Shi

Keyword(s):

Fractional Derivative ◽

Engineering Application ◽

Coupling Model ◽

Measured Data ◽

Thermomechanical Coupling ◽

Damping Characteristics ◽

Fractional Derivative Model ◽

Proposed Model ◽

Vibration Process ◽

Difference Form

In vibration process, viscoelastic isolators’ temperature will rise due to energy dissipation, especially when the isolators have high damping characteristics. First, for the arbitrary loadings, the thermomechanical coupling model and the corresponding difference form are established based on the five-parameter fractional derivative model. Then, for the steady-state harmonic inputs, which is very common in engineering application, the derived model is significantly simplified by Fourier transformation. Finally, the proposed model is verified by experiments and shows a reasonable agreement with measured data.

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Simulating chloride penetration in fly ash concrete by a fractal derivative model

Thermal Science ◽

10.2298/tsci181024331q ◽

2019 ◽

Vol 23 (Suppl. 1) ◽

pp. 67-78 ◽

Cited By ~ 1

Author(s):

Pengfei Qu ◽

Xiaoting Liu ◽

Dumitru Baleanu

Keyword(s):

Fractional Derivative ◽

Linear Regression Analysis ◽

Chemical Properties ◽

Mathematical Expression ◽

Fractal Model ◽

Chloride Ions ◽

Diffusion Behavior ◽

Fractional Derivative Model ◽

Fractal Derivative ◽

Diffusion Phenomena

In the real engineering field, the chloride ions behave abnormal diffusion phenomena in concrete caused by different compositions of the concrete which lead to the complex physical and chemical properties. This paper utilizes a fractal derivative model and a fractional derivative model to describe the diffusion phenomena. Furthermore, according to actual experimental data in the field, the fractional and fractal model can simulate the diffusion behavior of chloride ions in concrete. In comparison to the fractional derivative model, the fractal derivative model gives a simpler mathematical expression and lower calculation costs. In addition, the linear regression analysis method is used to establish an effective relationship between the internal composition of concrete and the parameters of fractal model such as fractal order, ?, and diffusion coefficient, D. As a result, the fractal model with the parameters estimated by previous relationship can predict the diffusion behavior of chloride ions.

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Variable-Order Fractional Models for Wall-Bounded Turbulent Flows

Entropy ◽

10.3390/e23060782 ◽

2021 ◽

Vol 23 (6) ◽

pp. 782

Author(s):

Fangying Song ◽

George Em Karniadakis

Keyword(s):

Fractional Derivative ◽

Turbulent Flows ◽

Fractional Order ◽

Variable Order ◽

Universal Curve ◽

Reynolds Stresses ◽

Continuous Change ◽

Mean Velocity ◽

Symmetric Variable ◽

Fractional Models

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

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On the sub–diffusion fractional initial value problem with time variable order

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0182 ◽

2021 ◽

Vol 10 (1) ◽

pp. 1301-1315

Author(s):

Eduardo Cuesta ◽

Mokhtar Kirane ◽

Ahmed Alsaedi ◽

Bashir Ahmad

Keyword(s):

Asymptotic Behavior ◽

Fractional Derivative ◽

Initial Value Problem ◽

Regularity Result ◽

Time Variable ◽

Variable Order ◽

Initial Value ◽

Integration By Parts ◽

Integration By Parts Formula ◽

Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

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