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Fractional Calculus and Applied Analysis

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Volume 16, Issue 1


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

Wen Chen
  • Institute of Soft Matter Mechanics Department of Engineering Mechanics, Hohai University, No. 1 Xikang Road, Nanjing, 210098, China
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/ Jianjun Zhang
  • Institute of Soft Matter Mechanics Department of Engineering Mechanics, Hohai University, No. 1 Xikang Road, Nanjing, 210098, China
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/ Jinyang Zhang
  • Institute of Soft Matter Mechanics Department of Engineering Mechanics, Hohai University, No. 1 Xikang Road, Nanjing, 210098, China
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Published Online: 2012-12-27 | DOI: https://doi.org/10.2478/s13540-013-0006-y


This 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.

MSC: 74H15; 74S20; 76R50; 81T80

Keywords: variable-order fractional derivative; sub-diffusion; time dependence; concrete; chloride ions; chloride binding; concentration dependence

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About the article

Published Online: 2012-12-27

Published in Print: 2013-03-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 16, Issue 1, Pages 76–92, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0006-y.

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