Multi-term time fractional diffusion equations and novel parameter estimation techniques for chloride ions sub-diffusion in reinforced concrete

Abstract

<jats:p>In this paper, searching for a better chloride ions sub-diffusion system, a multi-term time-fractional derivative diffusion model is proposed for the description of the time-dependent chloride ions penetration in reinforced concrete structures exposed to chloride environments. We prove the stability and convergence of the model. We use the modified grid approximation method (MGAM) to estimate the fractional orders and chloride ions diffusion coefficients in the reinforced concrete for the multi-term time fractional diffusion system. And then to verify the efficiency and accuracy of the proposed methods in dealing with the fractional inverse problem, two numerical examples with real data are investigated. Meanwhile, we use two methods of fixed chloride ions diffusion coefficient and variable diffusion coefficient with diffusion depth to simulate chloride ions sub-diffusion system. The result shows that with the new fractional orders and parameters, our multi-term fractional order chloride ions sub-diffusion system is capable of providing numerical results that agree better with the real data than other models. On the other hand, it is also noticed from the numerical solution of the chloride ions sub-diffusion system that setting the variable diffusion coefficient with diffusion depth is more reasonable. And it is also found that chloride ions diffusion coefficients in reinforced concrete should be decreased with diffusion depth which is completely consistent with the theory. In addition, the model can be used to predict the chloride profiles with a time-dependent property.</jats:p> <jats:p>This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.</jats:p>

Advanced materials modelling via fractional calculus: challenges and perspectives

“…Focusing on the transport of radioactive materials in fractures surrounded by porous matrices of fractal structure, the authors propose a novel form of the fractional differential equation where fractional derivatives account for contaminant exchange between the fracture and the surrounding porous matrix; exact and approximate expressions for solute concentration in fracture and porous medium are obtained. Further, chloride diffusion in the reinforced concrete is the subject of the contribution by Chen et al [81]. On observing that chloride ion penetration is generally slower than normal diffusion and exhibits anomalous characteristics as history dependence and long-range correlation, the authors propose a multiterm time-fractional model based on the Caputo fractional derivative and a pertinent numerical solution scheme, proving its convergence and stability.…”

Durability Investigation of Carbon Fiber Reinforced Concrete under Salt-Freeze Coupling Effect

“…The permeability of chloride ions in concrete is the lowest when the lateral compressive stress is close to 0.15 times the compressive strength of concrete. Chen et al proposed the diffusion model of chloride ions in reinforced concrete structures and proved the stability and convergence of the model [13].…”

Theoretical Analysis of Fractional Viscoelastic Flow in Circular Pipes: General Solutions

“…The comprehensive overview summarizing state-of-the-art practical applications of FC has been recently published by The Royal Society Publishing. The sixteen-paper issue entitled "Advanced materials modeling via fractional calculus: challenges and perspectives" [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] covers applications of constant-order (CO) and variable-order (VO) fractional differential operators to several fundamental phenomena. These include anomalous diffusion, [13,16] heat conduction [14,27], fractional viscoelasticity of fluids [19], and materials [12,18,22].…”

“…The sixteen-paper issue entitled "Advanced materials modeling via fractional calculus: challenges and perspectives" [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] covers applications of constant-order (CO) and variable-order (VO) fractional differential operators to several fundamental phenomena. These include anomalous diffusion, [13,16] heat conduction [14,27], fractional viscoelasticity of fluids [19], and materials [12,18,22]. The approach to model viscoelastic properties of materials with VO FC operators is undoubtedly among the most promising ones, as it allows for the consideration of fractional order dynamics with respect to time, space, and material variables [22].…”