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

Fractional Calculus and Applied Analysis

Volumn 16, Issue 1, 2013, Pages 76-92

  Chen, Wen ^a   Zhang, Jianjun ^a   Zhang, Jinyang ^a  

^a HOHAI UNIVERSITY   (China)

Author keywords

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

Indexed keywords

EID: 84871803760     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-013-0006-y     Document Type: Article

References (43)

1. C. Andrade, Calculation of chloride diffusion coefficients in concrete from ionic migration measurements. Cem. Concr. Res. 23, No 3 (1993), 724-742.
2. V. Baroghel-Bouny, T.Q. Nguyen, and P. Dangla, Assessment and prediction of RC structure service life by means of durability indicators and physical/chemical models. Cem. Concr. Compos. 31, No 8 (2009), 522-534.
3. A. Boddy, E. Bentz, M.D.A. Thomas, and R.D. Hooton, An overview and sensitivity study of a multimechanistic chloride transport model. Cem. Concr. Res. 29 (1999), 827-837.
4. S. Chatterji, On the applicability of Fick's second law to chloride ion migration through Portland cement concrete. Cem. Concr. Res. 25, No 2 (1995), 299-303.
5. A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Fractional diffusion in inhomogeneous media. J. Phys. A-Math. Gen. 38 (2005), 679-684.
6. W. Chen, L. Ye, and H. Sun, Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59, No 5 (2010), 1614-1620.
7. C.F.M. Coimbra, Mechanics with variable-order differential operators. Ann. Phys. (Leipzig). 12, No 11 (2003), 692-703.
8. M. Collepardi, A. Marcialis, and R. Tarriziani, The kinetics of penetration of chloride ions into the concrete. Il Cemento. 4 (1970), 157-164.
9. G.K. Glass and N.R. Buenfeld, The presentation of the chloride threshold level for corrosion of steel in concrete. Corros. Sci. 39, No 5 (1997), 1001-1013.
10. G.K. Glass and N.R. Buenfeld, The influence of chloride binding on the chloride induced corrosion risk in reinforced concrete. Corros. Sci. 42 (2000), 329-344.
11. T. Ishida, P.O.N. Iqbal, H. Thi, and L. Anh, Modeling of chloride diffusivity coupled with non-linear binding capacity in sound and cracked concrete. Cem. Concr. Res. 39 (2009), 913-923.
12. A. Khitab, S. Lorente, and J.P. Ollivier, Predictive model for chloride penetration through concrete. Mag. Concr. Res. 57, No 9 (2005), 511-520.
13. L.Y. Li and C.L. Page, Modelling of electrochemical chloride extraction from concrete: Influence of ionic activity coefficients. Comp. Mater. Sci. 9 (1998), 303-308.
14. R. Loser, B. Lothenbach, A. Leemann, and M. Tuchschmid, Chloride resistance of concrete and its binding capacity - Comparison between experimental results and thermodynamic modeling. Cem. Concr. Compos. 32, No 1 (2010), 34-42.
15. Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, No 1 (2012), 141-160; DOI:10.2478/s13540-012-0010-7; http://link.springer.com/article/10. 2478/s13540-012-0010-7.
16. R. Luo, Y. Cai, C. Wang, and X. Huang, Study of chloride binding and diffusion in GGBS concrete. Cem. Concr. Res. 33 (2003), 1-7.
17. M. Maage, S. Helland, E. Poulsen, O. Vennesland, and E. Carl, Service life prediction of existing concrete structures exposed to marine environment. ACI Mater. J. 93, No 6 (1996), 602-608.
18. J.A. Tenreiro Machado, And I say to myself: "What a fractional world!". Fract. Calc. Appl. Anal. 14, No 4 (2011), 635-654; DOI:10.2478/s13540-011-0037-1; http://link.springer.com/article/10.2478/s13540- 011-0037-1.
19. F. Mainardi, The fundamental solutions for the fractional diffusionwave equation. Appl. Math. Lett. 9, No 6 (1996), 23-28.
20. P.S. Mangat and B.T. Molloy, Prediction of long term chloride concentration in concrete. Mater. Struct. 27, (1994), 338-346.
21. B. Martin-Perez, H. Zibara, R. Hooton, and M.D.A. Thomas, A study of the effect of chloride binding on service life predictions. Cem. Concr. Res. 30 (2000), 1215-1223.
22. R. Metzler and J. Klafter, The random walks' guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1-77.
23. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. 133 (1986), 425-430.
24. Y. Pachepsky, D. Benson, and W. Rawls, Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation. Soil. Sci. Soc. Am. J. 64 (2000), 274-277.
25. I. Podlubny, Fractional Differential Equations. Academic Press, N. York (1999).
26. G. Sergi, S.W. Yu, and C.L. Page, Diffusion of chloride and hydroxyl ions in cementitious materials exposed to a saline environment. Mag.Concr. Res. 44, No 157 (1991), 63-69.
27. I. Sokolov and J. Klafter, Field-Induced dispersion in subdiffusion. Phys. Rev. Lett. 97 (2006), 140602.
28. M.G. Stewart and D.V. Rosowsky, Time-dependent reliability of deteriorating reinforced concrete bridge decks. Struct. Saf. 20 (1998), 91-109.
29. H. Sun, W. Chen, and Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A. 388 (2009), 4586-4592.
30. H. Sun, W. Chen, H. Sheng, and Y. Chen, On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys. Lett. A. 374 (2010), 906-910.
31. H. Sun, W. Chen, C. Li, and Y. Chen, Finite difference schemes for variable-order time fractional diffusion equation. Int. J.Bifurcat. Chaos. 22, No 4 (2012), 1250085.
32. A.K. Suryavanshi, J.D. Scantlebury, and S.B. Lyon, Mechanism of Friedels' salt formation in cements rich in tri-calcium aluminate. Cem. Concr. Res. 26, No 5 (1996), 717-727.
33. L. Tang, Concentration dependence of diffusion and migration of chloride ions: Part 1. Theoretical considerations. Cem. Concr. Compos. 29 (1999), 1463-1468.
34. L. Tang and J. Gulikers, On the mathematics of time-dependent apparent chloride diffusion coefficient in concrete. Cem. Concr. Res. 37, No 4 (2007), 589-595.
35. L. Tang, Guideline for practical use of methods for testing the resistance of concrete to chloride ingress, CHLORTEST - EU funded Research Project' Resistance of concrete to chloride ingress - from laboratory tests to in-field performance' G6RD-CT-2002-00855; Deliverable D23 (2008).
36. M.D.A. Thomas and P.B. Bamforth, Modelling chloride diffusion in concrete effect of fly ash and slag. Cem. Concr. Res. 29 (1999), 487-495.
37. W. Wyss, The fractional diffusion equation. J. Math. Phys. 27 (1986), 2782-2785.
38. Q. Yuan, C. Shi, G. De Schutter, K. Audenaert, and D. Deng, Chloride binding of cement-based materials subjected to external chloride environment - A review. Constr. Build. Mater. 23 (2009), 1-13.
39. G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002), 461-580.
40. J.J. Zhang, W. Chen, D.J. Shen, Time-fractional derivative model for chloride ion sub-diffusion in reinforced concrete. Constr. Build. Mater., Submitted.
41. Y. Zhang, M. Meerschaert, and B. Baeumer, Particle tracking for timefractional diffusion. Phys. Rev. E. 78, No 3 (2008), 1-7.
42. T. Zhang and O.E. Gjorv, Diffusion behavior of chloride ions in concrete. Cem. Concr. Res. 26, No 6 (1996), 907-917.
43. P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, No 3 (2009), 1760-1781.