Title:
Fractal Analysis of Defects in Concrete under Elevated Temperatures
Author(s):
Jiarong Shen, Qianjun Xu, Mingyi Liu
Publication:
Materials Journal
Volume:
119
Issue:
6
Appears on pages(s):
19-33
Keywords:
concrete; fractal; multifractal; defects; thermal damage; elevated temperature
DOI:
10.14359/51737183
Date:
11/1/2022
Abstract:
This study aims to quantitatively analyze the fractal and multifractal characteristics of concrete at elevated temperatures. Based on the fractal geometry theory, fractal dimensions and multi-fractal spectrum are used to characterize the fractal propagation rules of defects in concrete. The results show that the fractal dimension D (box-counting method), can quantitatively describe the overall defect propagation inside concrete materials. Thus, the more diverse the defects, the larger this fractal dimension. Moreover, the fractal dimension, D’ (island method), does not exhibit considerable variations with different concrete loading types and temperatures. In addition, the multifractal spectrum can reflect the defect characteristics at different levels (local and global) while varying with the defect configurations. The capacity dimension, D0 (f(α)max), the entropy dimension, D1, the holder exponent of order zero, α0, and the signs and values of L–R may reflect the range distribution, size distribution, the degree of mass concentration, and the heterogeneity of defects within concrete, respectively. Moreover, the relationship between the fractal dimension, D, and the thermal damage can be expressed by a quadratic function whose correlation coefficient exceeds 0.99897. Therefore, the thermal damage in concrete at elevated temperatures can be quantitatively described by the quadratic function using the fractal dimension, D. This study provides theoretical and experimental bases for the fractal and, multifractal characteristics and the thermal damage evolution of concrete at elevated temperatures.
Related References:
1. Vodák, F.; Trtík, K.; Kapičkova, O.; Hoskova, S.; and Demo, P., “The Effect of Temperature on Strength-Porosity Relationship for Concrete,” Construction and Building Materials, V. 18, No. 7, 2004, pp. 529-534. doi: 10.1016/j.conbuildmat.2004.04.009
2. Janotka, I., and Nürnbergerová, T., “Effect of Temperature on Structural Quality of the Cement Paste and High-Strength Concrete with Silica Fume,” Nuclear Engineering and Design, V. 235, No. 17-19, 2005, pp. 2019-2032. doi: 10.1016/j.nucengdes.2005.05.011
3. Dehdezi, P. K.; Hall, M. R.; Dawson, A. R.; and Casey, S. P., “Thermal, Mechanical and Microstructural Analysis of Concrete Containing Microencapsulated Phase Change Materials,” The International Journal of Pavement Engineering, V. 14, No. 5, 2013, pp. 449-462. doi: 10.1080/10298436.2012.716837
4. Fernandes, B.; Gil, A. M.; Bolina, F. L.; and Tutikian, B. F., “Microstructure of Concrete Subjected to Elevated Temperatures: Physico-Chemical Changes and Analysis Techniques,” Revista IBRACON de Estruturas e Materials, V. 10, No. 4, 2017, pp. 838-863. doi: 10.1590/s1983-41952017000400004
5. Ross, C. A.; Jerome, D. M.; Tedesco, J. W.; and Hughes, M. L., “Moisture and Strain Rate Effects on Concrete Strength,” ACI Materials Journal, V. 93, No. 3, May-June 1996, pp. 293-300.
6. Zhang, G. H.; Li, Z. L.; Nie, K. Y.; and Liu, M. H., “Experimental Study on Fracture Toughness of Concrete with Different Moisture Contents,” Journal of Hydraulic Engineering, V. 35, 2016, pp. 109-116.
7. Rossi, P., “Influence of Cracking in Presence of Free Water on the Mechanical Behavior of Concrete,” Magazine of Concrete Research, V. 43, No. 154, 1991, pp. 53-57. doi: 10.1680/macr.1991.43.154.53
8. Bartlett, F. M., and Macgregor, J. G., “Effect of Moisture Condition on Concrete Core Strengths,” ACI Materials Journal, V. 91, No. 3, May-June 1994, pp. 227-236.
9. Rossi, P., and Boulay, C., “Influence of Free Water in Concrete on the Cracking Process,” Magazine of Concrete Research, V. 42, No. 152, 1990, pp. 143-146. doi: 10.1680/macr.1990.42.152.143
10. Caré, S., “Effect of Temperature on Porosity and on Chloride Diffusion in Cement Pastes,” Construction and Building Materials, V. 22, No. 7, 2008, pp. 1560-1573. doi: 10.1016/j.conbuildmat.2007.03.018
11. Banthia, N., and Bhargava, A., “Permeability of Stressed Concrete and Role of Fiber Reinforcement,” ACI Materials Journal, V. 104, No. 1, Jan.-Feb. 2007, pp. 70-76.
12. Saouma, V. E.; Barton, C. C.; and Gamaleldin, N. A., “Fractal Characterization of Fracture Structures in Concrete,” Engineering Fracture Mechanics, V. 35, 1990, pp. 47-53. doi: 10.1016/0013-7944(90)90182-G
13. Gao, Y.; Jiang, J. Y.; De Schutter, G. D.; Ye, G.; and Sun, W., “Fractal and Multifractal Analysis on Pore Structure in Cement Paste,” Construction and Building Materials, V. 69, 2014, pp. 253-261. doi: 10.1016/j.conbuildmat.2014.07.065
14. Jin, S. S.; Zhang, J. X.; and Huang, B. S., “Fractal Analysis of Effect of Air Void on Freeze-Thaw Resistance of Concrete,” Construction and Building Materials, V. 47, 2013, pp. 126-130. doi: 10.1016/j.conbuildmat.2013.04.040
15. Ebrahimkhanlou, A.; Athanasiou, A.; Hrynyk, T.D.; Bayrak, O.; and Salamone, S., “Fractal and Multifractal Analysis of Crack Patterns in Prestressed Concrete Girder,” Journal of Bridge Engineering, ASCE, V. 24, 2019, p. 04019059.
16. Cao, M.; Ren, Q.; and Qiao, P., “Nondestructive Assessment of Reinforced Concrete Structures Based on Fractal Damage Characteristic Factors,” Journal of Engineering Mechanics, ASCE, V. 132, No. 9, 2006, pp. 924-931. doi: 10.1061/(ASCE)0733-9399(2006)132:9(924)
17. Valentini, L.; Artioli, G.; Voltolini, M.; and Dalconi, M. C., “Multifractal Analysis of Calcium Silicate Hydrate (C-S-H) Mapped by X-Ray Diffraction Microtomography,” Journal of the American Ceramic Society, V. 95, No. 8, 2012, pp. 2647-2652. doi: 10.1111/j.1551-2916.2012.05255.
18. Pia, G., and Sanna, U., “A Geometrical Fractal Model for the Porosity and Thermal Conductivity of Insulating Concrete,” Construction and Building Materials, V. 44, 2013, pp. 551-556. doi: 10.1016/j.conbuildmat.2013.03.049
19. Jin, H. Q.; Yao, X. L.; Fan, L. W.; Xu, X.; and Yu, Z. T., “Experimental Determination and Fractal Modeling of the Effective Thermal Conductivity of Autoclaved Aerated Concrete: Effect of Moisture Content,” International Journal of Heat and Mass Transfer, V. 92, 2016, pp. 589-602. doi: 10.1016/j.ijheatmasstransfer.2015.08.103
20. Carpinteri, A.; Lacidogna, G.; and Niccolini, G., “Fractal Analysis of Damage Detected in Concrete Structural Elements Under Loading,” Chaos, Solitons, and Fractals, V. 42, No. 4, 2009, pp. 2047-2056. doi: 10.1016/j.chaos.2009.03.165
21. Carpinteri, A.; Spagnoli, A.; and Vantadori, S., “A Multifractal Analysis of Fatigue Crack Growth and Its Application to Concrete,” Engineering Fracture Mechanics, V. 77, No. 6, 2010, pp. 974-984. doi: 10.1016/j.engfracmech.2010.01.019
22. Miranda, J. G. V.; Montero, E.; Alves, M. C.; and González, A. P., “Multifractal Characterization of Saprolite Particle-Size Distribution After Topsoil Removal,” Geoderma, V. 134, No. 3-4, 2006, pp. 373-385. doi: 10.1016/j.geoderma.2006.03.014
23. Xu, Q.; Chen, J. Y.; Li, J.; and Wang, M. M., “Multi-Scale Numerical Model for Simulating Concrete Material Based on Fractal Theory,” Guti Lixue Xuebao, V. 26, No. 4, 2013, pp. 344-352. doi: 10.1016/S0894-9166(13)60031-2
24. Liu, R.; Jiang, Y.; Li, B.; and Wang, X., “A Fractal Model for Characterizing Fluid Flow in Fractured Rock Masses Based on Randomly Distributed Rock Fracture Network,” Computers and Geotechnics, V. 65, 2015, pp. 45-55. doi: 10.1016/j.compgeo.2014.11.004
25. Guo, Z. H., Mechanical Properties of Concrete Materials and Components at Room Temperature and High Temperature, Tsinghua University Press, 2006.
26. ACI Committee 349, “Code Requirements for Nuclear Safety Related Concrete Structures (ACI 349-01) and Commentary (ACI 349R-01),” American Concrete Institute, Farmington Hills, MI, 2001, 134 pp.
27. Joint ACI-ASME Committee 359, “ASME Boiler and Pressure Vessel Code,” American Society of Mechanical Engineers, New York, 2002.
28. Maruyama, I.; Sasano, H.; Nishioka, Y.; and Igarashi, G., “Strength and Young’s Modulus Change in Concrete Due to Long-Term Drying and Heating Up to 90 °C,” Cement and Concrete Research, V. 66, 2014, pp. 48-63. doi: 10.1016/j.cemconres.2014.07.016
29. Gallé, C., “Effect of Drying on Cement-Based Materials Pore Structure as Identified by Mercury Intrusion Porosimetry: A Comparative Study Between Oven-, Vacuum-, and Freeze-Drying,” Cement and Concrete Research, V. 31, No. 10, 2001, pp. 1467-1477. doi: 10.1016/S0008-8846(01)00594-4
30. SL/T 352-2020, “Test Specification for Hydraulic Concrete,” Ministry of Water Resource of the People’s Republic of China, Beijing, China, 2006.
31. Shen, J. R.; Xu, Q. J.; and Li, Q., “Effect of Temperature on Pore Structure and Strength of Concrete,” ACI Materials Journal, V. 117, No. 1, Jan. 2020, pp. 85-95. doi: 10.14359/51718060
32. Loukili, A.; Khelidj, A.; and Richard, P., “Hydration Kinetics, Change of Relative Humidity, and Autogenous Shrinkage of Ultra-High-Strength Concrete,” Cement and Concrete Research, V. 29, No. 4, 1999, pp. 577-584. doi: 10.1016/S0008-8846(99)00022-8
33. EN 1992-1-2:2004, “Eurocode 2: Design of Concrete Structures - Part 1-2: General Rules - Structural Fire Design,” European Committee for Standardization, Brussels, Belgium, 2004.
34. ASCE Committee on Fire Protection, “Structural Fire Protection,” T. T. Lie, ed., American Society of Civil Engineers, Reston, VA, 1992.
35. Kodur, V. K. R.; Dwaikat, M. M. S.; and Dwaikat, M. B., “High Temperature Properties of Concrete for Fire Resistance Modeling of Structures,” ACI Materials Journal, V. 105, No. 5, Sept.-Oct., 2008, pp. 517-527.
36. Shen, J. R.; Xu, Q. J.; and Liu, M. Y., “Statistical Analysis of Defects Within Concrete Under Elevated Temperatures Based on SEM Image,” Construction and Building Materials, V. 293, 2021, p. 123503. doi: 10.1016/j.conbuildmat.2021.123503
37. Feder, J., Fractals, Plenum Press, New York, 1998.
38. Falconer, K. J., Fractal Geometry, second edition, John Wiley and Sons, New York, 2003.
39. Manderlbrot, B. B.; Passoja, D. E.; and Paullay, A. J., “Fractal Character of Fracture Surface of Metals,” Nature, V. 308, No. 5961, 1984, pp. 721-722. doi: 10.1038/308721a0
40. Mandelbrot, B. B., The Fractal Geometry of Nature, third edition, W. H. Freeman, New York, 1983.
41. Manderlbrot, B. B., Fractal: Form, Chance, and Dimension, W. H. Freeman, San Francisco, CA, 1977.
42. Bigerelle, M., and Lost, A., “Statistical Artefacts in the Determination of the Fractal Dimension by the Slit Island Method,” Engineering Fracture Mechanics, V. 71, No. 7-8, 2004, pp. 1081-1105. doi: 10.1016/S0013-7944(03)00136-X
43. Meisel, L. V., “Perimeter-Area Analysis the Slit Island Method and the Fractal Characterization of Metallic Fracture Surfaces,” Journal of Physics D: Applied Physics, V. 24, No. 6, 1991, pp. 942-952. doi: 10.1088/0022-3727/24/6/020
44. Papadopoulos, A.; Bird, N. R. A.; Mooney, S. J.; and Whitmore, A. P., “Fractal Analysis of Pore Roughness in Images of Soil Using the Slit Island Method,” Vadose Zone Journal, V. 7, No. 2, 2008, pp. 456-460. doi: 10.2136/vzj2007.0017
45. Chaudhari, A.; Yan, C. C. S.; and Lee, S. L., “Multifractal Analysis of Growing Surfaces,” Applied Surface Science, V. 238, No. 1-4, 2004, pp. 513-517. doi: 10.1016/j.apsusc.2004.05.247
46. Jeżewski, W., “Complex Multifractal Measures and A Generalized Multifractal Formalism,” Physica A: Statistical Mechanics and its Applications, V. 298, No. 3-4, 2001, pp. 419-430. doi: 10.1016/S0378-4371(01)00252-7
47. Miranda, J. G. V.; Montero, E.; Alves, M. C.; Paz-González, A.; and Vidal Vázquez, E., “Multifractal Characterization of Saprolite Particle-Size Distributions After Topsoil Removal,” Geoderma, V. 134, No. 3-4, 2006, pp. 373-385. doi: 10.1016/j.geoderma.2006.03.014
48. Wang, J. M.; Guo, L. L.; Bai, Z. K.; and Yang, L.L., “Using Computed Tomography (CT) Images and Multi-Fractal Theory to Quantify the Pore Distribution of Reconstructed Soils During Ecological Restoration in Opencast Coal-Mine,” Economic Geology and the Bulletin of the Society of Economic Geologists, V. 92, 2016, pp. 148-157.
49. Mandelbrot, B. B., “Multifractal Power Law Distributions: Negative and Critical Dimensions and Other Anomalies, Explained by A Simple Example,” Journal of Statistical Physics, V. 110, No. 3/6, 2003, pp. 739-774. doi: 10.1023/A:1022159802564
50. Davie, C. T.; Pearce, C. J.; and Bićanić, N., “A Fully Generalised, Coupled, Multi-Phase, Hygro-Thermo-Mechanical Model for Concrete,” Materials and Structures, V. 43, 2010, pp. 13-33. doi: 10.1617/s11527-010-9591-y
51. Francois, D., “Fracture and Damage Mechanics of Concrete,” Application of Fracture Mechanics to Cementitious Composites, S. P. Shah, ed., Springer, Dordrecht, the Netherlands, 1985, pp. 141-156.
52. Lemaitre, J., “Evaluation of Dissipation and Damage in Metals,” Proceedings of International Conference on Mechanical Behavior of Materials, Kyoto, Japan, 2003.