##
**Fractal analysis of damage detected in concrete structural elements under loading.**
*(English)*
Zbl 1198.94022

Summary: In Civil Engineering materials subjected to stress or strain states a quantitative evaluation of damage is of great importance due to the critical character of this phenomenon, which at a certain point suddenly turns into a catastrophic failure. An effective damage assessment criterion is represented by the statistical analysis of the amplitude distribution of acoustic emission (AE) signals emerging from the growing microcracks. The amplitudes of such signals are distributed according to the Gutenberg-Richter (GR) law and characterised through the \(b\)-value which decreases systematically with damage growth. On the other hand, the damage process is also characterised by the progressive coalescence of microcracks to form fracture surfaces. Geometrically the fractal dimension \(D\) of the damaged domain is expected to decrease from an initial value comprised between 2 and 3 towards a final value nearly equal to 2. The \(b\)-value and the fractal analysis, are here applied to two case studies of concrete specimens loaded up to failure, and the obtained results are compared and discussed. In particular, we emphasize that a single fractal dimension does not adequately describe a crack network, since two damaged domains with the same fractal dimension could have significantly different properties.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

37N35 | Dynamical systems in control |

74R99 | Fracture and damage |

PDFBibTeX
XMLCite

\textit{A. Carpinteri} et al., Chaos Solitons Fractals 42, No. 4, 2047--2056 (2009; Zbl 1198.94022)

Full Text:
DOI

### References:

[1] | Lemaitre, J.; Chaboche, J. L., Mechanics of solid materials (1990), University Press: University Press Cambridge |

[2] | Krajcinovic, D., Damage mechanics (1996), Elsevier · Zbl 1111.74491 |

[3] | Turcotte, D. L.; Newman, W. I.; Shcherbakov, R., Micro and macroscopic models of rock fracture, Geophys J Int, 152, 718-728 (2003) |

[4] | Shiotani, T.; Fujii, K.; Aoki, T.; Amou, K., Evaluation of progressive failure using AE sources and improved \(b\)-value on slope model tests, Prog Acoust Emission VII, 7, 529-534 (1994) |

[5] | Ohtsu, M., The history and development of acoustic emission in concrete engineering, Mag Concr Res, 48, 321-330 (1996) |

[6] | Colombo, S.; Main, I. G.; Forde, M. C., Assessing damage of reinforced concrete beam using “\(b\)-value” analysis of acoustic emission signals, J Mater Civil Eng, ASCE, 15, 280-286 (2003) |

[7] | Rao, M. V.M. S.; Lakshmi, K. J.P., Analysis of \(b\)-value and improved \(b\)-value of acoustic emissions accompanying rock fracture, Curr Sci, 89, 1577-1582 (2005) |

[8] | Kurz, J. H.; Finck, F.; Grosse, C. U.; Reinhardt, H. W., Stress drop and stress redistribution in concrete quantified over time by the \(b\)-value analysis, Struct Health Monitor, 5, 69-81 (2006) |

[9] | Carpinteri, A.; Lacidogna, G.; Niccolini, G., Critical behaviour in concrete structures and damage localization by acoustic emission, Key Eng Mater, 312, 305-310 (2006) |

[10] | Carpinteri, A.; Lacidogna, G.; Niccolini, G.; Puzzi, S., Critical defect size distributions in concrete structures detected by the acoustic emission technique, Meccanica, 43, 349-363 (2008) · Zbl 1163.74521 |

[11] | Lu, C.; Mai, Y. W.; Xie, H., A sudden drop of fractal dimension: a likely precursor of catastrophic failure in disordered media, Philos Mag Lett, 85, 33-40 (2005) |

[12] | Carpinteri, A.; Lacidogna, G.; Niccolini, G.; Puzzi, S., Morphological fractal dimension versus power-law exponent in the scaling of damaged media, Int J Damage Mech, 18, 259-282 (2009) |

[13] | Pollock, A. A., Acoustic emission-2: acoustic emission amplitudes, Non-Destruct Test, 6, 264-269 (1973) |

[14] | Grosse, C. U.; Reinhardt, H. W.; Dahm, T., Localization and classification of fracture types in concrete with quantitative acoustic emission measurement techniques, NDT Int, 30, 223-230 (1997) |

[15] | Shah, S. P.; Li, Z., Localisation of microcracking in concrete under uniaxial tension, ACI Mater J, 91, 372-381 (1994) |

[17] | Carpinteri, A.; Lacidogna, G.; Puzzi, S., From criticality to final collapse: evolution of the \(b\)-value from 1.5 to 1.0, Chaos Soliton Fract, 41, 843-853 (2009) |

[18] | Carpinteri, A., Scaling laws and renormalization groups for strength and toughness of disordered materials, Int J Solid Struct, 31, 291-302 (1994) · Zbl 0807.73050 |

[19] | Carpinteri, A.; Chiaia, B., Power scaling laws and dimensional transitions in solid mechanics, Chaos Soliton Fract, 7, 1343-1364 (1996) |

[20] | Carpinteri, A.; Chiaia, B., Multifractal scaling laws in the breaking behavior of disordered materials, Chaos Soliton Fract, 8, 135-150 (1997) · Zbl 0919.58058 |

[21] | Carpinteri, A.; Cornetti, P., A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos Soliton Fract, 13, 85-94 (2002) · Zbl 1030.74045 |

[22] | Carpinteri, A.; Pugno, N.; Puzzi, S., Strength vs. toughness optimization of microstructured composites, Chaos Soliton Fract, 39, 1210-1223 (2009) · Zbl 1197.74002 |

[23] | Carpinteri, A.; Cornetti, P.; Kolwankar, K. M., Calculation of the tensile and flexural strength of disordered materials using fractional calculus, Chaos Soliton Fract, 21, 623-632 (2004) · Zbl 1049.74790 |

[24] | Paggi, M.; Carpinteri, A., Fractal and multifractal approaches for the analysis of crack-size dependent scaling laws in fatigue, Chaos Soliton Fract, 40, 1136-1145 (2009) · Zbl 1197.74174 |

[25] | Carpinteri, A.; Paggi, M., A fractal interpretation of size-scale effects on strength, friction and fracture energy of faults, Chaos Soliton Fract, 39, 540-546 (2009) |

[26] | Carpinteri, A.; Chiaia, B.; Invernizzi, S., Applications of fractal geometry and renormalization group to the italian seismic activity, Chaos Soliton Fract, 14, 917-928 (2002) · Zbl 1047.86503 |

[27] | Bour, O.; Davy, P., Clustering and size distributions of fault patterns: theory and measurements, Geophys Res Lett, 26, 2001-2004 (1999) |

[28] | Bonnet, E.; Bour, O.; Odling, N. E.; Davy, P.; Main, I. G.; Cowie, P., Scaling of fracture systems in geological media, Rev Geophys, 39, 347-383 (2001) |

[29] | Carpinteri, A.; Pugno, N., A fractal comminution approach to evaluate the drilling energy dissipation, Int J Numer Anal Method Geomech, 26, 499-513 (2002) · Zbl 0995.74524 |

[30] | Carpinteri, A.; Lacidogna, G.; Pugno, N., A fractal approach for damage detection in concrete and masonry structures by the acoustic emission technique, Acoust Tech, 38, 31-37 (2004) |

[31] | Weyss, J., Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy, Int J Fract, 109, 365-381 (2001) |

[32] | Feder, J., Fractals (1988), Plenum Press · Zbl 0648.28006 |

[33] | Bour, O.; Davy, P.; Darcel, C.; Odling, N., A statistical scaling model for fracture network geometry, with validation on a multiscale mapping of a joint network (Hornelen Basin, Norway), J Geophys Res, 107, B6, 4/1-4/12 (2002) |

[35] | Posadas, A.; Gimenez, D.; Bittelli, M.; Vaz, C.; Flury, M., Multifractal characterization of soil particle-size distributions, Soil Sci Soc Am J, 65, 1361-1367 (2001) |

[36] | Grassberger, P.; Procaccia, I., Characterization of strange attractors, Phys Rev Lett, 50, 346-349 (1983) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.