Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06 54 C ONCLUSIONS n the present paper, a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth has been proposed in order to highlight and explain the deviations from the classical power-law equations used to characterize the fatigue behaviour of quasi-brittle materials. It has been theoretically demonstrated that both the parameters entering the Paris’ law and the Wöhler equation are microstructural-size, crack-size and size-scale dependent. From the theoretical point of view, these anomalous dependencies are due to incomplete self-similarities in the corresponding dimensionless numbers. More specifically, as far as the Paris’ law is concerned, it has been shown that the higher the structural size or the volumetric content of fibres, the lower the crack growth rate for a given stress-intensity factor range. Conversely, the higher the initial crack size or the loading ratio, the higher the crack growth rate. Regarding the S-N curves, the higher the structural size or the initial crack length, the lower the fatigue life for a given stress range. The opposite trend is noticed for the volumetric content of fibres and the loading ratio. Finally, we have also shown that the slopes of the S-N curves are dependent on the fibre aspect ratio. Similarly, the slope of the Paris’ curve is found to be dependent on the size of the specimen and on the ratio between the elastic modulus of concrete and its tensile strength. All these information are expected to be extremely useful for the design of experiments, since the role of the different dimensionless numbers governing the phenomenon of fatigue has been elucidated. R EFERENCES [1] D. A. Hordijk, Heron, 37 (1992) 1. [2] A. Wöhler, Z. Bauwesen (1860) 10. [3] O. H. Basquin, Proc. ASTM, 10 (1910) 625. [4] J. W. Murdock, C. E. Kesler, ACI J., 55 (1959) 221. [5] R. Tepfers, ACI J., 76 (1979) 919. [6] J. E. Butler, Strain, 26 (2008) 135. [7] V. Ramakrishnan, G. Oberling, P. Tatnall, SP-105-13, ACI Special Publication, ACI, Detroit (1987) 225. [8] C. D. Johnston, R. W. Zemp, ACI Mater. J., 88 (1991) 374. [9] J. Zhang, H. Stang, ACI Mat. J., 95 (1998) 58. [10] J. Zhang, H. Stang, V.C. Li, ASCE J. Mat. in Civ. Engrg., 12 (2000) 66. [11] J. Zhang, V.C. Li, H. Stang, ASCE J. Mater. Civil Engng., 13 (2001) 446. [12] S.P. Singh, Y. Mohammadi, S.K. Madan, J. Zhejiang University Science A, 7 (2006) 1329. [13] P. Paris, M. Gomez, W. Anderson, 13 (1961) 9. [14] P. Paris, F. Erdogan, J. Basic Eng. Trans. ASME, 58D (1963) 528. [15] Z.P. Bažant, K. Xu, ACI Mater. J., 88 (1991) 390. [16] Z.P. Bažant, W.F. Shell, ACI Mater. J., 90 (1993) 472. [17] S.V. Kolloru, E.F. O’Neil, J.S. Popovics, S.P. Shah, ASCE J. Engng. Mech., 126 (2000) 891. [18] Al. Carpinteri, M. Paggi, Engng. Fract. Mech., 74 (2007) 1041. [19] M. Paggi, Al. Carpinteri, Chaos, Solitons and Fractals, 40 (2009) 1136. [20] T. Matsumoto, V.C. Li, Cement & Concrete Composites, 21 (1999) 249-261. [21] K. L. Roe, T. Siegmund, Engng. Fract. Mech., 70 (2003) 209. [22] G. I. Barenblatt, L. R. Botvina, Fat. Fract. Engng. Mater. Struct., 3 (1980) 193. [23] G. I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics. Cambridge: Cambridge University Press, (1996). [24] M. Ciavarella, M. Paggi, Al. Carpinteri, J. Mech. Phys. Solids, 56 (2008) 3416. [25] Al. Carpinteri, M. Paggi, Int. J. Fatigue, in press, doi: 10.1016/j.ijfatigue.2009.04.014 [26] E. Buckingham, ASME Trans., 37 (1915) 263. [27] Al. Carpinteri, RILEM Mat. Struct., 14 (1981) 151. [28] Al. Carpinteri, Engng. Fract. Mech., 16 (1982) 467. [29] Al. Carpinteri, RILEM Mat. Struct., 16 (1983) 85. [30] K. Chan, Scripta Metal. Mater., 32 (1995) 235. [31] N. A. Fleck, K. J. Kang, M. F. Ashby, Acta Metall. Mater., 42 (1994) 365. [32] Al. Carpinteri, Mech. Mater., 18 (1994) 89. I