Abstract
This paper is devoted to the crack mechanics in heterogeneous materials with fractal (micro-)structures. Specifically, stress concentrations ahead of straight notches and self-affine cracks in fractal media are studied within a fractal continuum framework. A model of fractal continuum with fractal boundaries accounting for the metric, topological, and connectivity properties of the material microstructure and crack is employed for homogenization of crack problems in fractal media. It is found that the fractal nature of material heterogeneity can either delay or assist the crack initiation and propagation, depending on the interplay between metric and topological properties of the fractal domain.
Similar content being viewed by others
References
T.L. Anderson, Fracture Mechanics, Fundamentals and Applications, 3rd edn. (CRC Press, New York, 2005)
D. Cioranescu, P. Donato, An Introduction to Homogenization (Oxford University Press, Oxford, 1999)
M. Sahimi, Rev. Mod. Phys. 65, 1393 (1993)
J.J. Mecholsky, Key Eng. Mat. 409, 145 (2009)
E. Bouchaud, J. Phys.: Condens. Matter 9, 4319 (1997)
A. Hansen, J. Schmittbuhl, Phys. Rev. Lett. 90, 045504 (2003)
M. Ansari-Rad, S.M.V. Allaei, M. Sahimi, Phys. Rev. E 85, 021121 (2012)
A.S. Balankin, O. Susarrey, C.A. Mora Santos, J. Patiño, A. Yoguez, E.I. García, Phys. Rev. E 83, 015101(R) (2011)
G.P. Cherepanov, A.S. Balankin, V.S. Ivanova, Eng. Fract. Mech. 51, 997 (1995)
A.S. Balankin, Eng. Fract. Mech. 57, 135 (1997)
M.P. Wnuk, A. Yavari, Eng. Fract. Mech. 72, 2744 (2005)
M.P. Wnuk, A. Yavari, Eng. Fract. Mech. 75, 1127 (2008)
A. Yavari, H. Khezrzadeh, Eng. Fract. Mech. 77, 1516 (2010)
H. Khezrzadeh, M.P. Wnuk, A. Yavari, J. Phys. D 44, 395302 (2011)
S. Morel, J. Schmittbuhl, E. Bouchaud, G. Valentin, Phys. Rev. Lett. 85, 1678 (2000)
D. Vandembroucq, S. Roux, Phys. Rev. E 55, 6171 (1997)
V. Lazarus, J. Mech. Phys. Solids 59, 121 (2011)
A.S. Balankin, L.H. Hernandez, G. Urriolagoitia, O. Susarrey, J.M. Gonzáles, J. Martinez, Proc. Roy. Soc. A 455, 2565 (1999)
A.S. Balankin, Eur. Phys. J. B 88 90 (2015)
O. Panagouli, Chaos Solitons Fractals 8, 287 (1997)
H. Wallin, Manuscripta Mathematica 73, 117 (1991)
H. Wallin, Constr. Approx. 5, 137 (1989)
A. Jonsson, H. Wallin, in Function spaces on subsets of R n, Series: Math. Report (Harwood Acad. Publ., London, 1984), Vol. 2
P.D. Panagiotopoulos, O. Panagouli, Chaos Solitons Fractals 8, 253 (1997)
A. Carpinteri, B. Chiaia, Sadhana 27, 425 (2002)
Z.P. Bazant, A. Yavari, Eng. Fract. Mech. 72, 1 (2005)
E. Saouma, G. Fava, Int. J. Fracture 137, 231 (2006)
T. Nakayama, K. Yakubo, Rev. Mod. Phys. 66, 381 (1994)
S. Miyazima, H.E. Stanley, Phys. Rev. B 35, 8898 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Balankin, A.S. A continuum framework for mechanics of fractal materials II: elastic stress fields ahead of cracks in a fractal medium. Eur. Phys. J. B 88, 91 (2015). https://doi.org/10.1140/epjb/e2015-50703-8
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2015-50703-8