Abstract
We study anomalous scaling and multiscaling of two-dimensional crack profiles in the random fuse model using both periodic and open boundary conditions. Our large scale and extensively sampled numerical results reveal the importance of crack branching and coalescence of microcracks, which induce jumps in the solid-on-solid crack profiles. Removal of overhangs (jumps) in the crack profiles eliminates the multiscaling observed in earlier studies and reduces anomalous scaling. We find that the probability density distribution \({p(\Delta h(\ell))}\) of the height differences \({\Delta h(\ell) = [h(x+\ell) - h(x)]}\) of the crack profile obtained after removing the jumps in the profiles has the scaling form \({p(\Delta h(\ell)) = \langle\Delta h^2(\ell)\rangle^{-1/2} ~f\left(\frac{\Delta h(\ell)}{\langle\Delta h^2(\ell)\rangle^{1/2}}\right)}\) , and follows a Gaussian distribution even for small bin sizes ℓ. The anomalous scaling can be summarized with the scaling relation \({\left[\frac{\langle\Delta h^2(\ell)\rangle^{1/2}}{\langle\Delta h^2(L/2)\rangle^{1/2}}\right]^{1/\zeta_{loc}} + \frac{(\ell-L/2)^2}{(L/2)^2} = 1}\) , where \({\langle\Delta h^2(L/2)\rangle^{1/2}\sim L^{\zeta}}\) and L is the system size.
Similar content being viewed by others
References
Alava MJ, Nukala PKVV, Zapperi S (2006a) Statistical models of fracture. Adv Phys 55: 349–476
Alava MJ, Nukala PKVV, Zapperi S (2006b) Morphology of two-dimensional fracture surfaces. J Stat Mech: Theor Exp L10002
Bakke JOH, Bjelland J, Ramstad T, Stranden T, Hansen A, Schmittbuhl J (2003) Roughness of brittle fractures as a correlated percolation problem. Phys Scripta T 106: 65–69
Bakke JOH, Ramstad T, Hansen A (2007) Roughness exponent measurements for the central force model. Phys Rev B 76: 054110
Baldassarri A, Colaiori F, Castellano C (2003) Average shape of a fluctuation: Universality of excursions of stochastic processes. Phys Rev Lett 90:060601 (4 pp)
Boffa JM, Allain C, Hulin JP (1998) Experimental analysis of fracture rugosity in granular and compact rocks. Eur Phys J Appl Phys 2: 281–289
Bonamy D, Ponson L, Prades S, Bouchaud E, Guillot C (2006) Scaling exponents of fracture surfaces in homogeneous glass and glassy ceramics. Phys Rev Lett 97: 135504 (4 pp)
Bouchaud E (1997) Scaling properties of cracks. J Phys Condens Matter 9: 4319–4344
Bouchaud E (2003) The morphology of fracture surfaces, a tool to understand crack propagation in complex materials. Surf Sci Rev Lett 10: 797–814
Bouchaud E, Lapasset G, Planés J, Navéos S (1993) Statistics of branched fracture surfaces. Phys Rev B 48: 2917–2928
Bouchbinder E, Procaccia I, Santucci S, Vanel L (2006) Fracture surfaces as multiscaling graphs. Phys Rev Lett 96: 055509 (4 pp)
Bouchbinder E, Procaccia I, Sela S (2006) Statistical physics of fracture surfaces morphology. J Stat Phys 125: 1029–1068
Daguier P, Nghiem B, Bouchaud E, Creuzet F (1997) Pinning and depinning of crack fronts in heterogeneous materials. Phys Rev Lett 78: 1062–1065
de Arcangelis L, Redner S, Herrmann HJ (1985) A random fuse model for breaking processes. J Phys (Paris) Lett 46: 585–590
Engoy T, Maloy KJ, Hansen A, Roux S (1994) Roughness of two-dimensional cracks in wood. Phys Rev Lett 73: 834–837
Hansen A, Hinrichsen EL, Roux S (1991) Roughness of crack interfaces. Phys Rev Lett 66: 2476–2479
Herrmann HJ, Roux S (1990) Statistical models for the fracture of disordered media. North-Holland, Amsterdam
Kertész J, Horvath VK, Weber F (1993) Self-affine rupture lines in paper sheets. Fractals 1: 67
López JM, Schmittbuhl J (1998) Anomalous scaling of fracture surfaces. Phys Rev E 57: 6405–6408
López JM, Rodríguez MA, Cuerno R (1997) Superroughening versus intrinsic anomalous scaling of surfaces. Phys Rev E 56: 3993–3998
Mallick N, Cortet PP, Santucci S, Roux SG, Vanel L (2007) Discrepancy between subcritical and fast rupture roughness: a cumulant analysis. Phys Rev Lett 98: 255502 (4 pp)
Maloy KJ, Hansen A, Hinrichsen EL, Roux S (1992) Experimental measurements of the roughness of brittle cracks. Phys Rev Lett 68: 213–215
Mandelbrot BB, Passoja DE, Paullay AJ (1984) Fractal character of fracture surfaces of metals. Nature (London) 308: 721–722
Mecholsky JJ, Passoja DE, Feinberg-Ringel KS (1989) Quantitative analysis of brittle fracture surfaces using fractal geometry. J Am Ceram Soc 72: 60–65
Menezes-Sobrinho IL, Couto MS, Ribeiro IRB (2005) Anisotropy in rupture lines of paper sheets. Phys Rev E 71: 066121 (6 pp)
Mitchell SJ (2005) Discontinuities in self-affine functions lead to multiaffinity. Phys Rev E 72: 065103 (4 pp)
Morel S, Schmittbuhl J, López JM, Valentin G (1998) Anomalous roughening of wood fractured surfaces. Phys Rev E 58: 6999–7005
Nukala PKVV, Simunovic S (2003) An efficient algorithm for simulating fracture using large fuse networks. J Phys A: Math Gen 36: 11403–11412
Nukala PKVV, Zapperi S, Simunovic S (2006) Crack surface roughness in three-dimensional random fuse networks. Phys Rev E 74: 026105 (8 pp)
Nukala PKVV, Zapperi S, Alava MJ, Simunovic S (2008) preprint
Ponson L, Bonamy D, Auradou H, Mourot G, Morel S, Bouchaud E, Guillot C, Hulin JP (2006a) Anisotropic self-affine properties of experimental fracture surfaces. Int J Fract 140: 27–37
Ponson L, Bonamy D, Bouchaud E (2006b) Two-dimensional scaling properties of experimental fracture surfaces. Phys Rev Lett 96: 035506 (4 pp)
Ponson L, Auradou H, Pessel M, Lazarus V, Hulin JP (2007) Failure mechanisms and surface roughness statistics of fractured Fontainebleau sandstone. Phys Rev E 76: 036108 (7 pp)
Rosti J, Salminen LI, Seppälä ET, Alava MJ, Niskanen KJ (2001) Printing of cracks in two-dimensional disordered media. Eur Phys J B 19: 259–263
Salminen LI, Alava MJ, Niskanen KJ (2003) Analysis of long crack lines in paper webs. Eur Phys J B 32: 369–374
Santucci S, Maloy KJ, Delaplace A, Mathiesen J, Hansen A, Bakke JOH, Schmittbuhl J, Vanel L, Ray P (2007) Statistics of fracture surfaces. Phys Rev E 75: 016104
Schmittbuhl J, Roux S, Berthaud Y (1994) Development of roughness in crack propagation. Europhys Lett 28: 585–590
Schmittbuhl J, Schmitt F, Scholz C (1995a) Scaling invariance of crack surfaces. J Geophys Res 100: 5953–5973
Schmittbuhl J, Vilotte JP, Roux S (1995b) Reliability of self-affine measurements. Phys Rev E 51: 131–147
Seppälä ET, Räisänen VI, Alava MJ (2000) Scaling of interfaces in brittle fracture and perfect plasticity. Phys Rev E 61: 6312–6319
Zapperi S, Nukala PKVV, Simunovic S (2005) Crack roughness and avalanche precursors in random fuse model. Phys Rev E 71: 026106 (10 pp)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nukala, P.K.V.V., Zapperi, S., Alava, M.J. et al. Anomalous roughness of fracture surfaces in 2D fuse models. Int J Fract 154, 119–130 (2008). https://doi.org/10.1007/s10704-008-9298-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-008-9298-z