Mapping of the Roughness Exponent for the Fuse Model for Fracture Jan Øystein Haavig Bakke * and Alex Hansen † Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway (Received 14 July 2007; published 28 January 2008) The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches local 0:65 0:03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Le ´vy distribution and the Le ´vy characteristics contribute to the local roughness exponent. DOI: 10.1103/PhysRevLett.100.045501 PACS numbers: 62.20.M, 46.50.+a, 81.40.Np The fuse model [1] has been a workhorse in the numeri- cal studies of fractures and fracture surfaces. In the early 1990s Hansen et al. suggested in [2] that the two- dimensional fuse model shows universal self-affine scal- ing. The self-affine scaling was thought to be universal with respect to the disorder for narrow disorders in both two and three dimensions [2,3]. This universality in the fuse model made it an interesting numerical tool for re- searchers trying to reveal the mechanism behind the uni- versal roughness exponent found in experiments. See Ref. [4] for a review. Apart from the early work by Hansen et al. [2] and Batrouni and Hansen [3] little work has been done to measure systematically how the roughness exponent changes with the disorder in the model. It has been as- sumed the roughness exponent was constant for narrow disorders, and not well defined for large disorders, but no boundaries for these different regimes were set. Skjetne studied the two-dimensional and the three-dimensional beam model for different disorders and found the global roughness exponent to be a constant except for very small disorders [5]. The authors have done a systematical study of the central force model [6], and also for this model the roughness exponent was nonconstant for small disorders. We present results that restrict the universal value of the local roughness exponent for the fuse model to a limited disorder range and give the limits of this range based on numerical simulations. Outside of this range the local roughness exponent is found to be dependent on the lattice for narrow disorders and on the jumps in the surface for broad disorders. The fuse model is a quasistatic lattice model often used in the numerical studies of fracture. The system is modeled as a network of fuses that break irreversibly when the current through them increases beyond a threshold current. This is an electrical analogy of a fracture model with an elastic force field. The disorder in the material is introduced in the breaking thresholds of the fuses. To each fuse i we assign a threshold t i drawn at random from a power law distribution t/ t 1 ;t 20; 1 or t/ t 1 ;t 21; 1. The thresh- olds can then be made from t i r D i , where r i is a random number between 0 and 1 and D 1 or D 1 . controls the tail of very weak fuses, and controls the tail of very strong fuses. Values of jDj close to zero give narrow distributions of thresholds and a small degree of disorder in the system, while larger values of jDj give broader distributions of thresholds and a large degree of disorder in the system. In the simulations we use threshold distributions with either a tail toward weak fuses or toward strong fuses. We simulate the fractures on a square lattice of size L L, which is rotated 45 with the top and bottom border. This ensures that the initial currents are all equal. A voltage difference is applied between the top and the bottom row, and we allow the lattice to be periodic in the direction of the applied voltage difference. In the direction perpendicu- lar to the voltage difference the lattice can be both periodic and have free boundary conditions. We solve the Kirchhoff equations using a parallel conjugate gradient algorithm. When the currents in the system have been calculated, we pick the fuse j with the largest ration i j =t j and trip this fuse by setting its conductance to zero and increasing the cur- rent by a factor of t j =i j . Then the currents are calculated in this new configuration and we continue to trip fuses until the total conductivity of the system is zero. The roughness of surfaces can be described by the roughness exponent which is the self-affine scaling ex- ponent of the surface hx. It is defined from the scaling of the height-height distribution p h; l/ ph; l, where h is the height difference over the distance l, is a scaling factor, and is the anisotropic scaling exponent. The roughness exponent can be measured using several different methods; see, for example, Refs. [7,8]. Based on the tests done by the authors in [8] we chose to use scaling of the standard deviation of ph; l, l [9], and the averaged wavelet coefficients analysis [10] for the mea- surements of the local roughness exponent local . These two PRL 100, 045501 (2008) PHYSICAL REVIEW LETTERS week ending 1 FEBRUARY 2008 0031-9007= 08=100(4)=045501(4) 045501-1 2008 The American Physical Society