We compare the roughness of minimal energy (ME) surfaces and scalar "quasistatic" fracture (SQF) surfaces. Two-dimensional ME and SQF surfaces have the same roughness scaling, w ϳ L z (L is the system size) with z 2 3 . The 3d ME and SQF results at strong disorder are consistent with the random-bond Ising exponent z ͑d $ 3͒ ഠ 0.21͑5 2 d͒ (d is the bulk dimension). However, 3d SQF surfaces are rougher than ME surfaces due to a larger prefactor. ME surfaces undergo a "weakly rough" to "algebraically rough" transition in 3d, suggesting a similar behavior in fracture.[S0031-9007(97)04993-4] PACS numbers: 62.20.Mk, 03.40.Dz, 46.30.Nz, 81.40.Np Fracture [1] continues to attract the attention of the materials theory community, with the full spectrum of theoretical tools currently being applied to its analysis [1][2][3][4][5][6][7][8][9]. Cracks are usually self-affine and their roughness can be characterized by a roughness exponent ͑z ͒, which may take on a few distinct values [7-9] (here, we calculate the "outof-plane roughness" of fracture surfaces). However, a debate continues about whether or not fracture surfaces are ever generated by "quasistatic" processes [2,7,8,10]. A quasistatic fracture process is one in which the stress field is always close to equilibrium. For this to be true, damage must evolve much more slowly than the time required for the stress field to equilibrate. Slow crack growth and high cycle fatigue are expected to be in this limit. As well as their fundamental interest, the latter processes are of enormous industrial importance. Bouchaud et al. [8] argue that at short length scales (as probed by, e.g., scanning tunneling microscopy) quasistatic processes dominate, while at longer length scales dynamical processes are of primary importance. Here, we present extensive numerical results on the topology of quasistatic fracture surfaces in random fuse networks. We also compare these fracture surfaces with minimal energy surfaces in the same networks. Using fast optimization methods, we are able to simulate the latter interfaces for large system sizes.Surprisingly [11], the roughness exponent of minimal energy (ME) surfaces and scalar quasistatic brittle fracture (SQF) surfaces have been shown to be close in two dimensions. This is surprising because a minimal energy surface is the surface of minimum energy in, for example, an Ising model with random bonds (see below), which seems to have little to do with fracture. Nevertheless, there is some experimental evidence that this holds in two dimensions [12]. We present precise numerical confirmation of the equivalence of ME and SQF roughness exponents in two dimensions. We analyze networks with either continuous or discrete disorder and find the same size dependence in all cases, in contrast to previous suggestions that discrete disorder is special [13].In three dimensions, our calculations are for ME and SQF surfaces of the cubic lattice in the {100} direction. We choose the low energy, {100} direction as it is more typical of the orientation of frac...

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The shear-lag type model due to Cox (Br. J. Appl. Phys. 3, 72 (1952) is widely used to calculate the deformation properties of fibrous materials such as short fiber composites and random fiber networks. We compare the shear-lag stress transfer mechanism with numerical simulations at small, linearly elastic strains and conclude that the model does not apply to random fiber networks. Most of the axial stress is transferred directly from fiber to fiber rather than through intermediate shear-loaded segments as assumed in the Cox model. The implications for the elastic modulus and strength of random fiber networks are discussed.

The roughness properties of two-dimensional fracture surfaces as created by the slow failure of random fuse networks are considered and compared to yield surfaces of perfect plasticity with similar disorder. By studying systems up to a linear size L=350 it is found that in the cases studied the fracture surfaces exhibit self-affine scaling with a roughness exponent close to 2/3, which is asymptotically exactly true for plasticity though finite-size effects are evident for both. The overlap of yield or minimum energy and fracture surfaces with exactly the same disorder configuration is shown to be a decreasing function of the system size and to be of a rather large magnitude for all cases studied. The typical "overlap cluster" length between pairs of such interfaces converges to a constant with increasing L.