Mechanics of disordered solids. III. Fracture properties

Brittle fracture of disordered media are studied using Monte Carlo simulations in both two and three dimensions (3D). Elastic and superelastic percolation networks with central and bond-bending forces are used as models of disordered media. We find that the distribution of fracture strength in a solid with broadly distributed microscopic heterogeneities, and in randomly reinforced materials, is adequately described by the classical Weibull distribution, rather than the recently proposed Gumbel distribution. System-size dependence of the external stress F for fracture is also studied. We find that, contrary to recent claims, for a d-dimensional system of size L, F is given by F∼${\mathit{L}}^{\mathit{d}\mathrm{-}1}$/(lnL${)}^{\mathrm{\psi }}$, where 0≤ψ≤0.5. The fractal dimension of the cracks is found to be about 1.7 in 2D, close to that of fracture surfaces of natural rocks at small scales. The scaling of the fracture stress ${\mathrm{\sigma }}_{\mathit{f}}$ near the percolation threshold ${\mathit{p}}_{\mathit{c}}$ is found to obey, ${\mathrm{\sigma }}_{\mathit{f}}$∼(p-${\mathit{p}}_{\mathit{c}}$ ${)}_{\mathit{f}}^{\mathit{T}}$, where p is the fraction of intact springs (or the damage level$)— and, ${\mathit{T}}_{\mathit{f}}$≃2.42 in 2D and ${\mathit{T}}_{\mathit{f}}$≃2.64 in 3D. The 2D result is in agreement with the experimental estimate of ${\mathit{T}}_{\mathit{f}}$ for fracture of thin perforated metal foils. These values are also close to the lower bound, ${\mathit{T}}_{\mathit{f}}$≥f-ν${\mathit{d}}_{\min }$, where f is the critical exponent of the elastic moduli of the system, ν the correlation-length exponent of percolation, and ${\mathit{d}}_{\min }$ the fractal dimension of the shortest paths on a percolation cluster. Finally, we study the similarities and differences between fractured and percolation networks.