A Cellular Automaton Model of Catastrophic Failure

We introduce a two-dimensional cellular automaton model for studying the catastrophic failure of materials under stress. Our model is similar to the Olami-Feder-Christensen earthquake model [Z.\ Olami \emph{et al.}, Phys.\ Rev.\ Lett.\ \textbf{68}, 1244 (1992)] except that after a site fails $f$-times, it no longer can receive stress from its neighbors. In the limit that the interaction range, $R$, goes to infinity, our model is equivalent to the global load sharing fiber bundle model of Pierce [F.\ T.Pierce, J.\ Text.\ Ind.\ \textbf{17}, 355 (1926)] and Daniels [H.\ E.\ Daniels, Proc.\ Roy.\ Soc.\ London A \textbf{183}, 405 (1945)]. By varying the interaction range, we observe two qualitatively different failure modes. For $R \gg 1$ catastrophic failure resembles a nucleation-like event which grows symmetrically from a single initiating site and fails every site in the lattice. In contrast, for $R \approx 1$ a percolating cluster of failed sites spans the system despite the many active sites that persist, even after catastrophic failure. We use the stress-fluctuation metric to study the ergodicity of our model and hence the validity of equilibrium descriptions of fracture.