### Abstract

We present a two-dimensional, quasistatic model of fracture in disordered brittle materials that contains elements of first-passage percolation, i.e., we use a minimum-energy-consumption criterion for the fracture path. The first-passage model is employed in conjunction with a "semi-directed" Bernoulli percolation model, for which we calculate critical properties such as the correlation length exponent v^{sdir} and the percolation threshold p_{c}^{sdir}. Among other results, our numerics suggest that v^{sdir} is exactly 3/2, which lies between the corresponding known values in the literature for usual and directed Bernoulli percolation. We also find that the well-known scaling relation between the "wandering" and energy fluctuation exponents breaks down in the vicinity of the threshold for semi-directed percolation. For a restricted class of materials, we study the dependence of the fracture energy (toughness) on the width of the distribution of the specific fracture energy and lind that it is quadratic in the width for small widths for two different random fields, suggesting that this dependence may be universal.

Original language | English (US) |
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Pages (from-to) | 603-623 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 91 |

Issue number | 3-4 |

DOIs | |

State | Published - May 1998 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Brittle materials
- First-passage percolation
- Fracture
- Semi-directed percolation

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## Cite this

*Journal of Statistical Physics*,

*91*(3-4), 603-623. https://doi.org/10.1023/a:1023077627335