## Abstract

Engineering structures such as aircrafts, bridges, dams, nuclear containments and ships, as well as computer circuits, chips and MEMS, should be designed for failure probability < 10-6 to 10-7 per lifetime. However, the safety factors required to ensure it are still determined empirically, even though they represent much larger and much more uncertain corrections to deterministic calculations than do the typical errors of modern computer analysis of structures. Baẑant and Pang recently developed (and presented at previous ECF) a new theory for the cumulative distribution function (cdf) for the strength of quasibrittle structures failing at macro-fracture initiation, and offered its simplified justification in terms of thermally activated inter-atomic bond breaks. Presented here is a refined justification of this theory based on fracture mechanics of atomic lattice cracks advancing through the lattice by small jumps over numerous activation energy barriers on the surface of the free energy potential of the lattice. For the strength of the representative volume element (RVE) of material, simple statistical models based on chains and bundles are inadequate and a model consisting of a hierarchy of series and parallel couplings is adopted. The theory implies that the strength of one RVE must have a Gaussian cdf, onto which a Weibullian (or power-law) tail is grafted on the left at the failure probability of about 10-4 to 10-2. A positive-geometry structure of any size can be statistically modeled as a chain of RVEs. With increasing structure size, the Weibullian part of the cdf of structural strength expands from the left tail and the grafting point moves into the Gaussian core, until eventually, for a structure size exceeding about 104 equivalent RVEs, the entire cdf becomes Weibullian. Relative to the standard deviation, this transition nearly doubles the distance from the mean to the point of failure probability 10-6. Contrary to recent empirical models, it is found that the strength threshold must be zero. This finding and the size effect on cdf has a major effect on the required safety factor. The theory is further extended to model the lifetime distribution of quasibrittle structures under constant load (creep rupture). It is shown that, for quasibrittle materials, there exists a strong size effect on not only the structural strength but also the lifetime, and that the latter is stronger. Like the cdf of strength, the cdf of lifetime, too, is found to change from Gaussian with a remote power-law tail for small sizes, to Weibullian for large sizes. Furthermore, the theory provides an atomistic justification for the powerlaw form of Evans' law for crack growth rate under constant load and of Paris' law for crack growth under cyclic load. For various quasibrittle materials, such as industrial and dental ceramics, concrete and fibrous composites, it is finally demonstrated that the proposed theory correctly predicts the experimentally observed deviations of strength and lifetime histograms from the classical Weibull theory, as well as the deviations of the mean size effect curves from a power law.

Original language | English (US) |
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Title of host publication | 17th European Conference on Fracture 2008 |

Subtitle of host publication | Multilevel Approach to Fracture of Materials, Components and Structures |

Pages | 78-92 |

Number of pages | 15 |

State | Published - Dec 1 2008 |

Event | 17th European Conference on Fracture 2008: Multilevel Approach to Fracture of Materials, Components and Structures, ECF17 - Brno, Czech Republic Duration: Sep 2 2008 → Sep 5 2008 |

### Publication series

Name | 17th European Conference on Fracture 2008: Multilevel Approach to Fracture of Materials, Components and Structures |
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Volume | 1 |

### Other

Other | 17th European Conference on Fracture 2008: Multilevel Approach to Fracture of Materials, Components and Structures, ECF17 |
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Country/Territory | Czech Republic |

City | Brno |

Period | 9/2/08 → 9/5/08 |

## Keywords

- Lifetime; Probabilistic mechanics; Size effect; Scaling; Fracture; Extreme value statistics; Activation energy; Safety factors
- Structural reliability.
- Structural strength