Size- and Shape-Effects on Safety Factors of Quasibrittle Structures

Project summary:

The design of engineering structures such as bridges, dams, buildings, aircraft, and ships must ensure an extremely low failure probability. For structures that are made of quasi-brittle (brittle heterogeneous) materials, such as concrete, fiber composites, and toughened ceramics, the rational determination of the safety factors guarding against such a low failure probability remains a major challenge.

Recent studies have shown that the probability distribution of strength of quasi-brittle structures, which fail at the macrocrack initiation, depends on the structure size and geometry, varying from Gaussian distribution modified by a far-left Weibull tail for small-size structures, to Weibull distribution for large-size structures. The effects of structure size and geometry on the strength distribution imply that the corresponding safety factors must depend on the structure size and geometry.

This research reviewed a recently developed finite chain model for the weakest-link statistics of strength, lifetime, and size effect of quasi-brittle structures, which are structures in which the fracture-process zone size is not negligible compared to the cross-section size. The theory is based on the recognition that the failure probability is simple and clear only on the nanoscale, since the probability and frequency of interatomic bond failures must be equal. The paper outlines how a small set of relatively plausible hypotheses about the failure probability tail at nanoscale and its transition from nano- to macroscale makes it possible to derive the distribution of structural strength, the static crack growth rate, and the lifetime distribution, including the size and geometry effects [while an extension to fatigue crack growth rate and lifetime, published elsewhere (Le and Bazant, 2011, "Unified Nano-Mechanics Based Probabilistic Theory of Quasibrittle and Brittle Structures: II. Fatigue Crack Growth, Lifetime and Scaling," J. Mech. Phys. Solids, 1322-1337), is left aside]. A salient practical aspect of the theory is that, for quasibrittle materials, the chain model underlying the weakest-link statistics must be considered to have a finite number of links, which implies a major deviation from the Weibull distribution. Several new extensions of the theory are presented: (1) A derivation of the dependence of static crack growth rate on the structure size and geometry; (2) an approximate closed-form solution of the structure strength distribution; and (3) an effective method to determine the cumulative distribution function's of structure strength and lifetime based on the mean-size effect curve. Finally, as an example, a probabilistic reassessment of the 1959 Malpasset Dam failure is demonstrated.