Mechanical and Civil Engineering Seminar

The failure probability of engineering structures such as bridges, airframes and MEMS should be <10-6. This is a challenge. For perfectly brittle and ductile materials obeying the Weibull or Gaussian failure probability distribution functions (pdf) with the same coefficient of variation, the distances from the mean to 10-6 differ by about 2:1. For quasibrittle materials, which include concrete, composites, tough ceramics, rocks, ice, foams, bone or nacre, and various architected, meta- or bio-mimetic materials, this distance can be anywhere in-between. Hence, a new theory is needed. The lecture begins with a review of the recent formulation of Gauss-Weibull statistics derived from analytical nano-macro scale transitions and from probability-frequency equivalence of interatomic bond ruptures governed by activation energy. Extensions to the lifetime pdf based on subcritical crack growth is outlined. Then, motivated by the nanoscale imbricated lamellar architecture of nacre, a new probability model with alternating series and parallel links, resembling a diagonally-pulled fishnet, has been developed. After the weakest-link and fiber-bundle models, it is the third failure probability model tractable analytically. It describes a continuous transition between Gaussian and Weibull distributions, and is strongly size-dependent. The original fishnet model for strength of fishnet with brittle links is extended to quasibrittle links, handled by order statistics, and to 3D octet material architectures. The size effect on the mean fishnet strength is a new kind of Type 1 size effect. It represents an envelope of a series of intermediate asymptotes of decreasing slope and can be used for calibrating the fishnet distribution. It is emphasized that while increase of scatter and decrease of periodicity in the microstructure reduces the mean strength, it may drastically improve the strength at 10-6 failure risk, which is what usually matters. Finally, it is observed that all random particulate and fiber composites, including concrete, must partly follow the fishnet statistics on approach to 10-6. Comparisons with experimental histograms and size-effect tests support the theory.