General Fishnet Statistics of Strength: Nacreous, Biomimetic, Concrete, Octet-Truss, and Other Architected or Quasibrittle Materials

The fishnet probabilistic model was recently developed to characterize the strength distribution of nacre-like biomimetic materials. It reveals that the unique fishnet-like connectivity of the material microstructure brings about enormous safety gain at the extremely low failure probability level of one out of a million, desired for engineering structures. The gist of the theory is that the material microstructure plays a determining role in its failure probability tail. Therefore, a carefully designed connectivity for a material microstructure not only enhances its mean strength but also significantly reduces its marginal failure risk. Here, we first show that the initially introduced series expansion and the newer formulation based on order statistics are, in the fishnet model, essentially equivalent. From that we develop a neat general form of the fishnet statistics. Then, we extend our theoretical approach to the strength distributions of architected nanomaterials such as the printed octet-truss carbon nanolattices, as well as to quasibrittle particulate composites such as concrete, and formulate a unified general fishnet statistics. We demonstrate that the octet-truss system can be physically seen and statistically treated as a union of three fishnets with three mutually orthogonal orientations. We show that the three-dimensional assembly of fishnets further enhances the tail strength at the 10−6 probability quantile, compared to two-dimensional (2D) fishnet statistics. We compare the performance of different statistical strength models by fitting of the simulated and experimental histograms data for the octet-truss nanolattice. Finally, we argue that, at the extreme lower tail of failure probability, quasibrittle materials such as concrete or fiber composites should partially exhibit the fishnet-type statistical behavior.