A Precis of Fishnet Statistics for Tail Probability of Failure of Materials with Alternating Series and Parallel Links

Abstract

During the last dozen years it has been established that the Weibull statistical theory of structural failure and strength scaling does not apply to quasibrittle materials. These are heterogeneous materials with brittle constituents and a representative volume element that is not negligible compared to the structure dimensions. A new theory of quasibrittle strength statistics in which the strength distribution is a structure size dependent graft of Gaussian and Weibull distributions has been developed. The present article gives a precis of several recent studies, conducted chiefly at the writer’s home institution, in which the quasibrittle statistics has been refined to capture the statistical effect of alternating series and parallel links, which is exemplified by the material architecture of staggered platelets seen on the submicrometer scale in nacre. This architecture, which resembles a fishnet pulled diagonally, intervenes in many quasibrittle materials. The fishnet architecture is found to be advantageous for increasing the material strength at the tail of failure probability 10^—6, which represents the maximum tolerable risk for engineering structures and should be adopted as the basis of tail-risk design. Scaling analysis, asymptotic considerations, and cohesive fracture process zone, which were the hallmark of Barenblatt’s contributions, pervade the new theory, briefly called the “fishnet statistics”.