Asymptotically matched extrapolation of fishnet failure probability to continuum scale

Motivated by the extraordinary strength of nacre, which exceeds the strength of its fragile constituents by an order of magnitude, the fishnet statistics became in 2017 the only analytically solvable probabilistic model of structural strength other than the weakest-link and fiber -bundle models. These two models lead, respectively, to the Weibull and Gaussian (or normal) distributions at the large-size limit, which are hardly distinguishable in the central range of failure probability. But they differ enormously at the failure probability level of 10-6, considered as the maximum tolerable for engineering structures. Under the assumption that no more than three fishnet links fail prior to the peak load, the preceding studies led to exact solutions intermediate between Weibull and Gaussian distributions. Here massive Monte Carlo simulations are used to show that these exact solutions do not apply for fishnets with more than about 500 links. The simulations show that, as the number of links becomes larger, the likelihood of having more than three failed links up to the peak load is no longer negligible and becomes large for fishnets with many thousands of links. A differential equation is derived for the probability distribution of not-too-large fishnets, characterized by the size effect, the mean and the coefficient of variation. Although the large-size asymptotic distribution is beyond the reach of the Monte Carlo simulations, it can by illuminated by approximating the large-scale fishnet as a continuum with a crack or a circular hole. For the former, instability is proven via complex variables, and for the latter via a known elasticity solution for a hole in a continuum under antiplane shear. The fact that rows or enclaves of link failures acting as cracks or holes can form in the large-scale continuum at many random locations necessarily leads to the Weibull distribution of the large fishnet, given that these cracks or holes become unstable as soon they reach a certain critical size. The Weibull modulus of this continuum is estimated to be more than triple that of the central range of small fishnets. The new model is expected to allow spin-offs for printed materials with octet architecture maximizing the strength-weight ratio.