Fishnet model for failure probability tail of nacre-like imbricated lamellar materials

Nacre, the iridescent material of the shells of pearl oysters and abalone, consists mostly of aragonite (a form of CaCO3), a brittle constituent of relatively low strength (approximate to 10 MPa). Yet it has astonishing mean tensile strength (approximate to 150 MPa) and fracture energy (approximate to 350 to 1,240 J/m(2)). The reasons have recently become well understood: (i) the nanoscale thickness (approximate to 300 nm) of nacre's building blocks, the aragonite lamellae (or platelets), and (ii) the imbricated, or staggered, arrangement of these lamellea, bound by biopolymer layers only approximate to 25 nm thick, occupying <5 % of volume. These properties inspire manmade biomimetic materials. For engineering applications, however, the failure probability of <= 10(-6) is generally required. To guarantee it, the type of probability density function (pdf) of strength, including its tail, must be determined. This objective, not pursued previously, is hardly achievable by experiments alone, since >10(8) tests of specimens would be needed. Here we outline a statistical model of strength that resembles a fishnet pulled diagonally, captures the tail of pdf of strength and, importantly, allows analytical safety assessments of nacreous materials. The analysis shows that, in terms of safety, the imbricated lamellar structure provides a major additional advantage-similar to 10% strength increase at tail failure probability 10(-6) and a 1 to 2 orders of magnitude tail probability decrease at fixed stress. Another advantage is that a high scatter of microstructure properties diminishes the strength difference between the mean and the probability tail, compared with the weakest link model. These advantages of nacre-like materials are here justified analytically and supported by millions of Monte Carlo simulations.