Fishnet Statistics for the Strength Distribution of Nacre-like Biomimetic and Architected Quasibrittle Materials

Luo, Wen

doi:https://doi.org/10.21985/n2-m8xk-9996

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Fishnet Statistics for the Strength Distribution of Nacre-like Biomimetic and Architected Quasibrittle Materials

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The staggered (or imbricated) lamellar ``brick-and-mortar" nanostructure of nacre endows nacre with strength and fracture toughness values exceeding by an order of magnitude those of the constituents, and inspires the advent of new robust biomimetic materials. While many deterministic studies clarified these advantageous features in the mean sense, a closed-form statistical model is indispensable for determining the tail probability of failure in the range of 1 in a million, which is what is demanded for most engineering applications. So far, there is no undisputed theoretical basis for the probability distribution of strength of nacre. This study presents a numerical and theoretical study of the strength distribution and of the corresponding statistical size effect. After reasonable simplifications of the shear bonds, an axially loaded lamellar shell is statistically modeled as a square fishnet pulled diagonally. A finite element (FE) model is developed in Matlab (which has later been ported to C++) and used in Monte Carlo simulations of strength. An analytical model for failure probability of the fishnet is developed and matched to the computed statistical histograms of strength for various sizes. It appears that, with increasing size, the pdf of strength slowly transitions from Gaussian to Weibull distribution. Then, the fishnet statistical model is extended to capture the failure risk of nacre-like biomimetic materials with softening quasibrittle links. The probabilistic analysis is enabled by assuming the postpeak softening of a link to occur as a series of finite drops of stress and stiffness. The maximum load of the structure is approximated by the strength of the kth weakest link (order statistics), where k is in itself a random variable captured by the geometric-Poisson distribution. The analytically obtained probabilities are compared and verified by histograms of strength data obtained by millions of Monte Carlo simulations for each of many nacreous bodies with different link softening steepness and with various overall shapes. Finally, we 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. Using this general form of strength distribution, we apply it to architected nanomaterials such as the printed octet-truss carbon nanolattices, as well as to quasibrittle particulate composites such as concrete. We show that the three-dimensional assembly of fishnets further enhances the tail strength at the one-in-a-million probability quantile, compared to two-dimensional (2D) fishnet statistics.

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