# Project Details

### Description

The tensile strength of nacre is amazing—it is about 15-times higher than the strength of its main constituent (CaCO3). The reasons have recently become well understood: 1) the nanoscale thickness of nacre's building blocks, the aragonite lamellae (or platelets), and 2) their imbricated, or staggered, nanoscale architecture. Therefore, biomimetic materials inspired by nacre are of major interest. However, their recent studies have all been deterministic, while most engineering structures and devices must be designed for the maximum failure probability one in a million (1.E-6) per lifetime, This is what must be used to set the safety factor, instead of the coefficient of variation (CoV), since a material with superior mean strength and a lower CoV can have an inferior safety at 1.E-6, depending on the shape of the probability density function (pdf) (this point has mostly been overlooked by material scientists and engineers). The 1.E-6 probability level is a challenge---it can be determined neither by histogram testing (as one would need hundred million tests!) nor by stochastic finite element methods (which can deliver only the CoV). Therefore, the complete pdf of strength must be established by a new theory, which must then be verified experiments by other than repeated tests (i.e., histogram testing).

Only two analytical models for the pdf of material strength now exist—the weakest-link chain or series coupling (yielding Weibull pdf, 1939) and the fiber bundle or parallel coupling (yielding Gaussian or normal pdf, Daniels 1945). Their extrapolations from the mean to 1.E-6 differ enormously, almost as 2:1. The proposed new idea is to approximate the nanoscale nacreous architecture by a diagonally pulled fishnet with brittle or quasibrittle links, which promises to be the third ever pdf model solvable analytically. For a brittle fishnet, the pdf is obtained as a sum of probabilities that 0, 1, 2, … links have failed prior to maximum load. For a fishnet with quasibrittle links, it is proposed to decompose continuous softening into discrete stiffness jumps and use the methods of order statistics, in which the n-th minimum strength rather than the minimum itself matters.

As indicated by preliminary study, the fishnet architecture should offer an enormous additional safety advantage, increasing the tail strength probability by one to two orders of magnitude compared the Weibull distribution, the current standard for fracture safety. This advantage is of great practical interest both to the Army and to civilian industry. Million Monte Carlo random simulations will be run on a supercomputer for each of many fishnet models, to verify and calibrate the new theory and, importantly, link it to the size effect. Measurements of the structure size effect on the mean strength are then expected to allow inferring the pdf. Since the brittle fishnet statistics can provide a continuous transition between the weakest-link (Weibull) and fiber bundle (Gaussian) models, asymptotic matching will be used to obtain simple realistic approximations of the pdf, and will also be extended to simplify the order statistics of discretely softening fishnets.

The proposer will collaborate with an independently funded colleague, Prof. H. Espinosa, who conducts experiments with staggered systems of graphene oxide platelets. Various spinoffs are anticipated, e.g., to infer probability of inter-grain fracturing of coarse-grained ceramics. Recommendations for setting the safety factors for structural design will be made.

Only two analytical models for the pdf of material strength now exist—the weakest-link chain or series coupling (yielding Weibull pdf, 1939) and the fiber bundle or parallel coupling (yielding Gaussian or normal pdf, Daniels 1945). Their extrapolations from the mean to 1.E-6 differ enormously, almost as 2:1. The proposed new idea is to approximate the nanoscale nacreous architecture by a diagonally pulled fishnet with brittle or quasibrittle links, which promises to be the third ever pdf model solvable analytically. For a brittle fishnet, the pdf is obtained as a sum of probabilities that 0, 1, 2, … links have failed prior to maximum load. For a fishnet with quasibrittle links, it is proposed to decompose continuous softening into discrete stiffness jumps and use the methods of order statistics, in which the n-th minimum strength rather than the minimum itself matters.

As indicated by preliminary study, the fishnet architecture should offer an enormous additional safety advantage, increasing the tail strength probability by one to two orders of magnitude compared the Weibull distribution, the current standard for fracture safety. This advantage is of great practical interest both to the Army and to civilian industry. Million Monte Carlo random simulations will be run on a supercomputer for each of many fishnet models, to verify and calibrate the new theory and, importantly, link it to the size effect. Measurements of the structure size effect on the mean strength are then expected to allow inferring the pdf. Since the brittle fishnet statistics can provide a continuous transition between the weakest-link (Weibull) and fiber bundle (Gaussian) models, asymptotic matching will be used to obtain simple realistic approximations of the pdf, and will also be extended to simplify the order statistics of discretely softening fishnets.

The proposer will collaborate with an independently funded colleague, Prof. H. Espinosa, who conducts experiments with staggered systems of graphene oxide platelets. Various spinoffs are anticipated, e.g., to infer probability of inter-grain fracturing of coarse-grained ceramics. Recommendations for setting the safety factors for structural design will be made.

Status | Active |
---|---|

Effective start/end date | 1/7/19 → 1/6/22 |

### Funding

- Army Research Office (W911NF-19-1-0039)