On the application of Weibull statistics for describing strength of micro and nanostructures

Abstract Although the Weibull distribution has been used to describe strength data at the micro/nanoscale, its general applicability for micro/nanostructures has yet not been established. Most results are inconclusive due to insufficient data (an unavoidable challenge in micro/nanomechanical testing), and because they do not reveal the characteristics of the representative volume element. Macroscale structures are assumed to contain thousands of elements or more—the mathematical form of the Weibull distribution arises from this assumption. Micro/nanostructures, on the other hand, may contain much fewer elements. Another macroscale assumption is that the stress is far smaller than the strength of the representative element, i.e. the theoretical strength, but the strength of nanostructures approaches this value. The traditional mathematical form of the distribution also precludes calculation of the representative volume from strength data, preventing comparisons to defects or microstructural features that control failure (their size should be similar). Here, it is demonstrated that the Weibull distribution can mathematically describe the probability of failure with m > 2. Furthermore, using the exact form of the distribution, without the assumptions that lead to its traditional exponential form, equations are derived that allow the representative volume to be calculated from strength data, provided sufficient specimens and sizes are tested. Data for graphene and polysilicon are analyzed with the newly-derived equations, obtaining representative elements that agree well with the observed defects.