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 <5% error for a structure containing as few as 6 elements, provided the Weibull modulus $m$ > 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.

尽管威布尔分布已被用于描述微/纳米尺度的强度数据，但其对微/纳米结构的普遍适用性尚未建立。由于数据不足（微/纳米力学测试中不可避免的挑战），并且因为它们没有揭示代表性体积元素的特征，大多数结果是不确定的。假设宏观结构包含数千个或更多元素——威布尔分布的数学形式源于此假设。另一方面，微/纳米结构可能包含更少的元素。另一个宏观假设是应力远小于代表性元素的强度，即理论强度，但纳米结构的强度接近该值。分布的传统数学形式也排除了从强度数据中计算代表性体积的可能性，从而无法与控制失效的缺陷或微观结构特征（它们的大小应该相似）进行比较。在这里，证明了威布尔分布可以数学地描述包含少至 6 个元素的结构的失败概率，误差小于 5%，前提是威布尔模量$米$> 2. 此外，使用分布的精确形式，没有导致其传统指数形式的假设，推导出允许从强度数据计算代表性体积的方程，前提是测试了足够的样本和尺寸。石墨烯和多晶硅的数据使用新导出的方程进行分析，获得与观察到的缺陷非常吻合的代表性元素。