Weakest-link hypothesis

Weibull developed his statistical theory of brittle fracture on the basis of the weakest link hypothesis, i.e. the specimen fails if its weakest element fails [6, 7], In its simplest form and for an uniaxial homogenous and tensile stress state, ct, and for specimens of the volume, F, the so called Weibull distribution of the probability of failure, F, is given by ... 

In this paper the Weibull theory is applied to very small specimens. The analysis follows the ideas presented in [13]. The relationships between flaw population, size of the fracture initiating flaw and strength are discussed. It is shown that a limit for the applicability of the classical fracture statistics (i.e. Weibull statistics based on the weakest link hypothesis) exists for very small specimens (components). 

It is further assumed that the cracks do not interact (i.e., their separation is large enough for their stress fields not to overlap), an assumption which is essential for the following arguments and which is equivalent to the weakest-link hypothesis that the failure of a specimen is triggered by the weakest volume element or, in other words, by the largest flaw that it contains. 

There are certain prerequisites for the occurrence of a Weibull distribution [65, 67, 71], the most important being (i) that the structure fails if one single flaw becomes critical (weakest link hypothesis) and (ii) that dangerous flaws do not interact. 

Hypothesis the rupture of a chain with N undifferentiated Unks is set off by the rupture of the weakest link. Let be the survival probability of a link the survival probability of the chain can be written as PsN that is, by reasoning on the rupture probabilities 1 - = (1 - Pj) or also ... 

---