Analysis of the fiber length dependence of its strength by using the weakest-link approach 1. A family of weakest-link distribution functions

This paper presents a summary and further development of the ideas proposed in the previous papers of the authors, which were dedicated to investigating the fiber scale effect (strength-length relation). In the first part of the paper, some theoretical aspects of the problem are considered; in the second one, an application to the processing of test datasets is discussed. As distinct from our previous publications, two types of defects (“technological,” i.e., existing before loading, and load-dependent) and two types of influence of the number of defects on the fiber strength are considered; the probability of absence of defects is also taken into account. We consider a specimen as a sequence of n elements of the same length. It is supposed that there are defects in K of them, 0 ≤ K ≤ n. Two cases are considered: K is a random variable or a random process K (t). In the second case, the increase in K and the failure of a specimen is described as a Markov chain whose matrix of transition probabilities depends on the current value of the loading process, described as some (increasing to infinity) sequence {x₁, x₂, …, x_t,…}. Three versions of relationships between the specimen strength and the number of defective elements are considered for both the cases. Thus, six probability structures are introduced, and different versions of distribution functions and the corresponding models are considered. The methods for estimating the model parameters, the results obtained in processing glass, flax, carbon fiber, and carbon bundle datasets, as well as a comparison of different models, are presented in the second part of the paper.