Incorporation of statistical length scale into Weibull strength theory for composites

In this paper an extension of Weibull theory by the introduction of a statistical length scale is presented. The classical Weibull strength theory is self-similar; a feature that can be illustrated by the fact that the strength dependence on structural size is a power law (a straight line on a double logarithmic graph). Therefore, the theory predicts unlimited strength for extremely small structures. In the paper, it is shown that such a behavior is a direct implication of the assumption that structural elements have independent random strengths. By the introduction of statistical dependence in the form of spatial autocorrelation, the size dependent strength becomes bounded at the small size extreme. The local random strength is phenomenologically modeled as a random field with a certain autocorrelation function. In such a model, the autocorrelation length plays the role of a statistical length scale. The focus is on small failure probabilities and the related probabilistic distributions of the strength of composites. The theoretical part is followed by applications in fiber bundle models, chains of fiber bundle models and the stochastic finite element method in the context of quasibrittle failure.