Theory and application of Weibull distributions to 1D peridynamics for brittle solids

Abstract

Peridynamics is a continuum mechanics modelling method, which is emerging as a solution for – in particular – the modelling of brittle fracture. The inherent variability of brittle fracture is captured well by the Weibull distribution, which describes the probability of fracture of a given material at a given stress. Recreating a Weibull distribution in peridynamics involves adjusting for the fact that the body is made up of a large number of bonds, and the distribution of strengths associated with these bonds must be different to the distribution of strengths associated with the peridynamic body. In the local case, where the horizon ratio, m=1 is used, Weibull’s original simple size scaling gives exact results, but the overlapping nature of non-local bonds that occurs in higher m cases, typically used in the peridynamics literature (such as m=3), causes a significant distortion of Weibull distributions. The cause of these distortions is spurious toughening and partial component failures as a result of the reduced localisation associated with larger horizon ratios. In order to remove these distortions, appropriate size scaling is used for the bonds, and a methodology that is capable of reflecting the heterogeneity of the material in the model, is proposed. The methodology described means Weibull parameters measured at specimen or component level can be reproduced for higher values of m.