Energetic probabilistic size effect, its asymptotic properties and numerical applications

The paper reviews recent nonlocal and asymptotic models for the scaling of failure of quasibrittle materials, especially concrete, fiber-polymer composites, rock and ice, and presents a probabilistic generalization of the existing energetic deterministic scal- ing theory for failures of unnotched structures occuring at the initiation of macroscopic fracture growth. The presentation begins by asymptotic analysis of the scaling of the co- hesive crack model. Recent results on the scaling for kink-band failures of unidirectional fiber composites and its derivation from the J-integral are summarized. The energetic probabilistic theory is based on the nonlocal hypothesis that the failure probability at any point of the material is a power function of the average inelastic (or damage) strain of a certain neighborhood of the point whose size is determined by the characteristic length of the material. The averaging imposes spatial statistical correlation. A basic require- ment in constructing the theory is that the Weibull statistical theory of failure must be recovered as the limit case of a quasibrittle structure when the ratio of structure size to the characteristic length of the material tends to infinity. This property represents an important advantage of the proposed theory over the stochastic finite element method, which does not meet this asymptotic condition and is therefore unsuitable for determin- ing failure loads of extremely small probability. An approximate scaling law of asymptotic matching type is derived, with Weibull power scaling as the large size asymptotic limit. The theory is verified by numerical simulations as well as extensive comparisons with test data from the literature. Various numerical applications are demonstrated.