3.1. Is Weibull Statistical Theory Applicable to Quasibrittle Structures?

The statistical theory of size effect based on the concept of random strength was, in principle, completed by Weibull (1939) (also 1949, 1951, 1956). The Weibull theory has been enormously successful in applications to fine-grained ceramics and metal structures embrittled by fatigue. However, it took until the 1980s to realize that this theory does not really explain the size effect in quasibrittle structures failing after a large stable crack growth.

The Weibull theory rests on two basic hypotheses:

1. The structure fails as soon as one small element of the material representative volume attains the strength limit.

2. The strength limit is random and the probability P ₁ that the small element of material does not fail at a stress less than ? has a cumulative distribution with a power law tail:

   (3.1)

(Weibull 1939) where m, ? ₀, ? _u = material constants ( m = Weibull modulus, usually between 5 and 50; ? ₀ = scale parameter; ? _u = strength threshold, which may usually be taken as 0).

Based on Eq. (3.1), a three-dimensional continuous generalization of the weakest link model for the failure of a chain of links of random strength leads to the Weibull distribution:

(3.2)

which represents the probability that a structure that fails at nominal stress ? _N as soon as macroscopic fracture initiates from a microcrack (or a some flaw) anywhere in the structure; ?