Fracture of brittle materials, in general, involves statistical considerations. Materials have randomly distributed defects on their surfaces or in their interior. Fibrous materials, as we saw in Chapter 2, have a large surface area per unit volume. This makes it more likely for them to have surface defects than bulk materials. The presence of defects at random locations can lead to a scatter in the experimentally determined strength values of fibers, which calls for a statistical treatment of fiber strength. Clearly, such a scatter will be much more pronounced in brittle fibers than in ductile fibers such as metallic filaments. This is because ductile metals will yield plastically rather than fracture at a flaw of a critical size. Thus, most high performance fibers, with the exception of ductile metallic filaments, show a rather broad distribution of strength because they are highly flaw sensitive. Since the distribution of flaws is of statistical nature, the strength of a fiber must be treated as a statistical variable. To bring home this important point of variation in strength of a fiber as a function of fiber length, we show, in Figs. 11.1 through 11.3, the variation of tensile strength of some fibers as a function of gage length: high modulus carbon fiber (Fig. 11.1), boron fiber (Fig. 11.2), and Kevlar 49 aramid fiber (Fig. 11.3). In all cases, the strength decreases as the gage length increases. Intuitively, one can see that the probability of finding a critical flaw (which corresponds to the failure strength) increases as the volume of brittle material increases. In the case of a fiber, this translates into an increase in the probability of finding a critical flaw as the fiber length increases. Sometimes this phenomenon is termed the size effect, i.e., the average fiber strength decreases with increasing fiber gage length. An increase in fiber diameter can have similar effect. In this chapter, we provide a brief review of the statistical variation of fiber strength.

Variability of fiber strength

Variability of strength in brittle materials is analyzed by Weibull statistics, which is based on the assumption that we can regard the brittle material as consisting of a chain of links (Weibull, 1939, 1951). The failure of a material occurs when the weakest link in the chain fails. This is called the weakest-link assumption. We can regard a fiber as consisting of a chain of links.