Statistics of brittle fracture

Flaws, their shape, and their propagation are the central themes of this chapter. The various aspects of brittle failure are discussed from several viewpoints. The concepts of fracture toughness and flaw sensitivity are discussed first. The factors influencing the strengths of ceramics are dealt with in Sec. 11.3. Toughening mechanisms are dealt with in Sec. 11.4. Section 11.5 introduces the statistics of brittle failure and a methodology for design. 

Weibull developed his statistical theory of brittle fracture on the basis of the weakest link hypothesis, i.e. the specimen fails if its weakest element fails [6, 7], In its simplest form and for an uniaxial homogenous and tensile stress state, ct, and for specimens of the volume, F, the so called Weibull distribution of the probability of failure, F, is given by ... 

C. Lu, R. Danzer, and F. D. Fischer, Fracture Statistics of Brittle Materials Weibull or Normal Distribution, Physical Review E, 65, 1 - 4, (2002). 

S. B. Batdorf, Fracture statistics of brittle materials with intergranular cracks. Nuclear Engineering and Design, 35(3) 349-360, 1975. 

The Weibull s chain model of strength of materials, as today s foiandation of statistical theory of brittle fracture, been widely e plied and advanced due to its siitplicity in nathematical calculation has in the last decades. However, it has an obvious insufficiency. It does not take into account the interaction between various elements of calculation and in many cases, especially in the case of nonuniform stress field, it is the main cause which leads to ejnrors. The present paper, based on a large amount of ejq erimental data, attempts to put forward a hypothesis dealing with the mutual actions of the adjacent elements along coe direction and to correct the aforesaid insufficiency. The formulas obtained seem somevhat more ccmplicated than Weibull s, but they are more accurate vten verified by the ejqjerimental data. 

The mutual action between elements of a body must be considered on the statistical theory of brittle fracture. 

In predicting whether or not a candidate polymer will be ductile, it is recommended to err on the conservative side, and not to propose that the polymer will prefer shear yielding to brittle fracture unless ar(T)>l. 2-ay(T), instead of using the criterion af(T)>ay(T) suggested by Figure 11.9 for preference for shear yielding. There are several reasons for such caution. Firstly, when df(T) and ay(T) are very similar at a given T, mixed ductile/brittle failure modes and statistical... 

In summary, the fracture of brittle materials is a complex topic, with the fracture process being largely influenced by the local microstructure, and which renders the use of suitable statistics significant. This topic will be examined more closely in Section 12.5. 

In a series of fracture experiments on ceramic specimens, two important observations can be made, namely that the probabflity of failure increases with the load amplitude, and also with the size of the specimens [2-4,14]. This strength-size effect is the most prominent and relevant consequence of the statistical behavior of the strength of brittle materials. However, these observations carmot be explained in a deterministic way by using a simple model of a single crack in an elastic body rather, their interpretation requires an understanding of the behavior of many cracks distributed throughout a material. 

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